Small Sample Power of Tests of Normality when the Alternative is an α-stable Distribution
John C. Frain
TEP Working Paper No. 0207 February 2007
Trinity Economics Papers Department of Economics Trinity College Dublin
Small sample power of tests of normality when the alternative is an α-stable distribution John C. Frain.
∗
February 27, 2007
Abstract This paper is a Monte-Carlo study of the small sample power of six tests of a normality hypotheses when the alternative is an α-stable distribution with parameter values similar to those estimated for monthly total returns on equity indices. In these circumstances a sample size of 2oo is required to detect departures from normality. In most cases only small samples of consistent monthly data on such total returns are available and these are not sufficient to differentiate between normal and α-stable distributions.
Contents 1 Introduction
3
2 The Tests
5
2.1
Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
2.2
Lilliefors (Kolmogorov-Smirnov) Test . . . . . . . . . . . . . . . . . . . . .
6
2.3
Cramer-von Mises Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
2.4
Anderson-Darling Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
2
2.5
Pearson (χ Goodness of Fit) Test . . . . . . . . . . . . . . . . . . . . . .
7
2.6
Shapiro-Wilk Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
2.7
Jarque-Bera Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
∗ Comments are welcome. My email address is
[email protected]. This document is work in progress. Please consult me before quoting. Thanks are due to Prof. Antoin Murphy and to Michael Harrison for help and suggestions and to participants at a seminar in TCD for comments received. Any remaining errors in the paper are my responsibility. I would also like to thank my wife, Helen, for her great support and encouragement.
1
3 Results
10
3.1
Discussion of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.2
Application of tests to monthly Total Returns Equity Indices . . . . . . . 11
4 Summary and Conclusions
15
A Tables – Detailed Results
16
2
1
Introduction
In this paper I give an account of a series of simulations to measure the power of various tests of the null hypothesis of normality when the alternative is an α-stable distribution. Large samples of high frequency financial data generally reject this null (see, for example, Rachev and Mittnik (2000) and Frain (2006)). These tests applied to the smaller samples of monthly data, aggregated from the same daily data, do not always reject normality. For example, when the the six normality tests examined here are applied to one hundred months of daily observations of total returns on six equity indices the normality hypothesis is overwhelmingly rejected by tests for all six indices. When the six tests are applied to monthly aggregates derived from the six indices the null of normality is accepted in fifteen of the thirty six cases. A property of the α-stable distribution is that aggregated monthly data, derived from αstable distributed daily data, have an α-stable distribution with the same α parameter. The apparent failure of monthly data to reject the normality hypothesis has been taken as an indication that the daily data can not have an α-stable distribution. The tests examined here are shown to be of low power when applied to the short samples of monthly data typically available from aggregated daily data. Thus, failure to reject normality in these cases can not be seen as a rejection of the non-normal α-stable distribution. Mandelbrot, in a series of papers published in the early 60’s, was the first to suggest that financial returns had an α-stable distribution (Mandelbrot (1997) reprints many of his original papers and Mandelbrot and Hudson (2004) is a non-technical account of this work). The standard references on the mathematical properties of α-stable processes are Zolotarev (1986), Janicki and Weron (1994), Samorodnitsky and Taqqu (1994), Uchaikin and Zolotarev (1999) and Rachev and Mittnik (2000). An α-stable process depends on four parameters α The stability parameter which describes the weight of the tails of the distribution. (0 < α ≤ 2). The smaller the value of α the heavier the tails. β A skewness parameter. (−1 ≤ β ≤ 1). If β = 0 the distribution is symmetric otherwise it is skewed. γ A spread parameter similar to the variance of a normal distribution. (0 < γ) δ A location parameter. (−∞ < δ < ∞). A normal distribution is an α-stable distribution with α = 2. In this case the β parameter is redundant. γ 1 and δ correspond to the variance and mean of the normal. The distribution of high frequency financial returns has tails that are fatter than would be expected by a normal distribution (i.e. α < 2). The α-stable distribution appears to 1 For
the usual parametrization of α-stable and normal distributions
3
√
2γ = σ
fit the data well. In an examination of the distribution of total daily returns on 6 equity indices2 I found values of α in the range 1.65 to 1.73 and small negative values for the skew parameter. Section 2 gives details of the way the α-stable data were simulated and describes the six tests of normality that I have applied to sample sizes of 50, 100 and 200 and 3 values of each of the α β, γ and δ parameters. These sample sizes are typical of those that might be encountered when monthly data are derived from daily data. Detailed results are reported in Section 3 and in the Appendix. These results are summarized in Section 4. Section 3 also details the results of applying the normality tests to aggregated monthly series of 50, 100 and 200 observations derived from the daily returns used in the earlier analysis. The values of the β, γ and δ parameters used do not have a large effect on the analysis. In general the tests wrongly accept normality far too often and results are satisfactory only for α = 1.6. The Pearson and Cramer-von Mises tests are unsatisfactory in all cases while the Lilliefors (Kolmogorov-Smirnov) test is satisfactory only for a sample size of 200 and an α parameter of 1.6. The Jarque-Bera and Shapiro-Wilk test can differentiate with α = 1.6 and a sample size of greater than 100, with α = 1.7 and a sample size of 200. The Jarque-Bera can also detect the departure from normality for α = 1.8 and a sample size of 200. The measured relative power of these normality tests are specific to the alternative of an α-stable distribution and should not be regarded as measures of the relative merit of the tests against other alternatives.
2 The total returns indices examined included the ISEQ, CAC40, DAX30, FTSE100, Dow Jones Composite (DJC) and S&P500. The estimation period was from October 1959 to September 2005 for the DAX30 and from the late 1970s to September 2005 for the other indices
4
2
The Tests
2.1
Simulations
The α-stable random numbers used in this exercise were generated using the α-stable random number generator in the Rmetrics (Wuertz (2005)) package which is part of the R (R Development Core Team (2006)) statistical package. The method used is a variation of that proposed by Chambers et al. (1976) as extended by Weron (1996a,b). Let θ have a uniform distribution on (- π2 ,
π 2)
and w have an exponential distribution
with mean 1. If X = Cα,β
sin(α(θ + θ0 )) 1
cos(θ) α
cos(θ − α(θ + θ0 ) w
1−α α
where 1 πα 2α Cα,β = 1 + β 2 tan2 2 arctan(β tan πα 2 ) θ0 = α
and
then X has an α-stable distribution with stability parameter3 α for α 6= 1, skewness parameter β, spread parameter 1 and location parameter 0. The transformation of variables (Y = γX + δ) produces an α-stable variable with arbitrary spread (γ) and location (δ) parameters. For each of three values4 of the α-stable parameter (1.6, 1.7 and 1.8), three values of the skewness parameter, β, (0, -0.075 and -0.150), three values of the spread parameter, γ, (2.7, 3.6 and 4.5) and three values of the mean parameter δ (0.44, 0.88, 1.32) samples of 50, 100 and 200 observations were drawn. Each of these 243 experiments was replicated 1000 times. Six tests for normality were applied to each of the 243,000 samples. As a control on the process the simulations were repeated for a normal distribution with corresponding mean and variance. The tests used were 1. Anderson-Darling 2. Cramer-von Mises 3. Lilliefors (Kolmogorov-Smirnov) 4. Pearson (χ2 Goodness of Fit) 5. Shapiro-Wilk 3 when
α = 1 use " 2 π X= + βθ tan θ − β log π 2
4 The
π w cos θ 2 π + βθ 2
!#
ranges of values for each parameter are the monthly equivalent of those found in Frain (2006)
5
6. Jarque-Bera A brief summary of each test follows. For an extended account of testing for normality see Thode (2002)
2.2
Lilliefors (Kolmogorov-Smirnov) Test
The first three normality tests considered here are based on the difference between the empirical distribution function (EDF) and the normal distribution function. If the order statistics of a random sample of size n are given by x(1) , x(2) , . . . x(n) , the EDF is given by
Fn (x) =
0
x < x(1)
i/n x(i) ≤ x < x(i+1) 1 x(n) ≤ x
i = 1, . . . , n − 1
(1)
If Φ() is the standard normal distribution function and X has a normal distribution with mean µ and variance σ 2 the corresponding values of the distribution function are given by
qi = Φ([x(i) − µ]/σ)
(2)
The Kolmogorov-Smirnov test statistic is based on the maximum difference between the EDF and the qi . Thus if
D+ = max [i/n − qi ] i=1,...,n
D = max [qi − i/n] −
i=1,...,n
D = max[D+ , D− ]
(3)
The Kolmogorov-Smirnov test has been extended by Lilliefors (1967) to the case where the mean and variance are unknown and the estimated test statistic is based on the usual estimates of the mean and variance. See also Stephens (1974) and Thode (2002).
2.3
Cramer-von Mises Test
A class of EDF tests proposed by Anderson and Darling (1952) is defined by
Wn2 = n
Z
∞
|Fn (x) − F (x)|2 ψ[F (x)]dF
−∞
6
(4)
where F () is the hypothesized distribution function and ψ() is a non-negative weight function. For certain weight functions, including ψ = 1 and ψ(t) = 1/[t(1 − t)], it is possible to derive explicit limit distributions of this statistic. The Cramer-von Mises statistic uses the first of these weight functions and is given by X 1 2i − 1 2 + q(i) − W = 12n 2n
(5)
with the modification
W 2∗ = (1.0 + 0.5/n)W 2 accounting for differences in sample size when using tabulated critical values.
2.4
Anderson-Darling Test
The Anderson-Darling test uses the weighting function ψ(t) = 1/[t(1 − t)] in equation (4). This gives the test statistic A2 = −n − n−1
n X
[2i − 1][log(p(i) ) + log(1 − p(n−i+1) )]
(6)
i=1
where p(i) = Φ([x(i) − µ ˆ]/ˆ σ ) and µ ˆ and σ ˆ are estimated values of the mean and standard deviation. The modification A2∗ = (1.0 + 0.75/n + 2.25/n2)A2
(7)
allows the standard critical values to be applied to all sample sizes. The AndersonDarling test gives more weight to the tails of the distribution than the Cramer-von Mises test and may therefore be better able to differentiate between normal and αstable distributions.
2.5
Pearson (χ2 Goodness of Fit) Test
The Pearson test is the traditional test of goodness of fit. The observations are divided into k intervals. Let Oi and Ei be the observed and expected number in the ith interval. The Pearson test is P =
k X (Oi − Ei )2 i=1
(8)
Ei
The test is implemented here by dividing the samples of 50, 100 and 200 into 10, 13 and 17 equally probable intervals. P is distributed asymptotically as χ2 with k − 3 degrees of freedom, where k is the number of intervals used in the calculation of P . Since the advent of specific tests for a null of a normal distribution the Pearson test is not generally used. 7
2.6
Shapiro-Wilk Test
If the data are a good fit to a normal distribution then the plot of x(i) against Φ(i/n) will be close to a straight line. The Shapiro-Wilk test is a measure of this fit based on a generalized least squares regression using the covariance matrix of the order statistics. Due to difficulties in calculating this covariance matrix the Shapiro-Wilk test was originally available only for sample sizes up to 50. The difficulty being partially due to the fact that a separate covariance matrix had to be calculated for each sample size. Initially the Shapiro-Wilk test allowed smaller samples to be tested for normality than the previous Pearson test. Various approximations are now available that allow the test to be used for samples up to 5000. See Royston (1982a,b, 1995)
2.7
Jarque-Bera Test
The Jarque-Bera test is probably the normality test best known to economists and is often used as a test of the normality of residuals. If mi is the ith moment about the 1/2
mean of a sample then the skewness (b1 ) and kurtosis (b2 ) are defined by 1/2
b1
=
m3 3/2 m2
and b2 =
m4 m22
(9) 1/2
For a sample of size n from a normal distribution b1
is asymptotically normal with 1/2
mean zero and variance 6/n. For finite samples the variance of b1 c1 =
is better given by5
6(n − 2) (n + 1)(n + 3)
In the same circumstances the distribution of b2 is asymptotically normal with mean 3 and variance 24/n. For finite samples the mean c2 and variance c3 of b2 are given by 3(n − 1) (n + 1) 24n(n − 2)(n − 3) c3 = n + 1)2 (n + 3)(n + 5) c2 =
The Jarque-Bera statistic is given by ! 1/2 (b1 )2 (b2 − 3)2 JB = n + 6 24 which under the null hypothesis of normality has an asymptotic χ2 distribution with 2 degrees of freedom. In finite samples the skewness and kurtosis are not independent and the JB statistic converges slowly to it asymptotic limit. Two solutions have been proposed. First the JB statistic may be modified by replacing the asymptotic means 5 For
details see Thode (2002)
8
and variances by their values in finite samples and defining an adjusted Jarque-Bera (AJB) statistic. 1/2
AJB =
(b1 )2 (b2 − c2 )2 + c1 c3
!
The AJB and JB statistics have the same asymptotic distribution. For both the JB and AJB statistics critical values have been estimated by Weurtz and Katzgraber (2005) using a large sample Monte Carlo simulation. A comparison of the simulated and asymptotic critical values for the sample sizes used here is given in the table below. Critical Values of Jarque-Bera test of normality Simulated AJB
Sample Size
JB
Asymptotic
50
4.98
6.55
5.99
100 200
5.43 5.68
6.32 6.15
5.99 5.99
Thus inference based on the asymptotic distribution of the standard JB statistic will tend to accept normality to often. Inference based on the asymptotic distribution of the adjusted statistic tents to reject normality to often. In the simulations in this paper inferences were based on the simulated distribution of the standard Jarque-Bera statistic. To enable some comparisons to be made, both JB and AJB tests on total returns on equity indices both JB and AJB statistics are reported along with their finite sample probabilities as derived in Weurtz and Katzgraber (2005). Tables 1, 2 and 3 contain both JB and AJB tests on monthly returns and the significance levels given there are based on those in Weurtz and Katzgraber (2005). In all cases both tests lead to the same conclusion.
9
3
Results
The results of the simulations of the tests on the α-stable samples are shown in Tables 4 to 12 and summarized in Figures 1, 2 and 3. The control tests on the normal distribution are given in Table 13. Each of these 729 experiments described in Section 2.1 was replicated 1000 times. Each replication consisted of the generation of a pseudo random sample of the selected size from an α-stable distribution with the appropriate parameters. The six tests detailed in Section 2 were then applied to the random sample. The number of times that the normality assumption was accepted, at the test size specified, over 1000 replications is recorded in each case. Thus the figure of 318 at the top of column 5 of table 4 indicates that normality was accepted in 318 of the 1000 replications when an Anderson-Darling test of size 5% was used. The power of the test may be approximated as 68%. Similarly in 363, 423, 530, 280 and 225 from the 1000 replications normality was accepted at the the 5% size when, respectively, the Cramer-von Mises, Lilliefors, Pearson, Shapiro-Wilk and Jarque-Bera tests were applied. The numbers in these tables may be regarded as an estimate of the numbers of false acceptances of normality that may be found in applications of the test in the circumstances of the simulation. Smaller numbers are better. The results of applying the tests to simulated data drawn from a normal distribution are given in Table 13. The results in this table show that there are no significant size distortions in any of the tests examined at the sample sizes considered.
3.1
Discussion of Results
The data in the tables show that the power6 of the tests varies with α, the sample size and the test size. In the ranges examined the other three parameters are not as important. I have adopted the somewhat arbitrary definition of a satisfactory test as one of size 5% with power greater than 90%. A stricter definition would restrict the number of satisfactory tests while a more liberal approach would lead to a greater number of satisfactory outcomes. Using this definition no test is satisfactory for a sample size of 50. The Jarque-Bera test outperforms the others with an average power of 76% for α = 1.6 dropping to an average of under 50% for α = 1.8 For a sample size of 100 the Jarque-Bera test is again best in all cases. For α = 1.6 the average power of the test is 94%. This figure falls to 86% and 70% for α of 1.7 and 1.8, respectively. The Shapiro-Wilk and Anderson-Darling tests have power close to 90% when α = 1.6 and the size of the test is 5%. For a sample size of 200 and α = 1.6 the power of the Jarque-Bera, Shapiro-Wilk, Anderson-Darling and Lillifors (Kologmorov-Smirnov) tests are good, with average pow6 The
power of the test is estimated as 1 −
number normality accepted 1000
10
ers of 1.00, 0.99, 0.99, and 0.96 respectively. In this case the average power of the Pearson and Cramer-von Mises tests are 0.89 and 0.71 respectively. For a sample size of 200 and α = 1.7 the the Jarque-Bera, Shapiro-Wilk and AndersonDarling tests have powers of 0.98, 0.96, and 0.92 respectively. For a sample size of 200 and α = 1.8 the average power the Jarque-Bera test is just under 0.90. The Pearson and Cramer-von Mises tests are not satisfactory in any case. The JarqueBera test is the most satisfactory. The measured relative power of these normality tests are specific to the alternative of an α-stable distribution and should not be regarded as measures of the relative merit of the tests against other alternatives.
3.2
Application of tests to monthly Total Returns Equity Indices
Tables 1, 2 and 3 show the results of applying the 6 tests examined to monthly total returns on equity indices for periods of 50, 100 and 200 months, respectively, up to end August 2005. The total returns equity indices included are those for the CAC40, DAX30, FTSE100, ISEQ, Dow Composite (DCI) and the S&P500. Corresponding calculations for daily data show an overwhelming rejection of normality in all cases. For the samples of 50, 100 and 200 months there are 11, 15 and 9 acceptances of the null hypothesis of normality from the 36 tests completed in each case. Given the possible common trends in the series one can not regard them as independent samples but as an illustration of the application of the earlier results in this paper. Of the 9 acceptances of normality in the 200 month samples all but one are in the Pearson or Lilliefors tests which have been shown to have poor power. For the 100 month samples again the majority of rejections are in these two tests but, in this case, all tests show at least one acceptance of normality.
11
Figure 1: Power of Normality Tests when the alternative is α-Stable in sample size 50
1 AD CvM L P SW JB
test size 1%
0.9
0.8
0.7
0.6
0.5 D=1.6
D=1.7
D=1.8
D=1.6
D=1.7
D=1.8
D=1.6
D=1.7
D=1.8
1
test size 5%
0.9
0.8
0.7
0.6
0.5
1
test size 10%
0.9
0.8
0.7
0.6
0.5
12
Figure 2: Power of Normality Tests when the alternative is α-Stable in sample size 100
1 AD CvM L P SW JB
test size 1%
0.9
0.8
0.7
0.6
0.5 D=1.6
D=1.7
D=1.8
D=1.6
D=1.7
D=1.8
D=1.6
D=1.7
D=1.8
1
test size 5%
0.9
0.8
0.7
0.6
0.5
1
test size 10%
0.9
0.8
0.7
0.6
0.5
13
Figure 3: Power of Normality Tests when the alternative is α-Stable in sample size 200
1 AD CvM L P SW JB
test size 1%
0.9
0.8
0.7
0.6
0.5 D=1.6
D=1.7
D=1.8
D=1.6
D=1.7
D=1.8
D=1.6
D=1.7
D=1.8
1
test size 5%
0.9
0.8
0.7
0.6
0.5
1
test size 10%
0.9
0.8
0.7
0.6
0.5
14
4
Summary and Conclusions
If one regards a satisfactory test as one of size 5% with a power7 of 90% then the only satisfactory tests are Sample size 50 No test is satisfactory Sample size 100 • For α = 1.6 Jarque-Bera and Shapiro-Wilk tests are satisfactory. • For α = 1.7 No test is satisfactory • For α = 1.8 No test is satisfactory Sample size 200 • For α = 1.6 Jarque-Bera, Shapiro-Wilk, Anderson-Darling and Lilliefors tests are satisfactory. • For α = 1.7 Jarque-Bera, Shapiro-Wilk and Anderson-Darling tests are satisfactory. • For α = 1.8 The Jarque-Bera test was satisfactory in more than half the simulations at this level and close to satisfactory in the remailder At the parameter values likely to fit total returns on equity indices a sample size of the order of 200 is required in order to reliably detect departures from normality using common normality tests. The measured relative power of these normality tests do are specific to the alternative of an α-stable distribution and should not be regarded as measures of the relative merit of the tests against other alternatives.
7 The
power of a test is 1 − Prob(Type II Error). no of false acceptances/1000)
15
This is approximated by (1 −
A
Tables – Detailed Results
List of Tables 1
Normality Tests on Monthly Total Returns on Equity Indices for 50 months ending August, 2005 . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2
Normality Tests on Monthly Total Returns on Equity Indices for 100 months ending August, 2005 . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3
Normality Tests on Monthly Total Returns on Equity Indices for 200 months ending August, 2005 . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4
Simulation of 5% Normality tests on α-stable samples of size 50 (1000 replications) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
5
Simulation of 5% Normality tests on α-stable samples of size 100 (1000 replications) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
6
Simulation of 5% Normality tests on α-stable samples of size 200 (1000 replications) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
7
Simulation of 1% Normality tests on α-stable samples of size 50 (1000 replications) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
8
Simulation of 1% Normality tests on α-stable samples of size 100 (1000 replications) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
9
Simulation of 1% Normality tests on α-stable samples of size 200 (1000 replications) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
10
Simulation of 10% Normality tests on α-stable samples of size 50 (1000 replications) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
11
Simulation of 10% Normality tests on α-stable samples of size 100 (1000 replications) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
12
Simulation of 10% Normality tests on α-stable samples of size 200 (1000 replications) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
13
Simulation of Normality tests on a normal distribution (1000 replications) 47
16
Table 1: Normality Tests on Monthly Total Returns on Equity Indices for 50 months ending August, 2005 Equity
Summary Statistics
Normality Statistics
Index
AndersonDarling
Cranervon Mises
17
Obs.
Mean
St. dev
Skewness
Kurtosis
Lilliefors
Pearson
ShapiroWilk
Jarque-Bera (JB)
Jarque-Bera (AJB)
CAC40
50
-0.469
8.291
-0.876
2.485
0.732 (.053)
0.116 (.065)
0.094 (.327)
9.200 -0.239
0.949 (.032)
15.435 (.006)
20.877 (.006)
DAX30
50
-0.124
6.071
-0.775
1.512
1.067
0.193
0.154
23.200
0.943
7.952
10.523
FTSE
50
0.137
4.236
-1.024
1.592
(.008) 1.845
(.006) 0.285
(.004) 0.169
(.001) 36.700
(.018) 0.900
(.023) 11.842
(.023) 15.037
(.000) 0.246
(.001) 0.152
(.000) 16.800
(.000) 0.916
(.011) 7.093
(.012) 8.153
ISEQ
50
0.271
5.490
-0.947
0.326
(.000) 1.486
DCI
50
0.361
4.392
-1.106
2.186
(.001) 0.736
(.001) 0.095
(.005) 0.106
(.019) 8.000
(.002) 0.933
(.028) 16.755
(.036) 21.696
0.827
(.052) 0.833
(.129) 0.139
(.167) 0.129
(.333) 16.400
(.007) 0.963
(.005) 1.711
(.006) 2.560
(.030)
(.032)
(.036)
(.022)
(0.121)
(.284)
(0.189)
S&P
50
-0.091
4.380
-0.336
(Data in bold face indicate acceptance of normality hypothesis at 5% level)
Table 2: Normality Tests on Monthly Total Returns on Equity Indices for 100 months ending August, 2005 Equity Index
Summary Statistics Anderson-
Craner-
Normality Statistics Shapiro-
Jarque-Bera
Jarque-Bera
18
Obs.
Mean
St. dev
Skewness
Kurtosis
Darling
von Mises
Lilliefors
Pearson
Wilk
(JB)
(AJB)
CAC40
100
0.720
6.275
-0.628
0.527
0.901 (.021)
0.160 (.017)
0.102 (.012)
14.140 (.167)
0.970 (.021)
7.186 (.030)
7.979 (.033)
DAX30
100
0.332
7.747
-0.750
1.800
0.825 (.032)
0.126 (049)
0.071 (.247)
11.800 (.299)
0.966 (.010)
20.467 (.003)
23.819 (.003)
FTSE100
100
0.409
4.325
-0.711
0.600
1.121 (.006)
0.163 (.016)
0.091 (.041)
12.320 (.264)
0.961 (.005)
9.240 (.019)
10.242 (.021)
ISEQ
100
0.938
5.519
-0.822
0.993
1.165 (005)
0.195 (.006)
0.105 (.009)
12.060 (.281)
0.960 (.004)
14.188 (.007)
15.909 (.008)
DCI
100
0.626
4.424
-0.815
1.305
0.674 (.076)
0.073 (.253)
0.057 (.597)
4.780 (.905)
0.959 (.004)
16.559 (.005)
18.850 (.006)
S&P500
100
0.483
5.055
-0.499
0.293
0.536
0.072
0.077
15.440
0.978
4.222
4.644
(.166)
(.260)
(.153)
(.117)
(.094)
(.074)
(.083)
(Data in bold face indicate acceptance of normality hypothesis at 5% level)
Table 3: Normality Tests on Monthly Total Returns on Equity Indices for 200 months ending August, 2005 Equity
Summary Statistics
Normality Statistics
Index
AndersonDarling
Cranervon Mises
19
Obs.
Mean
St. dev
Skewness
Kurtosis
Lilliefors
Pearson
ShapiroWilk
Jarque-Bera (JB)
Jarque-Bera (AJB)
CAC40
200
0.745
5.689
-0.550
0.519
1.085 (.007)
0.195 (.006)
0.081 (.002)
17.600 (.226)
0.979 (.004)
11.832 (.010)
12.544 (.011)
DAX30
200
0.640
6.58
-0.908
2.710
2.073
0.319
0.086
24.400
0.952
83.942
90.586
FTSE100
200
0.850
4.271
-0.303
0.646
(.000) 1.083
(.000) 0.181
(.001) 0.062
(.041) 16.580
(.000) 0.986
(.000) 6.014
(.000) 6.668
(.009) 0.188
(.061) 0.062
(.279) 13.180
(.040) 0.974
(.044) 21.889
(.043) 23.996
ISEQ
200
1.013
5.287
-0.455
1.411
(.008) 1.236
DCI
200
0.924
4.060
-0.763
1.387
(.003) 1.177
(.007) 0.193
(.061) 0.066
(.512) 16.580
(.001) 0.978
(.002) 33.681
(.002) 36.110
0.911
(.004) 0.809
(.007) 0.118
(.032) 0.053
(.279) 19.779
(.000) 0.980
(.000) 15.929
(.000) 17.174
(.036)
(0.062)
(.187)
(.137)
(.006)
(.005)
(.005)
S&P500
200
0.871
4.311
-0.548
(Data in bold face indicate acceptance of normality hypothesis at 5% level)
Table 4: Simulation of 5% Normality tests on α-stable samples of size 50 (1000 replications) Number of replications where normality hypothesis accepted α-Stable Parameters α β γ δ
AndersonDarling
Cramervon Mises
Lilliefors
Pearson
ShapiroWilk
JarqueBera
1.6
0
2.7
0.44
318
363
423
530
280
225
1.6 1.6
0 0
2.7 2.7
0.88 1.32
325 353
386 399
448 430
555 565
286 303
238 251
1.6 1.6
0 0
3.6 3.6
0.44 0.88
301 362
348 416
400 461
532 576
262 303
207 234
1.6 1.6
0 0
3.6 4.5
1.32 0.44
329 354
377 397
418 458
526 554
288 283
244 226
1.6 1.6
0 0
4.5 4.5
0.88 1.32
323 313
377 372
431 433
546 529
274 265
225 208
1.6 1.6
-0.075 -0.075
2.7 2.7
0.44 0.88
309 305
371 359
420 390
522 512
282 265
232 215
1.6 1.6
-0.075 -0.075
2.7 3.6
1.32 0.44
344 316
395 370
432 418
535 539
298 264
231 216
1.6
-0.075
3.6
0.88
343
388
436
562
305
250
1.6 1.6
-0.075 -0.075
3.6 4.5
1.32 0.44
339 334
394 378
433 432
541 544
289 305
228 249
1.6 1.6
-0.075 -0.075
4.5 4.5
0.88 1.32
323 337
372 378
407 430
538 551
275 283
223 242
1.6 1.6
-0.15 -0.15
2.7 2.7
0.44 0.88
322 345
372 394
430 434
517 558
278 306
232 237
1.6 1.6
-0.15 -0.15
2.7 3.6
1.32 0.44
305 340
348 390
416 434
535 543
268 280
225 241
1.6 1.6
-0.15 -0.15
3.6 3.6
0.88 1.32
311 308
372 372
420 409
522 517
274 270
226 221
1.6 1.6
-0.15 -0.15
4.5 4.5
0.44 0.88
300 327
351 379
392 425
529 528
243 263
200 225
1.6 1.7
-0.15 0
4.5 2.7
1.32 0.44
305 500
345 538
407 583
522 656
267 415
221 345
1.7
0
2.7
0.88
477
522
576
693
423
351
1.7 1.7
0 0
2.7 3.6
1.32 0.44
464 440
510 498
560 536
668 650
394 392
343 328
1.7 1.7
0 0
3.6 3.6
0.88 1.32
473 470
508 521
559 591
664 671
414 419
338 351
1.7 1.7
0 0
4.5 4.5
0.44 0.88
464 481
529 523
576 588
667 674
403 417
336 346
Continued on next page
20
Simulation of 5% Normality tests on α-stable samples of size 50 (1000 replications) continued Number of replications where normality hypothesis accepted α-Stable Parameters α β γ δ
AndersonDarling
Cramervon Mises
Lilliefors
Pearson
ShapiroWilk
JarqueBera
1.7
0
4.5
1.32
477
526
577
696
435
360
1.7 1.7
-0.075 -0.075
2.7 2.7
0.44 0.88
470 479
505 532
569 574
669 664
410 410
332 345
1.7 1.7
-0.075 -0.075
2.7 3.6
1.32 0.44
454 468
496 514
549 581
662 664
407 406
348 332
1.7 1.7
-0.075 -0.075
3.6 3.6
0.88 1.32
442 479
483 522
553 582
655 680
386 399
330 342
1.7
-0.075
4.5
0.44
496
535
581
677
429
360
1.7 1.7
-0.075 -0.075
4.5 4.5
0.88 1.32
498 465
551 511
604 558
693 680
419 399
354 334
1.7 1.7
-0.15 -0.15
2.7 2.7
0.44 0.88
468 463
511 514
567 574
675 676
423 405
366 328
1.7 1.7
-0.15 -0.15
2.7 3.6
1.32 0.44
462 478
517 529
579 574
675 690
400 417
353 354
1.7 1.7
-0.15 -0.15
3.6 3.6
0.88 1.32
499 458
541 511
605 571
695 673
427 415
382 346
1.7 1.7
-0.15 -0.15
4.5 4.5
0.44 0.88
451 476
493 527
535 590
665 693
392 407
336 354
1.7 1.8
-0.15 0
4.5 2.7
1.32 0.44
482 601
530 650
575 707
694 773
422 534
364 469
1.8 1.8
0 0
2.7 2.7
0.88 1.32
625 649
661 691
704 727
767 785
566 573
506 512
1.8
0
3.6
0.44
631
685
721
782
559
497
1.8 1.8
0 0
3.6 3.6
0.88 1.32
658 634
693 676
717 718
797 782
589 568
534 525
1.8 1.8
0 0
4.5 4.5
0.44 0.88
626 624
668 671
707 699
774 783
561 549
490 470
1.8 1.8
0 -0.075
4.5 2.7
1.32 0.44
640 639
681 679
720 722
789 805
582 563
524 485
1.8 1.8
-0.075 -0.075
2.7 2.7
0.88 1.32
619 638
657 674
704 713
772 783
561 556
492 502
1.8 1.8
-0.075 -0.075
3.6 3.6
0.44 0.88
647 626
680 661
709 709
763 781
563 570
509 492
1.8 1.8
-0.075 -0.075
3.6 4.5
1.32 0.44
643 623
686 665
718 719
806 795
574 562
524 517
1.8
-0.075
4.5
0.88
654
688
728
782
578
520
Continued on next page 21
Simulation of 5% Normality tests on α-stable samples of size 50 (1000 replications) continued Number of replications where normality hypothesis accepted α-Stable Parameters α β γ δ
AndersonDarling
Cramervon Mises
Lilliefors
Pearson
ShapiroWilk
1.8
-0.075
4.5
1.32
650
690
714
789
582
521
1.8 1.8
-0.15 -0.15
2.7 2.7
0.44 0.88
654 598
679 646
724 692
798 769
584 542
530 483
1.8 1.8
-0.15 -0.15
2.7 3.6
1.32 0.44
657 623
690 656
727 691
785 762
583 555
507 492
1.8 1.8
-0.15 -0.15
3.6 3.6
0.88 1.32
641 648
685 688
746 714
786 770
578 580
510 497
1.8
-0.15
4.5
0.44
638
681
728
789
571
510
1.8 1.8
-0.15 -0.15
4.5 4.5
0.88 1.32
619 628
662 653
704 701
786 773
556 573
494 512
22
JarqueBera
Table 5: Simulation of 5% Normality tests on α-stable samples of size 100 (1000 replications) Number of replications where normality hypothesis accepted α-Stable Parameters α β γ δ
AndersonDarling
Cramervon Mises
Lilliefors
Pearson
ShapiroWilk
JarqueBera
1.6
0
2.7
0.44
134
229
217
343
98
67
1.6 1.6
0 0
2.7 2.7
0.88 1.32
110 103
215 216
177 181
307 314
74 78
52 55
1.6 1.6
0 0
3.6 3.6
0.44 0.88
93 113
201 229
177 189
319 309
68 81
51 51
1.6 1.6
0 0
3.6 4.5
1.32 0.44
117 104
227 242
198 195
324 332
77 76
49 52
1.6 1.6
0 0
4.5 4.5
0.88 1.32
94 95
209 225
159 176
281 313
68 75
49 55
1.6 1.6
-0.075 -0.075
2.7 2.7
0.44 0.88
119 110
249 216
186 199
317 323
90 78
68 58
1.6 1.6
-0.075 -0.075
2.7 3.6
1.32 0.44
100 121
225 242
180 187
318 304
69 97
47 71
1.6
-0.075
3.6
0.88
116
243
195
350
82
57
1.6 1.6
-0.075 -0.075
3.6 4.5
1.32 0.44
98 115
220 242
182 193
305 303
63 81
45 61
1.6 1.6
-0.075 -0.075
4.5 4.5
0.88 1.32
115 89
240 196
196 200
348 333
74 62
52 46
1.6 1.6
-0.15 -0.15
2.7 2.7
0.44 0.88
119 98
248 221
198 185
330 297
83 73
64 52
1.6 1.6
-0.15 -0.15
2.7 3.6
1.32 0.44
101 101
225 218
181 176
298 307
79 73
51 56
1.6 1.6
-0.15 -0.15
3.6 3.6
0.88 1.32
97 106
202 229
182 190
320 324
68 73
55 50
1.6 1.6
-0.15 -0.15
4.5 4.5
0.44 0.88
110 114
239 227
199 199
329 328
74 85
61 52
1.6 1.7
-0.15 0
4.5 2.7
1.32 0.44
113 254
241 351
169 366
297 506
82 202
52 146
1.7
0
2.7
0.88
273
354
371
505
202
141
1.7 1.7
0 0
2.7 3.6
1.32 0.44
215 246
317 351
333 367
482 497
155 175
116 122
1.7 1.7
0 0
3.6 3.6
0.88 1.32
240 231
320 332
352 358
483 513
158 169
109 130
1.7 1.7
0 0
4.5 4.5
0.44 0.88
240 264
338 342
353 373
493 528
170 180
127 135
Continued on next page
23
Simulation of 5% Normality tests on α-stable samples of size 100 (1000 replications) continued Number of replications where normality hypothesis accepted α-Stable Parameters α β γ δ
AndersonDarling
Cramervon Mises
Lilliefors
Pearson
ShapiroWilk
JarqueBera
1.7
0
4.5
1.32
262
348
375
498
183
128
1.7 1.7
-0.075 -0.075
2.7 2.7
0.44 0.88
234 257
318 317
347 338
486 498
164 191
120 138
1.7 1.7
-0.075 -0.075
2.7 3.6
1.32 0.44
250 246
342 334
377 350
519 494
188 187
153 148
1.7 1.7
-0.075 -0.075
3.6 3.6
0.88 1.32
240 263
344 353
344 359
504 497
181 170
137 126
1.7
-0.075
4.5
0.44
235
330
343
502
163
127
1.7 1.7
-0.075 -0.075
4.5 4.5
0.88 1.32
246 235
340 322
365 345
509 496
186 171
140 132
1.7 1.7
-0.15 -0.15
2.7 2.7
0.44 0.88
247 219
337 320
361 353
519 491
185 174
144 134
1.7 1.7
-0.15 -0.15
2.7 3.6
1.32 0.44
230 272
334 365
337 387
489 518
150 191
109 145
1.7 1.7
-0.15 -0.15
3.6 3.6
0.88 1.32
232 268
322 360
345 385
482 523
179 194
134 145
1.7 1.7
-0.15 -0.15
4.5 4.5
0.44 0.88
248 257
346 352
366 381
503 503
181 191
144 144
1.7 1.8
-0.15 0
4.5 2.7
1.32 0.44
254 453
358 531
360 572
494 670
169 345
124 274
1.8 1.8
0 0
2.7 2.7
0.88 1.32
472 419
538 505
577 548
672 656
364 331
269 270
1.8
0
3.6
0.44
465
525
563
686
365
304
1.8 1.8
0 0
3.6 3.6
0.88 1.32
445 472
511 546
543 579
665 670
341 361
287 284
1.8 1.8
0 0
4.5 4.5
0.44 0.88
463 447
533 509
575 556
682 669
370 346
298 291
1.8 1.8
0 -0.075
4.5 2.7
1.32 0.44
487 465
551 537
590 582
693 672
373 359
305 309
1.8 1.8
-0.075 -0.075
2.7 2.7
0.88 1.32
441 463
523 534
561 579
668 674
339 352
267 271
1.8 1.8
-0.075 -0.075
3.6 3.6
0.44 0.88
479 461
536 525
595 565
695 680
375 344
310 267
1.8 1.8
-0.075 -0.075
3.6 4.5
1.32 0.44
434 461
509 553
581 579
678 690
350 359
288 305
1.8
-0.075
4.5
0.88
447
521
559
668
352
288
Continued on next page 24
Simulation of 5% Normality tests on α-stable samples of size 100 (1000 replications) continued Number of replications where normality hypothesis accepted α-Stable Parameters α β γ δ
AndersonDarling
Cramervon Mises
Lilliefors
Pearson
ShapiroWilk
1.8
-0.075
4.5
1.32
445
535
578
671
349
270
1.8 1.8
-0.15 -0.15
2.7 2.7
0.44 0.88
472 452
534 525
581 578
699 677
365 348
285 278
1.8 1.8
-0.15 -0.15
2.7 3.6
1.32 0.44
457 449
534 525
585 549
693 661
368 334
297 267
1.8 1.8
-0.15 -0.15
3.6 3.6
0.88 1.32
472 461
543 530
581 559
708 663
359 353
295 294
1.8
-0.15
4.5
0.44
475
550
594
691
342
274
1.8 1.8
-0.15 -0.15
4.5 4.5
0.88 1.32
448 470
525 548
562 574
682 695
370 366
308 304
25
JarqueBera
Table 6: Simulation of 5% Normality tests on α-stable samples of size 200 (1000 replications) Number of replications where normality hypothesis accepted α-Stable Parameters α β γ δ
AndersonDarling
Cramervon Mises
Lilliefors
Pearson
ShapiroWilk
JarqueBera
1.6
0
2.7
0.44
11
271
47
102
5
1
1.6 1.6
0 0
2.7 2.7
0.88 1.32
16 15
283 287
34 37
112 99
11 9
6 8
1.6 1.6
0 0
3.6 3.6
0.44 0.88
9 10
280 288
33 30
84 106
5 7
3 4
1.6 1.6
0 0
3.6 4.5
1.32 0.44
15 13
294 270
34 42
96 104
6 9
3 6
1.6 1.6
0 0
4.5 4.5
0.88 1.32
14 12
288 270
26 34
83 98
9 3
5 3
1.6 1.6
-0.075 -0.075
2.7 2.7
0.44 0.88
10 9
279 276
38 30
102 106
4 7
3 3
1.6 1.6
-0.075 -0.075
2.7 3.6
1.32 0.44
7 9
286 288
22 31
88 91
4 6
1 4
1.6
-0.075
3.6
0.88
12
276
27
87
7
4
1.6 1.6
-0.075 -0.075
3.6 4.5
1.32 0.44
12 14
289 294
32 45
104 124
7 9
4 6
1.6 1.6
-0.075 -0.075
4.5 4.5
0.88 1.32
14 7
295 262
46 28
113 88
7 3
2 2
1.6 1.6
-0.15 -0.15
2.7 2.7
0.44 0.88
8 8
277 272
31 27
105 87
3 5
2 3
1.6 1.6
-0.15 -0.15
2.7 3.6
1.32 0.44
9 8
291 262
32 43
93 116
4 5
4 3
1.6 1.6
-0.15 -0.15
3.6 3.6
0.88 1.32
10 5
260 295
34 23
102 91
7 2
2 0
1.6 1.6
-0.15 -0.15
4.5 4.5
0.44 0.88
13 10
285 283
35 34
90 103
5 3
1 1
1.6 1.7
-0.15 0
4.5 2.7
1.32 0.44
6 56
294 224
28 128
93 259
4 30
2 19
1.7
0
2.7
0.88
58
244
141
264
35
20
1.7 1.7
0 0
2.7 3.6
1.32 0.44
64 61
256 233
132 133
272 275
33 29
20 21
1.7 1.7
0 0
3.6 3.6
0.88 1.32
59 59
238 217
151 125
267 234
26 28
20 25
1.7 1.7
0 0
4.5 4.5
0.44 0.88
68 57
236 241
138 131
253 259
34 26
21 18
Continued on next page
26
Simulation of 5% Normality tests on α-stable samples of size 200 (1000 replications) continued Number of replications where normality hypothesis accepted α-Stable Parameters α β γ δ
AndersonDarling
Cramervon Mises
Lilliefors
Pearson
ShapiroWilk
JarqueBera
1.7
0
4.5
1.32
61
244
128
269
30
18
1.7 1.7
-0.075 -0.075
2.7 2.7
0.44 0.88
75 66
211 242
156 147
308 288
44 41
29 26
1.7 1.7
-0.075 -0.075
2.7 3.6
1.32 0.44
72 66
226 224
140 135
272 258
38 30
18 13
1.7 1.7
-0.075 -0.075
3.6 3.6
0.88 1.32
66 52
228 211
130 111
245 246
37 28
21 15
1.7
-0.075
4.5
0.44
68
228
133
265
41
23
1.7 1.7
-0.075 -0.075
4.5 4.5
0.88 1.32
71 67
214 238
143 133
272 258
41 32
24 16
1.7 1.7
-0.15 -0.15
2.7 2.7
0.44 0.88
52 66
218 214
129 136
266 275
30 36
20 20
1.7 1.7
-0.15 -0.15
2.7 3.6
1.32 0.44
55 57
203 240
134 124
264 253
35 35
19 21
1.7 1.7
-0.15 -0.15
3.6 3.6
0.88 1.32
55 79
225 255
112 157
232 296
27 40
13 23
1.7 1.7
-0.15 -0.15
4.5 4.5
0.44 0.88
68 60
222 234
141 133
276 270
36 32
27 20
1.7 1.8
-0.15 0
4.5 2.7
1.32 0.44
57 225
219 348
131 364
267 505
28 128
14 84
1.8 1.8
0 0
2.7 2.7
0.88 1.32
247 230
375 365
360 349
508 515
151 133
107 94
1.8
0
3.6
0.44
241
351
370
551
149
101
1.8 1.8
0 0
3.6 3.6
0.88 1.32
230 232
345 347
366 360
541 513
133 156
98 114
1.8 1.8
0 0
4.5 4.5
0.44 0.88
240 243
367 354
363 349
503 519
131 142
101 101
1.8 1.8
0 -0.075
4.5 2.7
1.32 0.44
203 233
323 359
344 365
507 523
114 139
79 97
1.8 1.8
-0.075 -0.075
2.7 2.7
0.88 1.32
233 242
363 380
357 366
493 536
115 146
84 110
1.8 1.8
-0.075 -0.075
3.6 3.6
0.44 0.88
227 239
340 353
345 369
524 498
134 137
89 101
1.8 1.8
-0.075 -0.075
3.6 4.5
1.32 0.44
238 234
377 346
379 375
511 523
161 135
108 107
1.8
-0.075
4.5
0.88
208
335
306
477
118
84
Continued on next page 27
Simulation of 5% Normality tests on α-stable samples of size 200 (1000 replications) continued Number of replications where normality hypothesis accepted α-Stable Parameters α β γ δ
AndersonDarling
Cramervon Mises
Lilliefors
Pearson
ShapiroWilk
1.8
-0.075
4.5
1.32
200
315
342
504
119
90
1.8 1.8
-0.15 -0.15
2.7 2.7
0.44 0.88
217 226
341 362
366 354
523 515
133 138
101 96
1.8 1.8
-0.15 -0.15
2.7 3.6
1.32 0.44
195 191
309 312
313 343
474 509
123 106
90 72
1.8 1.8
-0.15 -0.15
3.6 3.6
0.88 1.32
219 239
347 354
347 354
496 526
126 151
93 107
1.8
-0.15
4.5
0.44
224
341
367
519
135
92
1.8 1.8
-0.15 -0.15
4.5 4.5
0.88 1.32
219 257
333 384
360 389
518 537
137 146
91 100
28
JarqueBera
Table 7: Simulation of 1% Normality tests on α-stable samples of size 50 (1000 replications) Number of replications where normality hypothesis accepted α-Stable Parameters α β γ δ
AndersonDarling
Cramervon Mises
Lilliefors
Pearson
ShapiroWilk
JarqueBera
1.6
0
2.7
0.44
417
489
541
642
368
312
1.6 1.6
0 0
2.7 2.7
0.88 1.32
440 447
505 493
570 555
695 674
381 389
327 335
1.6 1.6
0 0
3.6 3.6
0.44 0.88
403 470
456 525
540 593
647 697
345 401
300 340
1.6 1.6
0 0
3.6 4.5
1.32 0.44
417 448
481 494
543 549
652 652
365 371
319 318
1.6 1.6
0 0
4.5 4.5
0.88 1.32
438 432
499 481
563 552
667 668
360 351
302 298
1.6 1.6
-0.075 -0.075
2.7 2.7
0.44 0.88
426 411
489 465
560 538
662 668
366 335
316 291
1.6 1.6
-0.075 -0.075
2.7 3.6
1.32 0.44
443 432
503 485
546 551
660 662
383 358
323 293
1.6
-0.075
3.6
0.88
439
491
570
685
388
337
1.6 1.6
-0.075 -0.075
3.6 4.5
1.32 0.44
445 438
510 488
571 565
670 677
383 372
319 330
1.6 1.6
-0.075 -0.075
4.5 4.5
0.88 1.32
420 439
477 488
540 559
665 664
350 379
305 315
1.6 1.6
-0.15 -0.15
2.7 2.7
0.44 0.88
425 447
485 507
548 563
636 676
364 391
322 342
1.6 1.6
-0.15 -0.15
2.7 3.6
1.32 0.44
419 446
464 507
535 555
649 672
341 372
299 311
1.6 1.6
-0.15 -0.15
3.6 3.6
0.88 1.32
429 416
482 481
554 549
646 649
351 353
308 295
1.6 1.6
-0.15 -0.15
4.5 4.5
0.44 0.88
404 443
461 497
515 558
634 643
342 366
269 309
1.6 1.7
-0.15 0
4.5 2.7
1.32 0.44
415 590
476 640
535 694
637 764
349 511
300 436
1.7
0
2.7
0.88
589
646
704
806
531
456
1.7 1.7
0 0
2.7 3.6
1.32 0.44
571 567
614 613
676 660
766 758
478 490
420 434
1.7 1.7
0 0
3.6 3.6
0.88 1.32
566 585
611 642
684 715
787 780
503 504
441 435
1.7 1.7
0 0
4.5 4.5
0.44 0.88
584 596
635 645
698 698
764 777
499 512
434 440
Continued on next page
29
Simulation of 1% Normality tests on α-stable samples of size 50 (1000 replications) continued Number of replications where normality hypothesis accepted α-Stable Parameters α β γ δ
AndersonDarling
Cramervon Mises
Lilliefors
Pearson
ShapiroWilk
JarqueBera
1.7
0
4.5
1.32
597
649
708
805
520
460
1.7 1.7
-0.075 -0.075
2.7 2.7
0.44 0.88
576 580
624 638
676 687
773 772
512 490
444 439
1.7 1.7
-0.075 -0.075
2.7 3.6
1.32 0.44
562 585
600 636
671 690
767 779
484 515
432 438
1.7 1.7
-0.075 -0.075
3.6 3.6
0.88 1.32
561 584
598 638
672 689
762 783
486 500
411 435
1.7
-0.075
4.5
0.44
600
647
700
777
527
458
1.7 1.7
-0.075 -0.075
4.5 4.5
0.88 1.32
612 585
658 634
704 696
784 787
517 500
451 421
1.7 1.7
-0.15 -0.15
2.7 2.7
0.44 0.88
581 581
632 624
711 699
772 789
503 507
454 441
1.7 1.7
-0.15 -0.15
2.7 3.6
1.32 0.44
591 605
640 640
706 719
790 807
498 522
439 460
1.7 1.7
-0.15 -0.15
3.6 3.6
0.88 1.32
608 592
660 637
708 699
793 774
527 500
473 446
1.7 1.7
-0.15 -0.15
4.5 4.5
0.44 0.88
552 597
603 647
666 706
773 790
480 507
431 440
1.7 1.8
-0.15 0
4.5 2.7
1.32 0.44
588 725
641 763
693 816
788 874
507 629
456 570
1.8 1.8
0 0
2.7 2.7
0.88 1.32
731 751
765 792
812 833
858 876
664 663
594 607
1.8
0
3.6
0.44
750
792
820
871
664
591
1.8 1.8
0 0
3.6 3.6
0.88 1.32
754 738
781 773
813 819
864 861
675 666
618 601
1.8 1.8
0 0
4.5 4.5
0.44 0.88
741 720
773 757
817 811
854 864
648 640
594 578
1.8 1.8
0 -0.075
4.5 2.7
1.32 0.44
734 749
774 786
827 836
882 899
665 671
609 592
1.8 1.8
-0.075 -0.075
2.7 2.7
0.88 1.32
717 734
751 776
806 817
853 887
646 659
590 592
1.8 1.8
-0.075 -0.075
3.6 3.6
0.44 0.88
748 721
781 767
829 816
872 863
668 648
594 601
1.8 1.8
-0.075 -0.075
3.6 4.5
1.32 0.44
751 740
774 779
833 828
881 882
677 652
616 591
1.8
-0.075
4.5
0.88
741
777
821
870
673
607
Continued on next page 30
Simulation of 1% Normality tests on α-stable samples of size 50 (1000 replications) continued Number of replications where normality hypothesis accepted α-Stable Parameters α β γ δ
AndersonDarling
Cramervon Mises
Lilliefors
Pearson
ShapiroWilk
1.8
-0.075
4.5
1.32
748
768
808
864
664
622
1.8 1.8
-0.15 -0.15
2.7 2.7
0.44 0.88
758 715
789 745
835 793
879 857
675 625
613 568
1.8 1.8
-0.15 -0.15
2.7 3.6
1.32 0.44
755 716
784 749
829 803
873 863
678 642
622 583
1.8 1.8
-0.15 -0.15
3.6 3.6
0.88 1.32
746 733
783 776
831 807
876 858
670 665
604 596
1.8
-0.15
4.5
0.44
747
778
824
866
660
599
1.8 1.8
-0.15 -0.15
4.5 4.5
0.88 1.32
739 723
775 767
814 817
880 872
658 652
587 594
31
JarqueBera
Table 8: Simulation of 1% Normality tests on α-stable samples of size 100 (1000 replications) Number of replications where normality hypothesis accepted α-Stable Parameters α β γ δ
AndersonDarling
Cramervon Mises
Lilliefors
Pearson
ShapiroWilk
JarqueBera
1.6
0
2.7
0.44
6
191
20
268
3
52
1.6 1.6
0 0
2.7 2.7
0.88 1.32
14 10
181 184
27 27
247 245
7 7
42 37
1.6 1.6
0 0
3.6 3.6
0.44 0.88
3 7
167 189
14 20
256 240
2 3
45 39
1.6 1.6
0 0
3.6 4.5
1.32 0.44
10 6
195 204
24 18
260 257
4 4
38 38
1.6 1.6
0 0
4.5 4.5
0.88 1.32
7 9
179 193
13 21
218 251
7 3
38 35
1.6 1.6
-0.075 -0.075
2.7 2.7
0.44 0.88
6 7
206 181
21 22
250 258
3 4
49 43
1.6 1.6
-0.075 -0.075
2.7 3.6
1.32 0.44
4 5
193 209
16 16
250 233
2 4
36 50
1.6
-0.075
3.6
0.88
9
212
18
271
6
46
1.6 1.6
-0.075 -0.075
3.6 4.5
1.32 0.44
7 9
191 213
20 28
242 235
5 5
32 45
1.6 1.6
-0.075 -0.075
4.5 4.5
0.88 1.32
10 5
198 160
23 16
273 272
5 1
39 35
1.6 1.6
0.15 0.15
2.7 2.7
0.44 0.88
4 6
207 180
13 13
264 226
1 3
51 39
1.6 1.6
0.15 0.15
2.7 3.6
1.32 0.44
4 5
191 194
17 20
235 229
3 4
38 42
1.6 1.6
0.15 0.15
3.6 3.6
0.88 1.32
7 4
177 190
15 13
249 254
4 2
41 41
1.6 1.6
0.15 0.15
4.5 4.5
0.44 0.88
8 10
206 192
25 21
252 262
4 1
49 33
1.6 1.7
0.15 0
4.5 2.7
1.32 0.44
4 45
206 288
16 92
225 430
3 23
39 123
1.7
0
2.7
0.88
42
286
100
430
24
108
1.7 1.7
0 0
2.7 3.6
1.32 0.44
42 38
263 285
82 90
396 418
24 27
95 102
1.7 1.7
0 0
3.6 3.6
0.88 1.32
40 38
275 255
96 78
408 437
22 21
86 108
1.7 1.7
0 0
4.5 4.5
0.44 0.88
50 35
283 293
94 89
421 444
24 19
99 117
Continued on next page
32
Simulation of 1% Normality tests on α-stable samples of size 100 (1000 replications) continued Number of replications where normality hypothesis accepted α-Stable Parameters α β γ δ
AndersonDarling
Cramervon Mises
Lilliefors
Pearson
ShapiroWilk
JarqueBera
1.7
0
4.5
1.32
42
289
88
413
23
105
1.7 1.7
-0.075 -0.075
2.7 2.7
0.44 0.88
55 46
273 269
109 103
398 413
36 31
97 106
1.7 1.7
-0.075 -0.075
2.7 3.6
1.32 0.44
53 41
285 285
97 84
437 396
26 20
113 123
1.7 1.7
-0.075 -0.075
3.6 3.6
0.88 1.32
51 31
279 293
89 79
417 411
28 21
115 99
1.7
-0.075
4.5
0.44
53
272
98
402
31
100
1.7 1.7
-0.075 -0.075
4.5 4.5
0.88 1.32
49 52
279 260
101 91
423 417
30 19
115 102
1.7 1.7
0.15 0.15
2.7 2.7
0.44 0.88
31 47
290 259
85 95
438 413
19 25
116 111
1.7 1.7
0.15 0.15
2.7 3.6
1.32 0.44
39 43
276 307
87 87
392 448
22 26
92 116
1.7 1.7
0.15 0.15
3.6 3.6
0.88 1.32
41 54
265 300
78 111
404 440
16 30
112 122
1.7 1.7
0.15 0.15
4.5 4.5
0.44 0.88
50 45
279 286
89 92
422 423
29 25
119 121
1.7 1.8
0.15 0
4.5 2.7
1.32 0.44
43 175
298 456
87 285
426 595
21 101
104 222
1.8 1.8
0 0
2.7 2.7
0.88 1.32
200 182
471 423
283 288
606 573
125 107
228 234
1.8
0
3.6
0.44
189
455
308
605
122
271
1.8 1.8
0 0
3.6 3.6
0.88 1.32
185 182
448 472
293 278
587 594
107 125
240 235
1.8 1.8
0 0
4.5 4.5
0.44 0.88
187 197
464 442
297 271
611 586
110 116
254 237
1.8 1.8
0 -0.075
4.5 2.7
1.32 0.44
148 190
483 454
257 290
610 605
95 105
262 260
1.8 1.8
-0.075 -0.075
2.7 2.7
0.88 1.32
175 183
456 457
274 289
595 588
91 119
231 226
1.8 1.8
-0.075 -0.075
3.6 3.6
0.44 0.88
173 183
462 459
263 290
612 595
106 105
262 223
1.8 1.8
-0.075 -0.075
3.6 4.5
1.32 0.44
192 170
432 482
303 287
577 607
133 111
245 263
1.8
-0.075
4.5
0.88
168
451
246
597
97
251
Continued on next page 33
Simulation of 1% Normality tests on α-stable samples of size 100 (1000 replications) continued Number of replications where normality hypothesis accepted α-Stable Parameters α β γ δ
AndersonDarling
Cramervon Mises
Lilliefors
Pearson
ShapiroWilk
1.8
-0.075
4.5
1.32
167
457
255
602
98
230
1.8 1.8
0.15 0.15
2.7 2.7
0.44 0.88
158 184
458 454
289 272
614 619
110 111
243 237
1.8 1.8
0.15 0.15
2.7 3.6
1.32 0.44
156 146
465 448
231 257
616 579
103 84
261 227
1.8 1.8
0.15 0.15
3.6 3.6
0.88 1.32
171 188
462 457
263 291
634 598
103 126
249 248
1.8
0.15
4.5
0.44
164
476
289
611
104
239
1.8 1.8
0.15 0.15
4.5 4.5
0.88 1.32
169 207
462 481
278 319
601 618
107 114
269 268
34
JarqueBera
Table 9: Simulation of 1% Normality tests on α-stable samples of size 200 (1000 replications) Number of replications where normality hypothesis accepted α-Stable Parameters α β γ δ
AndersonDarling
Cramervon Mises
Lilliefors
Pearson
ShapiroWilk
JarqueBera
1.6
0
2.7
0.44
24
298
90
177
11
6
1.6 1.6
0 0
2.7 2.7
0.88 1.32
28 31
313 302
80 79
183 160
16 17
10 10
1.6 1.6
0 0
3.6 3.6
0.44 0.88
24 26
301 314
69 74
148 184
8 15
5 11
1.6 1.6
0 0
3.6 4.5
1.32 0.44
29 28
313 304
74 85
159 164
7 20
6 15
1.6 1.6
0 0
4.5 4.5
0.88 1.32
24 22
302 290
75 74
155 170
13 13
9 3
1.6 1.6
-0.075 -0.075
2.7 2.7
0.44 0.88
25 19
307 300
82 75
165 177
13 10
9 9
1.6 1.6
-0.075 -0.075
2.7 3.6
1.32 0.44
19 18
309 306
63 58
161 161
6 10
3 7
1.6
-0.075
3.6
0.88
23
291
62
154
12
7
1.6 1.6
-0.075 -0.075
3.6 4.5
1.32 0.44
22 31
311 322
73 91
186 191
13 16
9 7
1.6 1.6
-0.075 -0.075
4.5 4.5
0.88 1.32
30 23
326 286
89 62
188 165
14 6
9 4
1.6 1.6
-0.15 -0.15
2.7 2.7
0.44 0.88
25 17
302 293
70 75
179 165
11 11
8 9
1.6 1.6
-0.15 -0.15
2.7 3.6
1.32 0.44
23 34
317 295
70 86
155 192
8 16
5 7
1.6 1.6
-0.15 -0.15
3.6 3.6
0.88 1.32
20 14
277 317
82 70
176 151
11 3
9 2
1.6 1.6
-0.15 -0.15
4.5 4.5
0.44 0.88
24 26
309 304
69 72
181 170
13 9
6 3
1.6 1.7
-0.15 0
4.5 2.7
1.32 0.44
18 106
316 294
62 228
164 371
10 51
5 33
1.7
0
2.7
0.88
116
304
227
392
59
37
1.7 1.7
0 0
2.7 3.6
1.32 0.44
109 103
317 307
242 246
381 387
56 51
34 35
1.7 1.7
0 0
3.6 3.6
0.88 1.32
111 107
304 279
247 212
377 355
47 55
29 33
1.7 1.7
0 0
4.5 4.5
0.44 0.88
115 103
296 306
219 219
377 364
56 47
42 27
Continued on next page
35
Simulation of 1% Normality tests on α-stable samples of size 200 (1000 replications) continued Number of replications where normality hypothesis accepted α-Stable Parameters α β γ δ
AndersonDarling
Cramervon Mises
Lilliefors
Pearson
ShapiroWilk
JarqueBera
1.7
0
4.5
1.32
109
306
219
372
60
35
1.7 1.7
-0.075 -0.075
2.7 2.7
0.44 0.88
128 131
273 316
259 256
428 390
62 63
47 46
1.7 1.7
-0.075 -0.075
2.7 3.6
1.32 0.44
121 114
297 284
235 228
377 380
60 50
40 32
1.7 1.7
-0.075 -0.075
3.6 3.6
0.88 1.32
93 97
279 274
224 212
357 367
52 47
39 29
1.7
-0.075
4.5
0.44
108
294
226
379
71
39
1.7 1.7
-0.075 -0.075
4.5 4.5
0.88 1.32
127 108
286 297
247 223
389 378
72 62
42 31
1.7 1.7
-0.15 -0.15
2.7 2.7
0.44 0.88
99 111
288 279
237 220
377 381
50 53
32 36
1.7 1.7
-0.15 -0.15
2.7 3.6
1.32 0.44
94 104
269 305
235 231
389 370
47 52
34 40
1.7 1.7
-0.15 -0.15
3.6 3.6
0.88 1.32
95 127
280 330
197 262
359 404
49 62
33 46
1.7 1.7
-0.15 -0.15
4.5 4.5
0.44 0.88
111 104
282 290
252 240
394 381
53 50
41 30
1.7 1.8
-0.15 0
4.5 2.7
1.32 0.44
112 336
279 468
220 497
395 622
45 187
30 129
1.8 1.8
0 0
2.7 2.7
0.88 1.32
351 343
474 477
520 518
633 626
213 193
156 139
1.8
0
3.6
0.44
344
470
506
663
193
151
1.8 1.8
0 0
3.6 3.6
0.88 1.32
350 337
476 456
530 498
674 638
201 211
145 159
1.8 1.8
0 0
4.5 4.5
0.44 0.88
343 331
466 462
501 499
616 646
197 195
140 146
1.8 1.8
0 -0.075
4.5 2.7
1.32 0.44
300 336
437 454
486 494
619 647
170 191
116 135
1.8 1.8
-0.075 -0.075
2.7 2.7
0.88 1.32
330 346
476 497
492 530
612 662
195 199
125 154
1.8 1.8
-0.075 -0.075
3.6 3.6
0.44 0.88
322 335
464 463
500 495
639 623
194 185
135 137
1.8 1.8
-0.075 -0.075
3.6 4.5
1.32 0.44
364 341
491 472
504 505
636 647
214 197
162 141
1.8
-0.075
4.5
0.88
301
438
452
608
176
123
Continued on next page 36
Simulation of 1% Normality tests on α-stable samples of size 200 (1000 replications) continued Number of replications where normality hypothesis accepted α-Stable Parameters α β γ δ
AndersonDarling
Cramervon Mises
Lilliefors
Pearson
ShapiroWilk
1.8
-0.075
4.5
1.32
305
449
492
636
171
129
1.8 1.8
-0.15 -0.15
2.7 2.7
0.44 0.88
330 330
472 463
511 498
631 625
190 192
145 138
1.8 1.8
-0.15 -0.15
2.7 3.6
1.32 0.44
277 299
415 429
440 484
595 643
169 164
124 120
1.8 1.8
-0.15 -0.15
3.6 3.6
0.88 1.32
316 337
457 467
481 501
630 657
174 206
137 153
1.8
-0.15
4.5
0.44
321
459
493
630
179
134
1.8 1.8
-0.15 -0.15
4.5 4.5
0.88 1.32
318 361
459 488
503 520
642 637
182 218
130 150
37
JarqueBera
Table 10: Simulation of 10% Normality tests on α-stable samples of size 50 (1000 replications) Number of replications where normality hypothesis accepted α-Stable Parameters α β γ δ
AndersonDarling
Cramervon Mises
Lilliefors
Pearson
ShapiroWilk
JarqueBera
1.6
0
2.7
0.44
259
311
332
442
233
195
1.6 1.6
0 0
2.7 2.7
0.88 1.32
271 285
317 338
374 367
467 476
253 253
197 210
1.6 1.6
0 0
3.6 3.6
0.44 0.88
234 286
287 336
327 382
445 487
221 246
185 197
1.6 1.6
0 0
3.6 4.5
1.32 0.44
270 283
321 335
343 372
438 486
249 232
211 182
1.6 1.6
0 0
4.5 4.5
0.88 1.32
266 257
302 304
353 341
471 449
234 220
189 168
1.6 1.6
-0.075 -0.075
2.7 2.7
0.44 0.88
255 252
295 299
350 324
447 431
231 224
189 172
1.6 1.6
-0.075 -0.075
2.7 3.6
1.32 0.44
279 253
322 293
359 356
463 460
237 221
188 176
1.6
-0.075
3.6
0.88
289
325
357
466
262
215
1.6 1.6
-0.075 -0.075
3.6 4.5
1.32 0.44
284 275
332 314
354 349
458 461
238 257
192 207
1.6 1.6
-0.075 -0.075
4.5 4.5
0.88 1.32
259 271
303 315
334 359
459 464
229 243
180 198
1.6 1.6
-0.15 -0.15
2.7 2.7
0.44 0.88
263 287
307 339
346 365
446 487
243 246
200 191
1.6 1.6
-0.15 -0.15
2.7 3.6
1.32 0.44
261 277
303 320
337 358
454 456
232 243
179 195
1.6 1.6
-0.15 -0.15
3.6 3.6
0.88 1.32
255 262
307 307
335 329
443 443
223 224
187 178
1.6 1.6
-0.15 -0.15
4.5 4.5
0.44 0.88
235 252
275 306
318 338
435 462
195 226
152 179
1.6 1.7
-0.15 0
4.5 2.7
1.32 0.44
250 428
291 483
330 513
435 572
223 361
181 298
1.7
0
2.7
0.88
405
446
490
611
359
302
1.7 1.7
0 0
2.7 3.6
1.32 0.44
392 386
437 431
473 470
582 572
354 343
289 283
1.7 1.7
0 0
3.6 3.6
0.88 1.32
407 401
443 445
495 501
574 589
339 366
275 308
1.7 1.7
0 0
4.5 4.5
0.44 0.88
385 412
425 459
490 496
583 600
346 364
278 294
Continued on next page
38
Simulation of 10% Normality tests on α-stable samples of size 50 (1000 replications) continued Number of replications where normality hypothesis accepted α-Stable Parameters α β γ δ
AndersonDarling
Cramervon Mises
Lilliefors
Pearson
ShapiroWilk
JarqueBera
1.7
0
4.5
1.32
419
460
501
609
372
309
1.7 1.7
-0.075 -0.075
2.7 2.7
0.44 0.88
410 403
451 441
489 502
591 605
353 365
279 298
1.7 1.7
-0.075 -0.075
2.7 3.6
1.32 0.44
387 408
431 448
474 493
580 584
360 346
309 279
1.7 1.7
-0.075 -0.075
3.6 3.6
0.88 1.32
375 401
406 455
469 505
571 613
337 349
275 290
1.7
-0.075
4.5
0.44
426
463
508
612
361
300
1.7 1.7
-0.075 -0.075
4.5 4.5
0.88 1.32
414 401
463 444
525 476
629 613
370 342
299 287
1.7 1.7
-0.15 -0.15
2.7 2.7
0.44 0.88
409 395
454 441
481 489
580 592
373 346
315 273
1.7 1.7
-0.15 -0.15
2.7 3.6
1.32 0.44
387 412
425 449
487 494
577 600
349 360
299 289
1.7 1.7
-0.15 -0.15
3.6 3.6
0.88 1.32
423 397
472 437
521 478
606 592
376 346
317 289
1.7 1.7
-0.15 -0.15
4.5 4.5
0.44 0.88
386 407
427 456
462 510
585 602
344 353
275 317
1.7 1.8
-0.15 0
4.5 2.7
1.32 0.44
421 532
465 562
502 625
607 702
371 471
300 418
1.8 1.8
0 0
2.7 2.7
0.88 1.32
561 574
593 612
627 647
705 715
507 515
453 448
1.8
0
3.6
0.44
551
601
643
703
502
428
1.8 1.8
0 0
3.6 3.6
0.88 1.32
583 568
621 605
641 633
717 692
530 522
471 465
1.8 1.8
0 0
4.5 4.5
0.44 0.88
561 550
603 595
632 627
701 718
504 493
435 425
1.8 1.8
0 -0.075
4.5 2.7
1.32 0.44
583 538
614 586
651 625
714 729
535 502
463 435
1.8 1.8
-0.075 -0.075
2.7 2.7
0.88 1.32
553 555
587 608
630 633
698 694
502 505
443 445
1.8 1.8
-0.075 -0.075
3.6 3.6
0.44 0.88
569 558
610 597
642 639
698 713
511 515
444 447
1.8 1.8
-0.075 -0.075
3.6 4.5
1.32 0.44
577 563
614 585
633 629
719 713
528 515
461 468
1.8
-0.075
4.5
0.88
577
617
660
725
517
450
Continued on next page 39
Simulation of 10% Normality tests on α-stable samples of size 50 (1000 replications) continued Number of replications where normality hypothesis accepted α-Stable Parameters α β γ δ
AndersonDarling
Cramervon Mises
Lilliefors
Pearson
ShapiroWilk
1.8
-0.075
4.5
1.32
573
610
640
721
523
457
1.8 1.8
-0.15 -0.15
2.7 2.7
0.44 0.88
587 535
619 574
650 611
731 711
525 480
472 424
1.8 1.8
-0.15 -0.15
2.7 3.6
1.32 0.44
577 554
607 596
649 628
703 691
521 502
451 419
1.8 1.8
-0.15 -0.15
3.6 3.6
0.88 1.32
575 581
606 619
650 642
727 702
514 522
448 446
1.8
-0.15
4.5
0.44
569
611
655
717
525
447
1.8 1.8
-0.15 -0.15
4.5 4.5
0.88 1.32
548 554
585 589
634 629
713 702
501 508
437 447
40
JarqueBera
Table 11: Simulation of 10% Normality tests on α-stable samples of size 100 (1000 replications) Number of replications where normality hypothesis accepted α-Stable Parameters α β γ δ
AndersonDarling
Cramervon Mises
Lilliefors
Pearson
ShapiroWilk
JarqueBera
1.6
0
2.7
0.44
99
191
163
268
69
52
1.6 1.6
0 0
2.7 2.7
0.88 1.32
86 81
181 184
129 130
247 245
58 58
42 37
1.6 1.6
0 0
3.6 3.6
0.44 0.88
70 79
167 189
128 137
256 240
50 60
45 39
1.6 1.6
0 0
3.6 4.5
1.32 0.44
82 75
195 204
137 145
260 257
61 52
38 38
1.6 1.6
0 0
4.5 4.5
0.88 1.32
72 73
179 193
124 125
218 251
54 59
38 35
1.6 1.6
-0.075 -0.075
2.7 2.7
0.44 0.88
88 76
206 181
145 137
250 258
70 60
49 43
1.6 1.6
-0.075 -0.075
2.7 3.6
1.32 0.44
75 95
193 209
133 141
250 233
52 79
36 50
1.6
-0.075
3.6
0.88
97
212
144
271
67
46
1.6 1.6
-0.075 -0.075
3.6 4.5
1.32 0.44
70 88
191 213
135 129
242 235
50 64
32 45
1.6 1.6
-0.075 -0.075
4.5 4.5
0.88 1.32
78 60
198 160
144 142
273 272
53 48
39 35
1.6 1.6
-0.15 -0.15
2.7 2.7
0.44 0.88
90 71
207 180
150 117
264 226
68 58
51 39
1.6 1.6
-0.15 -0.15
2.7 3.6
1.32 0.44
76 81
191 194
129 132
235 229
53 55
38 42
1.6 1.6
-0.15 -0.15
3.6 3.6
0.88 1.32
69 76
177 190
130 142
249 254
52 54
41 41
1.6 1.6
-0.15 -0.15
4.5 4.5
0.44 0.88
82 89
206 192
145 149
252 262
61 68
49 33
1.6 1.7
-0.15 0
4.5 2.7
1.32 0.44
85 203
206 288
131 287
225 430
63 165
39 123
1.7
0
2.7
0.88
193
286
288
430
162
108
1.7 1.7
0 0
2.7 3.6
1.32 0.44
167 190
263 285
266 291
396 418
126 142
95 102
1.7 1.7
0 0
3.6 3.6
0.88 1.32
184 187
275 255
268 279
408 437
130 136
86 108
1.7 1.7
0 0
4.5 4.5
0.44 0.88
192 211
283 293
283 290
421 444
139 155
99 117
Continued on next page
41
Simulation of 10% Normality tests on α-stable samples of size 100 (1000 replications) continued Number of replications where normality hypothesis accepted α-Stable Parameters α β γ δ
AndersonDarling
Cramervon Mises
Lilliefors
Pearson
ShapiroWilk
JarqueBera
1.7
0
4.5
1.32
196
289
295
413
148
105
1.7 1.7
-0.075 -0.075
2.7 2.7
0.44 0.88
179 201
273 269
254 281
398 413
128 147
97 106
1.7 1.7
-0.075 -0.075
2.7 3.6
1.32 0.44
191 212
285 285
297 290
437 396
155 152
113 123
1.7 1.7
-0.075 -0.075
3.6 3.6
0.88 1.32
189 196
279 293
276 297
417 411
143 142
115 99
1.7
-0.075
4.5
0.44
183
272
278
402
136
100
1.7 1.7
-0.075 -0.075
4.5 4.5
0.88 1.32
197 185
279 260
282 278
423 417
147 139
115 102
1.7 1.7
-0.15 -0.15
2.7 2.7
0.44 0.88
199 181
290 259
278 269
438 413
152 143
116 111
1.7 1.7
-0.15 -0.15
2.7 3.6
1.32 0.44
178 207
276 307
270 301
392 448
130 160
92 116
1.7 1.7
-0.15 -0.15
3.6 3.6
0.88 1.32
188 196
265 300
269 309
404 440
154 159
112 122
1.7 1.7
-0.15 -0.15
4.5 4.5
0.44 0.88
186 203
279 286
284 299
422 423
147 163
119 121
1.7 1.8
-0.15 0
4.5 2.7
1.32 0.44
191 376
298 456
298 483
426 595
138 288
104 222
1.8 1.8
0 0
2.7 2.7
0.88 1.32
381 360
471 423
500 455
606 573
308 287
228 234
1.8
0
3.6
0.44
380
455
470
605
316
271
1.8 1.8
0 0
3.6 3.6
0.88 1.32
386 393
448 472
468 500
587 594
302 305
240 235
1.8 1.8
0 0
4.5 4.5
0.44 0.88
393 373
464 442
502 460
611 586
314 297
254 237
1.8 1.8
0 -0.075
4.5 2.7
1.32 0.44
411 404
483 454
495 498
610 605
326 314
262 260
1.8 1.8
-0.075 -0.075
2.7 2.7
0.88 1.32
380 391
456 457
482 502
595 588
281 302
231 226
1.8 1.8
-0.075 -0.075
3.6 3.6
0.44 0.88
403 389
462 459
510 481
612 595
320 304
262 223
1.8 1.8
-0.075 -0.075
3.6 4.5
1.32 0.44
374 384
432 482
474 490
577 607
299 322
245 263
1.8
-0.075
4.5
0.88
371
451
479
597
305
251
Continued on next page 42
Simulation of 10% Normality tests on α-stable samples of size 100 (1000 replications) continued Number of replications where normality hypothesis accepted α-Stable Parameters α β γ δ
AndersonDarling
Cramervon Mises
Lilliefors
Pearson
ShapiroWilk
1.8
-0.075
4.5
1.32
392
457
495
602
296
230
1.8 1.8
-0.15 -0.15
2.7 2.7
0.44 0.88
400 392
458 454
498 477
614 619
307 302
243 237
1.8 1.8
-0.15 -0.15
2.7 3.6
1.32 0.44
392 364
465 448
495 470
616 579
315 286
261 227
1.8 1.8
-0.15 -0.15
3.6 3.6
0.88 1.32
392 374
462 457
495 477
634 598
320 309
249 248
1.8
-0.15
4.5
0.44
397
476
505
611
296
239
1.8 1.8
-0.15 -0.15
4.5 4.5
0.88 1.32
391 414
462 481
463 499
601 618
320 324
269 268
43
JarqueBera
Table 12: Simulation of 10% Normality tests on α-stable samples of size 200 (1000 replications) Number of replications where normality hypothesis accepted α-Stable Parameters α β γ δ
AndersonDarling
Cramervon Mises
Lilliefors
Pearson
ShapiroWilk
JarqueBera
1.6
0
2.7
0.44
6
262
20
84
3
1
1.6 1.6
0 0
2.7 2.7
0.88 1.32
14 10
276 278
27 27
89 73
7 7
3 6
1.6 1.6
0 0
3.6 3.6
0.44 0.88
3 7
269 278
14 20
65 77
2 3
2 1
1.6 1.6
0 0
3.6 4.5
1.32 0.44
10 6
287 263
24 18
68 74
4 4
1 3
1.6 1.6
0 0
4.5 4.5
0.88 1.32
7 9
282 267
13 21
58 77
7 3
4 2
1.6 1.6
-0.075 -0.075
2.7 2.7
0.44 0.88
6 7
269 271
21 22
75 73
3 4
2 1
1.6 1.6
-0.075 -0.075
2.7 3.6
1.32 0.44
4 5
284 282
16 16
67 63
2 4
1 2
1.6
-0.075
3.6
0.88
9
272
18
69
6
4
1.6 1.6
-0.075 -0.075
3.6 4.5
1.32 0.44
7 9
282 281
20 28
77 80
5 5
3 5
1.6 1.6
-0.075 -0.075
4.5 4.5
0.88 1.32
10 5
281 252
23 16
81 60
5 1
1 1
1.6 1.6
-0.15 -0.15
2.7 2.7
0.44 0.88
4 6
266 265
13 13
74 62
1 3
2 2
1.6 1.6
-0.15 -0.15
2.7 3.6
1.32 0.44
4 5
283 251
17 20
72 86
3 4
3 2
1.6 1.6
-0.15 -0.15
3.6 3.6
0.88 1.32
7 4
250 284
15 13
69 61
4 2
2 0
1.6 1.6
-0.15 -0.15
4.5 4.5
0.44 0.88
8 10
273 265
25 21
65 79
4 1
0 0
1.6 1.7
-0.15 0
4.5 2.7
1.32 0.44
4 45
288 204
16 92
63 199
3 23
1 18
1.7
0
2.7
0.88
42
217
100
195
24
15
1.7 1.7
0 0
2.7 3.6
1.32 0.44
42 38
225 208
82 90
211 213
24 27
14 16
1.7 1.7
0 0
3.6 3.6
0.88 1.32
40 38
211 200
96 78
215 186
22 21
17 14
1.7 1.7
0 0
4.5 4.5
0.44 0.88
50 35
215 220
94 89
206 203
24 19
16 12
Continued on next page
44
Simulation of 10% Normality tests on α-stable samples of size 200 (1000 replications) continued Number of replications where normality hypothesis accepted α-Stable Parameters α β γ δ
AndersonDarling
Cramervon Mises
Lilliefors
Pearson
ShapiroWilk
JarqueBera
1.7
0
4.5
1.32
42
215
88
200
23
14
1.7 1.7
-0.075 -0.075
2.7 2.7
0.44 0.88
55 46
175 208
109 103
243 218
36 31
19 19
1.7 1.7
-0.075 -0.075
2.7 3.6
1.32 0.44
53 41
206 200
97 84
217 207
26 20
15 9
1.7 1.7
-0.075 -0.075
3.6 3.6
0.88 1.32
51 31
204 193
89 79
195 192
28 21
13 14
1.7
-0.075
4.5
0.44
53
211
98
205
31
16
1.7 1.7
-0.075 -0.075
4.5 4.5
0.88 1.32
49 52
192 216
101 91
215 203
30 19
15 10
1.7 1.7
-0.15 -0.15
2.7 2.7
0.44 0.88
31 47
189 185
85 95
208 213
19 25
18 17
1.7 1.7
-0.15 -0.15
2.7 3.6
1.32 0.44
39 43
179 217
87 87
219 196
22 26
14 18
1.7 1.7
-0.15 -0.15
3.6 3.6
0.88 1.32
41 54
206 234
78 111
164 230
16 30
13 16
1.7 1.7
-0.15 -0.15
4.5 4.5
0.44 0.88
50 45
197 203
89 92
212 224
29 25
22 11
1.7 1.8
-0.15 0
4.5 2.7
1.32 0.44
43 175
193 295
87 285
199 433
21 101
11 66
1.8 1.8
0 0
2.7 2.7
0.88 1.32
200 182
311 297
283 288
426 443
125 107
89 77
1.8
0
3.6
0.44
189
297
308
469
122
84
1.8 1.8
0 0
3.6 3.6
0.88 1.32
185 182
282 298
293 278
465 425
107 125
77 86
1.8 1.8
0 0
4.5 4.5
0.44 0.88
187 197
306 306
297 271
419 434
110 116
86 86
1.8 1.8
0 -0.075
4.5 2.7
1.32 0.44
148 190
266 306
257 290
419 439
95 105
68 72
1.8 1.8
-0.075 -0.075
2.7 2.7
0.88 1.32
175 183
312 322
274 289
432 460
91 119
69 85
1.8 1.8
-0.075 -0.075
3.6 3.6
0.44 0.88
173 183
278 289
263 290
427 427
106 105
75 83
1.8 1.8
-0.075 -0.075
3.6 4.5
1.32 0.44
192 170
320 280
303 287
435 448
133 111
94 87
1.8
-0.075
4.5
0.88
168
285
246
405
97
72
Continued on next page 45
Simulation of 10% Normality tests on α-stable samples of size 200 (1000 replications) continued Number of replications where normality hypothesis accepted α-Stable Parameters α β γ δ
AndersonDarling
Cramervon Mises
Lilliefors
Pearson
ShapiroWilk
1.8
-0.075
4.5
1.32
167
267
255
422
98
73
1.8 1.8
-0.15 -0.15
2.7 2.7
0.44 0.88
158 184
281 304
289 272
445 437
110 111
81 84
1.8 1.8
-0.15 -0.15
2.7 3.6
1.32 0.44
156 146
262 248
231 257
396 424
103 84
71 52
1.8 1.8
-0.15 -0.15
3.6 3.6
0.88 1.32
171 188
285 296
263 291
436 448
103 126
70 87
1.8
-0.15
4.5
0.44
164
271
289
437
104
78
1.8 1.8
-0.15 -0.15
4.5 4.5
0.88 1.32
169 207
274 322
278 319
429 470
107 114
73 85
46
JarqueBera
Table 13: Simulation of Normality tests on a normal distribution (1000 replications) Number of replications where normality hypothesis accepted Simulation details
Test
sample
test
Anderson-
Cramer-
Shapiro-
Jarque-
size
size
st.dev
mean
Darling
von Mises
Lilliefors
Pearson
Wilk
Bera
50
5
3.8
0.44
946
950
942
946
934
947
50 50
5 5
3.8 3.8
0.88 1.32
946 938
939 938
955 947
955 944
951 933
940 928
50
5
5.1
0.44
935
934
932
933
934
933
50 50
5 5
5.1 5.1
0.88 1.32
946 957
942 951
945 953
958 944
944 964
946 949
50 50
5 5
6.4 6.4
0.44 0.88
937 938
935 926
937 942
944 938
940 948
941 951
50 100
5 5
6.4 3.8
1.32 0.44
951 953
951 955
953 958
954 952
958 951
951 944
100 100
5 5
3.8 3.8
0.88 1.32
948 949
948 948
944 952
950 947
952 945
943 938
100 100
5 5
5.1 5.1
0.44 0.88
954 953
954 954
954 965
949 953
945 958
936 941
100 100
5 5
5.1 6.4
1.32 0.44
953 944
953 942
944 941
961 942
956 945
949 933
100 100
5 5
6.4 6.4
0.88 1.32
942 952
931 947
936 947
960 948
941 957
935 950
200 200
5 5
3.8 3.8
0.44 0.88
949 940
953 947
954 943
956 946
953 948
948 943
200
5
3.8
1.32
949
951
953
939
949
937
200 200
5 5
5.1 5.1
0.44 0.88
952 952
952 953
956 953
941 944
945 957
954 948
200 200
5 5
5.1 6.4
1.32 0.44
970 956
967 954
961 949
951 949
952 963
953 955
200 200
5 5
6.4 6.4
0.88 1.32
947 947
945 946
950 948
938 943
952 953
949 950
50 50
1 1
3.8 3.8
0.44 0.88
983 990
982 992
983 994
985 993
987 986
983 989
50 50
1 1
3.8 5.1
1.32 0.44
988 981
990 982
991 986
992 986
987 981
982 984
50 50
1 1
5.1 5.1
0.88 1.32
985 992
986 991
989 992
993 989
992 994
982 989
50 50
1 1
6.4 6.4
0.44 0.88
985 986
983 981
989 987
988 987
984 990
984 989
Continued on next page
47
Simulation of Normality tests on a normal distribution (1000 replications)
continued
Number of replications where normality hypothesis accepted Simulation details
Test
sample
test
Anderson-
Cramer-
Shapiro-
Jarque-
size
size
st.dev
mean
Darling
von Mises
Lilliefors
Pearson
Wilk
Bera
50
1
6.4
1.32
991
990
991
994
992
990
100 100
1 1
3.8 3.8
0.44 0.88
993 993
993 991
994 991
992 993
992 993
987 985
100
1
3.8
1.32
990
990
989
993
986
986
100 100
1 1
5.1 5.1
0.44 0.88
989 992
989 992
988 990
988 994
989 991
977 985
100 100
1 1
5.1 6.4
1.32 0.44
985 989
986 986
987 992
993 988
988 983
987 980
100 100
1 1
6.4 6.4
0.88 1.32
988 988
987 992
985 989
990 989
982 990
988 986
200 200
1 1
3.8 3.8
0.44 0.88
992 988
993 991
991 993
992 993
992 987
987 985
200 200
1 1
3.8 5.1
1.32 0.44
993 989
990 989
993 986
993 988
991 992
986 977
200 200
1 1
5.1 5.1
0.88 1.32
992 994
992 986
992 998
994 993
992 990
985 987
200 200
1 1
6.4 6.4
0.44 0.88
994 991
986 987
996 991
988 990
994 988
980 988
200 50
1 10
6.4 3.8
1.32 0.44
989 891
992 893
991 906
989 895
991 885
986 893
50
10
3.8
0.88
896
894
898
896
902
889
50 50
10 10
3.8 5.1
1.32 0.44
870 880
868 885
881 873
892 879
880 884
877 869
50 50
10 10
5.1 5.1
0.88 1.32
887 898
881 906
905 895
912 900
878 907
889 907
50 50
10 10
6.4 6.4
0.44 0.88
890 886
877 881
886 870
893 885
886 896
886 906
50 100
10 10
6.4 3.8
1.32 0.44
906 909
907 909
910 904
906 901
906 906
915 888
100 100
10 10
3.8 3.8
0.88 1.32
894 894
891 895
897 894
896 892
906 894
888 885
100 100
10 10
5.1 5.1
0.44 0.88
899 916
907 917
901 921
892 909
894 917
879 887
100 100
10 10
5.1 6.4
1.32 0.44
903 898
893 891
891 877
910 895
912 899
910 889
Continued on next page
48
Simulation of Normality tests on a normal distribution (1000 replications)
continued
Number of replications where normality hypothesis accepted Simulation details
Test
sample
test
Anderson-
Cramer-
Shapiro-
Jarque-
size
size
st.dev
mean
Darling
von Mises
Lilliefors
Pearson
Wilk
Bera
100
10
6.4
0.88
885
884
881
908
894
889
100 200
10 10
6.4 3.8
1.32 0.44
895 913
901 909
891 905
886 901
905 902
903 888
200
10
3.8
0.88
893
891
879
896
896
888
200 200
10 10
3.8 5.1
1.32 0.44
900 909
895 907
892 903
892 892
889 904
885 879
200 200
10 10
5.1 5.1
0.88 1.32
908 922
917 893
913 911
909 910
913 913
887 910
200 200
10 10
6.4 6.4
0.44 0.88
895 895
891 884
899 893
895 908
896 905
889 889
200
10
6.4
1.32
905
901
894
886
908
903
49
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Weron, R. (1996b). On the Chambers-Mallows-Stuck method for simulating skewed random variables. Statistics and Probability Letters 28, 16–171. Weurtz, D. and H. G. Katzgraber (2005). Precise finite-sample quantiles of the JarqueBera adjusted Lagrange multiplier test. Swiss Federal Institute of Technology, Institute for Theoretical Physics, ETH H¨ onggerberg, C-8093 Zurich. Wuertz, D. (2005). Rmetrics - an environment for teaching financial engineering and computational finance with R. Institute for Theoretical Physics, Swiss Federal Institute of Technology, Zurich. Zolotarev, V. M. (1986). One-dimensional Stable Distributions. Translations of Mathematical Monographs, Volume 65, American Mathematical Society.
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