On More Robust Estimation of Skewness and … kurtosis. In the finance literature, there has been...

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On More Robust Estimation of Skewness and Kurtosis: Simulation and

Application to the S&P500 Index

Tae-Hwan Kim

School of Economics, University of Nottingham University Park, Nottingham NG7 2RD, UK

(Phone) 44-115-951-5466 (Fax) 44-115-951-4159 [email protected]

Halbert White

Department of Economics, University of California, San Diego 9500 Gilman Drive, La Jolla, CA 92093-0508 (Phone) 858-534-3502 (Fax) 858-534-7040

[email protected]

September, 2003

Abstract: For both the academic and the financial communities it is a familiar stylized fact that stock

market returns have negative skewness and excess kurtosis. This stylized fact has been supported by a vast

collection of empirical studies. Given that the conventional measures of skewness and kurtosis are computed

as an average and that averages are not robust, we ask, “How useful are the measures of skewness and

kurtosis used in previous empirical studies?” To answer this question we provide a survey of robust

measures of skewness and kurtosis from the statistics literature and carry out extensive Monte Carlo

simulations that compare the conventional measures with the robust measures of our survey. An application

of the robust measures to daily S&P500 index data indicates that the stylized facts might have been accepted

too readily. We suggest that looking beyond the standard skewness and kurtosis measures can provide

deeper insight into market returns behaviour.

Keywords : Skewness, Kurtosis, Quantiles, Robustness.

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1. Introduction

It has long been recognized that the behavior of stock market returns does not agree with the

frequently assumed normal distribution. This disagreement is often highlighted by showing the

large departures of the skewness and kurtosis of returns from normal distribution counterparts. For

both the academic and the financial communities it has become a firm and indisputable stylized

fact that stock market returns have negative skewness and excess kurtosis. This stylized fact has

been supported by a huge collection of empirical studies. Some recent papers on this issue include

Bates (1996), Jorion (1988), Hwang and Satchell (1999), and Harvey and Siddique (1999, 2000).

The role of higher moments has become increasingly important in the literature mainly because

the traditional measure of risk, variance (or standard deviation), has failed to capture fully the "true

risk" of the distribution of stock market returns. For example, if investors prefer right-skewed

portfolios, then more reward should be given to investors willing to invest in left-skewed portfolios

even though both portfolios have the same standard deviation. This suggests that the "true risk"

may be a multi-dimensional concept and that other measures of distributional shape such as higher

moments can be useful in obtaining a better description of multi-dimensional risk. In this context,

Harvey and Siddique (2000) proposed an asset pricing model that incorporates skewness and

Hwang and Satchell (1999) developed a CAPM for emerging markets taking into account

skewness and kurtosis.

Given this emerging interest in skewness and kurtosis in financial markets, one should ask the

following question: how useful are the measures of skewness and kurtosis used in previous

empirical studies? Practically all of the previous work concerning skewness and kurtosis in

financial markets has used the conventional measures of skewness and kurtosis (that is, the

standardized third and fourth sample moments or some variants of these). It is well known that the

sample mean (also its regression version, the least squares estimator) is very sensitive to outliers.

Since the conventional measures of skewness and kurtosis are essentially based on sample

averages, they are also sensitive to outliers. Moreover, the impact of outliers is greatly amplified in

the conventional measures of skewness and kurtosis due to the fact that they are raised to the third

and fourth powers.

In the statistics literature, a great deal of effort has been taken to overcome the non-robustness

of the conventional measures of location and dispersion (i.e. mean and variance), and some

attention has been paid to the non-robustness of the conventional measures of skewness and

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kurtosis. In the finance literature, there has been some concern with non-robustness of conventional

measures of location and dispersion, but almost no attention to the non-robustness of conventional

measures of skewness and kurtosis. In this paper, we consider certain robust alternative measures

of skewness and kurtosis based on quantiles that have been previously developed in the statistics

literature, and we conduct extensive Monte Carlo simulations to evaluate and compare the

conventional measures of skewness and kurtosis and their robust counterparts. Our simulation

results demonstrate that the conventional measures are extremely sensitive to single outliers or

small groups of outliers, comparable to those observed in U.S. stock returns. An application of

robust measures to the daily S&P500 index indicates that the familiar stylized facts (negative

skewness and excess kurtosis in financial markets) may have been too readily accepted.

2. Review of Robust Measures of Skewness and Kurtosis

We consider a process Ntty ,...,2,1}{ = and assume that the ty ’s are independent and identically

distributed with a cumulative distribution function F . The conventional coefficients of skewness

and kurtosis for ty are given by:

,3

1

−=

σµty

ESK 34

1 −

−=

σµty

EKR ,

where )( tyE=µ and ,)( 22 µσ −= tyE and expectation E is taken with respect to F . Given the

data Ntty ,...,2,1}{ = , 1SK and 1KR are usually estimated by the sample averages

,ˆ

ˆ

1

31

1 ∑=

−∧

−=

N

t

tyTSK

σµ

∑=

−∧

−

−=

N

t

tyTKR

1

41

1 3ˆ

ˆσ

µ,

where ,ˆ1

1∑=

−=N

ttyTµ .)ˆ(ˆ

1

212 ∑=

− −=N

ttyT µσ

Due to the third and fourth power terms in 1SK and 1KR , the values of these measures can be

arbitrarily large, especially when there are one or more large outliers in the data. For this reason, it

can sometimes be difficult to give a sensible interpretation to large values of these measures simply

because we do not know whether the true values are indeed large or there exist some outliers. One

seemingly simple solution is to eliminate the outliers from the data. Two problems arise in this

approach. One is that the decision to eliminate outliers is taken usually after visually inspecting the

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data; this can invalidate subsequent statistical inference. The other is that deciding which

observations are outliers can be somewhat arbitrary.

Hence, eliminating outliers manually is not as simple as it may appear, and it is desirable to

have non-subjective robust measures of skewness and kurtosis that are not too sensitive to outliers.

We turn now to a description of a number of more robust measures of skewness and kurtosis that

have been proposed in the statistics literature. It is interesting to note that among the robust

measures we discuss below, only one requires the second moment and all other measures do not

require any moments to exist.

2.1 Robust Measures of Skewness

Robust measures of location and dispersion are well known in the literature. For example, the

median can be used for location and the interquartile range for dispersion. Both the median and the

interquartile range are based on quantiles. Following this tradition, Bowley (1920) proposed a

coefficient of skewness based on quantiles:

,2

13

2132 QQ

QQQSK

−−+

=

where iQ is the thi quartile of ty : that is, ),25.0(11

−= FQ )5.0(12

−= FQ and ).75.0(13

−= FQ It

is easily seen that for any symmetric distribution, the Bowley coefficient of skewness is zero. The

denominator 13 QQ − re-scales the coefficient so that the maximum value for 2SK is 1,

representing extreme right skewness and the minimum value for 2SK is –1, representing extreme

left skewness.

The Bowley coefficient of skewness has been generalized by Hinkley (1975):

)()1(2)()1(

)(11

211

3 αααα

α−−

−−

−−−+−

=FF

QFFSK ,

for any α between 0 and 0.5. Note that the Bowley coefficient of skewness is a special case of

Hinkley's coefficient when α = 0.25. This measure, however, depends on the value of α and it is

not clear what value should be used for α . One way of removing this dependence is to integrate

out ,α as done in Groeneveld and Meeden (1984):

5

{ }{ }

.||

)()1(

2)()1(

2

2

5.0

0

11

5.0

0 211

3

QyEQ

dFF

dQFFSK

t −−

=

−−

−+−=

∫∫

−−

−−

µ

ααα

ααα

This measure is also zero for any symmetric distributions and is bounded by –1 and 1.

Noting that the denominator || 2QyE t − is a kind of dispersion measure, we observe that the

Pearson coefficient of skewness [Kendall and Stuart (1977)] is obtained by replacing the

denominator with the standard deviation as follows:

.24 σ

µ QSK

−=

Groeneveld and Meeden (1984) have put forward the following four properties that any

reasonable coefficient of skewness )( tyγ should satisfy: (i) for any ),0( ∞∈a and ),,( ∞−∞∈b

)( tyγ = )( bay t +γ ; (ii) if ty is symmetrically distributed, then )( tyγ = 0; (iii) )( tyγ− = )( ty−γ ;

(iv) if F and G are cumulative distribution functions of ty and tx , and GF c< , then )( tyγ ≤

)( txγ where c< is a skewness-ordering among distributions (see Zwet (1964) for the definition).

Groeneveld and Meeden proved that 1SK , 2SK and 3SK satisfy (i)-(iv), but 4SK satisfies (i)-(iii)

only.

2.2 Robust Measures of Kurtosis

Moors (1988) showed that the conventional measure of kurtosis ( 1KR ) can be interpreted as a

measure of the dispersion of a distribution around the two values σµ ± . Hence, 1KR can be large

when probability mass is concentrated either near the mean µ or in the tails of the distributions.

Based on this interpretation, Moors (1988) proposed a robust alternative to 1KR :

,)()(

26

1357

EEEEEE

−−+−

where iE is the thi octile: that is, )8/(1 iFEi−= for .7...,,2,1=i Moors

justified this estimator on the ground that the two terms, )( 57 EE − and )( 13 EE − , are large (small)

if relatively little (much) probability mass is concentrated in the neighbourhood of 6E and 2E ,

corresponding to large (small) dispersion around σµ ± . The denominator is a scaling factor

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ensuring that the statistic is invariant under linear transformation. As we do for ,1KR we center the

Moors coefficient of kurtosis at the value for the standard normal distribution. It is easy to

calculate that ,15.171 −=−= EE ,68.062 −=−= EE 32.053 −=−= EE and 04 =E for )1,0(N

and therefore the Moors coefficient of kurtosis is 1.23. Hence, the centered coefficient is given by:

.23.1)()(

26

13572 −

−−+−

=EE

EEEEKR

While investigating how to test light-tailed distributions against heavy-tailed distributions, Hogg

(1972, 1974) found that the following measure of kurtosis performs better than the traditional

measure 1KR in detecting heavy-tailed distributions: ββ

αα

LULU

−−

where )( αα LU is the average of

the upper (lower) α quantiles defined as: ,)(1 1

1

1∫ −

−=αα α

dyyFU ,)(1

0

1∫ −=α

α αdyyFL for

).1,0(∈α According to Hogg’s simulation experiments, α = 0.05 and β = 0.5 gave the most

satisfactory results. Here, we adopt these values for α and β . For ),1,0(N we have

06.205.005.0 =−= LU and ,80.05.05.0 =−= LU implying that the Hogg coefficient with α = 0.05

and β = 0.5 is 2.59. Hence, the centered Hogg coefficient is given by:

.59.25.05.0

05.005.03 −

−−

=LULU

KR

Another interesting measure based on quantiles has been used in Crow and Siddiqui (1967),

which is given by )()1()()1(

11

11

ββαα

−−

−−

−−+−

FFFF

for ).1,0(, ∈βα Their choices for α and β are 0.025 and

0.25 respectively. For these values, we obtain 96.1)025.0()975.0( 11 =−= −− FF and

68.0)25.0()75.0( 11 −=−= −− FF for N(0,1) and the coefficient is 2.91. Hence, the centered

coefficient is:

.91.2)25.0()75.0(

)025.0()975.0(11

11

4 −−+

=−−

−−

FFFF

KR

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3. Monte Carlo Simulations

In this section we conduct Monte Carlo simulations designed to investigate how robust the

alternative measures of skewness and kurtosis are in finite samples. The simulations were carried

out on a 700MHz PC using MATLAB. The random number generator used in the simulations is

that from the MATLAB Statistics Toolbox.

We choose three symmetric distributions and one non-symmetric distribution for our simulation

study. Our symmetric distributions are the standard normal distribution N(0,1), and the Student

−t distributions with 10 and 5 degrees of freedom [T-10, T-5]. These represent moderate, heavy

and very heavy tailed distributions. For the non-symmetric distribution we use the log-normal

distribution with µ = 1, σ = 0.4 (shifted by 2)4.0(5.01+− e so that the mean is zero) denoted Log-

N(1,0.4). For a range of values for N (50, 250, 500, 1000, 2500 and 5000), we generate Ntty ,...,2,1}{ =

using the four distributions and calculate the various measures of skewness and kurtosis discussed

in the previous section. The number of replications for each experiment is 1000. The true values of

the skewness and kurtosis measures for the various distributions are provided in Table 1. If our

statistics are consistent, then their Monte Carlo distributions should collapse around these values as

∞→N .

The simulation results are reported in Figures 1-8. Each figure is divided into two sections: A

and B. Figure A shows smoothed histograms of estimated coefficients of skewness or kurtosis for

the 5 different sample sizes and 4 data generating distributions. We use a kernel density method to

smooth the histograms. In order to provide additional information, Figure B displays box-plots for

the corresponding smoothed histograms. As usual, each box represents the lower quartile, median,

and upper quartile values. The whiskers are lines extending from each end of the box and their

length is chosen to be the same as the length of the corresponding box (i.e. the inter-quartile range).

The number at the end of each whisker is the number of observations beyond the end of the

whiskers.

As expected, the performance of 1SK (Figure 1) deteriorates as the distribution moves from

N(0,1) to T-10 and T-5. The sampling distributions tend to have large dispersion and the number of

observations outside the whiskers is noticeably increasing. For the lognormal case, 1SK is not a

good measure when N is small, which is indicated by the fact that the center of the box-plot is

quite different from the limiting value (1.32), although it does move towards the limiting value as

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∞→N . These problems, however, are not present in the other robust measures, 2SK (Figure 2),

3SK (Figure 3), 4SK (Figure 4). That is, the sampling distributions are fairly stable and similar

across various distributions and also, for the lognormal case, the center of the box-plot stays near

the limiting value even for N = 50.

The performance of 1KR (Figure 5) is even worse than 1SK as the distribution moves from

N(0,1) to T-10 and T-5. For the T-5 case, the center of the box-plot is still far away from the true

value 6 even for N = 5000. The small sample bias of 1KR for Log-N(1,0.4) is also evident in the

box-plot. Other robust estimates ( 2KR and 4KR ) do not exhibit these problems at all. The

sampling distribution of 3KR indicates that there is a small finite sample bias when the number of

observations is less than 1000.

Next we add a single outlier to the same set of generated random numbers Ntty ,...,2,1}{ = and

calculate the same set of measures in order to see the impact of the single outlier on the various

measures. The outlier is constructed to occur at time )1,0(∈τ . In order to inject some realism into

the simulation study, we use the daily S&P500 index to calculate the size and location of the

outlier. The sample period is January 1, 1982 through June 29, 2001 with 5085. The largest outlier

(-20.41%) is caused by the 1987 stock market crash. The timing of the crash in this sample period

is τ = 0.3 (the location of the outlier in the sample divided by the total number of observations).

We calculate the 25th percentile of the sampling distribution of the S&P index. It turns out that the

25th percentile is -0.42. The size of the outlier relative to the 25th percentile is then given by

62.4842.0/41.20)25.0(/ 1][ =−−== −Fym Tτ . Therefore, we first generate random numbers

Ntty ,...,2,1}{ = and calculate the 25th percentile ).25.0(1−F The outlier is then )25.0(1−mF and we

replace the observation ]3.0[ Ty with the outlier.

The results are reported in Figures 9-16. The impact of one outlier on 1SK is clearly visible in

Figure 9. Note that the center of the sampling distributions of 1SK is moving toward to zero (the

true value for all symmetric distributions) once N is greater than 500, but even for N = 5000, the

center is far from zero. No whiskers from the box-plots contain the true value 0 in their upper tails.

The maximum impact occurs approximately when N = 500. In that case the center of the box-

plots is around between -13 and -10. In contrast, the outlier has no impact on 2SK at all. This is

expected since 2SK is based on quantiles whose values are not changed by a single observation.

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On the other hand, we see the moving-box type of convergence for the other robust skewness

estimators ( 3SK and 4SK ). This is because these measures involve µ and σ , and the outlier must

have some impact on the sample mean and sample standard deviation. Even for N = 5000 the

medians of the sampling distributions of 3SK and 4SK for the symmetric generating distributions

are slightly less than zero. If one does not want to have any impact of a single outlier when

measuring skewness, then 2SK is preferable. However, if one wishes to take into account the

outlier in the calculation, but does not want to have as severe a distortion as appears in 1SK , then

3SK or 4SK might be a better measure.

The impact of the single outlier on 1KR is truly spectacular (Figure 13). The effect is

maximized around at N = 1000 in which the center of box-plots are between 150 and 300 for

various distributions when the true value should be between 0 and 6. Once N becomes larger than

1000, 1KR converges towards its true value, but even with N = 5000 the medians are still between

80 and 200. The results indicate that it may not be possible to attach any meaningful interpretation

to a large value of 1KR even when there are sufficiently many observations. The Moors coefficient

2KR , on the other hand, is not influenced by the outlier at all (Figure 14). For the same reason as

for 2SK , this is because 2KR is based on octiles, which are not affected by a single observation.

The centered Hogg coefficient 3KR is mildly influenced by the outlier, especially when the number

of observations is very small, but the influence disappears quickly as N increases (Figure 15). The

source of the influence is the terms 05.0L and 5.0L . As explained before, αL is the average of the

lower α percentile tails. Hence, when N is small (e.g. 50), the outlier becomes one of the

percentiles and enters the calculation of αL . On the other hand, the impact on 4KR is present only

when N = 50 (Figure 16). This is because the term )025.0(1−F in the formula is equal to the

outlier when the number of observations is very small. The same comment can be applied to the

choice of kurtosis measures: 2KR completely ignores a single outlier while 3KR and 4KR reflects

to some degree the presence of an outlier without much distortion.

If an outlier occurs only once and its size does not depend on the sample size (N ), then its

impact on any statistic will eventually disappear as the sample size goes to infinity. This must be

true for even 1SK or 1KR , despite their being severely degraded in finite samples. In reality,

however, we tend to find that large outliers recur through time: for example, the 1987 stock market

10

crash and the 1998 Asian crisis. One way of modelling this phenomenon is to use a mixture

distribution. Suppose that a process }{ ty is generated from ),( 11 σµD with probability p and

from ),( 122 γσσµ =D with probability p−1 . If the probability p is very close to one and 2σ is

fairly large compared to 1σ , then the process has recurring outliers with probability p−1 through

time. In our simulations we determine the values of p and γ , again using the daily S&P index.

We treat a change larger than 7% per day as an outlier. Over the sample period, there are six

observations whose absolute values are greater than 7%. Our estimate for p is then given by

9988.05085/5079ˆ ==p and p̂1− = 0.0012. The sample standard devia tion of the sample

without the six outliers is 0.94, and the sample standard deviation of the six outliers is 9.99. Hence,

the ratio is 63.1094.0/99.9 = . The sample mean of the six outliers is -7.322. Taking into account

these estimates, we choose the following set of parameter values for our simulations: ,01 =µ

11 =σ , ,72 −=µ 10=γ and .9988.0=p Hence, the random numbers are generated by

)1,0(9988.0 D + )10,7(0012.0 −D using the same four distributions for D . We present the true

values of the skewness and kurtosis measures for these mixture distributions in Table 2. We

provide a short discussion of how to obtain these values in the Appendix.

The simulation results are displayed in Figures 17-24. The behaviour of 1SK (Figure 17) is

quite different from that of 1SK with a single outlier in that the dispersion of sampling distributions

increases as the sample size becomes larger. This may seem surprising because we usually expect

the dispersion of a sampling distribution to shrink as N increases. In the single outlier case we

guaranteed the occurrence of the outlier in each sample regardless of the value of N . In the

mixture distribut ion case, on the other hand, when N is small, the sample may not have any

outliers due to the high value of .9988.0=p This may explain why the dispersion of the sampling

distribution of 1SK is small when N is small, and becomes larger as N increases. In most

distributions, the center of box-plots converges to the true value rather slowly: with N = 5000 it is

still far from the true value; -2.27 for N(0,1), -2.00 for T-10, -1.71 for T-5 and 0.52 for Log-

N(1,0.4). The other robust skewness measures ( ,2SK ,3SK 4SK ) are in general not influenced by

this type of outlier: the dispersion is quite small even for small N and there is no finite sample

bias. As for 1SK , the sampling distributions of 1KR (Figure 21) tend to have larger dispersion as

N approaches 5000. Two other measures ( ,2KR 4KR ) are robust to these recurring outliers and

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their medians converge to the true limiting values reasonably quickly (Figures 22 and 24). The

Hogg coefficient 3KR (Figure 23) has a finite sample bias for small N , but this disappears once

the number of observations becomes larger than 500.

4. Application: Skewness and Kurtosis of the S&P500 Index

In this section we apply conventional and robust measures of skewness and kurtosis to the same

S&P500 index data described in the previous section. The sample period of January 1, 1982

through Jun 29, 2001 yields 5085 observations, and the unit is percent return.

First we compute the conventional measures of skewness and kurtosis using all observations.

For this sample period, we have 1SK = -2.39 and 1KR = 53.62, which is consistent with the

previous findings in the literature; i.e. negative skewness and excess kurtosis. Next we use robust

measures to estimate skewness and kurtosis. The results are displayed in the first column of Table

3. The outcomes are interesting in that all but one robust estimate support the opposite

characterization of skewness and kurtosis. All the robust skewness measures are pretty close to

zero and hence indicate that there is little skewness in the distribution of the S&P500 index. Given

that all robust kurtosis measure iKR ( i = 2,3,4) are centered by the values for N(0,1), positive

values ( 2KR = 0.28, 3KR = 0.77, 4KR = 1.16) indicate that there exists excess kurtosis, but this is

considerably more mild than usually thought.

Next we re-compute all the statistics after removing the single observation corresponding the

1987 stock market crash. The results are reported in the second column of Table 3. The values for

1SK and 1KR are dramatically reduced to 1SK = -0.26 and 1KR = 6.80, but the other robust

measures hardly change. This clearly shows that the single observation must be very influential in

the calculation of 1SK and 1KR , which is consistent with what we find in our simulations. On that

basis, we may argue that it is difficult to attach a meaningful interpretation to the value of 1KR

(53.62) calculated using all observations. Finally, as we have done in the simulations, we remove

the six observations whose absolute values are larger than 7%. The results are in the third column

of Table 3. Not only the conventional measures become substantially smaller ( 1SK = -0.04 and

1KR = 3.43) as in the case where the single crash observation is removed, but also they indicate

that there exist no negative skewness and the magnitude of kurtosis is not as large as previously

12

believed. The implication of all other robust measures are qualitatively the same; that is, no

negative skewness and quite mild kurtosis.

5. Conclusion

The use of robust measures of skewness and kurtosis reveals interesting evidence, which is in sharp

contrast to what heretofore has been firmly regarded as true in the finance literature. Rather than

arguing that we have obtained definite evidence to refute the long-believed stylized facts about

skewness and kurtosis, we hope that the current paper will serve as a starting point for further

constructive research on these important issues. We do propose however, that the standard

measures of skewness and kurtosis be viewed with skepticism, that the robust measures described

here be routinely computed, and, finally, that it may be more productive to think of the S&P500

index returns studied here as being better described as a mixture containing a predominant

component that is nearly symmetric with mild kurtosis and a relatively rare component that

generates highly extreme behaviour. Viewing financial markets in this way further suggests that

useful extensions of asset pricing models now embodying only skewness and kurtosis may be

obtained by accommodating mixtures similar to those discussed here.

13

Appendix

Everitt and Hand (1981) provided the first five central moments of a two-component univariate

normal mixture. Extending their results, we consider the following mixture distribution.

21 )1( gppgf −+=

where ig has mean iµ and variance ,2iσ and is assumed to have up to 4th moment. Let

∫= dxxxf )(µ and ∫ −= dxxfxv rr )()( µ . Then it can be proved that

21 )1( µµµ pp −+=

))(1()( 22

22

21

212 δσδσ +−++= ppv

)3)(1()3( 32

222

322

31

211

3113 δσδσδσδσ ++−+++= SpSpv

)64)3)((1()64)3(( 42

22

22

3222

422

41

21

21

3111

4113 δσδσδσδσδσδσ ++++−+++++= SKpSKpv

where µµδ −= ii , and iS and iK are the skewness and kurtosis of ig respectively. Moreover, let

1X and 2X be the random variables governed by 1g and 2g . In our simulation set-up, we have

222 µγ −= XX . Then it is easily seen that both skewness and kurtosis are invariant under this

linear transformation. Hence, we have 21 SS = and 21 KK = , and the values of 1S and 1K are in

Table 1. All central moments up to 4v can be now calculated in order to obtain the skewness and

kurtosis of the corresponding mixture distribution.

14

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Financial Studies, 1, 27-445.

Kendall, M.G. and Stuart, A. (1977), The Advanced Theory of Statistics, Vol. 1, London: Charles

Griffin and Co.

Moors, J. J. A. (1988), "A Quantile Alternative for Kurtosis," The Statistician, 37, 25-32.

Zwet, W.R. van (1964), "Convex Transformations of Random Variables," Math. Centrum,

Amsterdam.

15

Table 1. Values of skewness and kurtosis for various distributions

(no outlier and single outlier cases)

N(0,1) T-10 T-5 Log-N(1,0.4)

1SK 0 0 0 1.32

2SK 0 0 0 0.13

3SK 0 0 0 0.25

4SK 0 0 0 0.18

1KR 3 1 6 3.26

2KR 1.23 0.04 0.10 0.04

3KR 2.59 0.20 0.46 0.19

4KR 2.91 0.28 0.63 0.27

Table 2. Values of skewness and kurtosis for various distributions

(mixture distribution case)

N(0,1) T-10 T-5 Log-N(1,0.4)

1SK -2.27 -2.00 -1.71 0.52

2SK 0 0 0 0.14

3SK -0.01 -0.01 -0.01 0.24

4SK -0.01 -0.01 -0.01 0.17

1KR 52.76 57.33 101.47 49.18

2KR 0.01 0.04 0.10 0.40

3KR 0.09 0.29 0.53 0.27

4KR 0.01 0.28 0.66 0.27

16

Table 3. Skewness and kurtosis of the S&P500 Index

Using all observations Without the 1987

crash observation

Without the 6

observations ( ≥ 7%)

1SK -2.39 -0.26 -0.04

2SK 0.08 0.08 0.08

3SK 0.04 0.04 0.05

4SK 0.02 0.03 0.03

1KR 53.62 6.80 3.44

2KR 0.28 0.28 0.28

3KR 0.77 0.73 0.67

4KR 1.16 1.16 1.13

1

Figure 1A. Sampling distributions of 1SK (smoothed histograms): no outlier case

Figure 1B. Sampling distributions of 1SK (Box-plots): no outlier case

-2 0 20

5

10N(0,1)

N=5

0

-2 0 20

5

10T-10

-2 0 20

5

10T-5

-2 0 20

5

10Log-N(1,0.4)

-2 0 20

5

10

N=250

-2 0 20

5

10

-2 0 20

5

10

-2 0 20

5

10

-2 0 20

5

10

N=500

-2 0 20

5

10

-2 0 20

5

10

-2 0 20

5

10

-2 0 20

5

10

N=1

000

-2 0 20

5

10

-2 0 20

5

10

-2 0 20

5

10

-2 0 20

5

10

N=2500

-2 0 20

5

10

-2 0 20

5

10

-2 0 20

5

10

-2 0 20

5

10

N=5

000

-2 0 20

5

10

-2 0 20

5

10

-2 0 20

5

10

-2 -1 0 1 2 3

50

250

500

1000

2500

5000

N(0,1)

29 27

19 23

14 16

17 19

20 23

20 25

-2 -1 0 1 2 3

50

250

500

1000

2500

5000

T-10

29 38

52 39

19 32

31 36

35 21

26 21

-2 -1 0 1 2 3

50

250

500

1000

2500

5000

T-5

52 43

47 46

52 63

52 62

42 64

43 51

-2 -1 0 1 2 3

50

250

500

1000

2500

5000

Log-N(1,0.4)

3 62

3 61

11 72

7 46

8 51

7 47

2

Figure 2A. Sampling distributions of 2SK (Smoothed histograms): no outlier case


-1 -0.5 0 0.5 1

50

250

500

1000

2500

5000

N(0,1)

23 16

19 25

23 21

19 23

26 22

21 21

-1 -0.5 0 0.5 1

50

250

500

1000

2500

5000

T-10

11 11

16 9

19 26

27 22

26 26

29 26

-1 -0.5 0 0.5 1

50

250

500

1000

2500

5000

T-5

15 28

29 28

19 22

20 16

23 24

32 32

-1 -0.5 0 0.5 1

50

250

500

1000

2500

5000

Log-N(1,0.4)

19 19

20 13

14 29

19 20

31 32

16 21

-0.5 0 0.50

10

20N(0,1)

N = 50

-0.5 0 0.50

10

20T-10

-0.5 0 0.50

10

20T-5

-0.5 0 0.50

10

20Log-N(1,0.4)

-0.5 0 0.50

10

20

N = 250

-0.5 0 0.50

10

20

-0.5 0 0.50

10

20

-0.5 0 0.50

10

20

-0.5 0 0.50

10

20

N = 500

-0.5 0 0.50

10

20

-0.5 0 0.50

10

20

-0.5 0 0.50

10

20

-0.5 0 0.50

10

20

N = 100

0

-0.5 0 0.50

10

20

-0.5 0 0.50

10

20

-0.5 0 0.50

10

20

-0.5 0 0.50

10

20

N = 250

0

-0.5 0 0.50

10

20

-0.5 0 0.50

10

20

-0.5 0 0.50

10

20

-0.5 0 0.50

10

20

N = 500

0

-0.5 0 0.50

10

20

-0.5 0 0.50

10

20

-0.5 0 0.50

10

20

3



-1 -0.5 0 0.5 1

50

250

500

1000

2500

5000

N(0,1)

15 18

24 21

18 32

17 27

29 25

11 14

-1 -0.5 0 0.5 1

50

250

500

1000

2500

5000

T-10

16 17

18 20

9 23

25 35

16 22

9 16

-1 -0.5 0 0.5 1

50

250

500

1000

2500

5000

T-5

29 16

20 18

23 23

22 21

28 38

22 20

-1 -0.5 0 0.5 1

50

250

500

1000

2500

5000

Log-N(1,0.4)

25 8

27 22

26 22

23 19

16 25

22 30

-0.5 0 0.50

10

20

30N(0,1)

N = 50

-0.5 0 0.50

10

20

30T-10

-0.5 0 0.50

10

20

30T-5

-0.5 0 0.50

10

20

30Log-N(1,0.4)

-0.5 0 0.50

10

20

30

N = 250

-0.5 0 0.50

10

20

30

-0.5 0 0.50

10

20

30

-0.5 0 0.50

10

20

30

-0.5 0 0.50

10

20

30

N = 500

-0.5 0 0.50

10

20

30

-0.5 0 0.50

10

20

30

-0.5 0 0.50

10

20

30

-0.5 0 0.50

10

20

30

N = 100

0

-0.5 0 0.50

10

20

30

-0.5 0 0.50

10

20

30

-0.5 0 0.50

10

20

30

-0.5 0 0.50

10

20

30

N = 250

0

-0.5 0 0.50

10

20

30

-0.5 0 0.50

10

20

30

-0.5 0 0.50

10

20

30

-0.5 0 0.50

10

20

30

N = 500

0

-0.5 0 0.50

10

20

30

-0.5 0 0.50

10

20

30

-0.5 0 0.50

10

20

30

4



-0.5 0 0.50

20

N(0,1)

N = 50

-0.5 0 0.50

20

T-10

-0.5 0 0.50

20

T-5

-0.5 0 0.50

20

Log-N(1,0.4)

-0.5 0 0.50

20

N = 250

-0.5 0 0.50

20

-0.5 0 0.50

20

-0.5 0 0.50

20

-0.5 0 0.50

20

N = 500

-0.5 0 0.50

20

-0.5 0 0.50

20

-0.5 0 0.50

20

-0.5 0 0.50

20

N = 100

0

-0.5 0 0.50

20

-0.5 0 0.50

20

-0.5 0 0.50

20

-0.5 0 0.50

20

N = 250

0

-0.5 0 0.50

20

-0.5 0 0.50

20

-0.5 0 0.50

20

-0.5 0 0.50

20

N = 500

0

-0.5 0 0.50

20

-0.5 0 0.50

20

-0.5 0 0.50

20

-0.5 0 0.5

50

250

500

1000

2500

5000

N(0,1)

16 19

23 21

19 34

18 27

29 26

11 13

-0.5 0 0.5

50

250

500

1000

2500

5000

T-10

17 20

16 20

9 22

26 35

16 22

9 17

-0.5 0 0.5

50

250

500

1000

2500

5000

T-5

23 16

20 17

22 20

22 20

28 37

22 17

-0.5 0 0.5

50

250

500

1000

2500

5000

Log-N(1,0.4)

27 8

28 17

23 20

19 18

16 29

21 30

5

Figure 5A. Sampling distributions of 1KR (Smoothed histograms): no outlier case

Figure 5B. Sampling distributions of 1KR (Box-plots): no outlier case

-2 0 2 4 6 80

5

N(0,1)

N=5

0

-2 0 2 4 6 80

5

T-10

-2 0 2 4 6 80

5

T-5

-2 0 2 4 6 80

5

Log-N(1,0.4)

-2 0 2 4 6 80

5

N=2

50

-2 0 2 4 6 80

5

-2 0 2 4 6 80

5

-2 0 2 4 6 80

5

-2 0 2 4 6 80

5

N=5

00

-2 0 2 4 6 80

5

-2 0 2 4 6 80

5

-2 0 2 4 6 80

5

-2 0 2 4 6 80

5

N=1

000

-2 0 2 4 6 80

5

-2 0 2 4 6 80

5

-2 0 2 4 6 80

5

-2 0 2 4 6 80

5

N=2

500

-2 0 2 4 6 80

5

-2 0 2 4 6 80

5

-2 0 2 4 6 80

5

-2 0 2 4 6 80

5

N=5

000

-2 0 2 4 6 80

5

-2 0 2 4 6 80

5

-2 0 2 4 6 80

5

-2 0 2 4 6 8

50

250

500

1000

2500

5000

N(0,1)

0 78

5 54

3 33

10 29

13 30

17 23

-2 0 2 4 6 8

50

250

500

1000

2500

5000

T-10

0 73

1 93

0 78

2 80

4 76

2 57

-2 0 2 4 6 8

50

250

500

1000

2500

5000

T-5

0 101

0 92

0 124

0 122

0 93

0 93

-2 0 2 4 6 8

50

250

500

1000

2500

5000

Log-N(1,0.4)

0 104

0 90

0 100

0 74

0 82

1 79

6



-1 0 10

5

10

15N(0,1)

N=50

-1 0 10

5

10

15T-10

-1 0 10

5

10

15T-5

-1 0 10

5

10

15Log-N(1,0.4)

-1 0 10

5

10

15

N=2

50

-1 0 10

5

10

15

-1 0 10

5

10

15

-1 0 10

5

10

15

-1 0 10

5

10

15

N=500

-1 0 10

5

10

15

-1 0 10

5

10

15

-1 0 10

5

10

15

-1 0 10

5

10

15

N=1000

-1 0 10

5

10

15

-1 0 10

5

10

15

-1 0 10

5

10

15

-1 0 10

5

10

15

N=2500

-1 0 10

5

10

15

-1 0 10

5

10

15

-1 0 10

5

10

15

-1 0 10

5

10

15

N=5000

-1 0 10

5

10

15

-1 0 10

5

10

15

-1 0 10

5

10

15

-1 -0.5 0 0.5 1

50

250

500

1000

2500

5000

N(0,1)

4 41

10 29

13 28

10 29

22 31

32 31

-1 -0.5 0 0.5 1

50

250

500

1000

2500

5000

T-10

8 42

13 27

10 26

14 26

22 34

14 28

-1 -0.5 0 0.5 1

50

250

500

1000

2500

5000

T-5

0 39

5 30

8 19

14 26

24 26

20 22

-1 -0.5 0 0.5 1

50

250

500

1000

2500

5000

Log-N(1,0.4)

8 47

10 30

15 30

18 35

14 24

13 25

7



-1 0 10

10

N(0,1)

N=50

-1 0 10

10

T-10

-1 0 10

10

T-5

-1 0 10

10

Log-N(1,0.4)

-1 0 10

10

N=2

50

-1 0 10

10

-1 0 10

10

-1 0 10

10

-1 0 10

10

N=500

-1 0 10

10

-1 0 10

10

-1 0 10

10

-1 0 10

10

N=1

000

-1 0 10

10

-1 0 10

10

-1 0 10

10

-1 0 10

10

N=2

500

-1 0 10

10

-1 0 10

10

-1 0 10

10

-1 0 10

10

N=5

000

-1 0 10

10

-1 0 10

10

-1 0 10

10

-1 -0.5 0 0.5 1

50

250

500

1000

2500

5000

N(0,1)

15 36

8 25

15 15

21 24

28 19

19 19

-1 -0.5 0 0.5 1

50

250

500

1000

2500

5000

T-10

6 36

10 27

14 27

16 16

27 24

25 22

-1 -0.5 0 0.5 1

50

250

500

1000

2500

5000

T-5

4 40

18 37

16 41

19 34

17 28

13 25

-1 -0.5 0 0.5 1

50

250

500

1000

2500

5000

Log-N(1,0.4)

6 39

8 31

19 24

22 16

22 28

28 17

8



-1 0 1 2

50

250

500

1000

2500

5000

N(0,1)

4 43

14 41

16 28

15 23

21 28

16 11

-1 0 1 2

50

250

500

1000

2500

5000

T-10

2 48

15 42

7 26

12 16

21 24

21 16

-1 0 1 2

50

250

500

1000

2500

5000

T-5

1 65

8 32

13 34

16 25

9 23

18 30

-1 0 1 2

50

250

500

1000

2500

5000

Log-N(1,0.4)

0 40

13 56

8 31

18 21

22 18

14 15

-1 0 1 20

5

N(0,1)

N=5

0

-1 0 1 20

5

T-10

-1 0 1 20

5

T-5

-1 0 1 20

5

Log-N(1,0.4)

-1 0 1 20

5

N=2

50

-1 0 1 20

5

-1 0 1 20

5

-1 0 1 20

5

-1 0 1 20

5

N=5

00

-1 0 1 20

5

-1 0 1 20

5

-1 0 1 20

5

-1 0 1 20

5

N=1

000

-1 0 1 20

5

-1 0 1 20

5

-1 0 1 20

5

-1 0 1 20

5

N=2

500

-1 0 1 20

5

-1 0 1 20

5

-1 0 1 20

5

-1 0 1 20

5

N=5

000

-1 0 1 20

5

-1 0 1 20

5

-1 0 1 20

5

1

Figure 9A. Sampling distributions of 1SK (smoothed histograms): single outlier case

Figure 9B. Sampling distributions of 1SK (Box-plots): single outlier case

-15 -10 -50

1

2

N(0,1)

N=5

0

-15 -10 -50

1

2

T-10

-15 -10 -50

1

2

T-5

-15 -10 -50

1

2

Log-N(1,0.4)

-15 -10 -50

1

2

N=2

50

-15 -10 -50

1

2

-15 -10 -50

1

2

-15 -10 -50

1

2

-15 -10 -50

1

2

N=5

00

-15 -10 -50

1

2

-15 -10 -50

1

2

-15 -10 -50

1

2

-15 -10 -50

1

2

N=1

000

-15 -10 -50

1

2

-15 -10 -50

1

2

-15 -10 -50

1

2

-15 -10 -50

1

2

N=2

500

-15 -10 -50

1

2

-15 -10 -50

1

2

-15 -10 -50

1

2

-15 -10 -50

1

2

N=5

000

-15 -10 -50

1

2

-15 -10 -50

1

2

-15 -10 -50

1

2

-15 -10 -5

50

250

500

1000

2500

5000

N(0,1)

0 103

3 40

22 42

15 19

15 25

20 16

-15 -10 -5

50

250

500

1000

2500

5000

T-10

0 83

17 52

10 32

15 25

28 26

19 14

-15 -10 -5

50

250

500

1000

2500

5000

T-5

0 102

5 34

13 35

13 20

19 36

32 44

-15 -10 -5

50

250

500

1000

2500

5000

Log-N(1,0.4)

0 87

7 40

7 26

12 24

23 34

29 34

2

Figure 10A. Sampling distributions of 2SK (Smoothed histograms): single outlier case


-0.5 0 0.50

10

20

N(0,1)

N=5

0

-0.5 0 0.50

10

20

T-10

-0.5 0 0.50

10

20

T-5

-0.5 0 0.50

10

20

Log-N(1,0.4)

-0.5 0 0.50

10

20

N=2

50

-0.5 0 0.50

10

20

-0.5 0 0.50

10

20

-0.5 0 0.50

10

20

-0.5 0 0.50

10

20

N=5

00

-0.5 0 0.50

10

20

-0.5 0 0.50

10

20

-0.5 0 0.50

10

20

-0.5 0 0.50

10

20

N=1

000

-0.5 0 0.50

10

20

-0.5 0 0.50

10

20

-0.5 0 0.50

10

20

-0.5 0 0.50

10

20

N=2

500

-0.5 0 0.50

10

20

-0.5 0 0.50

10

20

-0.5 0 0.50

10

20

-0.5 0 0.50

10

20

N=5

000

-0.5 0 0.50

10

20

-0.5 0 0.50

10

20

-0.5 0 0.50

10

20

-0.4 -0.2 0 0.2 0.4 0.6

50

250

500

1000

2500

5000

N(0,1)

21 14

18 22

24 19

21 27

26 22

22 21

-0.4 -0.2 0 0.2 0.4 0.6

50

250

500

1000

2500

5000

T-10

14 21

18 11

20 21

23 21

25 24

32 29

-0.4 -0.2 0 0.2 0.4 0.6

50

250

500

1000

2500

5000

T-5

24 37

25 21

19 21

22 18

26 28

33 33

-0.4 -0.2 0 0.2 0.4 0.6

50

250

500

1000

2500

5000

Log-N(1,0.4)

19 12

20 12

15 29

21 19

30 30

16 20

3



-1 -0.5 0 0.50

10

20

30N(0,1)

N = 50

-1 -0.5 0 0.50

10

20

30T-10

-1 -0.5 0 0.50

10

20

30T-5

-1 -0.5 0 0.50

10

20

30Log-N(1,0.4)

-1 -0.5 0 0.50

10

20

30

N = 250

-1 -0.5 0 0.50

10

20

30

-1 -0.5 0 0.50

10

20

30

-1 -0.5 0 0.50

10

20

30

-1 -0.5 0 0.50

10

20

30

N = 50

0

-1 -0.5 0 0.50

10

20

30

-1 -0.5 0 0.50

10

20

30

-1 -0.5 0 0.50

10

20

30

-1 -0.5 0 0.50

10

20

30

N = 100

0

-1 -0.5 0 0.50

10

20

30

-1 -0.5 0 0.50

10

20

30

-1 -0.5 0 0.50

10

20

30

-1 -0.5 0 0.50

10

20

30

N = 250

0

-1 -0.5 0 0.50

10

20

30

-1 -0.5 0 0.50

10

20

30

-1 -0.5 0 0.50

10

20

30

-1 -0.5 0 0.50

10

20

30

N = 500

0

-1 -0.5 0 0.50

10

20

30

-1 -0.5 0 0.50

10

20

30

-1 -0.5 0 0.50

10

20

30

-1 -0.5 0 0.5

50

250

500

1000

2500

5000

N(0,1)

16 44

25 28

14 31

18 25

29 27

11 10

-1 -0.5 0 0.5

50

250

500

1000

2500

5000

T-10

10 29

23 25

11 21

23 27

18 21

10 14

-1 -0.5 0 0.5

50

250

500

1000

2500

5000

T-5

17 39

16 24

20 30

18 20

28 38

22 19

-1 -0.5 0 0.5

50

250

500

1000

2500

5000

Log-N(1,0.4)

10 29

17 19

21 19

26 17

17 26

22 30

4



-0.5 0 0.50

20

40

N(0,1)

N = 50

-0.5 0 0.50

20

40

T-10

-0.5 0 0.50

20

40

T-5

-0.5 0 0.50

20

40

Log-N(1,0.4)

-0.5 0 0.50

20

40

N = 25

0

-0.5 0 0.50

20

40

-0.5 0 0.50

20

40

-0.5 0 0.50

20

40

-0.5 0 0.50

20

40

N = 50

0

-0.5 0 0.50

20

40

-0.5 0 0.50

20

40

-0.5 0 0.50

20

40

-0.5 0 0.50

20

40

N = 10

00

-0.5 0 0.50

20

40

-0.5 0 0.50

20

40

-0.5 0 0.50

20

40

-0.5 0 0.50

20

40

N = 25

00

-0.5 0 0.50

20

40

-0.5 0 0.50

20

40

-0.5 0 0.50

20

40

-0.5 0 0.50

20

40

N = 50

00

-0.5 0 0.50

20

40

-0.5 0 0.50

20

40

-0.5 0 0.50

20

40

-0.5 0 0.5

50

250

500

1000

2500

5000

N(0,1)

26 43

29 32

21 34

19 26

29 27

11 10

-0.5 0 0.5

50

250

500

1000

2500

5000

T-10

19 36

19 27

11 21

23 28

17 21

9 15

-0.5 0 0.5

50

250

500

1000

2500

5000

T-5

18 40

20 27

22 31

18 20

29 38

22 18

-0.5 0 0.5

50

250

500

1000

2500

5000

Log-N(1,0.4)

12 39

13 31

19 17

18 16

18 30

21 30

5

Figure 13A. Sampling distributions of 1KR (Smoothed histograms): single outlier case

Figure 13B. Sampling distributions of 1KR (Box-plots): single outlier case

0 2000

0.1

0.2

N(0,1)

N=50

0 2000

0.1

0.2

T-10

0 2000

0.1

0.2

T-5

0 2000

0.1

0.2

Log-N(1,0.4)

0 2000

0.1

0.2

N=2

50

0 2000

0.1

0.2

0 2000

0.1

0.2

0 2000

0.1

0.2

0 2000

0.1

0.2

N=5

00

0 2000

0.1

0.2

0 2000

0.1

0.2

0 2000

0.1

0.2

0 2000

0.1

0.2

N=1

000

0 2000

0.1

0.2

0 2000

0.1

0.2

0 2000

0.1

0.2

0 2000

0.1

0.2

N=2

500

0 2000

0.1

0.2

0 2000

0.1

0.2

0 2000

0.1

0.2

0 2000

0.1

0.2

N=5

000

0 2000

0.1

0.2

0 2000

0.1

0.2

0 2000

0.1

0.2

0 50 100 150 200 250 300 350

50

250

500

1000

2500

5000

N(0,1)

102 0

37 3

42 24

15 18

24 15

16 24

0 50 100 150 200 250 300 350

50

250

500

1000

2500

5000

T-10

82 0

48 19

28 15

21 16

23 31

15 23

0 50 100 150 200 250 300 350

50

250

500

1000

2500

5000

T-5

99 0

31 5

27 15

8 14

18 23

14 32

0 50 100 150 200 250 300 350

50

250

500

1000

2500

5000

Log-N(1,0.4)

84 0

35 7

25 9

22 14

28 27

23 28

6


Figure 14B. Sampling distribut ions of 2KR (Box-plots): single outlier case

-1 0 10

10

N(0,1)

N=5

0

-1 0 10

10

T-10

-1 0 10

10

T-5

-1 0 10

10

Log-N(1,0.4)

-1 0 10

10

N=2

50

-1 0 10

10

-1 0 10

10

-1 0 10

10

-1 0 10

10

N=5

00

-1 0 10

10

-1 0 10

10

-1 0 10

10

-1 0 10

10

N=1

000

-1 0 10

10

-1 0 10

10

-1 0 10

10

-1 0 10

10

N=2

500

-1 0 10

10

-1 0 10

10

-1 0 10

10

-1 0 10

10

N=5

000

-1 0 10

10

-1 0 10

10

-1 0 10

10

-1 -0.5 0 0.5 1

50

250

500

1000

2500

5000

N(0,1)

1 35

10 37

16 31

11 26

20 29

28 27

-1 -0.5 0 0.5 1

50

250

500

1000

2500

5000

T-10

7 45

11 20

7 28

19 29

22 33

15 27

-1 -0.5 0 0.5 1

50

250

500

1000

2500

5000

T-5

2 44

6 26

10 28

13 27

25 23

18 21

-1 -0.5 0 0.5 1

50

250

500

1000

2500

5000

Log-N(1,0.4)

7 50

10 28

20 31

19 38

14 24

12 26

7



0 2 40

5

10

N(0,1)

N=5

0

0 2 40

5

10

T-10

0 2 40

5

10

T-5

0 2 40

5

10

Log-N(1,0.4)

0 2 40

5

10

N=2

50

0 2 40

5

10

0 2 40

5

10

0 2 40

5

10

0 2 40

5

10

N=5

00

0 2 40

5

10

0 2 40

5

10

0 2 40

5

10

0 2 40

5

10

N=1

000

0 2 40

5

10

0 2 40

5

10

0 2 40

5

10

0 2 40

5

10

N=2

500

0 2 40

5

10

0 2 40

5

10

0 2 40

5

10

0 2 40

5

10

N=5

000

0 2 40

5

10

0 2 40

5

10

0 2 40

5

10

-1 0 1 2 3 4

50

250

500

1000

2500

5000

N(0,1)

48 8

35 37

19 28

24 31

29 22

22 19

-1 0 1 2 3 4

50

250

500

1000

2500

5000

T-10

40 0

16 22

16 15

16 15

28 23

24 26

-1 0 1 2 3 4

50

250

500

1000

2500

5000

T-5

47 13

15 23

11 36

17 26

17 31

12 23

-1 0 1 2 3 4

50

250

500

1000

2500

5000

Log-N(1,0.4)

38 16

11 18

20 34

19 18

21 25

26 14

8



0 5 100

5

N(0,1)

N=5

0

0 5 100

5

T-10

0 5 100

5

T-5

0 5 100

5

Log-N(1,0.4)

0 5 100

5

N=2

50

0 5 100

5

0 5 100

5

0 5 100

5

0 5 100

5

N=5

00

0 5 100

5

0 5 100

5

0 5 100

5

0 5 100

5

N=1

000

0 5 100

5

0 5 100

5

0 5 100

5

0 5 100

5

N=2

500

0 5 100

5

0 5 100

5

0 5 100

5

0 5 100

5

N=5

000

0 5 100

5

0 5 100

5

0 5 100

5

-2 0 2 4 6 8 10

50

250

500

1000

2500

5000

N(0,1)

20 31

7 31

13 22

17 22

21 27

15 11

-2 0 2 4 6 8 10

50

250

500

1000

2500

5000

T-10

19 29

23 42

8 26

11 19

22 23

19 15

-2 0 2 4 6 8 10

50

250

500

1000

2500

5000

T-5

23 41

8 36

15 30

17 30

8 22

19 28

-2 0 2 4 6 8 10

50

250

500

1000

2500

5000

Log-N(1,0.4)

8 38

14 53

11 33

18 20

22 18

15 15

1

Figure 17A. Sampling distributions of 1SK (smoothed histograms): mixture distribution case

Figure 17B. Sampling distributions of 1SK (Box-plots): mixture distribution case

-6 -4 -2 0 2 40

1

N(0,1)

N=5

0

-6 -4 -2 0 2 40

1

T-10

-6 -4 -2 0 2 40

1

T-5

-6 -4 -2 0 2 40

1

Log-N(1,0.4)

-6 -4 -2 0 2 40

1

N=2

50

-6 -4 -2 0 2 40

1

-6 -4 -2 0 2 40

1

-6 -4 -2 0 2 40

1

-6 -4 -2 0 2 40

1

N=5

00

-6 -4 -2 0 2 40

1

-6 -4 -2 0 2 40

1

-6 -4 -2 0 2 40

1

-6 -4 -2 0 2 40

1

N=1

000

-6 -4 -2 0 2 40

1

-6 -4 -2 0 2 40

1

-6 -4 -2 0 2 40

1

-6 -4 -2 0 2 40

1

N=2

500

-6 -4 -2 0 2 40

1

-6 -4 -2 0 2 40

1

-6 -4 -2 0 2 40

1

-6 -4 -2 0 2 40

1

N=5

000

-6 -4 -2 0 2 40

1

-6 -4 -2 0 2 40

1

-6 -4 -2 0 2 40

1

-6 -4 -2 0 2 4

50

250

500

1000

2500

5000

N(0,1)

54 29

174 29

194 38

136 10

63 3

44 3

-6 -4 -2 0 2 4

50

250

500

1000

2500

5000

T-10

53 32

139 43

199 38

164 28

99 15

66 8

-6 -4 -2 0 2 4

50

250

500

1000

2500

5000

T-5

70 60

132 42

157 40

124 38

104 30

93 25

-6 -4 -2 0 2 4

50

250

500

1000

2500

5000

Log-N(1,0.4)

46 59

140 82

194 42

82 39

22 47

27 69

2

Figure 18A. Sampling distributions of 2SK (Smoothed histograms): mixture distribution case


-0.5 0 0.50

10

20N(0,1)

N=5

0

-0.5 0 0.50

10

20T-10

-0.5 0 0.50

10

20T-5

-0.5 0 0.50

10

20Log-N(1,0.4)

-0.5 0 0.50

10

20

N=2

50

-0.5 0 0.50

10

20

-0.5 0 0.50

10

20

-0.5 0 0.50

10

20

-0.5 0 0.50

10

20

N=5

00

-0.5 0 0.50

10

20

-0.5 0 0.50

10

20

-0.5 0 0.50

10

20

-0.5 0 0.50

10

20

N=1

000

-0.5 0 0.50

10

20

-0.5 0 0.50

10

20

-0.5 0 0.50

10

20

-0.5 0 0.50

10

20

N=2

500

-0.5 0 0.50

10

20

-0.5 0 0.50

10

20

-0.5 0 0.50

10

20

-0.5 0 0.50

10

20

N=5

000

-0.5 0 0.50

10

20

-0.5 0 0.50

10

20

-0.5 0 0.50

10

20

-0.5 0 0.5

50

250

500

1000

2500

5000

N(0,1)

10 15

26 31

18 15

27 29

33 29

17 16

-0.5 0 0.5

50

250

500

1000

2500

5000

T-10

18 20

21 24

23 21

20 25

32 22

21 19

-0.5 0 0.5

50

250

500

1000

2500

5000

T-5

25 29

23 17

26 27

19 24

18 13

25 21

-0.5 0 0.5

50

250

500

1000

2500

5000

Log-N(1,0.4)

21 14

16 19

23 15

17 13

23 18

21 17

3



-0.5 0 0.50

10

20N(0,1)

N=5

0

-0.5 0 0.50

10

20T-10

-0.5 0 0.50

10

20T-5

-0.5 0 0.50

10

20Log-N(1,0.4)

-0.5 0 0.50

10

20

N=2

50

-0.5 0 0.50

10

20

-0.5 0 0.50

10

20

-0.5 0 0.50

10

20

-0.5 0 0.50

10

20

N=5

00

-0.5 0 0.50

10

20

-0.5 0 0.50

10

20

-0.5 0 0.50

10

20

-0.5 0 0.50

10

20

N=1

000

-0.5 0 0.50

10

20

-0.5 0 0.50

10

20

-0.5 0 0.50

10

20

-0.5 0 0.50

10

20

N=2

500

-0.5 0 0.50

10

20

-0.5 0 0.50

10

20

-0.5 0 0.50

10

20

-0.5 0 0.50

10

20

N=5

000

-0.5 0 0.50

10

20

-0.5 0 0.50

10

20

-0.5 0 0.50

10

20

-0.5 0 0.5

50

250

500

1000

2500

5000

N(0,1)

21 17

20 31

29 31

12 19

16 16

21 25

-0.5 0 0.5

50

250

500

1000

2500

5000

T-10

28 27

26 26

14 28

12 24

24 12

20 17

-0.5 0 0.5

50

250

500

1000

2500

5000

T-5

14 13

17 23

21 26

23 23

27 30

16 20

-0.5 0 0.5

50

250

500

1000

2500

5000

Log-N(1,0.4)

21 13

24 24

21 20

24 23

31 31

27 16

4



-0.5 0 0.50

10

20N(0,1)

N=5

0

-0.5 0 0.50

10

20T-10

-0.5 0 0.50

10

20T-5

-0.5 0 0.50

10

20Log-N(1,0.4)

-0.5 0 0.50

10

20

N=2

50

-0.5 0 0.50

10

20

-0.5 0 0.50

10

20

-0.5 0 0.50

10

20

-0.5 0 0.50

10

20

N=5

00

-0.5 0 0.50

10

20

-0.5 0 0.50

10

20

-0.5 0 0.50

10

20

-0.5 0 0.50

10

20

N=1

000

-0.5 0 0.50

10

20

-0.5 0 0.50

10

20

-0.5 0 0.50

10

20

-0.5 0 0.50

10

20

N=2

500

-0.5 0 0.50

10

20

-0.5 0 0.50

10

20

-0.5 0 0.50

10

20

-0.5 0 0.50

10

20

N=5

000

-0.5 0 0.50

10

20

-0.5 0 0.50

10

20

-0.5 0 0.50

10

20

-0.5 0 0.5

50

250

500

1000

2500

5000

N(0,1)

27 27

16 17

22 20

24 14

11 12

9 24

-0.5 0 0.5

50

250

500

1000

2500

5000

T-10

19 23

21 25

20 19

13 26

16 24

26 19

-0.5 0 0.5

50

250

500

1000

2500

5000

T-5

20 30

8 19

18 30

12 23

20 17

22 20

-0.5 0 0.5

50

250

500

1000

2500

5000

Log-N(1,0.4)

38 12

46 14

37 12

26 9

34 12

23 12

5

Figure 21A. Sampling distributions of 1KR (Smoothed histograms): mixture distribution case

Figure 21B. Sampling distributions of 1KR (Box-plots): mixture distribution case

0 20 400

0.5N(0,1)

N=5

0

0 20 400

0.5T-10

0 20 400

0.5T-5

0 20 400

0.5Log-N(1,0.4)

0 20 400

0.5

N=2

50

0 20 400

0.5

0 20 400

0.5

0 20 400

0.5

0 20 400

0.5

N=5

00

0 20 400

0.5

0 20 400

0.5

0 20 400

0.5

0 20 400

0.5

N=1

000

0 20 400

0.5

0 20 400

0.5

0 20 400

0.5

0 20 400

0.5

N=2

500

0 20 400

0.5

0 20 400

0.5

0 20 400

0.5

0 20 400

0.5

N=5

000

0 20 400

0.5

0 20 400

0.5

0 20 400

0.5

0 50 100

50

250

500

1000

2500

5000

N(0,1)

0 90

0 190

0 201

0 150

0 74

0 78

0 50 100

50

250

500

1000

2500

5000

T-10

0 108

0 182

0 211

0 159

0 117

0 96

0 50 100

50

250

500

1000

2500

5000

T-5

0 119

0 160

0 177

0 170

0 120

0 125

0 50 100

50

250

500

1000

2500

5000

Log-N(1,0.4)

0 115

0 175

0 189

0 109

0 66

0 78

6



-1 0 10

5

10

N(0,1)

N=5

0

-1 0 10

5

10

T-10

-1 0 10

5

10

T-5

-1 0 10

5

10

Log-N(1,0.4)

-1 0 10

5

10

N=2

50

-1 0 10

5

10

-1 0 10

5

10

-1 0 10

5

10

-1 0 10

5

10

N=5

00

-1 0 10

5

10

-1 0 10

5

10

-1 0 10

5

10

-1 0 10

5

10

N=1

000

-1 0 10

5

10

-1 0 10

5

10

-1 0 10

5

10

-1 0 10

5

10

N=2

500

-1 0 10

5

10

-1 0 10

5

10

-1 0 10

5

10

-1 0 10

5

10

N=5

000

-1 0 10

5

10

-1 0 10

5

10

-1 0 10

5

10

-1 -0.5 0 0.5 1

50

250

500

1000

2500

5000

N(0,1)

7 45

7 38

7 28

9 32

13 18

27 26

-1 -0.5 0 0.5 1

50

250

500

1000

2500

5000

T-10

5 48

11 34

10 23

18 32

22 23

21 27

-1 -0.5 0 0.5 1

50

250

500

1000

2500

5000

T-5

10 49

11 36

11 35

22 28

9 19

22 20

-1 -0.5 0 0.5 1

50

250

500

1000

2500

5000

Log-N(1,0.4)

6 39

8 33

10 25

23 30

20 27

32 28

7



-1 0 10

5

N(0,1)

N=5

0

-1 0 10

5

T-10

-1 0 10

5

T-5

-1 0 10

5

Log-N(1,0.4)

-1 0 10

5

N=2

50

-1 0 10

5

-1 0 10

5

-1 0 10

5

-1 0 10

5

N=5

00

-1 0 10

5

-1 0 10

5

-1 0 10

5

-1 0 10

5

N=1

000

-1 0 10

5

-1 0 10

5

-1 0 10

5

-1 0 10

5

N=2

500

-1 0 10

5

-1 0 10

5

-1 0 10

5

-1 0 10

5

N=5

000

-1 0 10

5

-1 0 10

5

-1 0 10

5

-1 -0.5 0 0.5 1

50

250

500

1000

2500

5000

N(0,1)

9 51

9 82

0 74

1 63

4 40

13 44

-1 -0.5 0 0.5 1

50

250

500

1000

2500

5000

T-10

13 61

6 88

4 76

3 46

13 46

4 39

-1 -0.5 0 0.5 1

50

250

500

1000

2500

5000

T-5

3 48

3 53

4 50

9 51

7 37

15 36

-1 -0.5 0 0.5 1

50

250

500

1000

2500

5000

Log-N(1,0.4)

5 60

2 75

1 56

7 38

14 28

15 30

8



-1 0 1 20

5

N(0,1)

N=5

0

-1 0 1 20

5

T-10

-1 0 1 20

5

T-5

-1 0 1 20

5

Log-N(1,0.4)

-1 0 1 20

5

N=2

50

-1 0 1 20

5

-1 0 1 20

5

-1 0 1 20

5

-1 0 1 20

5

N=5

00

-1 0 1 20

5

-1 0 1 20

5

-1 0 1 20

5

-1 0 1 20

5

N=1

000

-1 0 1 20

5

-1 0 1 20

5

-1 0 1 20

5

-1 0 1 20

5

N=2

500

-1 0 1 20

5

-1 0 1 20

5

-1 0 1 20

5

-1 0 1 20

5

N=5

000

-1 0 1 20

5

-1 0 1 20

5

-1 0 1 20

5

-1 0 1 2

50

250

500

1000

2500

5000

N(0,1)

1 57

12 45

14 30

19 31

25 40

32 27

-1 0 1 2

50

250

500

1000

2500

5000

T-10

1 54

9 45

14 37

15 22

19 15

16 26

-1 0 1 2

50

250

500

1000

2500

5000

T-5

0 58

5 34

14 31

22 34

8 26

19 28

-1 0 1 2

50

250

500

1000

2500

5000

Log-N(1,0.4)

0 72

9 35

8 25

15 33

12 27

22 26

On More Robust Estimation of Skewness and … kurtosis. In the finance literature, there has been...

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Transcript of On More Robust Estimation of Skewness and … kurtosis. In the finance literature, there has been...