the Meaning of Zero Skewness and Zero Kurtosis of the Discrete

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Section 3 the Meaning of Zero Skewness and Zero Kurtosis of the Discrete Probability Distributions

Some may think the normal approximated discrete probability distribution can be

characterized as having both skewness and kurtosis values of 0. However, the above

characterization may not be reasonable. It can be show as following example. If we

symmetrically divided a normal distribution between -3 and 3 into equal-interval

categories probability distributions with number of categories from 3 to 20, the

skewness are all 0 but none of the kurtosis is 0. The values of kurtosis are presented in

Table 6. It can be seen that all the values of kurtosis are smaller than 0 except in the

case that the number of categories is 3. The more categories of the discrete probability

distribution, the closer the value of kurtosis gets to 0. The range of kurtosis is -.26 in

the 4 categories probability distribution and -.17 in the 20 categories probability

distribution. Therefore, the discrete probability distribution may not be normal

approximated when the discrete probability distribution with both the values of

skewness and kurtosis are 0.

In addition, the probability distribution might not be symmetric even with zero

skewness. While relatively large value of kurtosis is specified for a discrete

probability distribution, in order to satisfying the constraint, the probability might be

highly concentrated in 1 category. This makes the even categories probability

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distributions might be asymmetric when skewness is 0. When the value of skewness is

set at 0 and the value of kurtosis is set at a relatively high value (such as 0 in the 4

categories probability distributions), large probability could appear in only 1 category

near center, and thus the discrete probability distribution would be asymmetric.

Although the MEP-2 and MEP-4 don’t tend to generate the probability distributions

with peaks since the maximum entropy distributions are the smoothest one, they still

could not guarantee the symmetric with unsupported information.

Section 4 the Uniqueness of the Solutions of the MEP-4 and MEP-2

The uniqueness of the solutions with the maximum entropy with the first 4

moments has been discussed and proved (Kapur & Kesavan, 1992; Kesavan & Kapur,

1989; Zellner & Highfield, 1988). However, the uniqueness of the solutions of MEP-2

is still in question. In this research, it could be seen through empirically that in some

conditions that the solutions of MEP-2 are not unique.

First, it has been proven that the solutions on the boundary of skewness and

kurtosis would be solved if and only if the number of nonempty categories are 2

(Wilkins, 1944). Therefore, in a k categories probability distribution, there are 

 



 2 k

solutions with the same value of information entropy would be generated when the

values of skewness and kurtosis are on the boundary. It’s obvious there’s no unique

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solution in this situation. A more efficient R package for this situation is proposed in

Appendix C. Since this package is used for the boundary, the only information needed

is the value of skewness. The solutions obtained through this package are quite

accurate. It’s suggested to use the package when the set value of skewness and

kurtosis are on the boundary.

Second, there are some discrete maximum entropy distributions with zero

skewness are not symmetric as mentioned. These probability distributions could be

flipped over and the skewness, kurtosis and entropy would be still the same. It shows

that if the estimated discrete maximum entropy distributions are not symmetric with 0

skewness, there are at least 2 different maximum entropy distributions exist. Through

above discussion, it can be seen that the uniqueness of the solutions in this situation

does not hold.

Through discussion above, it can be seen that although the uniqueness of the

MEP-4 is proved, the uniqueness of the MEP-2 is still in question. The uniqueness of

solutions of MEP-2 would not be held in at least 2 situations. The situations are 1. the

values of skewness and kurtosis are on the boundary. 2. The maximum entropy

distributions are not symmetric when skewness is 0.

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Section 5 the Maximum Entropy Distributions with Prior Probability Distributions

The MEP presented in this research is restricted the known information as the

parameters of the target probability distribution. With more information of the shape

other than the parameters of the target probability distribution, another procedure of

maximum entropy family might be more useful. The probability of each category of

the probability distribution need to be set and the probability distribution is defined as

prior probability distribution. With probability of each category of the prior discrete

probability distribution are set, the measure Kullback–Leibler divergence introduced

by Kullback and Leiber (1951) could be used to choosing the maximum entropy

distribution satisfying the parameters and with the smallest divergence to the prior

probability distribution. The measure Kullback–Leibler divergence for the discrete

probability distributions is defined as follows,

 k

i i

i q

p p

ln (24)

where p_i and q_i are the probability of ith category of the estimated probability

distribution A and the prior probability distribution B respectively. The minimization

of formula (24) entails the maximization of the entropy of distribution A. The

Kullback–Leibler divergence is also defined as cross entropy. It seeks to determine the

probability distribution A that satisfies all the constraints and with the closest

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probability values to the given probability distribution B. It means that the

probabilities of A would be modified to the closest values to the corresponding

probabilities of B and fulfilled the constraints. Therefore, with a clear picture of the

shape of probability distribution in mind, the cross entropy could be used and a proper

probability distribution could be suggested.

Section 6 Conclusion

The discrete variables are quite common in psychological researches, but the

robustness researches on this topic are relatively scant. When conducting the Monte

Carlo researches for understanding the effect of the parameters, there are 2 difficulties

would be encountered. The first difficulty is to estimate the discrete probability

distribution with the specified parameters precisely and the second difficulty is to

choose the discrete probability distribution from infinite discrete probability

distribution with the same parameters when the number of constraints is smaller than

the number of categories. Therefore a procedure to estimate the discrete probability

distribution is necessary and practical. The Maximum Entropy Procedure (MEP)

suggests to estimating the discrete maximum entropy distribution with specified

parameters as constraints. The maximum entropy distribution is the greatest number

distribution from all discrete probability distributions satisfying the same constraints.

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Therefore, it’s a rationale choice if only the values of the parameters are known

information. In addition, the discrete maximum entropy distribution chosen through

MEP are smooth and have no empty categories unless necessary. These properties of

maximum entropy distributions make MEP a suitable procedure to follow in

simulating the discrete probability distribution of psychological research with the

specified parameters as the only information. The Maximum Entropy Procedure with

4 Parameters (MEP-4) and Maximum Entropy Procedure with 2 Parameters (MEP-2)

are presented, evaluated and discussed. Moreover, the R packages of these procedures

are also presented in this research for the implementation of these procedures.

Furthermore, the accuracy of these procedures also the R packages is confirmed by

three simulation studies in this research and the possible generated parameter spaces

of these procedures are consistent with our knowledge. The R packages could

estimate the discrete probability distribution with specified information and also

sample specified number of samples from this estimated distribution with specified

sample size. Therefore, theses programs are convenient and practical for the Monte

Carlo researches of the robustness of the statistics with the univarate non-normal

approximated discrete data.

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Table 5 the Descriptive Statistics and Root Mean Square Errors of Difference between the Specified and Obtained 2 Parameters of 4 and 5 (in brackets) Categories Discrete Measures of the Study 3

N=925

(N=925) Descriptive Statistics Root Mean Squared Difference

Difference Min Max Mean Variance Mean Variance

skewness -.001(-.001) .001(.001) -.000(-.000) .000(.000) .000(.000) .000(.000) kurtosis -.001(-.001) .001(.001) -.000(-.000) .000(.000) .000(.000) .000(.000)

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Table 6 the Values of Kurtosis of 3 to 20 Equal-Interval Categories Discrete Measures from the Normal Distribution

Number of Categories Values of Kurtosis

3 .170

4 -.264

5 -.217

6 -.203

7 -.194

8 -.189

9 -.185

10 -.182

11 -.180

12 -.179

13 -.178

14 -.177

15 -.176

MEP4=function(points,mean,variance,skewness,kurtosis,size=0,number=0){

center=points/2+0.5 scale=-(1-center)

X=matrix(0,5,points) for (i in 1:nrow(X)){

X[i,]=rep(1,points)*((seq(1:points)-(points/2+0.5))/scale)^(i-1) }

mea=(mean-center)/scale var=variance/scale^2 skew=skewness kurt=kurtosis+3

x1=mea

x2=(var)+(mea)^2

x3=((var)^(3/2))*(skew)+(3*x2*x1-2*(x1^3))

x4=((var)^2)*(kurt)+((4*x3*x1)-(6*x2*(x1^2))+3*(x1^4)) moment=rbind(x1,x2,x3,x4)

M=c(1,moment)

lambda=matrix(c(-log(1/points),0,0,0,0),1,5)

for (t in 1:1000){

P=exp(-1*lambda%*%X)

gnk=matrix(0,5,5) for (i in 1:5){

for (j in 1:5){

gnk[i,j]=P%*%as.matrix(X[2,]^(i+j-2)) }}

v=X%*%t(P)-M

z=try(solve(gnk),silent=T)

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if(is(z,"try-error")) break else { delta=solve(gnk)%*%v

if (sum(is.na(abs(delta)))>0) break else { if (sum(abs(delta))<1e-20) break else { lambda=lambda+t(delta)}

}}

}

mean.est=(X[2,]%*%t(P))*scale+center

var.est=(X[3,]%*%t(P)-(X[2,]%*%t(P))^2)*scale^2

skew.est=(X[4,]%*%t(P)-3*X[3,]%*%t(P)*X[2,]%*%t(P)+2*(X[2,]%*%t(P))^3)/(X[

3,]%*%t(P)-(X[2,]%*%t(P))^2)^(3/2)

kurt.est=(X[5,]%*%t(P)-4*X[4,]%*%t(P)*X[2,]%*%t(P)+6*(X[3,]%*%t(P))*(X[2,]

%*%t(P))^2-3*(X[2,]%*%t(P))^4)/((X[3,]%*%t(P)-(X[2,]%*%t(P))^2)^2)-3

P1=P

for (c in 1:ncol(P1)){

if (P1[1,c]==0) P1[1,c]=1 }

entropy=sum(log(P1)*P1)*-1

diff=c(mean-mean.est,variance-var.est,skewness-skew.est,kurtosis-kurt.est) abs=abs(diff)

output=c(mean,variance,skewness,kurtosis,P,mean.est,var.est,skew.est,kurt.est,entropy ,diff,abs)

names(output)=c("set.mean","set.var","set.skew","set.kurt",paste("p",1:points,sep="") ,"est.mean","est.var","est.skew","est.kurt","entropy","diff.mean","diff.var","diff.skew

","diff.kurt","abs.mean","abs.var","abs.skew","abs.kurt")

write.table(t(as.matrix(output)),file=paste("c:/output",paste(mean,",",variance,",",ske wness,",",kurtosis,sep=""),".txt",sep="",collapse=""),row.names=F)

error=0

if (is.na(mean.est)==T|is.na(var.est)==T|is.na(skew.est)==T|is.na(kurt.est)==T){

error=T} else {

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if (abs(mean.est-mean)>0.001 & abs(var.est-variance)>0.001 &

abs(skew.est-skewness)>0.001 & abs(kurt.est-3-kurtosis)>0.001 & is.na(entropy)==F) { error=T }}

n=size m=number

if (n==0){

m=0}

if (n>0){

if (m==0){

m=1 }}

if (error==T) list(error="improper solution") else {

if (n>0){

if (m>0){

data=matrix(0,n,m) for (i in 1:m){

pro=runif(n,min=0,max=sum(output[5:(points+4)])) for (j in 1:n){

for (k in 1:points){

if (k==1){

if (pro[j]<=output[5]) data[j,i]=1}

else

if (pro[j]>sum(output[5:(k+3)]) & pro[j]<=sum(output[5:(k+4)])) data[j,i]=k }}}

colnames(data)=paste("sample",1:m,sep="") write.table(data,file="C:/data.dat")

# save the data generated

list(output=output,data=data) }} else list(output=output) }}

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Appendix B the Code of the R Package for the MEP-2

MEP2=function(points,skewness,kurtosis,maxit=500,st=10, size=0,number=0){

par=rep(1/points,points-1) center=points/2+0.5 scale=-(1-center)

X=matrix(0,5,points) for (i in 1:nrow(X)){

X[i,]=rep(1,points)*((seq(1:points)-(points/2+0.5))/scale)^(i-1) }

MEP.P=function(par,points,skew,kurt){

p=1-sum(par)

if (all.equal(0,p)==T) p=0

P=c(as.matrix(par),p)

skew.est=(X[4,]%*%P-3*X[3,]%*%P*X[2,]%*%P+2*(X[2,]%*%P)^3)/(X[3,]%*%P -(X[2,]%*%P)^2)^(3/2)

kurt=kurt+3

kurt.est=(X[5,]%*%P-4*(X[4,]%*%P)*(X[2,]%*%P)+6*(X[3,]%*%P)*((X[2,]%*%

P)^2)-3*(X[2,]%*%P)^4)/((X[3,]%*%P-(X[2,]%*%P)^2)^2)

for (i in 1:points){

if (P[i]==0){P[i]=1}}

if (is.na(skew.est)==T) skew.est=100 if (is.na(kurt.est)==T) kurt.est=100

return(sum(log(P)*P)+abs(skew.est-skew)+abs(kurt.est-kurt)) }

for (j in 1:maxit){

p.est=optim(par,MEP.P,points=points,skew=skewness,kurt=kurtosis,control=list(ndep s=1e-4,maxit=10000,reltol=1e-20))$par

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P=c(p.est,1-sum(p.est))

skew.est=(X[4,]%*%P-3*X[3,]%*%P*X[2,]%*%P+2*(X[2,]%*%P)^3)/(X[3,]%*%P -(X[2,]%*%P)^2)^(3/2)

kurt.est=(X[5,]%*%P-4*X[4,]%*%P*X[2,]%*%P+6*(X[3,]%*%P)*(X[2,]%*%P)^2 -3*(X[2,]%*%P)^4)/((X[3,]%*%P-(X[2,]%*%P)^2)^2)

for (i in 1:points){

if (P[i]==0){P[i]=1}}

entropy=sum(log(P)*P)

p=1-sum(p.est) if (p<0) p=0

if (abs(skew.est-skewness)<0.001 & abs(kurt.est-3-kurtosis)<0.001 &

is.na(entropy)==F) break else { if (j<=11){

par=p.est[1:points-1]+rnorm((points-1))/1000}

if (j>11 & j<=20){

par=p.est[1:points-1]+rnorm((points-1))/100}

if (j>20){

par=abs(rnorm(points-1)/3)}

for (b in 1:(points-1)){

if(par[b]<0) par[b]=0 if(par[b]>1) par[b]=1 }

par=par/sum(c(par,p)) }}

st_en=matrix(0,(st+1),(points+5)) st_en[1,1]=skewness

st_en[1,2]=kurtosis st_en[1,3:(points+2)]=P st_en[1,(points+3)]=skew.est st_en[1,(points+4)]=kurt.est-3

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st_en[1,(points+5)]=entropy

for (k in 2:(st+1)){

for (l in 1:500){

par=abs(rnorm(points-1)/3)

for (b in 1:(points-1)){

if(par[b]>1) par[b]=1 }

par=par/sum(c(par,p))

p.est=optim(par,MEP.P,points=points,skew=skewness,kurt=kurtosis,control=list(ndep s=1e-4,maxit=10000,reltol=1e-20))$par

P=c(p.est,1-sum(p.est))

skew.est=(X[4,]%*%P-3*X[3,]%*%P*X[2,]%*%P+2*(X[2,]%*%P)^3)/(X[3,]%*%P -(X[2,]%*%P)^2)^(3/2)

kurt.est=(X[5,]%*%P-4*X[4,]%*%P*X[2,]%*%P+6*(X[3,]%*%P)*(X[2,]%*%P)^2 -3*(X[2,]%*%P)^4)/((X[3,]%*%P-(X[2,]%*%P)^2)^2)

for (i in 1:points){

if (P[i]==0){P[i]=1}}

entropy=sum(log(P)*P)

if (abs(skew.est-skewness)<0.001 & abs(kurt.est-3-kurtosis)<0.001 &

is.na(entropy)==F) break p=1-sum(p.est)

if (p<0) p=0 }

st_en[k,1]=skewness st_en[k,2]=kurtosis st_en[k,3:(points+2)]=P st_en[k,(points+3)]=skew.est

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st_en[k,(points+4)]=kurt.est-3 st_en[k,(points+5)]=entropy }

colnames(st_en)=c("set.skew","set.kurt",paste("p",1:points,sep=""),"est.skew","est.ku rt","entropy")

diff=cbind((skewness-st_en[,(points+3)]),(kurtosis-st_en[,(points+4)])) group=abs(diff[,1])<0.001&abs(diff[,2])<0.001

st_en[,(points+5)]=st_en[,(points+5)]*-1

output=st_en[(order(group,st_en[,points+5],decreasing=T)[1]),]

names(output)=c("set.skew","set.kurt",paste("p",1:points,sep=""),"est.skew","est.kurt"

,"entropy")

write.table(t(as.matrix(output)),file=paste("c:/mep2/validation/output4/MEP2_v4/outp ut",paste(skewness,",",kurtosis,sep=""),".txt",sep="",collapse=""),row.names=F) write.table(st_en,file=paste("c:/mep2/validation/output4/MEP2_v4/diss_st",paste(ske wness,",",kurtosis,sep=""),".txt",sep="",collapse=""),row.names=F)

n=size m=number

if (n==0){

m=0 }

if (n>0){

if (m==0){

m=1 }}

if (n>0){

if (m>0){

data=matrix(0,n,m) for (i in 1:m){

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pro=runif(n,min=0,max=sum(output[3:(points+4)])) for (j in 1:n){

for (k in 1:points){

if (k==1){

if (pro[j]<=output[3]) data[j,i]=1}

else

if (pro[j]>sum(output[3:(k+1)]) & pro[j]<=sum(output[3:(k+2)])) data[j,i]=k }}}

# generate variables of different sample size and number of samples with specified parameters

list(output=output,data=data) }} else list(output=output) }

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Na tiona

l Ch engchi University

Appendix C the Code of the R Package for the Boundary of Skewness and Kurtosis MEP.bon=function(skew){

MEP.P=function(par,skew){

P=c(par,1-par)

X=matrix(0,5,2) for (i in 1:nrow(X)){

X[i,]=rep(1,2)*(seq(1:2)^(i-1)) }

skew.est=(X[4,]%*%P-3*X[3,]%*%P*X[2,]%*%P+2*(X[2,]%*%P)^3)/(X[3,]%*%P -(X[2,]%*%P)^2)^(3/2)

for (i in 1:points){

if (P[i]==0){P[i]=1}}

return(sum(log(P)*P)+100*abs(skew.est-skew)) }

p1=optimize(MEP.P,interval=c(0,1),skew=skew)$minimum probability=c(p1,1-p1)

list(probability=probability) }

在文檔中非常態間斷隨機變數的產生 - 政大學術集成 (頁 59-0)

the Meaning of Zero Skewness and Zero Kurtosis of the Discrete

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Section 3 the Meaning of Zero Skewness and Zero Kurtosis of the Discrete Probability Distributions

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Section 4 the Uniqueness of the Solutions of the MEP-4 and MEP-2

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Section 5 the Maximum Entropy Distributions with Prior Probability Distributions

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Section 6 Conclusion

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Reference

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