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1 Math Tibergs Simulating the Sampling Distribution of the t-statistic Name: Eample May 20, 2011 How well does a statistic behave when the assumptions about the underlying population fail to hold? We shall eplo the t-statistic, for small and large samples. For small samples, the population is assumed to be approimately normal, so that the statistic itself follows a t- distribution and that hypothesis tests can me made using critical t-scos. We simulate random samples taken from eponential and Cauchy distributions and look at the empirical distributions of the t-statistic. This study follows the description in Verzani s SimpleR: Using R for Introductory Statistics, from the Appendi A sample R session: a simulation eample. R Session: R version ( ) Copyright (C) 2009 The R Foundation for Statistical Computing ISBN R is fe softwa and comes with ABSOLUTELY NO WARRANTY. You a welcome to distribute it under certain conditions. Type license() or licence() for distribution details. Natural language support but running in an English locale R is a collaborative project with many contributors. Type contributors() for mo information and citation() on how to cite R or R packages in publications. Type demo() for some demos, help() for on-line help, or help.start() for an HTML browser interface to help. Type q() to quit R. [R.app GUI 1.31 (5538) powerpc-apple-darwin8.11.1] [Workspace stod from /Users/andjstibergs/.RData] ##################################################################### ### SIMULATION TO SEE HOW ROBUST THE t-statistic IS TO CHANGES OF ### ### DISTRIBUTION ### ##################################################################### # This study follows the description in Verzani s "SimpleR: Using R for # Introductory Statistics", from the Appendi "A sample R session: a # simulation eample." # We a intested in how much the distribution of the t-statistic # is influenced by the shape of the population distribution. # For various background distributions of mean mu, we take 500 # random samples of size n. For each we compute the t-statistic and # then analyze the distribution by plotting its histogram, boplot # QQ-normal plot. 1

2 ######################### FUNCTION GIVES t-statistic ################ make.t <- function(,mu) (mean()-mu)/(sqrt(var()/length())) mu <- 2; <- rnorm(100,mu,1) make.t(,mu) [1] mu <- 16; <- p(100,1/mu);make.t(,mu) [1] mu <- 16; <- p(100,1/mu);make.t(,mu) [1] mu <- 16; <- p(100,1/mu);make.t(,mu) [1] ################## LOOP TO TAKE 500 SAMPLES OF SIZE n ################ # Set up four plots per page. Save old graphics parameters. Resto at end. opar <- par(mfrow=c(2,2)) n<- 150;mu<-1 for(i in 1:500)[i]<- make.t(p(n,1/mu),mu) hist(,main="dist t-values for samples of",fq=f) curve(dt(,n-1),add=t,col=4) boplot(,main=paste("ep vars of size n =",n,", mu=",mu)) qqnorm();qqline(,col=4) curve(dep(,1/mu),lim=c(0,4),main=paste("ep dist with mu=",mu),col=4) abline(h=0);abline(v=0) 2

3 Dist t-values for samples of Ep vars of size n = 150, mu= 1 Density Ep dist with mu= 1 dep(, 1/mu) Theotical Quantiles # With a large sample selected from eponential distribution, the # t-statistic looke ptty nortmal: it is symmetric, and the QQ plot # is ptty much normal. We have added the standard t-dist with n-1 df # superimposed across the histogram. 3

4 # The eponential distribution is asymmetric and one tail is light, # the other s heavy. Do these properties show up in the simulation? # Let s try samples of size 15. The distribution of the statistic is # skewed and no longer normal. n<- 15;mu<-10 for(i in 1:500)[i]<- make.t(p(n,1/mu),mu) for(i in 1:500)[i]<- make.t(p(n,1/mu),mu) hist(,main="dist t-values for samples of") boplot(,main=paste("eponentials of size n =",n,", mu=",mu));qqnorm();qqline(,col=2) curve(dep(,1/mu),lim=c(0,4),main=paste("ep dist with mu=",mu));abline(h=0);abline(v=0) Dist t-values for samples of eponentials of size n = 15, mu= 10 Fquency ep dist with mu= dep(, 1/mu) Theotical Quantiles

5 # Let s try samples of size 8 and change the parameter. The # distribution of the statistic looks worse. n<- 8;mu<-4 for(i in 1:500)[i]<- make.t(p(n,1/mu),mu) hist(,main="dist t-values for samples of",fq=f);curve(dt(,n-1),add=t,col=4) legend(-18,.19,legend=paste("t-dist, df=",n-1),col=4,pch=19) boplot(,main=paste("eponentials of size n =",n,", mu=",mu));qqnorm();qqline(,col=4) curve(dep(,1/mu),lim=c(0,4),main=paste("ep dist with mu=",mu));abline(h=0);abline(v=0) Dist t-values for samples of eponentials of size n = 8, mu= 4 Density t-dist, df= ep dist with mu= dep(, 1/mu) Theotical Quantiles

6 # Let s try samples of size 5 and change the parameter. The # distribution of the statistic looks even worse. n <- 5; mu <- 2 for(i in 1:500)[i]<- make.t(p(n,1/mu),mu) hist(,main="dist t-values for samples of",fq=f);curve(dt(,n-1),add=t,col=6) legend(-14,.19,legend=paste("t-dist, df=",n-1),col=6,pch=19) boplot(,main=paste("eponentials of size n =",n,", mu=",mu));qqnorm();qqline(,col=4) curve(dep(,1/mu),lim=c(0,4),main=paste("ep dist with mu=",mu));abline(h=0);abline(v=0) Dist t-values for samples of eponentials of size n = 5, mu= 2 Density t-dist, df= ep dist with mu= dep(, 1/mu) Theotical Quantiles

7 # Following Venables, we eperiment with a symmetric, heavy-tailed # distribution, the Cauchy distribution. Take a look at large sample # first. Its t-statistic looks fairly normal, if not slightly short. n <- 100 mu <- 0; for(i in 1:500)[i]<- make.t(rcauchy(n),mu) hist(,main="dist t-values for samples of",fq=f) curve(dt(,n-1),add=t,col=2) legend(.5,.43,legend=paste("t-dist, df=",n-1),col=2,pch=19,bty="n") boplot(,main=paste("cauchy vars of size n =",n,", mu=",mu)) qqnorm() qqline(,col=2) curve(dcauchy(),lim=c(-4,4),main=paste("cauchy dist with mu=",mu),col=2) abline(h=0) abline(v=0) 7

8 Dist t-values for samples of Cauchy vars of size n = 100, mu= 0 Density t-dist, df= cauchy dist with mu= dcauchy() Theotical Quantiles

9 # Take a look at a small sample (n=7). Its t-statistic still looks fairly # normal. Thus the t-statistic is fairly robust with spect to this # kind of population. n <- 7; mu <- 0 for(i in 1:500)[i]<- make.t(rcauchy(n),mu) hist(,main="dist t-values for samples of",fq=f) curve(dt(,n-1),add=t,col=3) legend(1.6,.28,legend=paste("t-dist, df=",n-1),col=3,pch=19,bty="n") boplot(,main=paste("cauchy vars of size n =",n,", mu=",mu)) qqnorm() qqline(,col=3) curve(dcauchy(),lim=c(-4,4),main=paste("cauchy dist with mu=",mu),col=3) abline(h=0) abline(v=0) # Return to pvious settings. par(opar) 9

10 Dist t-values for samples of Cauchy vars of size n = 7, mu= 0 Density t-dist, df= Cauchy dist with mu= dcauchy() Theotical Quantiles