3. (10 points) Suppose that measurements of a response are a realization of X1,

Question

3. (10 points) Suppose that measurements of a response are a realization of X1, . . . , Xn, where all the Xi ’s have the same distribution and are dependent. In particular, suppose that Xi = µ + T + Ui , i = 1, . . . , n, where T N(0, 2 ), (0, 1), U1, . . . , Un are iid N(0,(1 ) 2 ) and are independent of T. In this model, Xi N(µ, 2 ) for i = 1, . . . , n and cor(Xi , Xj ) = for i 6= j. We will see what happens when we use this sample in the one-sample (t-distribution-based) confidence interval for µ Perform a simulation study that makes a simulation-based 99% score approximate confidence interval for the coverage probability of the random 95% confidence interval for µ: X¯ ± t0.975,n1S/ n, where X¯ = n 1 Pn i=1 Xi and S = q (n 1)1 Pn i=1(Xi X¯) 2. Set µ = 68 and = 3. Use 10,000 replications and report the score approximate 99% confidence interval for the coverage probability of X¯ ± t0.975,n1S/ n for each (n, ) {10, 50, 100} × {0.1, 0.5, 0.9}. What happens to the coverage probability as increases? What happens to the coverage probability as n increases?

Explanation / Answer

Coverage probabilities:

n=10,     rho=0.1

Coverage probability of 95 % CI, CP95=0.8676

Coverage probability of 99 % CI, CP99=0.9699

n=10,     rho=0.5

Coverage probability of 95 % CI, CP95=0.4924

Coverage probability of 99 % CI, CP99=0.6581

n=10,     rho=0.9

Coverage probability of 95 % CI, CP95=0.1845

Coverage probability of 99 % CI, CP99=0.2564

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n=50,     rho=0.1

Coverage probability of 95 % CI, CP95=0.5658

Coverage probability of 99 % CI, CP99=0.7019

n=50,     rho=0.5

Coverage probability of 95 % CI, CP95=0.2136

Coverage probability of 99 % CI, CP99=0.2818

n=50,     rho=0.9

Coverage probability of 95 % CI, CP95=0.074

Coverage probability of 99 % CI, CP99=0.1003

n=100,     rho=0.1

Coverage probability of 95 % CI, CP95=0.4242

Coverage probability of 99 % CI, CP99=0.5421

n=100,     rho=0.5

Coverage probability of 95 % CI, CP95=0.1558

Coverage probability of 99 % CI, CP99=0.2028

n=100,     rho=0.9

Coverage probability of 95 % CI, CP95=0.0507

Coverage probability of 99 % CI, CP99=0.066

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R-code

mu=68

sig=3

n=100 # 10 # 50

rho=0.9   # 0.9 # 0.5

R=10000

L95<-NULL

R95<-NULL

L99<-NULL

R99<-NULL

X_bar<-NULL

for(i in 1:R){

T=rnorm(1,0,sqrt(rho*sig^2))

U=rnorm(n,0,sqrt((1-rho)*sig^2))

X=mu+T+U

X_bar[i]=mean(X)

X_sd=sd(X)

t95=qt(0.975,n-1)

t99=qt(0.995,n-1)

L95[i]=X_bar[i]-t95*X_sd/sqrt(n)

R95[i]=X_bar[i]+t95*X_sd/sqrt(n)

L99[i]=X_bar[i]-t99*X_sd/sqrt(n)

R99[i]=X_bar[i]+t99*X_sd/sqrt(n)

}

l95=mean(L95)

r95=mean(R95)

cp95=0

for(i in 1:R){

if((l95<=X_bar[i]) && (r95>=X_bar[i])){

cp95=cp95+1

}}

CP95=cp95/R

l99=mean(L99)

r99=mean(R99)

cp99=0

for(i in 1:R){

if((l99<=X_bar[i]) && (r99>=X_bar[i])){

cp99=cp99+1

}}

CP99=cp99/R

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