91Ó°ÊÓ

The statistic of interest is the sample correlation. The date is given in one of the previous exercises and consists of n=14 observations for two years - 1970. and 1980. The goal is to create\(\nu = 1000\) bootstrap samples and to find the bootstrap estimate of the variance of the sample correlation. First, the sample correlation is

\(R = \frac{{\sum\limits_{i = 1}^n {\left( {{X_i} - \bar X} \right)} \left( {{Y_i} - \bar Y} \right)}}{{\sqrt {\sum\limits_{i = 1}^n {{{\left( {{X_i} - \bar X} \right)}^2}} \sum\limits_{i = 1}^n {{{\left( {{Y_i} - \bar Y} \right)}^2}} } }}\)

However, use the build-in function in R to compute the sample correlation. The first code generated the\(\nu = 1000\)bootstrap samples and their sample correlation. After generating the sample correlation of the bootstrap samples, the bootstrap estimate of the variance of the sample correlation is the sample variance of the generated bootstrap samples of the sample correlation.

Hence, compute\({R^{(i)}},i = 1,2, \ldots ,\nu \)and then find the variance of the\(\nu = 1000\)sample correlation. The simulation gives an approximate value of 0.0741. This is the bootstrap estimate of the variance of the sample correlation.

\#read data

Users

Exercise.txt",

\(header = TRUE,\left. {sep = m,dec = {L^{\prime \prime }}} \right)\)

\(\begin{aligned}{l}Year1970 = matrix(unlist(data(1)))\\Year1980 = matrix(unlist(data(2)))\end{aligned}\)

#The number of bootstrap samples

\(nu = 1000\)

Correlation. 1970.1980= numeric (nu)

#Generate sample correlation

for \(\left( {i in \left( {1: n u} \right)} \right)\) {

#Generate the bootstrap sample with replacement

Sample.1970 = sample (Year1970, length (Year1970), replace = T)

#Generate the bootstrap sample with replacement

Sample.1980 = sample (Year1980, length (Year1980), replace = T)

#Correlation

Correlation.1970.1980(i) = cor (Sample.1970, Sample.1980)

}

Bootstrap Estimate = var (Correlation.1970.1980)}

In this case, one way to get the bootstrap estimate of the sample is to subtract from\({R^{(i)}},i = 1,2, \ldots ,\nu \)the correlation between the two initial samples. Then, just average the values to obtain the bootstrap estimate of the bias. Using the following code, one obtains the bootstrap estimate of the bias of approximately -0.9456

\#read data

data = read. delim(file = "C:

Users

Exercise.txt",

\(\begin{aligned}{l}Year1970 &= matrix(unlist(data(1)))\\Year1980 &= matrix(unlist(data(2)))\end{aligned}\)

#The number of bootstrap samples

nu = 1000

\(Correlation.1970.1980.bias = numeric(nu)\)

x = cor(Year1970, Year1980)

\#generate sample correlation

for (i in (1: n u)) {

#Generate the bootstrap sample with replacement

\(Sample.1970 = sample(Year1970,length(Year1970),replace = T)\)

#Generate the bootstrap sample with replacement

\(Sample.1980 = sample(Year1980,length(Year1980),replace = T)\)

#Correlation

Correlation.1970. 1980.bias(i) = cor (Sample.1970, Sample.1980) – cor (Year1970, Year1980}

Bootstrap estimate = mean (Correlation.1970. 1980.bias)

\(Estimate Of The Simulation SD = sqrt(var(Correlation.1970.1980.bias)/nu)\)

Let's first find the solution for (a), that is, for the bootstrap estimate of variance. Such simulation of the standard error of the sample variance uses the following approach. Denote with\({R^{(i)}},i = 1,2, \ldots ,\nu \)the bootstrap sample correlation coefficients. To estimate the variance, one uses the sample variance

\(Z = \frac{1}{\nu }\sum\limits_{i = 1}^\nu {{{\left( {{R^{(i)}} - \bar R} \right)}^2}} \)

The goal is now to find the simulation variance of this $Z$. The delta method can be used. By denoting

\({W^{(i)}} = {R^{(i)2}}\)

the sample variance Z may be written as

\(Z = \frac{1}{\nu }\sum\limits_{i = 1}^\nu {{{\left( {{R^{(i)}} - \bar R} \right)}^2}} = \bar W - {\bar Y^2}\)

Let V be the sample variance of\({W^{(i)}},i = 1,2, \ldots \ldots ,\nu .\)And lastly, the required covariance between the initial bootstrap sample\({R^{(i)}},i = 1,2, \ldots ,\nu \)and the squared\({W^{(i)}},i = 1,2, \ldots ,\nu \) is given by

\(C = \frac{1}{\nu }\sum\limits_{i = 1}^\nu {\left( {{R^{(i)}} - \bar R} \right)} \left( {{W^{(i)}} - \bar W} \right)\)

Finally, the variance of the necessary Z is

\(\hat \sigma _Z^2 = \frac{1}{\nu }\left( {4{{\bar Y}^2}Z - 4\bar YC + V} \right)\)

This is the formula used in the code in the end. Note that the standard deviation is the square root of the variance. The result given by the code below is $0.0031$.

The solution for part (b) is more straightforward, and it is only required to find the sample variance of\({R^{(i)}},i = 1,2, \ldots ,\nu \)minus the sample correlation coefficient as given in the last line of code of part (b). The standard simulation error is 0.0086.

#Read data

\(data = read.delim(file{ = ^{\prime \prime }}C:\)

Users

Exercise. txt",

Year1970 = matrix (unlisted(data (1)))

Year1980 = matrix (unlisted (data (2))

nu =1000

R = numeric(nu)

W = numeric(nu)

#Generate sample correlation

for (i in (1: n u)) {

#Generate the bootstrap sample with replacement

Sample.1970 = sample (Year1970, length (Year1970), replace =T)

Sample.1980 = sample (Year1980, length (Year1980), replace =T)

#Correlation

\(\begin{aligned}{l}R(i) &= cor(Sample.1970,Sample.1980)\\W(i) &= R{(i)^ \wedge }2\end{aligned}\)

}

\(\begin{aligned}{l}Z &= mean(W) - mean{(R)^ \wedge }2\\V &= var(W)\end{aligned}\)

\(C.for.sum = numeric(nu)\)

for ( i in (1: n u)) {

\(C.for.sum(i) = (R(i) - mean(R))*(W(i) - mean(W))\)

\(C = mean(C.for.sum)\)

\(Bootstrap Eestimate Variance = \left( {4*mean{{(R)}^ \wedge }2*Z - 4*mean(R)*C + V} \right)/nu\)

\(Bootstrap Eestimate SD = sqrt(Bootstrap Eestimate Variance)\)