Let X1, . . . , Xn be a sample from cdf F and denote the

Chapter , Problem 10

(choose chapter or problem)

Let \(X_{1}, \ldots, X_{n}\) be a sample from cdf F and denote the order statistics by \(X_{(1)}, X_{(2)}, \ldots, X_{(n)}\). We will assume that F is continuous, with density function f. From Theorem A in Section 3.7, the density function of \(X_{(k)}\) is

\(f_{k}(x)=n\left(\begin{array}{l}n-1 \\ k-1\end{array}\right)[F(x)]^{k-1}[1-F(x)]^{n-k} f(x)\)

a. Find the mean and variance of \(X_{(k)}\) from a uniform distribution on [0, 1]. You will need to use the fact that the density of \(X_{(k)}\) integrates to 1. Show that

\(\begin{aligned}\text { Mean } & =\frac{k}{n+1} \\ \text { Variance } & =\frac{1}{n+2}\left(\frac{k}{n+1}\right)\left(1-\frac{k}{n+1}\right)\end{aligned}\)

b. Find the approximate mean and variance of \(Y_{(k)}\), the kth-order statistic of a sample of size n from F. To do this, let

\(X_{i}=F\left(Y_{i}\right)\)

or

\(Y_{i}=F^{-1}\left(X_{i}\right)\)

The \(X_i\) are a sample from a U[0, 1] distribution (why?). Use the propagation of error formula,

\(\begin{aligned}Y_{(k)} & =F^{-1}\left(X_{(k)}\right) \\ & \approx F^{-1}\left(\frac{k}{n+1}\right)+\left.\left(X_{(k)}-\frac{k}{n+1}\right) \frac{d}{d x}F^{-1}(x)\right|_{k /(n+1)}\end{aligned}\)

An urged that

\(\begin{aligned}E Y_{(k)} & \approx F^{-1}\left(\frac{k}{n+1}\right) \\ \operatorname{Var}\left(Y_{(k)}\right) & \approx \frac{k}{n+1}\left(1-\frac{k}{n+1}\right)\frac{1}{\left(f\left\{F^{-1}[k /(n+1)]\right\}\right)^{2}}\left(\frac{1}{n+2}\right)\end{aligned}\)

c. Use the results of parts (a) and (b) to show that the variance of the pth sample quantile is approximately

\(\frac{1}{n f^{2}\left(x_{n}\right)} p(1-p)\)

where \(x_p\) is the pth quantile.

d. Use the result of part (c) to find the approximate variance of the median of a sample of size n from a \(N\left(\mu, \sigma^{2}\right)\) distribution. Compare to the variance of the sample mean.