Let $X_{1}, X_{2}, \cdots, X_{n}$ be a random sample from $\operatorname{Normal}\left(\mu, \sigma^{2}\right)$ distribution. Suppose we wish to test the hypothesis that the largest order statistic $X_{n: n}$ is an outlier, and that we wish to use the test statistic
$$
T=\frac{X_{n: n}-X_{n-1: n}}{S},
$$
where $S$ is the sample standard deviation.
(a) Under the null hypothesis that $X_{n: n}$ is not an outlier, show that the statistic $T$ is ancillary;
(b) Then, using Basu’s theorem, derive an expression for the mean and variance of $T$, under $H_{0}$;
(c) Using the tables of means, variances and covariances of order statistics (see, for example, H.L. Harter and N. Balakrishnan, CRC Handbook of Tables for the Use of Order Statistics in Estimation, CRC Press, Florida, 1996), compute the mean and variance of $T$ for sample size $n=10$, using the expressions derived in Part (b);
(d) Using 1,000 Monte Carlo simulation runs, simulate the values of mean and variance of $T$ under $H_{0}$, and compare them with those determined in Part (c) and comment;
(e) Explain what the critical region will be for the test based on $T$ for testing whether $X_{n: n}$ is a large outlier;
(f) Then, determine the upper $5 \%$ critical value for $T$, for sample size $n=10$, through Monte Carlo simulations (use 1,000 simulation runs);
(g) With the logarithms of the number of trees in orchards being assumed to be normal, Singh et al. (1982) presented the following observed values of logarithms of the number of trees:
$$
1.7918,2.3026,2.7726,3.2581,3.5264,3.8067,3.9703,4.0943,4.2905,4.5747
$$
Then, using the statistic $T$, test whether the largest observation $4.5747$ is an outlier or not, at $5 \%$ level of sienificance.

Suppose $\mathbf{C}=\left(\left(c_{i j}\right){i, j=1}^{k}\right.$ is a symmetric non-singular product-decomposable matrix with $c{i j}=a_{i} b_{j}$. Then, show that $\mathrm{C}^{-1}$ is a symmetric tri-diagonal matrix with (for $i \leq j$ )
$$
c^{i j}=\left{\begin{array}{cll}
\frac{a_{2}}{a_{1}\left(a_{2} b_{1}-a_{1} b_{2}\right)} & \text { for } & i=j=1 \
\frac{a_{i+1} b_{i-1}-a_{i-1} b_{i+1}}{\left(a_{i} b_{i-1}-a_{i-1} b_{i}\right)\left(a_{i+1} b_{i}-a_{i} b_{i+1}\right)} & \text { for } & 2 \leq i=j \leq k-1 \
\frac{b_{k-1}}{b_{k}\left(a_{k} b_{k-1}-a_{k-1} b_{k}\right)} & \text { for } & i=j=k \
-\frac{1}{\left(a_{i+1} b_{i}-a_{i} b_{i+1}\right)} & \text { for } & j=i+1 \text { and } 1 \leq i \leq k-1 \
0 & \text { for } & j>i+1
\end{array}\right.
$$

Let $X_{1: n}<X_{2 n n}<\cdots<X_{n: n}$ be the order statistics obtained from a random sample of size $n$ from Uniform $(0, \theta)$ distribution. Then:
(a) Using the results of the last exercise, derive the Generalized Least Squares Estimator (Best Linear Unbiased Estimator) of $\theta$ and its variance;
(b) Make some comments about how it compares with the Uniformly Minimum Variance Unbiased Estimator derived from Lehmann-Scheffé theorem.

Consider a location family of distributions with density function $f_{X}(x ; \mu)=f(x-\mu)$, for $x, \mu \in \mathbf{R}$. Let $X_{1: n}<X_{2: n}<\cdots<X_{n: n}$ be the order statistics obtained from a random sample of size $n$ from this location family of distributions.
(a) Then, by adopting the Lagrangian multiplier method, derive an expression for the Best Linear Unbiased Estimator of $\mu$;
(b) Obtain an expression for the variance of that estimator.

Let $X_{1: n}<X_{2 n}<\cdots<X_{n: n}$ be the order statistics obtained from a random sample of size $n$ from Uniform $(\theta, \theta+1)$ distribution. By using the results of the last exercise, derive the Best Linear Unbiased Estimator of $\theta$ and its variance.