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#### ISI MStat 2020 PSB Problem 8 Solution

##### Problem

Assume that $X_{1}, \ldots, X_{n}$ is a random sample from $N(\mu, 1)$, with $\mu \in \mathbb{R}$. We want to test $H_{0}: \mu=0$ against $H_{1}: \mu=1$. For a fixed integer $m \in{1, \ldots, n}$, the following statistics are defined:

\begin{aligned} T_{1} &= \frac{\left(X_{1}+\ldots+X_{m}\right)}{m} \\ T_{2} &= \frac{\left(X_{2}+\ldots+X_{m+1}\right)} {m} \\ \vdots &=\vdots \\ T_{n-m+1} &= \frac{\left(X_{n-m+1}+\ldots+X_{n}\right)}{m} . \end{aligned}

Fix $\alpha \in(0,1)$.

Consider the test

Reject $H_{0}$ if $\max \{T_{i}: 1 \leq i \leq n-m+1\}>c_{m, \alpha}$

Find a choice of $c_{m, \alpha} \in \mathbb{R}$ in terms of the standard normal distribution function $\Phi$ that ensures that the size of the test is at most $\alpha$.

##### Hint 1

Show that the problem is equivalent to finding that $P_{\mu = 0}(\max \{T_{i}: 1 \leq i \leq n-m+1\}>c_{m, \alpha}) \leq \alpha$

##### Hint 2

$P_{\mu = 0}(\max \{T_{i}: 1 \leq i \leq n-m+1\}>c_{m, \alpha}) = P_{\mu = 0}( T_1 > c_{m, \alpha} \cup T_2 > c_{m, \alpha} \cdots T_{n-m+1} > c_{m, \alpha})$

##### Hint 3

Use Boole's Inequality o get $P_{\mu = 0}( T_1 > c_{m, \alpha} \cup T_2 > c_{m, \alpha} \cdots T_{n-m+1} > c_{m, \alpha}) \leq \sum_{i = 1}^{n-m+1} P(T_i > c_{m, \alpha}) = \alpha$

##### Hint 4

Show that under $H_0$, $T_i$ ~ $N(0,\frac{1}{m})$. Hence, find $c_{m, \alpha}$

See the full solution below.

#### Food For Thoughts

• What if, $\lim _{n \rightarrow \infty} c_{m, \alpha}$?
• What if strict $c_{m, \alpha}$ was required to find? Can you solve this using Jacobian. (You can give an approximate solution).