0

Let $X_1, X_2,..., X_n$ be a sample of size n from the PMF $$P_N(x) = {1 \over N},\ \ \ \ \ \ \ \ \ x = 1,2,...,N;N \in \mathbb{N} $$

Show that $$ \varphi(x_1, x_2, ..., x_n) = \begin{cases} 1 & x_{(n)} > N_0 \text{ or } x_{(n)} \leq α^{{1 \over n}}N_0 \\ 0 & \text{otherwise} \end{cases} $$ is a UMP size $α$ test of $H'_0 : N = N_0$ against $H'_1 : N = N_0$.

Work from my side:

  • Proved that $P_N(X)$ has Monotone Likelihood ratio in statistic $T(x) = \max(x_1, x_2, ..., x_n) = x_{(n)}$
  • Proved that the following test $$ \varphi(x_1, x_2, ..., x_n) = \begin{cases} 1 & x_{(n)} > N_0 \\ \alpha & x_{(n)} \leq N_0 \end{cases} $$ is UMP size α for testing for $H_0 : N ≤ N_0 \text{ against } H_1 : N > N_0$.
AxyuS
  • 101
  • 3
  • 1
    Please add the `self-study` tag if this is some sort of homework. To show this is UMP against $H_1':N\ne N_0$, one approach is to show $\varphi$ is UMP against both the alternatives $N>N_0$ and $N – StubbornAtom Aug 16 '20 at 21:01
  • Perhaps see [this Q & A](https://stats.stackexchange.com/questions/117844/ump-for-u0-theta-simple-x-simple-hypothesis/441875#441875), a continuous analogue. – BruceET Aug 17 '20 at 00:07
  • @StubbornAtom No it's not a part of any homework. I am just studying for an examination by myself. – AxyuS Aug 17 '20 at 14:29

0 Answers0