Problem: Suppose X1,X2,...Xn follows exponential (mean=1). Only the largest values of Xi are recorded and denoted by T. To test H0:n=5 vs H1:n=10, show that the most powerful test of size 0.05 rejects H0 for T>c, for some c. Also find c.
Here my question is that I can use NP lemma for joint pdf but can I use the same for order statistics. Alternatively I can just redefine the problem as a RV following the pdf of Max. Of the the RV and then I can use NP lemma to solve the problem. In this case, can we say that my redefined problem and the above stated one are the same?
What you have to do is, find $c$ such that \begin{equation} P_0\left(\max\limits_{1\leq i\leq n}X_i>c\right)=0.05. \end{equation} Now, under hypothesis $H_0$, we know that $n=5$. Thus, the probability term on the left hand side of the above equation is \begin{align} P\left(\max\limits_{1\leq i\leq 5}X_i>c \right)&=1-P\left(\max\limits_{1\leq i\leq 5}X_i\leq c \right)\\ & \stackrel{(a)}{=} 1-P(X_1\leq c)^5\\ & =1-(1-e^{-c})^5, \end{align} where in writing $(a)$ above, I have assumed that $X_1,\ldots,X_n$ are independent. Equate the above expression to find out the value of $c$.