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Let $X$ be an integer valued random variable supported on $\{0.1.2...,12\}$ whose pmf is either $g(x)=1/13; x=0,1,...,12$ or

$ f(x)=\dfrac x {36} 1_{\{0,1,...,6\}} + (\dfrac 13 - \dfrac x{36})1_{\{7,8,...,12\}}; x =0,1,...,12$. (where $1_A$ denotes the indicator function of a set $A$)

On the basis of one single observation $X$, how to construct the most powerful test of

$H_0: $ pmf of $X$ is $f$ vs. $H_1:$ pmf of $X$ is $g$ , at a $0.25$ level of significance ?

I don't know, have never encountered, testing hypothesis for pmf before ... I have only known testing hypothesis for parameter.

A detailed solution of this, with some relevant reading material will be much appreciated.

Thanks in advance.

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  • $\begingroup$ @StubbornAtom: not homework but a past exam question ... $\endgroup$ – user521337 Jan 8 at 6:26
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    $\begingroup$ Even then, please add self-study. Also, please tell us what you have tried so far. $\endgroup$ – StatsStudent Jan 8 at 6:46
  • $\begingroup$ @StatsStudent: as I said ... I only know how to use Neyman-Pearson for parameter hypothesis testing ... I don't know how to apply N-P to test pmf or pdf hypothesis ... $\endgroup$ – user521337 Jan 8 at 7:18
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    $\begingroup$ The distribution is fully specified under both models, so you can directly calculate likelihood of any observation under null and alternative and so calculate the likelihood ratio as a function of $x$. Indeed since the null is uniform, one can proceed directly from inspection of the alternative. Consequently, the form of the rejection region (in terms of $x$) is obvious - the only question is where the boundary is placed, which is determined by $\alpha$ (keeping in mind that the location of the cutoff is determined under the null) $\endgroup$ – Glen_b Jan 9 at 3:23
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Recall the construction of most powerful test using the fundamental Neyman-Pearson lemma.

Consider the problem of testing a simple null against a simple alternative, namely $$H_0:X\sim f_0\quad\text{ vs }\quad H_1:X\sim f_1\,(\ne f_0)$$

, where $X$ is a random variable/vector, $f_0$ (or $f_{H_0}$) is the null distribution which is completely specified, and $f_1$ (or $f_{H_1}$) is the alternative distribution which is also completely specified.

Then any test $\varphi$ of the form

$$\varphi(x)=\begin{cases}1&,\text{ if }f_1(x)>kf_0(x)\\ \gamma&,\text{ if }f_1(x)=kf_0(x)\\0&,\text{ if }f_1(x)<kf_0(x)\end{cases}$$

where $\gamma\in(0,1)$ and $k>0$ are such that $\varphi$ is of size $\alpha$, is a most powerful size $\alpha$ test for testing $H_0$ against $H_1$.

Since you are familiar with hypothesis testing of parameters using N-P lemma, the above would not come as much of a surprise. After all, if your distribution $f$ was indexed by the parameter $\theta$ you would do the same thing for testing, say, $H_0:\theta=\theta_0$ vs $H_1:\theta=\theta_1\,(\ne\theta_0)$. In that case, $f_0$ becomes $f_{\theta_0}$ and $f_1$ becomes $f_{\theta_1}$.


To get you started, let $f$ be the pmf of $X$.

Since both the null and alternative distributions are discrete, a most powerful level $\alpha=0.25$ test for testing $H_0$ against $H_1$ is simply of the form

$$\varphi(x)=\begin{cases}1&,\text{ if }x\in E\\0&,\text{ otherwise }\end{cases}$$

, where $$E_{H_0}\varphi(X)\le 0.25$$

To choose the set of points in $E$, compute the likelihood ratio $\lambda(x)=\frac{f_{H_1}(x)}{f_{H_0}(x)}$ for each $x$, and write the different values of $\lambda(x)$ in decreasing order (because we reject $H_0$ for large values of $\lambda(x)$).

For this question, the order looks something like $$\lambda(0)=\lambda(12)>\lambda(1)=\lambda(11)>\ldots\tag{*}$$

The critical region $E$ would consist of sample points taken according to the preference rule $(*)$. So here $0,12$ would be the first values to enter into $E$, and so on until $E_{H_0}\varphi(X)$ exceeds $0.25$.

[The test $\varphi$ might not be an exact size $\alpha$ test, in which case you would have to use randomisation.]

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  • $\begingroup$ as I said ... I only know how to use Neyman-Pearson for parameter hypothesis testing ... could you please add some reference as to where to find application of N-P to test pmf or pdf hypothesis ? $\endgroup$ – user521337 Jan 8 at 7:17
  • $\begingroup$ @user521337 This is a direct application of N-P lemma and is a more general question than testing of parameters. So I would guess a standard mathematical statistics textbook would discuss these problems, worked out or as exercise. If you are testing say, $H_0:\theta=\theta_0$ vs $H_1:\theta=\theta_1 (\ne \theta_0)$, you would use the ratio $\lambda(x)=f_{H_1}(x)/f_{H_0}(x)=f_{\theta_1}(x)/f_{\theta_0}(x)$. Does this look more familiar? $\endgroup$ – StubbornAtom Jan 8 at 7:32
  • $\begingroup$ @user521337 I have updated my answer a little to focus on this particular exercise. Let me know if things are not clear. $\endgroup$ – StubbornAtom Jan 12 at 14:52
  • $\begingroup$ thank you for your solution here very much ... could you also possibly take a look at these two questions stats.stackexchange.com/questions/386254/… and stats.stackexchange.com/questions/386815/… $\endgroup$ – user521337 Jan 12 at 15:51

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