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Let $X$ be a random sample from a discrete distribution with the probability mass function $f(x, \theta) =\frac{1}{\theta} , x=1,2,...,\theta;= 0 \ \text{otherwise} $ where $\theta \ \text {is either 20 or 40} $ is the unknown parameter. Consider testing $H_{0}: \theta = 40$ against $H_{1}: \theta = 20$ Find the uniformly most powerful level $\alpha=0.1$ test for testing $H_{0}$ vs $H_{1}$

I am new to construction of MP tests, and I was trying to use Neyman Pearson Lemma to construct the test but however the ratio is meaningless here as the support of the two distributions are different in the two hypotheses, so how will I tackle this problem?

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  • $\begingroup$ Note that there are other ways of defining the ratio in a likelihood-ratio test that might be more suitable for some cases: en.wikipedia.org/wiki/Likelihood-ratio_test . But here the ratio, with $H_1$ in the denominator, should make sense? $\endgroup$
    – Winther
    Sep 23, 2018 at 15:54
  • $\begingroup$ How did you study the likelihood ratio and how is it meaningless? $\endgroup$ Sep 23, 2018 at 16:12
  • $\begingroup$ @StubbornAtom When we take the ratio of likelihoods under null and alternative in most cases ratio that pops up is in the form $x\ge k$ or the other way around. But in this case, it's just constant I think he meant that, $\endgroup$
    – Daman
    Dec 16, 2018 at 13:32

2 Answers 2

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We have the distribution of a single observation $X$ :

\begin{align} f_{\theta}(x)&=\frac{1}{\theta}\mathbf1_{x\in\{1,2,\ldots,\theta\}}\quad,\,\theta\in\{20,40\} \end{align}

By NP lemma, an MP test of level $\alpha$ for testing $H_0:\theta=40$ against $H_1:\theta=20$ is of the form

\begin{align} \varphi(x)&=\begin{cases}1&,\text{ if }\lambda(x)>k\\\gamma&,\text{ if }\lambda(x)=k\\0&,\text{ if }\lambda(x)<k\end{cases} \end{align}

, where $$\lambda(x)=\frac{f_{H_1}(x)}{f_{H_0}(x)}$$

and $\gamma\in[0,1]$ and $k(> 0)$ are so chosen that $$E_{H_0}\,\varphi(X)\leqslant 0.1$$

Now,

\begin{align} \lambda(x)&=2\frac{\mathbf1_{x\in\{1,2,\ldots,20\}}}{\mathbf1_{x\in\{1,2,\ldots,40\}}} \\\\&=\begin{cases}2&,\text{ if }x=1,2,\ldots,20 \\0&,\text{ if }x=21,22,\ldots,40 \end{cases} \end{align}

Therefore, for some $c$, $$\lambda(x)\gtrless k\implies x\lessgtr c$$

And the level restriction gives $$P_{H_0}(X<c)+\gamma P_{H_0}(X=c)\leqslant 0.1\tag{1}$$

Taking different values of $c$ (namely $c=2,3,4,5$) and finding the corresponding tail probability $P_{H_0}(X<c)$ subject to $(1)$, I end up with $$c=4\quad,\quad \gamma=1$$

So the required test is $$\varphi(x)=\mathbf1_{x\leqslant 4}$$

This is UMP because $\varphi$ obviously does not depend on the value of $\theta$ under $H_1$.

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  • $\begingroup$ Can you tell me from which book you have read your testing with same notations and all ? $\endgroup$
    – Daman
    Dec 16, 2018 at 13:36
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    $\begingroup$ @Damandeep You already know the books. Nothing new. $\endgroup$ Dec 16, 2018 at 14:54
  • $\begingroup$ hmmmmm got ya . $\endgroup$
    – Daman
    Dec 16, 2018 at 15:22
  • $\begingroup$ Since this ratio is monotone so the way I solved this question without following all this method is by just calculating the probability of type 1 error and I thought what should be my rejection region so that probability of Type 1 error is $.1$ and that is $P(X \le 4 | \theta = 40 )$. So my question is can there be a situation where the probability of type error not equal to $\alpha$ because I ve read this notation $E(\phi) \le \alpha$ and I am confused, can you tell me what exactly does it mean? And did I think correctly about calculating P(type 1 error) for finding my region? $\endgroup$
    – Daman
    Dec 16, 2018 at 15:31
  • $\begingroup$ Never assumed 'everything'. I assumed more than me but anyways thought it would be a good topic for discussion you are not interested so I will post it as a question. $\endgroup$
    – Daman
    Dec 16, 2018 at 15:59
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The domains of the two distributions are not really different. In a hypothesis-testing scenario, the domains can't be different, otherwise the hypothesis testing doesn't really make sense - if the experiment is producing outputs in an entirely different domain depending on the two hypotheses, then surely it must be pretty easy to tell which hypothesis is true.

You can imagine both of these distributions as being on $\{1, ..., 40\}$, or on $\mathbb N$ or even $\mathbb R$ - any measurable space which contains the supports of the two distributions, really. In any case, the likelihood ratio of the null distribution to the alternative distribution comes out to be $\frac 1 2$ on $\{1, ..., 20\}$ and $0$ everywhere else. Notice that if you take the common domain of the distributions $H_0$ and $H_1$ to be, say, $\mathbb N$, then the ratio is undefined for most of $\mathbb N$ since you'd be dividing by zero. But it's well-defined $H_0$-almost everywhere, which is all that matters for the Neyman-Pearson lemma.

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