This is a really beautiful sample problem from ISI MStat PSB 2008 Problem 10. It is based on testing simple hypothesis. This problem teaches me how observation, makes life simple. Go for it!

**Problem**– ISI MStat PSB 2008 Problem 10

Consider a population with three kinds of individuals labelled 1,2 and 3. Suppose the proportion of individuals of the three types are given by \(f(k, \theta)\), k=1,2,3 where 0< \(\theta\)<1.

\(f(k, \theta) = \begin{cases} {\theta}^2 & k=1 \\ 2\theta(1-\theta) & k=2 \\ (1-\theta)^2 & k=3 \end{cases}\)

Let \(X_1,X_2,….,X_n\) be a random sample from this population. Find the most powerful test for testing \( H_o : \theta =\theta_o\) versus \(H_1: \theta = \theta_1\). (\(\theta_o< \theta_1< 1\)).

**Prerequisites**

Binomial Distribution.

Neyman-Pearson Lemma.

Test function and power function.

Hypothesis Testing.

## Solution :

This is a quite beautiful problem, only when you observe it closely. Here the distribution of X may seem non-trivial ( non-theoretical), but if one observes the distribution of Y=X-1 (say), instead of X , one will find that \( Y \sim binomial( 2, 1-\theta) \) .

so, now let, p= 1-\( \theta\) , so, 0<p<1, and let, \( p_o= 1-\theta_o \) and \(p_1=1-\theta_1\).

and since , \( \theta_o< \theta_1 so, p_0>p_1 \), and our hypotheses, reduces to,

\( H_o : p = p_o\) versus \(H_1: p = p_1, where 1> p_o> p_1\).

so, under \(H_o\) , our joint pmf ( of Y=X-1), is \( f_o( \vec{y}) = \prod_{i=1}^n {2 \choose y_i} {(p_o)^{y_i}(1-p_0)^{2-y_i}}\) ; where \(y_i=x_i-1 , i=1,…,n \)

and under \(H_1\), our joint pmf is, \( f_o( \vec{y}) = \prod_{i=1}^n{2 \choose y_i}{(p_1)^{y_i}(1-p_1)^{2-y_i}} \) ; where \( y_i=x_i-1, i=1,…,n \)

So, now we can use, widely used Neyman-Pearson Lemma , and end up with,

\(\lambda (\vec{y})\)=\(\frac{f_1(\vec{y})}{f_o(\vec{y})}\)=\(\frac{\prod_{i=1}^{n} {2 \choose y_i} {p_1}^{y_i} {(1-p_0)}^{2-y_i}}{\prod_{i=1}^n {2 \choose y_i}{p_1}^{y_i}{(1-p_1)}^{2-y_i}}\)=\( {(\frac{p_1}{p_0})}^{\sum{y_i}} {(\frac{1-p_1}{1-p_o})}^{2-\sum{y_i}}\) .

now we define a test function, \(\phi(\vec{x})= \begin{cases} 1& \lambda*(\vec{x})> k \\ 0 &\lambda*(\vec{x}) \le k \end{cases}\). for some positive constant k.

Where \(\lambda(\vec{y})=\lambda*(\vec{x}), \vec{x}= ( X_1,….,X_n)\)

so, our test rule is, we reject \(H_o\) if \(\phi(\vec{x})=1\), and we choose k such that the for a give level \(\alpha\),

\(E_{H_o}(\phi(\vec{x})) \le \alpha\), for a given \(0<\alpha<1 \),

with a power function , \( \beta(\theta)= E(\phi(\vec{x})) \). Can you find the more subtle condition when,\( \lambda^*(\vec{x}) \le k \) ? Try It!

## Food For Thought

Suppose, \(\theta_o \le \theta_1\), can you verify, that there for any constant c, \(P_{\theta_1}(X>c) \le P_{\theta_1}(X>c) \) . Can you generalize the situation, what kind distribution must X follow ?? Think over it, until we meet again !

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