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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!

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 \(H_1: \theta = \theta_1\). (\(\theta_o< \theta_1< 1\)).

Binomial Distribution.

Neyman-Pearson Lemma.

Test function and power function.

Hypothesis Testing.

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!

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 !

What to do to shape your Career in Mathematics after 12th?

From the video below, let's learn from Dr. Ashani Dasgupta (a Ph.D. in Mathematics from the University of Milwaukee-Wisconsin and Founder-Faculty of Cheenta) how you can shape your career in Mathematics and pursue it after 12th in India and Abroad. These are some of the key questions that we are discussing here:

- What are some of the best colleges for Mathematics that you can aim to apply for after high school?
- How can you strategically opt for less known colleges and prepare yourself for the best universities in India or Abroad for your Masters or Ph.D. Programs?
- What are the best universities for MS, MMath, and Ph.D. Programs in India?
- What topics in Mathematics are really needed to crack some great Masters or Ph.D. level entrances?
- How can you pursue a Ph.D. in Mathematics outside India?
- What are the 5 ways Cheenta can help you to pursue Higher Mathematics in India and abroad?

Cheenta has taken an initiative of helping College and High School Passout Students with its "Open Seminars" and "Open for all Math Camps". These events are extremely useful for students who are really passionate for Mathematic and want to pursue their career in it.

Cheenta is a knowledge partner of Aditya Birla Education Academy

Advanced Mathematical Science. Taught by olympians, researchers and true masters of the subject.

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