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## ISI MStat 2020 PSB Problem 6

This post discuses the problem 6 of the ISI MStat 2020 PSB Entrance Exam.

Suppose individuals are classified into three categories C1,C2$C_{1}, C_{2}$ and C3$C_{3}$.

Let p2,(1p)2$p^{2},(1-p)^{2}$ and 2p(1p)$2 p(1-p)$ be the respective population proportions, where p∈(0,1)$p \in (0,1)$. A random sample of N$N$ individuals is selected from the population and the category of each selected individual recorded.

For i=1,2,3$i=1,2,3$, let Xi$X_{i}$ denote the number of individuals in the sample belonging to category Ci$C_{i}$. Define U=X1+X32$U=X_{1}+\frac{X_{3}}{2}$.

• Is $U$ sufficient for $p$ ? Justify your answer.
• Show that the mean squared error of $\frac{U}{N}$ is $\frac{p(1-p)}{2 N}$.

## Hints, Solution, and More

• Prove that the joint distribution of $(X_1,X_2,X_3)$ follows Multinomial Distribution.
• Write the Likelihood of the data.
• Use Neymann Factorization to prove the sufficiency of $U$.
• Show that $\frac{U}{N}$ is unbiased.
• Show that $2U$ follows Binomial Distribution.

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• Prove that $\frac{U}{N}$ is the UMVUE of $p$.
• Find the minimal sufficient and complete statistic of $p$.
• For other Food for Thought, refer to the youtube video for full solution.

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