ISI MStat 2020 PSB Problem 6 Problem & Solution

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 $C_{1}, C_{2}$ and $C_{3}$ .

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

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

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.