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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 and C3.

Let p2,(1p)2 and 2p(1p) be the respective population proportions, where p∈(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 Xi denote the number of individuals in the sample belonging to category Ci. Define U=X1+X32.

  • 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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