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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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