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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,(1−p)2 and 2p(1−p) 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 sufficient for ? Justify your answer.
Show that the mean squared error of is .
Hints, Solution, and More
Prove that the joint distribution of follows Multinomial Distribution.
Write the Likelihood of the data.
Use Neymann Factorization to prove the sufficiency of .
Show that is unbiased.
Show that follows Binomial Distribution.
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Prove that is the UMVUE of .
Find the minimal sufficient and complete statistic of .
For other Food for Thought, refer to the youtube video for full solution.