ISS Statistics Paper II (Old Subjective Pattern): Questions 38 - 39 of 39

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Question number: 38

» Hypotheses Testing » Uniformly Most Powerful Test

Appeared in Year: 2015

Essay Question▾

Describe in Detail

Fins a Most powerful test for testing the simple hypothesis Equation against the simple alternative hypothesis Equation based on n random observations from N (µ, σ 2) where µ is know. Show that this MP test is UMP (Uniformly most powerful).

Explanation

Let X i ’s are n random observations from N (µ, σ 2) and the pdf is

Equation

The hypothesis is

Equation

To find the most powerful test, we use following step

Equation

The N-P test is

Equation

Given α, we can find k by solving Equation if

Equation<span class="more">… (59 more words) …</span>

Question number: 39

» Multivariate Analysis » Wishart's Distribution » Reproductive Properties

Appeared in Year: 2015

Essay Question▾

Describe in Detail

(i) Let A i be distributed as Wishart Equation , i = 1, 2 and A 1, A 2 be independent. Show that A 1 +A 2 is distributed as Equation .

(ii) If A is distributed as Equation , then CAC’ is distributed as Equation where C is a nonsingular matrix of order m.

Explanation

(i) We have Equation that is

Equation where Equation) and Y i ’s are independent.

Similarly, Equation that is

Equation where Equation) and Z i ’s are independent.

We know that A 1, A 2 are independent. So, Y i ’s are independent of Z i ’s… (55 more words) …

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