(1-alpha) Confidence Intervals. Let Y_1, Y_2,...,Y_n be i.d.d. random variables from a gamma distribution with shape parameter alpha_0>0 and unknown scale parameter beta.

IJzerboor07

IJzerboor07

Answered question

2022-09-10

( 1 α ) Confidence Intervals
Let Y 1 , Y 2 , . . . , Y n be i.d.d. random variables from a gamma distribution with shape parameter                   α 0 > 0 and unknown scale parameter β. Find a ( 1 α ) confidence interval for the parameter β. Note that the minimum variance unbiased estimator for β is β ^ = Y ¯ α 0 where Y ¯ = 1 n i = 1 n Y i .
I have no clue where to start here. Any hints?

Answer & Explanation

Ashlee Ramos

Ashlee Ramos

Beginner2022-09-11Added 20 answers

Step 1
Well, I bet you have to use the central limit theorem. The random variables Y 1 , Y 2 , Y 3 , . . . , Y n satisfy the condition of this theorem (they have the finite variance and same expectation μ. Thus, their sum will have the following distribution (for any sufficiently large number n):
P ( i = 1 n Y n n ) = N ( 0 , σ 2 )
Step 2
Thus, β ¯ = 1 α 0 Y ¯ will have the following distribution: P ( β ¯ ) = N ( 0 , σ 2 α 0 ). From the latter, you can get the confidence interval.

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