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7.3. NEGATIVE BINOMIAL DISTRIBUTION PROBLEM 3. Let {Y₁,...,, Yn} be a random sample from the negative binomial dis- trib
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7.3. NEGATIVE BINOMIAL DISTRIBUTION PROBLEM 3. Let {Y₁,...,, Yn} be a random sample from the negative binomial dis- tribution. Assume first that r> 0 is known but not necessarily an integer. In this situation we know that the population pmf belongs to a regular 1-parameter exponential family, S = 1 Y; is sufficient for p, PML = Ânom = r(r+n−¹S) and n =1 (7.7) √n(p − p) → N (1, r¯¹p²(1 − p)). n a) Construct a confidence interval based on (7.7) with approximate (1-a) coverage probability where a is some prescribed level. b) Find the population information matrix from the formulae ] = -E8² log fy(Y\r, p)/²p, and check your answer with (7.7). Assume that r> 0 is unknown and that Y has pmf given by (B.10). c) Describe how you can find a value for Avar (nom) given that you know (r, p). Do the same when the parameters are unknown but you have a random sample. Hint: