Answer Happy

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 l-parameter exponential family, S = {}=1Y; is sufficient for p, ÔML = Anom =r(r+n-'S) and (7.7) VnCô - p) N(1,r=p?(1 – p)). = 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 I = -E22 log fy(Y|r,p)/?p, 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(Anom) given that you know (r,p). Do the same when the parameters are unknown but you have a random sample. Hint: d) Show that log likehood equations for (r,p) can be written as 1 = 0, nrp=1 – S(1 – p)-460 Ž 91();, r) + n log(P) n = 0 Ir,p j=1 with, in the notation of (B.11), 1 91(y,r) = -1(y > 1). r+k-1 k=1 e) Solve the first log likehood equation with respect to pand insert the result in the second equation. If you have done correctly, P r F+Y = (1+--17) r n (7.8) A(r), de del [9ı(Y;,r) + n log(), = A(n) = 0. = j=1 f) Choose a set of the 27 possible values of (r, p, n) from Table 2 and get a data set by simulation. Then caluclate the moment estimators for (r,p) and use Tinom as initial value for a Newton Raphson (NR) iteration as described below with y 2 1 92(y,r) = - (+k-1)+1(y21), k=1 Σ() Σ η) n A7(n) = 92(Y;,r), = j=1

() and the NR update is r'rr – A(T)/A1(r). Report your findings and do this experiment at least 3 times. The different input values should be taken from Table 2. For each simulation insert the estimated parameters and find an expression for the asymptotic variance. Compute the the relative increase in asymptotic variance for p going from known to unknown r. This can be done by first computing the asymptotic correlation between the estimators. All formulas that you need for computing the information matrix are stated below. We write the population information matrix as Irr Irp I= Ipar g) Show by computations that information matrix is given by 1 -E 92(Y,r) р 1 р p2(1-P) h) Explain that Irr = -E92(Y,r) ~ -n-1A10. i) Use general theory for MLE and verify that 12 Avar(?) = Avar(Ộo)(1+ Ippler - Ihr where Avar(Po) is the asymptotic variance of p with known r. j) Prove that 1 Avar(P) = Avar(Po) 1-p2 where p is the correlation between the population score components. P k) Explain that Avar(Tom) >r-p²(1 - p). Do you think this is a strict inquality? I- T pr = Table 2: Values for (r,p) with ne {10, 100, 1000} rip 0.01 0.1 0.5 0.3 من نه نه TL 13