Question

Q2 (15%): Suppose that the number of items one purchased online last month Yi depended on one’s salary Xi follows a Poisson distribution Yi ∼ Poisson(β0 + β1Xi). We have n pairs of independent observations (x1, y1),(x2, y2), . . . ,(xn, yn) so that Yi are mutually independent. Please use Newton’s method to find the maximum likelihood estimation (MLE) for (β0, β1). Hint: Xis are fixed and Yis are random. Please first write out the probability mass function for Yi , then derive the likelihood function, and find the MLE.

Answer 1

. Anguer Yir poi (Rot B, X;), Isisa indenendently, so, f (Yil Ru, R1, Xi) =é Bot Bixi) ( But Bixi) Ti So, the likelihood is, 1 (Bo, Bil, x) = f(ni (Xi, fa, pu). che chi = exp ? (Bot Bixi) a T (B+BiX;? 2) L (Bo, Bily,F) = log2 (Roß, l7,4) nBooB, Exit I Yilog (Bot Bixi) los (43) ant Î . . . . Mit einem Buki لایه 3 ) , soal (Bo, Bily, and olema, Riti, B) Х; ч It ك OB, cs Scanned with CamScanner

I So; MLE of ßo and Bi are solution of the following given by the system, I Ri 2030 - 0 .pl BotBiti Now support, F: R2 R² be is defined as, F (Ru, Ri) == hai-, Ex]'. so that (x) is equivalent to = Fi (P., B1), FalPoR)] F (Bo, Ri) = 0 . nto -- or F (L) = 0 ; where. B = [Ro, B,]? So, by Newton-Raphson method, Bo = lin Bin is the role of (t) wherre, Bntle Pr - FJ (2x)] F(La). and Jione sofa, BV) ; of (PR) | o FM - 8) of-fe, 8) | OB OBO. P - È xili . ist (Rot Bixi)? Trot RiX:)2 (Rot Bixi) 2. TE & yixiz pi TRAB, X;)2 (Bot B, X;) 2 Scanned with Cambeanner Woo

Now it it is difficult (or impossible) iretored to find the limit in (t) in in closed form, it is advisable to stp advices to stop the algorithn, when, ll Bati-rol) is small enough. :llfollit Scanned with 1. CamScanner