Question

2 Gamma waiting times (frequentist) Suppose we model a sample of times between arrivals of the 1 train of the New York City subway at the 116th Street station, y1, . . . , Yn, as IID random variables Y1, ... , Yn sampled from a Gam(v, 1) distribution, for some unknown v and X. 1. What is the joint log-likelihood, In fy ...Y|0,1(91, ... , Yn | v, 4)? [5 mark(s)] 2. For a fixed value of v, what is the maximum likelihood estimator of , 1? [6 mark(s)] 3. Using 1, what equation must be solved to obtain the maximum likelihood estimator of v, 7? (Recall that the digamma function is y(t) = n(T(z)). Furthermore, 4(x) > 0 for x > 0.) [6 mark(s)] 4. How can the equation in the previous part be solved? (Which R function?) [2 mark(s)] 5. Show that the expected (i.e., Fisher) information matrix, In (v, 1) is In(v, 1) = ( ("% ve n). [4 mark(s)]

Question 5-7

Answer 1

Solu ha M The likelihood function is given L = ñ flyi luid) e re jal n こ I dr y, exp(-14) xil n s- ny exp(-x & i) n can ebolnu = 7hol ih bol 3 (1-1)+ aj bolu- bolnu ich u - Eye so 12? = nlogy - n460) (4cul = log ro] 2²1 -na 4 (5) U. duz Also ә0 adav u Y n nr Eye pub 2 - sut 2P 다. zre as In (Sid = -El como los 2) (=(01) - d²e "Exbectation Elremoved ارد rene del because here double derivative ordd has no yi's 7ve na 4 (u) sk 18 U P ov al's ZP

a înd taking VO 6. By asymptotic property of MLE ů v No, Var (ů) N(o, vor (6) The variance of ŕ and i can obtained by Inverse of of In Cuid). The diagonal of this Inverse matrix denotes the vor (ů) vor (a) and and vascal respectively. How if " Hlo, Ucol) there exists soch that P(-2012 <ô - 0220,2) 21-2 To < 80 < 26, 8(01) = = 1-2 Thin Zan PL- P 2212 or or as Pl 8-20, STO 2020 + 2014 51101 )=1-2 In samk way by butting o=d com obtain the (1-a7% 6 Confidence interval. Motice that here a=0.05. = CI for v (W - 2a suar(ſ), 5 + 2a Juvai (0) we e CI for x = ( A - Za svoneft, Å + Za svarcas) If CI for v x's (Un, val then it means that 95% (as (I is 95%) confident that the true value of u will lie in that in terual.

We are Similarly xt 95% Clot d es (A2, du), then 95% confident that the true value of d will lie, in that interval. Hotice that [ Ve Sul and (il, du] is given in question.