# Question 5-7 2 Gamma waiting times (frequentist) Suppose we model a sample of times between arrivals...

###### Question:

Question 5-7

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)]
5. Show that the expected (i.e., Fisher) information matrix, In (v, 1) is L.(v, 1) = (183 m) n [4 mark(s)] 6. For a specific sample yı, ... , Yn, describe (i.e., do not calculate any matrix inverses) the steps to calculate approximate 95% confidence intervals for each of v and based on the previous parts. You can assume that inference will be performed individually for each parameter rather than jointly. [4 mark(s)] 7. n = 37, y = 3.7215, and 121-1 In(y) = 1.2583, the following maximum likelihood estimates and approximate 95% confidence interval for , based on the large-sample properties of maximum likelihood, can be obtained: Parameter ML estimate Approximate 95% CI û = 9.115 [5.0354, 13.195) À = 2.449 [1.3223, 3.5764 How do we interpret the confidence intervals? [3 mark(s)]

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