Statistics 149 Spring 2016 Assignment 1 Due Monday February 8, 2016 Homework is to be handed in either at the Monday lecture, or directly into the Stat 149 dropbox on the seventh oor of the Science Center Monday afternoon by 4pm. Readings: \"Introductory lecture\" and \"Basic theory of GLMs\" course notes. Faraway: Chapter 1 (for review), Section 6.1, and 6.2 (just scan this section). Written assignment 1. In the following situations, identify whether the described random variable plausibly has a Bernoulli, Binomial, Poisson, or normal distribution, or whether it has some other type of probability distribution (you do not need to be specic about the alternative - just whether it is one of the four named distributions). Briey justify your answer. (a) Let Y be the number of iPhone owners out of a random sample of 100 that download an app today. (b) Let Y be the number of babies born on a single day in Boston. (c) Let Y be the initial weight (in pounds) of a randomly selected male enrolling in the Biggest Loser diet. (d) Let Y be the initial weight (in pounds) of a randomly selected person on the Biggest Loser diet. (e) Let Y be 1 if a randomly selected person who saw the movie Star Wars: Episode VII and liked it, and 0 if not. (f) Let Y be 1 if you ip a coin and it lands tails, and 2 if it lands heads. (g) Let Y be the number of hairs found in a randomly chosen McDonald's Big Mac. (h) Let Y be 1 if at least one hair is found in a randomly chosen McDonald's Big Mac, and 0 if not. (i) Let Y be the actual amount of Pepsi (in ounces) in a randomly selected 16 ounce can of Pepsi. (j) Let Y be the number of times Donald Trump says the words \"make America great again\" in a randomly selected half-hour footage of his presidential campaign. (k) Let Y be the salary (in dollars) of a randomly selected \"quant\" working on Wall Street. (l) Let Y be the number of hospital workers in a random sample of 20 who have had the u this season. 2. Suppose you observe two independent values, y1 and y2 , from a Poisson distribution with mean , that is, e y p(y) = . y! (a) The likelihood function for given the two values y1 and y2 is the product of their probabilities, ( y1 ) ( y2 ) e e L() = . y1 ! y2 ! Write out the expressions for the log-likelihood for , () = log L(), the rst derivative of the log-likelihood, (), and the second derivative, (). (b) Find the maximum likelihood estimate (MLE) for as a function of y1 and y2 using the information from part (a) by setting () = 0, and checking that () is negative. (c) Determine the update formula for the Newton-Raphson algorithm that computes the MLE iteratively based on the information in part (a). (d) The expected second derivative of the log-likelihood function with respect to turns out to be 2/. With this piece of information, write out the update formula for the Fisher scoring algorithm, using information from part (a) as needed. (e) Suppose you observe y1 = 3 and y2 = 5. What is the MLE based on your answer to part (b)? Now using a starting value of (0) = 2.0, perform the iterative computation for the Newton-Raphson algorithm for about 10 iterations. Does the algorithm converge to the same solution? What about the convergence of the Fisher scoring algorithm? 3. Suppose Y follows an EDF model, and the single value y is observed. We want to show that probability density function (or the probability mass function) p(y) for Y is maximized when = E(Y ) is set to the observed value y. (a) Let the density/mass function be ( p(y) = exp ) y b() + c(y, ) . Write down the expression for log p(y). Why is maximizing log p(y) as a function of parameters the same as maximizing p(y)? (b) Take the derivative of log p(y) with respect to , and set the resulting expression to 0. Show this implies that = y is either the minimum or the maximum of log p(y). (c) To determine that = y is the maximum, you need to show that the second derivative of log p(y) at = y is negative. Find the second derivative, and use the fact that Var(Y ) = b () where Var(Y ) 0 and > 0. 4. Let Y have an exponential distribution with probability density p(y | ) = ey for y > 0, and with unknown parameter > 0. (a) Show that the exponential distribution is a member of the exponential dispersion family by rewriting the density in the appropriate form. Letting be the natural parameter, show that = . What is the dispersion parameter, ? (b) What are b() and c(y, )? (c) Determine E(Y ) and Var(Y ) as functions of through differentiating b(). Recognizing from part (a) that = , what are E(Y ) and Var(Y ) in terms of ? What is the variance function, V (), where = E(Y )? (d) Suppose that J predictor variables xi1 , xi2 , . . . , xiJ are observed for all n observations, and let 1 , . . . , J be their effects. Let the i-th linear predictor be i = 0 + 1 xi1 + + J xiJ and suppose the link function is given by i = g(i ) = log(i ) Write the probability density function for the generalized linear model for Yi in terms of 0 , 1 , . . . , J and xi1 , xi2 , . . . , xiJ . 5. Faraway, problem 5, page 131 (end of Chapter 6). Hint: You should assume for this problem that is a known constant rather than a parameter that would be estimated. 6. This question concerns the choice of link functions. (a) Suppose a probability distribution has an unknown mean that is restricted to be greater than 1. Why is g() = log() not a sensible link function? What would be a more reasonable link function? (b) Suppose that a probability distribution has an unknown mean that is restricted to be between 1 and 1. What would be a reasonable link function for