Question

In this problem, we will model the likelihood of a particular client of a financial firm defaulting on his or her loans based on previous transactions. There are only two outcomes, "Yes" or "No", depending on whether the client eventually defaults or not. It is believed that the client's current balance is a good predictor for this outcome, so that the more money is spent without paying, the more likely it is for that person to default. For each x, we will write Y as the 0-1 outcome of defaulting/not defaulting, given a particular current balance r. In other words, we will model the distribution of Y as a Bernoulli distribution with with a parameter z, which is reasonable given that there are only two possible outcomes. First, recall the likelihood of a Bernouli RV Y in terms of the parameter p: Rewrite this in terms of an exponential family P (Y = y) = h (y) exp 17(p) T (y) _ B (p)| Since this representation is only unique up to re-scaling by constants, take the convention that T (y) y We can write this in canonical form, e.g. as P (Y = y) = h (y) exp lyn-b (n)] . What is b (η)? b (n) = Recall that the mean of a Bernoulli(p) distribution is p. What is the canonical link function g() associated with this exponential family, where μ-E [Y] ? Write your answer in terms of p. g (A) Return to the original model. We now introduce a bias parameter pz person's account. for every possible amount of money that measures the balance in a Denote the parameter () that gives the canonical exponential family representation as above by 0. We choose to employ a linear model connecting the balance a with the canonical parameter 6 of the Bernoulli distribution above, i.e., In other words, we choose a generalized linear model with the Bernoulli distribution and its canonical link function. That also means that conditioned on , we assume the Y to be independent. Imagine we observe the following data: 1100 0 (Never defaulted) 2-2000 0 (Never defaulted) 3-2000 1 (Defaulted s- 5000 1 (Defaulted) We want to produce a maximum likelihood estimator for (a, b). To this end, write down the log likelihood& (a, b) of the model for the provided four observations at xi , X2-T3 and X4 (plug in their values). t (a, b) What is its gradient? Enter your answer as a pair of derivatives. a.1 (a, b)- A1 (a, b) Assume that we can reasonably estimate the likelihood estimator by using numerical methods to solve V(ab0. Consider the scenario where, using many more samples, we obtain the estimates 0.0012,0.00035 Given these results, what would be the predicted expected outcome E Y] for 4000? Round your answer to the nearest 0.001.

Answer 1

K(n)i-p