Question

(9) 112 pts] An exponentially distributed random variable, call it X, has the following probability density function: f(x)-Be-ex , x > 0, θ > 0. Note that E(X) and VX-สั่ For the rest of this question, assume that you have a data set xn1 consisting of a random sample of N observations of X (a) Derive two different Method of Moments estimators for θ. HINT: remember that the MOM is based on the analogy principle, or the idea that sample moments are related to their analogous population moments. (b) Derive the Maximum Likelihood estimator for θ. HINT 1: remember that the ML esti- mator seeks to choose a value of θ that will maximize the likelihood function. The likelihood of each individual observation xn is f(xn), and since the data are a random sample, we simply multiply the individual likelihoods to get their joint likelihood (i.e., the likelihood unction) HINT 2: t is always easiest to maximize the log of the likelihood function, rather than the function in its original form. Recall that this doesn't change the answer, because maximizers/ minimizers are invariant to non-decreasing transformations of the objective function. (c) Is the ML estimator for θ biased? If so, is it biased upward or downward? Provide a say mathematical proof of your response. about it? t might Jensen's inequality have to (d) Is the ML estimator for 8 consistent? Why? HINT: What does the continuous mapping theorem have to say about it?

Answer 1