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how long he Question 5 (13 marks) On his way to work in the mornings, Damjan records to wait for his train at the station. Over a series of days, he observes the following times (in minutes): 4.8 1.2 3.7 0.9 0.7 0.3 0.9 3.2 1.4 cides to use an exponential distribution with mean θ as a model for these data. He would like to estimate his median waiting time, m. For these data we have -1.9 and s 1.58. Damjan de (a) Express rn in terms of θ. (b) Damjan decides to use the sample median, M, as his estimator. (i) What is the asymptotic sampling distribution of M? (ii) What is Damjans estimate of m for this dataset? (iii) Calculate a standard error for Damjans estimate. (c) Damjan now considers using X as his estimator of m. (i) Show that this estimator is biased. (ii) Let T-cX be an adjusted estimator. Find c so that T is unbiased. (iii) Determine var(T). (iv) Which of M and T is the better estimator? (v) What is the estimate, t, based on the data above? (vi) Calculate a standard error for this estimate.
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