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ity are ying the most dangerous ctions in Sta Colle The city records the number of car accidents at each of n intersections over the course of the year. Let's assume that the number of accidents for a particular intersection i is denoted by X₁. We are provided with a simple random sample of accidents at n intersections, X₁,..., Xn. Let's assume the following model, Xi.i.d. Pois (A) where A is an unknown parameter of interest (i.e., the unknown population accident rate). (a) Show that the MLE of A is X. Find its sufficient statistic. Is the MLE a function of the sufficient statistic? Find the bias, variance, and MSE of ÂMLE. Does the MLE achieve the Cramér-Rao's lower bound? (b) (c)

ity are ying the most dangerous ctions in Sta Colle The city records the number of car accidents at each of n intersections over the course of the year. Let's assume that the number of accidents for a particular intersection i is denoted by X₁. We are provided with a simple random sample of accidents at n intersections, X₁,..., Xn. Let's assume the following model, Xi.i.d. Pois (A) where A is an unknown parameter of interest (i.e., the unknown population accident rate). (a) Show that the MLE of A is X. Find its sufficient statistic. Is the MLE a function of the sufficient statistic? Find the bias, variance, and MSE of ÂMLE. Does the MLE achieve the Cramér-Rao's lower bound? (b) (c)

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter1: Functions
Section1.2: The Least Square Line
Problem 8E
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Please solve for questions d e and f Please solve for questions d e and f

Please solve for questions d e and f Please solve for questions d e and f

Please solve for questions d e and f Please solve for questions d e and f

2. City officials are interested in identifying the most dangerous intersections in State College. The city records the number of car accidents at each of n intersections over the course of the year. Let's assume that the number of accidents for a particular intersection i is denoted by X₁. We are provided with a simple random sample of accidents at n intersections, X₁,..., Xn. Let's assume the following model, Xi Pois (A) where A is an unknown parameter of interest (i.e., the unknown population accident rate). i.i.d. Show that the MLE of A is X. Find its sufficient statistic. Is the MLE a function of the sufficient statistic? (a) Find the bias, variance, and MSE of ÂMLE. Does the MLE achieve the Cramér-Rao's lower bound? Find the asymptotic distribution of the MLE for A. Now, the officials are interested in the total number of accidents expected in those n intersections, that is, E(X) = nλ. Let us denote this expected total as (= nλ). Find the MLE of 0. (Hint: invariance property) (f) Find the asymptotic distribution of MLE for 0. (Hint: Delta method) (b) (c) (d) (e)
Transcribed Image Text:2. City officials are interested in identifying the most dangerous intersections in State College. The city records the number of car accidents at each of n intersections over the course of the year. Let's assume that the number of accidents for a particular intersection i is denoted by X₁. We are provided with a simple random sample of accidents at n intersections, X₁,..., Xn. Let's assume the following model, Xi Pois (A) where A is an unknown parameter of interest (i.e., the unknown population accident rate). i.i.d. Show that the MLE of A is X. Find its sufficient statistic. Is the MLE a function of the sufficient statistic? (a) Find the bias, variance, and MSE of ÂMLE. Does the MLE achieve the Cramér-Rao's lower bound? Find the asymptotic distribution of the MLE for A. Now, the officials are interested in the total number of accidents expected in those n intersections, that is, E(X) = nλ. Let us denote this expected total as (= nλ). Find the MLE of 0. (Hint: invariance property) (f) Find the asymptotic distribution of MLE for 0. (Hint: Delta method) (b) (c) (d) (e)
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