This exercise gives a less standard example of MLE. Let m be a positive integer, and U₁, U2,,Um are i.i.d. Uniform([0, 1]) RV's. Let X = max(U₁, U2, ···,Um), i.e. X is the highest outcome among m random numbers independently and uniformly chosen between 0 and 1. (i) What is the p.d.f. of the random variable X? (ii) Suppose we don't know the value of m and wish to determine it from an i.i.d. sample ,xn) of the random variable X. Verify that the log-likelihood function x = (x1, x2, equals l(x; m) = = n log m - n (m-1) log(1/x₂). i=1 (iii) Given x1,x2,...,xn, find a formula for the real number m* such that the function l above is maximized at m = m*. (However, m* is not the MLE estimator for m because it is not necessarily an integer.)
This exercise gives a less standard example of MLE. Let m be a positive integer, and U₁, U2,,Um are i.i.d. Uniform([0, 1]) RV's. Let X = max(U₁, U2, ···,Um), i.e. X is the highest outcome among m random numbers independently and uniformly chosen between 0 and 1. (i) What is the p.d.f. of the random variable X? (ii) Suppose we don't know the value of m and wish to determine it from an i.i.d. sample ,xn) of the random variable X. Verify that the log-likelihood function x = (x1, x2, equals l(x; m) = = n log m - n (m-1) log(1/x₂). i=1 (iii) Given x1,x2,...,xn, find a formula for the real number m* such that the function l above is maximized at m = m*. (However, m* is not the MLE estimator for m because it is not necessarily an integer.)
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.
Chapter12: Probability
Section12.CR: Chapter 12 Review
Problem 84CR
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Can you show me the answer to (i)(ii) and (III). Thank you so much for your help
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