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Let X₁,..., Xn be a random sample of size n from the U(0, 0) distribution, where > 0 is an unknown parameter. Recall that the pdf fof the U(0, 0) distribution is of the form 0-¹ if 0

Let X₁,..., Xn be a random sample of size n from the U(0, 0) distribution, where > 0 is an unknown parameter. Recall that the pdf fof the U(0, 0) distribution is of the form 0-¹ if 0

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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Let X₁, Xn be a random sample of size n from the U(0, 0) distribution, where @ > 0 is an unknown parameter. Recall that the pdf fof the U(0, 0) distribution is of the form f(x) = { 0 Note that the information about contained in the random sample X₁, ...,. information about contained in the statistic if 0<x<0 otherwise. T - max(X₁,..., Xn). manner, To understand why so, let's think of the random sample as being obtained in a sequential that is, you obtain X₁ and pause before obtaining X₂. What does X₁ tell you about 0? It tells you that 0 > X₁. Once you have X₁ and the information that > X₁, obtain X₂. If X2 > X₁, then you know a bit more about 9, namely, 0 > X₂; however, if X2X₁, then it does not contribute anything, above and beyond what you already know from X₁, to your knowledge about 0. In other words, when you have obtained X₁ and X2, what you know about is that it is greater than the maximum of X₁ and X₂. As such, any reasonable estimator of 0 should be a function of T. Xn equals the (a) Carefully argue that T is the maximum likelihood estimator of 0. Recall that the likelihood function L of 0, given a data sample ₁,, n, is the product of f(x₁), · ·., f (xn). (b) Construct an unbiased estimator of 0 which is not a function of T and calculate its variance. To start with, you may like to calculate the expected value and the variance of the U(0, 0) distribution.
Transcribed Image Text:Let X₁, Xn be a random sample of size n from the U(0, 0) distribution, where @ > 0 is an unknown parameter. Recall that the pdf fof the U(0, 0) distribution is of the form f(x) = { 0 Note that the information about contained in the random sample X₁, ...,. information about contained in the statistic if 0<x<0 otherwise. T - max(X₁,..., Xn). manner, To understand why so, let's think of the random sample as being obtained in a sequential that is, you obtain X₁ and pause before obtaining X₂. What does X₁ tell you about 0? It tells you that 0 > X₁. Once you have X₁ and the information that > X₁, obtain X₂. If X2 > X₁, then you know a bit more about 9, namely, 0 > X₂; however, if X2X₁, then it does not contribute anything, above and beyond what you already know from X₁, to your knowledge about 0. In other words, when you have obtained X₁ and X2, what you know about is that it is greater than the maximum of X₁ and X₂. As such, any reasonable estimator of 0 should be a function of T. Xn equals the (a) Carefully argue that T is the maximum likelihood estimator of 0. Recall that the likelihood function L of 0, given a data sample ₁,, n, is the product of f(x₁), · ·., f (xn). (b) Construct an unbiased estimator of 0 which is not a function of T and calculate its variance. To start with, you may like to calculate the expected value and the variance of the U(0, 0) distribution.
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