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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 if 0 X₁. Once you have X₁ and the information that > X₁, obtain X₂. If X2 > X₁, then you know a bit more about 0, 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 should be a function of T. Construct an unbiased estimator of which is a function of T and calculate its variance. To start with, you should calculate, in that order, the cdf, the pdf, the expected value, and the variance of T.

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 if 0 X₁. Once you have X₁ and the information that > X₁, obtain X₂. If X2 > X₁, then you know a bit more about 0, 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 should be a function of T. Construct an unbiased estimator of which is a function of T and calculate its variance. To start with, you should calculate, in that order, the cdf, the pdf, the expected value, and the variance of T.

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) = { S0-1 Note that the information about contained in the random sample X₁,..., Xn equals the information about contained in the statistic T if 0 < x < 0 otherwise. = max(X₁, , Xn). To understand why so, let's think of the random sample as being obtained in a sequential manner, that is, you obtain X₁ and pause before obtaining X₂. What does X₁ tell you about #? It tells you that > X₁. Once you have X₁ and the information that > X₁, obtain X₂. If X₂ > X₁, then you know a bit more about 0, 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 should be a function of T. Construct an unbiased estimator of 0 which is a function of T and calculate its variance. To start with, you should calculate, in that order, the cdf, the pdf, the expected value, and the variance of T.
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) = { S0-1 Note that the information about contained in the random sample X₁,..., Xn equals the information about contained in the statistic T if 0 < x < 0 otherwise. = max(X₁, , Xn). To understand why so, let's think of the random sample as being obtained in a sequential manner, that is, you obtain X₁ and pause before obtaining X₂. What does X₁ tell you about #? It tells you that > X₁. Once you have X₁ and the information that > X₁, obtain X₂. If X₂ > X₁, then you know a bit more about 0, 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 should be a function of T. Construct an unbiased estimator of 0 which is a function of T and calculate its variance. To start with, you should calculate, in that order, the cdf, the pdf, the expected value, and the variance of T.
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