Problem 1 Let X₁, Xn be a random sample of size n from the U(0, 0) distribution, where 0 > 0 is an unknown parameter. Recall that the pdf fof the U(0, 0) distribution is of the form 1 f(x) = { 0-¹ if 0 < x < 0 otherwise. Note that the information about contained in the random sample X₁, information about contained in the statistic T = max(X₁, ..., Xn equals the •, 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. (a) Carefully argue that T is the maximum likelihood estimator of 0. Recall that the likelihood function of 0, given a data sample x₁,, n, is the product of f(x₁), ..., f (x₂). (b) Construct an unbiased estimator of 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. (c) 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. Page 1 of 2 (d) Now consider a biased estimator of 0 of the form cT. Find c* that minimizes the MSE in the class of estimators of the form cT and explicitly verify that the MSE of c*T is less than that of the estimator constructed in part (c).
Problem 1 Let X₁, Xn be a random sample of size n from the U(0, 0) distribution, where 0 > 0 is an unknown parameter. Recall that the pdf fof the U(0, 0) distribution is of the form 1 f(x) = { 0-¹ if 0 < x < 0 otherwise. Note that the information about contained in the random sample X₁, information about contained in the statistic T = max(X₁, ..., Xn equals the •, 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. (a) Carefully argue that T is the maximum likelihood estimator of 0. Recall that the likelihood function of 0, given a data sample x₁,, n, is the product of f(x₁), ..., f (x₂). (b) Construct an unbiased estimator of 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. (c) 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. Page 1 of 2 (d) Now consider a biased estimator of 0 of the form cT. Find c* that minimizes the MSE in the class of estimators of the form cT and explicitly verify that the MSE of c*T is less than that of the estimator constructed in part (c).
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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finish abcd
Transcribed Image Text:Problem 1
Let X₁,
Xn be a random sample of size n from the U(0, 0) distribution, where 0 > 0
is an unknown parameter. Recall that the pdf fof the U(0, 0) distribution is of the form
1
f(x) = { 0-¹
if 0 < x < 0
otherwise.
Note that the information about contained in the random sample X₁,
information about contained in the statistic
T = max(X₁,
..., Xn equals the
•, 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.
(a) Carefully argue that T is the maximum likelihood estimator of 0. Recall that the
likelihood function of 0, given a data sample x₁,, n, is the product of
f(x₁), ..., f (x₂).
(b) Construct an unbiased estimator of 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:(c) 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.
Page 1 of 2
(d) Now consider a biased estimator of 0 of the form cT. Find c* that minimizes the MSE
in the class of estimators of the form cT and explicitly verify that the MSE of c*T is less
than that of the estimator constructed in part (c).
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