Let X₁,..., Xn be a random sample of size n from the U(0, 0) distribution, where @ > 0is an unknown parameter. Recall that the pdf fof the U(0, 0) distribution is of the formf(x) = { 0-1if 0 < x < 0otherwise.Note that the information about contained in the random sample X₁, Xn equals theinformation about contained in the statisticT = max(X₁, ..',Xn).To understand why so, let's think of the random sample as being obtained in a sequentialmanner, that is, you obtain X₁ and pause before obtaining X₂. What does X₁ tell youabout? 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 8, namely, 0 > X₂; however, ifX2 X₁, then it does not contribute anything, above and beyond what you already knowfrom X₁, to your knowledge about 0. In other words, when you have obtained X₁ andX2, what you know about is that it is greater than the maximum of X₁ and X2₂. As such,any reasonable estimator of should be a function of T.(c) Construct an unbiased estimator of which is a function of T and calculate itsvariance. To start with, you should calculate, in that order, the cdf, the pdf, the expectedvalue, and the variance of T.(d) Now consider a biased estimator of 0 of the form cT. Find c* that minimizes the MSEin the class of estimators of the form cT and explicitly verify that the MSE of c*T is lessthan that of the estimator constructed in part (c).

Given that be a random sample of size from the distribution, where is an unknown parameter.

Answered: Let X₁, Xn be a random sample of size n…

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 f(x) = { {0¹ if 0

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Question

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
-1
if 0 < x < 0
otherwise.
Note that the information about contained in the random sample X₁, Xn equals the
information about contained in the statistic
T = 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 8, namely, 0 > X₂; however, if
X2 X₁, 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 X2₂. As such,
any reasonable estimator of should be a function of T.
(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.
(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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