Let X1,, X, denote a random sample of size n from the population with probability density function 16) = {** 38-4, r2 B 0, otherwise where 3> 0 is the population parameter we want to estimate. Consider the estimator B = min(X1,,X„). [The estimator is the 1st-order order statistic of the random sample.] (a) Derive the bias of the estimator 3. (b) Derive the mean squared error of 3, MSE(3).
Let X1,, X, denote a random sample of size n from the population with probability density function 16) = {** 38-4, r2 B 0, otherwise where 3> 0 is the population parameter we want to estimate. Consider the estimator B = min(X1,,X„). [The estimator is the 1st-order order statistic of the random sample.] (a) Derive the bias of the estimator 3. (b) Derive the mean squared error of 3, MSE(3).
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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![Let \( X_1, \ldots, X_n \) denote a random sample of size \( n \) from the population with probability density function
\[
f(x) =
\begin{cases}
3\beta^3 x^{-4}, & x \geq \beta \\
0, & \text{otherwise}
\end{cases}
\]
where \( \beta > 0 \) is the population parameter we want to estimate. Consider the estimator \(\hat{\beta} = \min(X_1, \ldots, X_n)\). [The estimator \(\hat{\beta}\) is the 1st-order order statistic of the random sample.]
(a) Derive the bias of the estimator \(\hat{\beta}\).
(b) Derive the mean squared error of \(\hat{\beta}\), \( \text{MSE}(\hat{\beta}) \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe9ad9a17-86a2-4d23-8aeb-c537a78d9db8%2F5f0dd013-0da4-4100-aa27-ab1be174c1fd%2F76b7dyf_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Let \( X_1, \ldots, X_n \) denote a random sample of size \( n \) from the population with probability density function
\[
f(x) =
\begin{cases}
3\beta^3 x^{-4}, & x \geq \beta \\
0, & \text{otherwise}
\end{cases}
\]
where \( \beta > 0 \) is the population parameter we want to estimate. Consider the estimator \(\hat{\beta} = \min(X_1, \ldots, X_n)\). [The estimator \(\hat{\beta}\) is the 1st-order order statistic of the random sample.]
(a) Derive the bias of the estimator \(\hat{\beta}\).
(b) Derive the mean squared error of \(\hat{\beta}\), \( \text{MSE}(\hat{\beta}) \).
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