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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

1993 • Question 10

Question

Math 6010 • Linear Models 1993 • Question 10

Let \(X_1,X_2,\ldots ,X_n\) be independent identically distributed random variables with \(\mathbb {P}\{X_i=1\}=p\), \(\mathbb {P}\{X_i=0\}=1-p\).

(a) Compute the maximum likelihood estimator of \(\sigma ^2=p(1-p)\).

(b) Compute the bias, the variance and the mean-square error of the estimator.

(c) Is the estimator asymptotically efficient?

(d) Find the uniformly minimum variance unbiased estimator for \(\sigma ^2\).

Raw LaTeX

Let \(X_1,X_2,\ldots,X_n\) be independent identically distributed
random variables with \(\Prob\{X_i=1\}=p\), \(\Prob\{X_i=0\}=1-p\).

(a) Compute the maximum likelihood estimator of \(\sigma^2=p(1-p)\).

(b) Compute the bias, the variance and the mean-square error of the estimator.

(c) Is the estimator asymptotically efficient?

(d) Find the uniformly minimum variance unbiased estimator for \(\sigma^2\).

Topic Tags

Covariance and Independence Estimator Distributions Maximum Likelihood Estimation Sufficiency, Completeness, and UMVUE