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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

1998 • Question 7

Question

Math 6010 • Linear Models 1998 • Question 7

Let \(X_1,X_2,\ldots ,X_n\) denote a random sample from a Poisson distribution with parameter \(\theta >0\).

(a) Show that \((-1)^{X_1}\) is unbiased for \(e^{-2\theta }\), and use the Lehmann–Scheffé theorem to deduce a UMVUE of \(e^{-2\theta }\) based on the sample. (Hint: The conditional distribution of \(X_1\) given \(X_1+\cdots +X_n\) has a simple form.) (5 points)

(b) Argue that the UMVUE in part (a) is a consistent estimator of \(e^{-2\theta }\). (5 points)

Raw LaTeX

Let \(X_1,X_2,\ldots,X_n\) denote a random sample from a Poisson
distribution with parameter \(\theta>0\).

(a) Show that \((-1)^{X_1}\) is unbiased for \(e^{-2\theta}\), and use the
Lehmann--Scheff\'e theorem to deduce a UMVUE of \(e^{-2\theta}\) based on the
sample. (Hint: The conditional distribution of \(X_1\) given
\(X_1+\cdots+X_n\) has a simple form.) (5 points)

(b) Argue that the UMVUE in part (a) is a consistent estimator of
\(e^{-2\theta}\). (5 points)

Topic Tags

Conditional Expectation and Conditional Independence Estimator Distributions Sufficiency, Completeness, and UMVUE Uniform Integrability and Truncation Methods