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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

1996 • Question 9

Question

Math 6010 • Linear Models 1996 • Question 9

Let \(X_1,\ldots ,X_n\) be an i.i.d. sample which is uniformly chosen from the interval \((\theta _1,\theta _2)\).

(a) Prove that there are no one-dimensional (i.e., not vector-valued) sufficient statistics for \(h(\theta _1,\theta _2)\) where \(h\) is a one-to-one measurable function.

(b) Is there a one-dimensional (i.e., not vector-valued) sufficient statistics for \(\mu =(\theta _1+\theta _2)/2\)?

Raw LaTeX

Let \(X_1,\ldots,X_n\) be an i.i.d. sample which is uniformly chosen
from the interval \((\theta_1,\theta_2)\).

(a) Prove that there are no one-dimensional (i.e., not vector-valued) sufficient
statistics for \(h(\theta_1,\theta_2)\) where \(h\) is a one-to-one measurable
function.

(b) Is there a one-dimensional (i.e., not vector-valued) sufficient statistics
for \(\mu=(\theta_1+\theta_2)/2\)?

Topic Tags

Estimator Distributions Measure-Theoretic Probability Sufficiency, Completeness, and UMVUE Uniform Integrability and Truncation Methods