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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Fall 2007 • Question 7

Question

Math 6010 • Linear Models Fall 2007 • Question 7

Let Let \(X_1,X_2,\ldots ,X_n\) be independent identically distributed random variables with \(\E X_1=\mu \), \(\operatorname {Var} X_1=\sigma ^2\). We assume that \(\E X_4<\infty \). The random variable \[ \hat \sigma _n^2=\frac {1}{2n}\sum _{2\leq i\leq n}(X_i-X_{i-1})^2 \] is suggested to estimate \(\sigma ^2\).

(i) Establish some basic properties of \(\hat \sigma _n^2\). (7 points)

(ii) Is \(\hat \sigma _n^2\) a better estimator than the sample variance? Justify your answer. (7 points)

Raw LaTeX

Let Let \(X_1,X_2,\ldots,X_n\) be independent identically distributed
random variables with \(\E X_1=\mu\), \(\var X_1=\sigma^2\). We assume that
\(\E X_4<\infty\). The random variable
\[
  \hat\sigma_n^2=\frac{1}{2n}\sum_{2\leq i\leq n}(X_i-X_{i-1})^2
\]
is suggested to estimate \(\sigma^2\).

(i) Establish some basic properties of \(\hat\sigma_n^2\). (7 points)

(ii) Is \(\hat\sigma_n^2\) a better estimator than the sample variance? Justify
your answer. (7 points)

Topic Tags

Covariance and Independence Estimator Distributions Probability Inequalities and Tail Bounds Uniform Integrability and Truncation Methods