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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

1993 • Question 7

Question

Math 6010 • Linear Models 1993 • Question 7

Let \(Y_i=\alpha ^{i/n}+\epsilon _i\), \(1\le i\le n\). We assume that \(\epsilon _1,\epsilon _2,\ldots ,\epsilon _n\) are independent, identically distributed random variables with \(\E \epsilon _i=0\), \(0<\sigma ^2=\operatorname {Var}\epsilon _i<\infty \) and \(\E \epsilon _i^4<\infty \).

(a) Find the least-squares estimator for \(\alpha \).

(b) Show that the estimator is asymptotically normal.

(c) Find an estimator for \(\sigma ^2\).

Raw LaTeX

Let \(Y_i=\alpha^{i/n}+\epsilon_i\), \(1\le i\le n\). We assume that
\(\epsilon_1,\epsilon_2,\ldots,\epsilon_n\) are independent, identically
distributed random variables with \(\E\epsilon_i=0\),
\(0<\sigma^2=\var\epsilon_i<\infty\) and \(\E\epsilon_i^4<\infty\).

(a) Find the least-squares estimator for \(\alpha\).

(b) Show that the estimator is asymptotically normal.

(c) Find an estimator for \(\sigma^2\).

Topic Tags

Estimator Distributions Generalized Least Squares Ordinary Least Squares Probability Inequalities and Tail Bounds