Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

2006 • Question 4

Question

Math 6010 • Linear Models 2006 • Question 4

Let \(X_1, X_2, \ldots \) denote an i.i.d. sample with common distribution function \(F\). Define \[ F_n(x) = n^{-1}\sum _{j=1}^{n} I\{X_j \leq x\} \] for all \(x \in (-\infty , \infty )\); i.e., \(F_n\) is the empirical distribution function based on the \(X_i\)’s. Prove that for all \(x \in (-\infty , \infty )\) fixed, \[ \operatorname {Var}\bigl (F_n(x)-F(x)\bigr ) \leq \frac {1}{4n}. \] Use this to conclude that \(F_n(x)\) is a consistent estimator for \(F(x)\).

From here on let \(X_1, X_2, \ldots \) denote an i.i.d. sample from an exponential density with mean \(1/\theta \).

Let \(X_1, X_2, \ldots\) denote an i.i.d. sample with common distribution function \(F\). Define
\[
F_n(x) = n^{-1}\sum_{j=1}^{n} I\{X_j \leq x\}
\]
for all \(x \in (-\infty, \infty)\); i.e., \(F_n\) is the empirical distribution function based on the \(X_i\)'s. Prove that for all \(x \in (-\infty, \infty)\) fixed,
\[
\operatorname{Var}\bigl(F_n(x)-F(x)\bigr) \leq \frac{1}{4n}.
\]
Use this to conclude that \(F_n(x)\) is a consistent estimator for \(F(x)\).

From here on let \(X_1, X_2, \ldots\) denote an i.i.d. sample from an exponential density with mean \(1/\theta\).

Topic Tags

Estimator Distributions Multivariate Normal Distributions Probability Inequalities and Tail Bounds Uniform Integrability and Truncation Methods