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  • Limiting Distributions
  • STA258 Lecture 05

Consistency in Statistics

• $T_n\stackrel{P}{\to }\theta$

• $T_n$ is a function of the data.

• $\theta$ is an estimator of the data.

• By Weak Law of Large Numbers

• Definition: An estimator ${\hat{\theta }}_n$ is consistent for a parameter $\theta$ if the sequence of estimators converges in probability to $\theta$ as the sample size $n$ approaches infinity.

  • Notation: ${\hat{\theta }}_n\stackrel{P}{\to }\theta$.
  • Formal condition: $\underset{n\to \mathrm{\infty }}{lim}P(|{\hat{\theta }}_n-\theta |>ϵ)=0$ for any $ϵ>0$.

• Sufficient Conditions for Consistency:

  • An estimator is consistent if:
    1. It is unbiased (or bias $\to 0$): $\underset{n\to \mathrm{\infty }}{lim}\mathrm{E}[{\hat{\theta }}_n]=\theta$.
    2. Its variance goes to zero: $\underset{n\to \mathrm{\infty }}{lim}\mathrm{Var}({\hat{\theta }}_n)=0$.

• Examples:

  • The sample mean $\overline{X}$ is a consistent estimator for $\mu$ (Weak Law of Large Numbers).
  • The sample variance $S^2$ is a consistent estimator for ${\sigma }^2$.

• Footnotes