Action space

Often \(\Theta\).

Decision rule

In the context of point estimation, the decision rule is called a statistical estimator (also termed statistic or estimator) of \(\theta\), usually designed as \(\widehat{\theta}\), is any function of the data (sample of the random variable) \(\widehat{\theta}=d(x)\).

Nuisance parameter

Parameters that we do not want to estimate.

Sampling distribution of a statistic \(\widehat{\theta}\)

The distribution of \(\widehat{\theta}\) is called the sampling distribution.

Bias of \(\widehat{\theta}\)

\[bias(\widehat{\theta}) = E[\widehat{\theta}] - \theta\]

Standard error \(se\) of \(\widehat{\theta}\)

\[se(\widehat{\theta})=\sqrt{V(\widehat{\theta})}\] Often \(se\) is difficult to calculate as the variance depends on the unknown distribution \(F\). A typical solution is to aproximate \(F\) with the empirical distribution \(\widehat{F}\) and use bootstrap.

Consistency

\(\widehat{\theta}_n\) is consistent if it converges in probability to \(\theta\).

Asymptotic normality

\(\widehat{\theta}_n\) is asymptotically normal if \(\frac{\widehat{\theta}_n - \theta}{se}\) converges in distribution to \(N(0,1)\).

Efficiency

Robustness

Loss and risk functions

Squared error

\(L(\theta,\widehat{\theta})=(\theta - \widehat{\theta})^2\)

The risk function is called MSE.

\(MSE = R(\theta,\widehat{\theta})=E[L(\theta,\widehat{\theta})]= \int (\theta - \widehat{\theta})^2 f(x;\theta) \, dx\)

Properties

  • \(MSE = bias(\widehat{\theta})^2 + var(\widehat{\theta})\)

    Demonstration: …

Examples

Example 1

\(X \sim N(\mu,1)\)

\(\widehat{\mu}_1 = 4\)

\(MSE_1 = bias(\widehat{\mu}_1)^2 + var(\widehat{\mu}_1)= \left( E[\widehat{\mu}_1] - \mu \right)^2+ var(\widehat{\mu}_1) = \left(4-\mu \right)^2 + 0 = \left(4-\mu \right)^2\)

\(\widehat{\mu}_2 = X\)

\(MSE_2 = bias(\widehat{\mu}_2)^2 + var(\widehat{\mu}_2)= \left( E[\widehat{\mu}_2] - \mu \right)^2+ var(\widehat{\mu}_2) = \left(E[X]-\mu \right)^2 + var(X) = \left(\mu-\mu \right)^2 + 1 = 1\)

If the parameter happens to be close to 4 the risk for the first estimator is better than for the second. Otherwise, the second si better.

Absolute error

\(L(\theta,\widehat{\theta})=|\theta - \widehat{\theta}|\)

Kullback-Leibler

\(L(\theta,\widehat{\theta})=\int \log \left( \frac{f(x;\theta)}{f(x;\widehat{\theta})} \right) f(x;\theta) dx\)