## 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) $$\widehat{\theta}=d(x)$$.

We might want to estimate some parameter of $$f$$ or some feature such as $$\mu$$.

### 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 $$N(0,1)$$.

## 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

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