Statistics

Estimation

Method of Moments Estimator

Sometimes referred to as MME or MMO

Calculate your population moments in terms of your parameters
- \(\displaystyle E(X) = g(\theta)\)
Then invert to get your parameters as a function of your moments
- \(\displaystyle \theta = g^{-1}(E(X))\)
Replace population moments with sample moments
- \(\displaystyle E(X) \rightarrow \bar{x}\)
- \(\displaystyle E(X^2) \rightarrow \frac{1}{n}\sum(x_i - \bar{x})^2\)
- \(\displaystyle \hat{\theta} = g^{-1}(\bar{x})\)

Maximum Likelihood Estimator

(MLE) Maximum Likelihood Estimator

Write out the likelihood function \(\displaystyle L(\theta; \mathbf{x}) = f(\mathbf{x}; \theta)\)
(Optional) Write out the log-likelihood function \(\displaystyle l(\theta) = \log L(\theta; \mathbf{x})\)
Take the derivative of the log-likelihood function w.r.t \(\displaystyle \theta\)
Find the maximum of the log-likelihood function by setting the first derivative to 0
(Optional) Make sure it is the maximum by checking that the Hessian is positive definite
Your MLE \(\displaystyle \hat{\theta}\) is the value which maximizes \(\displaystyle L(\theta)\)
Note if the derivative is always 0, then any value is the MLE. If it is always positive, then take the largest possible value.

Notes

If \(\displaystyle \hat{\theta}\) is the MLE for \(\displaystyle \theta\) then the MLE for \(\displaystyle g(\theta)\) is \(\displaystyle g(\hat{\theta})\)

Uniformly Minimum Variance Unbiased Estimator (UMVUE)

UMVUE, sometimes called MVUE or UMVU.
See Wikipedia: Lehmann–Scheffé theorem
An unbiased estimator of a complete-sufficient statistics is a UMVUE.
In general, you should find a complete sufficient statistic using the property of exponential families.
Then make it unbiased with some factors to get the UMVUE.

Properties

Unbiased

An estimator \(\displaystyle \hat{\theta}\) is unbiased for \(\displaystyle \theta\) if \(\displaystyle E[\hat{\theta}] = \theta\)

\(\displaystyle X_n\) is unbiased for \(\displaystyle E[X]\) but is not consistent

Consistent

An estimator \(\displaystyle \hat{\theta}\) is consistent for \(\displaystyle \theta\) if it converges in probability to \(\displaystyle \theta\)

Example: \(\displaystyle \frac{1}{n}\sum (X-\bar{X})^2\) is a consistent estimator

for \(\displaystyle \sigma^2\) for \(\displaystyle N(\mu, \sigma^2\) but is not unbiased.

Efficiency

Fisher Information

\(\displaystyle I(\theta) = E[ (\frac{\partial}{\partial \theta} \log f(X; \theta) )^2 | \theta]\)
or if \(\displaystyle \log f(x)\) is twice differentiable \(\displaystyle I(\theta) = -E[ \frac{\partial^2}{\partial \theta^2} \log f(X; \theta) | \theta]\)
\(\displaystyle I_n(\theta) = n*I(\theta)\) is the fisher information of the sample. Replace \(\displaystyle f\) with your full likelihood.

Cramer-Rao Lower Bound

Given an estimator \(\displaystyle T(X)\), let \(\displaystyle \psi(\theta)=E[T(X)]\). Then \(\displaystyle Var(T) \geq \frac{(\psi'(\theta))^2}{I(\theta)}\)

Notes

If \(\displaystyle T(X)\) is unbiased then \(\displaystyle \psi(\theta)=\theta \implies \psi'(\theta) = 1\)

Our lower bound will be \(\displaystyle \frac{1}{I(\theta)}\)

The efficiency of an unbiased estimator is defined as \(\displaystyle e(T) = \frac{I(\theta)^{-1}}{Var(T)}\)

Sufficient Statistics

Auxiliary Statistics

Tests

Basic Tests

T-test

Used to test the mean.

F-test

Use to test the ratio of variances.

Likelihood Ratio Test

See Wikipedia: Likelihood Ratio Test

\(\displaystyle LR = -2 \log \frac{\sup_{\theta \in \Theta_0} L(\theta)}{\sup_{\theta \in \Theta} L(\theta)}\)

Uniformly Most Powerful Test

UMP Test
See Wikipedia: Neyman-Pearson Lemma

\(\displaystyle R_{NP} = \left\{x : \frac{L(\theta_0 | x)}{L(\theta_1 | x)} \leq \eta\right\}\)

Anova

Confidence Sets

Confidence Intervals

Relationship with Tests

Regression

Quadratic Forms

Bootstrapping

Wikipedia
Boostrapping is used to sample from your sample to get a measure of accuracy of your statistics.

Nonparametric Bootstrapping

In nonparametric bootstrapping, you resample from your sample with replacement.
In this scenario, you don't need to know the family of distributions that your sample comes from.

Parametric Bootstrapping

In parametric bootstrapping, you learn the distribution parameters of your sample, e.g. with MLE.
Then you can generate samples from that distribution on a computer.