Parametric Inference

Point Estimation

Point Estimators for Mean and Variance

Maximum Likelihood Estimation

Asymptotic Properties of MLEs

We end this section by mentioning that MLEs have some nice asymptotic properties. By asymptotic properties we mean properties that are true when the sample size becomes large. Here, we state these properties without proofs. 

Interval Estimation (Confidence Intervals)

The General Framework of Interval Estimation

Finding Interval Estimators

Confidence Intervals for Normal Samples

Chi-Squared Distribution

The t-Distribution

The next distribution that we need is the Student's t-distribution (or simply the t-distribution). Here, we provide the definition and some properties of the t-distribution.  

Confidence Intervals for the Mean of Normal Random Variables

Confidence Intervals for the Variance of Normal Random Variables

Hypothesis Testing

General Setting and Definitions

Hypothesis Testing for the Mean

Two-sided Tests for the Mean:

One-sided Tests for the Mean:

P-Values

Likelihood Ratio Tests

So far we have focused on specific examples of hypothesis testing problems. Here, we would like to introduce a relatively general hypothesis testing procedure called the likelihood ratio test.

The likelihood-ratio test assesses the goodness of fit of two competing statistical models, specifically one found by maximization over the entire parameter space and another found after imposing some constraint, based on the ratio of their likelihoods. If the constraint (i.e., the null hypothesis) is supported by the observed data, the two likelihoods should not differ by more than sampling error. Thus the likelihood-ratio test tests whether this ratio is significantly different from one, or equivalently whether its natural logarithm is significantly different from zero.

The likelihood-ratio test, also known as Wilks test, is the oldest of the three classical approaches to hypothesis testing, together with the Lagrange multiplier test and the Wald test.

Review of the Likelihood Function: