Before diving into equations, we must understand the fundamental shift in thinking:
L(θ)=∏i=1nf(Xi;θ)cap L open paren theta close paren equals product from i equals 1 to n of f of open paren cap X sub i ; theta close paren
Var(θ̂)≥1nI(θ)Var open paren theta hat close paren is greater than or equal to the fraction with numerator 1 and denominator n cap I open paren theta close paren end-fraction
Your current (e.g., undergraduate student, graduate researcher, working data scientist) The specific textbook or curriculum you are following
: A fundamental tool for finding the "best" test in simple hypothesis scenarios. The null hypothesis is generally rejected when the likelihood ratio—the joint PDF under the null divided by the joint PDF under the alternative—is small. Sampling Distributions
The MLE is the parameter value that maximizes the likelihood function, meaning it makes the observed data most probable.Given a joint probability density function , the likelihood function is:
. Completeness is vital for establishing the uniqueness of optimal estimators. 3. Point Estimation Theory
Lectures teach you standard algorithmic ways to construct estimators from data. Maximum Likelihood Estimation (MLE)
Finds the parameter values that maximize the likelihood function, making the observed data most probable under the assumed statistical model. 4. Interval Estimation
: An estimator is unbiased if its expected value equals the true parameter value.
