This module introduces the maximum likelihood estimator. We show how the MLE implements the likelihood principle. Methods for computing th MLE are covered. Properties of the MLE are discussed including asymptotic e ciency and invariance under reparameterization. The maximum likelihood estimator (MLE) is an alternative to the minimum variance unbiased estimator (MVUE). For many estimation proble...

1 Summary of Lecture 12 In the last lecture we derived a risk (MSE) bound for regression problems; i.e., select an f ∈ F so that E[(f(X)− Y )]− E[(f∗(X)− Y )] is small, where f∗(x) = E[Y |X = x]. The result is summarized below. Theorem 1 (Complexity Regularization with Squared Error Loss) Let X = R, Y = [−b/2, b/2], {Xi, Yi}i=1 iid, PXY unknown, F = {collection of candidate functions}, f : R → ...

In this paper we present a visual learning approach that uses non-parametric probability estimators. We use entropy analysis over the training set in order to select the features that best represent the pattern class of faces, and set up discrete probability models. These models are tested in the context of maximum likelihooddetection of faces. Excellent results are reported in terms of the cor...

Summary The classical likelihood ratio test based on the asymptotic chi-squared distribution of log-likelihood is one fundamental tools statistical inference. A recent universal approach sample splitting provides valid hypothesis tests and confidence sets in any setting for which we can compute split statistic, or, more generally, an upper bound null maximum likelihood. finite samples without r...

The restricted maximum likelihood is preferred by many to the full maximum likelihood for estimation with variance component and other random coefficient models, because the variance estimator is unbiased. It is shown that this unbiasedness is accompanied in some balanced designs by an inflation of the mean squared error. An estimator of the cluster-level variance that is uniformly more efficie...