The problem of estimating the parameters for continuous-time partially observed systems is discussed. New exact lters for obtaining Maximum Likelihood (ML) parameter estimates via the Expectation Maximization algorithm are derived. The methodology exploits relations between incomplete and complete data likelihood and gradient of likelihood functions, which are derived using Girsanov's measure t...

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 → ...

We introduce an estimator for the population mean based on maximizing likelihoods formed by parameterizing a kernel density estimate. Due to these origins, we have dubbed the estimator the maximum kernel likelihood estimate (mkle). A speedy computational method to compute the mkle based on binning is implemented in a simulation study which shows that the mkle at an optimal bandwidth is decidedl...

Looking myopically at the larger features of the likelihood function, absent some fine detail, can theoretically improve maximum likelihood estimation. Such estimators are, in fact, used routinely, since numerical techniques for maximizing a computationally expensive likelihood function or for maximizing a Monte Carlo approximation to a likelihood function may be unable to investigate small sca...

where p above is the density function if X is continuous and the mass function if X is discrete. The MLE is denoted θ̂ or θ̂n if we wish to emphasize the sample size. Above, we suppress the dependency of L on X1, . . . , X (n) to emphasize that we are treating the likelihood as a function of θ. Note that both X and θ may be scalars or vectors (not necessarily of the same dimension) and that L may...