The Sharpe ratio function is a commonly used risk/return measure in financial econometrics. To estimate this function, most existing methods take two-step procedure that first estimates the mean and volatility functions separately then applies plug-in method. In paper, we propose direct method via local maximum likelihood to simultaneously negative log-volatility as well their derivatives. We e...

A decision theoretic foundation of the maximum likelihood estimation method is provided. In regular parametric models, the maximum likelihood estimator θ̂ is shown to be finite sample efficient for the parameter θ, with respect to the mean squared error of the scores and within a large class C of estimates; C includes for some of these models the unbiased, as well as the equivariant estimators o...

Mutual information is useful in various data processing tasks such as feature selection or independent component analysis. In this paper, we propose a new method of approximating mutual information based on maximum likelihood estimation of a density ratio function. Our method, called Maximum Likelihood Mutual Information (MLMI), has several attractive properties, e.g., density estimation is not...

In the statistical literature, the conditional density model specification is commonly used to study regression effects. One attractive model is the semiparametric density ratio model, under which the conditional density function is the product of an unknown baseline density function and a known parametric function containing the covariate information. This model has a natural connection with g...

A very popular form of hypothesis test is the likelihood ratio test, which is a generalization of the optimal test for simple null and alternative hypotheses that was developed by Neyman and Pearson (We skipped Neyman-Pearson lemma because we are short of time). The likelihood ratio test is based on the likelihood function fn(X−1, · · · , Xn|θ), and the intuition that the likelihood function te...

The behavior of maximum likelihood estimates (MLEs) and the likelihood ratio statistic in a family of problems involving pointwise nonparametric estimation of a monotone function is studied. This class of problems differs radically from the usual parametric or semiparametric situations in that the MLE of the monotone function at a point converges to the truth at rate n (slower than the usual √ ...