Martingale estimation functions for Bessel processes

In this paper we derive martingale estimating functions for the dimensionality parameter of a Bessel process based on the eigenfunctions of the diffusion operator. Since a Bessel process is non-ergodic and the theory of martingale estimating functions is developed for ergodic diffusions, we use the space-time transformation of the Bessel process and formulate our results for a modified Bessel process. We deduce consistency, asymptotic normality and discuss optimality. It turns out that the martingale estimating function based of the first eigenfunction of the modified Bessel process coincides with the linear martingale estimating function for the Cox Ingersoll Ross process. Furthermore, our results may also be applied to estimating the multiplicity parameter of a one-dimensional Dunkl process and some related polynomial processes.

Improved Doubly Robust Estimation in Marginal Mean Models for Dynamic Regimes

Doubly robust (DR) estimators are an important class of statistics derived from a theory of semiparametric efficiency. They have become a popular tool in causal inference, including applications to dynamic treatment regimes. The doubly robust estimators for the mean response to a dynamic treatment regime may be conceived through the augmented inverse probability weighted (AIPW) estimating function, defined as the sum of the inverse probability weighted (IPW) estimating function and an augmentation term. The IPW estimating function of the causal estimand via marginal structural model is defined as the complete-case score function for those subjects whose treatment sequence is consistent with the dynamic regime in question divided by the probability of observing the treatment sequence given the subject's treatment and covariate histories. The augmentation term is derived by projecting the IPW estimating function onto the nuisance tangent space and has mean-zero under the truth. The IPW estimator of the causal estimand is consistent if (i) the treatment assignment mechanism is correctly modeled and the AIPW estimator is consistent if either (i) is true or (ii) nested functions of intermediate and final outcomes are correctly modeled. Hence, the AIPW estimator is doubly robust and, moreover, the AIPW is semiparametric efficient if both (i) and (ii) are true simultaneously. Unfortunately, DR estimators can be inferior when either (i) or (ii) is true and the other false. In this case, the misspecified parts of the model can have a detrimental effect on the variance of the DR estimator. We propose an improved DR estimator of causal estimand in dynamic treatment regimes through a technique originally developed by [4] which aims to mitigate the ill-effects of model misspecification through a constrained optimization. In addition to solving a doubly robust system of equations, the improved DR estimator simultaneously minimizes the asymptotic variance of the estimator under a correctly specified treatment assignment mechanism but misspecification of intermediate and final outcome models. We illustrate the desirable operating characteristics of the estimator through Monte Carlo studies and apply the methods to data from a randomized study of integrilin therapy for patients undergoing coronary stent implantation. The methods proposed here are new and may be used to further improve personalized medicine, in general.

Comparison of Optim, Nleqslv and MaxLik to Estimate Parameters in Some of Regression Models

Our main goal in this article is to present the approaches and examples of three functions in R consist of optim, nleqslv and maxLik function to detect the optimization solution of the estimating function in the regression models. We then compare the results with numerous sample sizes (n=150, 300 and 500), the execution time of R code, as well as Normal Q - Q plots of three approaches through some of regression models such as the zero-inflated Binomial (ZIB) regression model, logistic regression model, the zero-inflated Poisson (ZIP) regression model and the zero-inflated Bernoulli (ZIBer) regression model. Finally, we discuss potential research directions in the coming times. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium provided the original work is properly cited.

Double Poisson-Tweedie Regression Models

In this paper, we further extend the recently proposed Poisson-Tweedie regression models to include a linear predictor for the dispersion as well as for the expectation of the count response variable. The family of the considered models is specified using only second-moments assumptions, where the variance of the count response has the form $\mu + \phi \mu^p$, where µ is the expectation, ϕ and p are the dispersion and power parameters, respectively. Parameter estimations are carried out using an estimating function approach obtained by combining the quasi-score and Pearson estimating functions. The performance of the fitting algorithm is investigated through simulation studies. The results showed that our estimating function approach provides consistent estimators for both mean and dispersion parameters. The class of models is motivated by a data set concerning CD4 counting in HIV-positive pregnant women assisted in a public hospital in Curitiba, Paraná, Brazil. Specifically, we investigate the effects of a set of covariates in both expectation and dispersion structures. Our results showed that women living out of the capital Curitiba, with viral load equal or larger than 1000 copies and with previous diagnostic of HIV infection, present lower levels of CD4 cell count. Furthermore, we detected that the time to initiate the antiretroviral therapy decreases the data dispersion. The data set and R code are available as supplementary materials.

Flexible quasi-beta regression models for continuous bounded data

We propose a flexible class of regression models for continuous bounded data based on second-moment assumptions. The mean structure is modelled by means of a link function and a linear predictor, while the mean and variance relationship has the form [Formula: see text], where [Formula: see text], [Formula: see text] and [Formula: see text] are the mean, dispersion and power parameters respectively. The models are fitted by using an estimating function approach where the quasi-score and Pearson estimating functions are employed for the estimation of the regression and dispersion parameters respectively. The flexible quasi-beta regression model can automatically adapt to the underlying bounded data distribution by the estimation of the power parameter. Furthermore, the model can easily handle data with exact zeroes and ones in a unified way and has the Bernoulli mean and variance relationship as a limiting case. The computational implementation of the proposed model is fast, relying on a simple Newton scoring algorithm. Simulation studies, using datasets generated from simplex and beta regression models show that the estimating function estimators are unbiased and consistent for the regression coefficients. We illustrate the flexibility of the quasi-beta regression model to deal with bounded data with two examples. We provide an R implementation and the datasets as supplementary materials.