Semiparametric Maximum Likelihood Estimation Research Articles

Using a monotonic density ratio model to increase the power of the goodness-of-fit test for logistic regression models with case-control data.

Logistic regression models are widely used in case-control data analysis, and testing the goodness-of-fit of their parametric model assumption is a fundamental research problem. In this article, we propose to enhance the power of the goodness-of-fit test by exploiting a monotonic density ratio model, in which the ratio of case and control densities is assumed to be a monotone function. We show that such a monotonic density ratio model is naturally induced by the retrospective case-control sampling design under the alternative hypothesis. The pool-adjacent-violator algorithm is adapted to solve for the constrained nonparametric maximum likelihood estimator under the alternative hypothesis. By measuring the discrepancy between this estimator and the semiparametric maximum likelihood estimator under the null hypothesis, we develop a new Kolmogorov-Smirnov-type statistic to test the goodness-of-fit for logistic regression models with case-control data. A bootstrap resampling procedure is suggested to approximate the -value of the proposed test. Simulation results show that the type I error of the proposed test is well controlled and the power improvement is substantial in many cases. Three real data applications are also included for illustration.

Dynamic Autoregressive Liquidity (DArLiQ)

We introduce a new class of semiparametric dynamic autoregressive models for the Amihud illiquidity measure, which captures both the long-run trend in the illiquidity series with a nonparametric component and the short-run dynamics with an autoregressive component. We develop a generalized method of moments (GMM) estimator based on conditional moment restrictions and an efficient semiparametric maximum likelihood (ML) estimator based on an iid assumption. We derive large sample properties for our estimators. Finally, we demonstrate the model fitting performance and its empirical relevance on an application. We investigate how the different components of the illiquidity process obtained from our model relate to the stock market risk premium using data on the S&P 500 stock market index.

An accelerated failure time regression model for illness-death data: A frailty approach.

This work presents a new model and estimation procedure for the illness-death survival data where the hazard functions follow accelerated failure time (AFT) models. A shared frailty variate induces positive dependence among failure times of a subject for handling the unobserved dependency between the nonterminal and the terminal failure times given the observed covariates. The motivation behind the proposed modeling approach is to leverage the well-known interpretability advantage of AFT models with respect to the observed covariates, while also benefiting from the simple and intuitive interpretation of the hazard functions. A semiparametric maximum likelihood estimation procedure is developed via a kernel smoothed-aided expectation-maximization algorithm, and variances are estimated by weighted bootstrap. We consider existing frailty-based illness-death models and place particular emphasis on highlighting the contribution of our current research. The breast cancer data of the Rotterdam tumor bank are analyzed using the proposed as well as existing illness-death models. The results are contrasted and evaluated based on a new graphical goodness-of-fit procedure. Simulation results and data analysis nicely demonstrate the practical utility of the shared frailty variate with the AFT regression model under the illness-deathframework.

Uniform and [formula omitted] convergences for nonparametric continuous time regressions with semiparametric applications

We obtain uniform and Lp convergence rates of kernel type nonparametric estimators for the instantaneous conditional mean and variance functions of continuous time regressions, where the regressor is assumed to be a general recurrent diffusion. Our asymptotics are developed under a general set-up, with a shrinking sampling interval and an increasing time span, and without the stationarity assumption. Based on our convergence results, we develop a semiparametric inferential procedure for continuous time predictive regressions. In particular, a robust semiparametric likelihood ratio test for linear predictability is proposed, with its limit distribution established. We also apply our convergence results to obtain the asymptotics of a semiparametric maximum likelihood estimator of the drift of recurrent diffusions. In our simulation study, we examine the finite sample performance of our robust test against several existing tests in the literature. An empirical illustration is presented to test the predictability of the excess returns of two major stock indices using the commonly used dividend–price ratio and earnings–price ratio as the predictor.

Asymptotic properties of semiparametric [formula omitted]-estimators with multiple change points

Statistical models with multiple change points are used in many fields; however, the theoretical properties of semiparametric M-estimators of such models have received relatively little attention. The main purpose of the present work is to investigate the asymptotic properties of semiparametric M-estimators with non-smooth criterion functions of the parameters of a multiple change-point model for a general class of models in which the form of the distribution can change from segment to segment and in which, possibly, there are parameters that are common to all segments. Consistency of the semiparametric M-estimators of the change points is established and the rate of convergence is determined. The asymptotic normality of the semiparametric M-estimators of the parameters of the within-segment distributions is established under quite general conditions. These results, together with a generic paradigm for studying semiparametric M-estimators with multiple change points, provide a valuable extension to previous related research on (semi)parametric maximum-likelihood estimators. For illustration, the classification with missing data in the model is investigated in detail and a short simulation result is provided.

Semiparametric current status regression model using kernel smoothing with application to HIV study

In current status regression analysis, the responses are indirectly observed and the regression coefficients can not be estimated using the commonly used least squares estimate, instead semiparametric maximum likelihood estimate is used. As the nonparametric component estimate is a non-smooth step function, the regression coefficients are bundled inside it, so the asymptotic distribution of the estimated coefficients is an open problem. To overcome this issue, smoothed versions are proposed in the literature; however, in these versions the score equation for the regression coefficients is still not continuous, the estimate is not an exact solution of the score equation, and zero-crossing is used instead. Here we propose a modified smoothed version to make the score equation a continuous function of the regression coefficients, and thus the estimator is the exact solution of the score equation. Asymptotic distribution of the resulting estimators can be shown as in our related papers. Simulation studies are conducted to evaluate the performance of the method, and it demonstrates superior finite sample behavior. For illustration, we applied the method to the analysis of a real data set from a HIV study.

Maximum Likelihood Estimation for Shape-restricted Single-index Hazard Models.

Single-index models are becoming increasingly popular in many scientific applications as they offer the advantages of flexibility in regression modeling as well as interpretable covariate effects. In the context of survival analysis, the single-index hazards models are natural extensions of the Cox proportional hazards models. In this paper, we propose a novel estimation procedure for single-index hazard models under a monotone constraint of the index. We apply the profile likelihood method to obtain the semiparametric maximum likelihood estimator, where the novelty of the estimation procedure lies in estimating the unknown monotone link function by embedding the problem in isotonic regression with exponentially distributed random variables. The consistency of the proposed semiparametric maximum likelihood estimator is established under suitable regularity conditions. Numerical simulations are conducted to examine the finite-sample performance of the proposed method. An analysis of breast cancer data is presented for illustration.

Semiparametric partial linear modeling of risk factors for ear infections: the Early Childhood Longitudinal Study

The Early Childhood Longitudinal Study–Kindergarten Class of 2010–2011 (ECLS-K:2011) ascertained timing of ear infections within age specified intervals and parent's/caregiver's report of medically diagnosed hearing loss. In this nationally representative, school-based sample of children followed from kindergarten entry through fifth grade, academic performance in reading, mathematics, and science was assessed longitudinally. Prior investigations of this ECLS-K:2011 cohort showed that age has a non-linear, monotonically increasing functional relationship with academic performance. Because of this knowledge, a semiparametric partial linear model is proposed, in which the effect of age is modeled by an unknown monotonically increasing function along with other regression parameters. The parameters are estimated by a semiparametric maximum likelihood estimator. A test of a constant effect of age is also proposed. Simulation studies are conducted to evaluate the performance of the proposed method, as compared with the commonly used linear model; the former outperforms the latter based on several criteria. We then analyzed ECLS-K:2011 data to compare results of the partial linear parametric model estimation with that of classical linear regression models.

Robust estimates of regional treatment effects in multiregional randomized clinical trials with ordinal responses

ABSTRACT Global clinical trials involving multiple regions are common in current drug development processes. Determining the regional treatment effects of a new therapy over an existing therapy is important to both the sponsors and the regulatory agencies in the regions. Existing methods are mainly for continuous primary endpoints and use subjectively specified models, which may deviate from the true model. Here, we consider trials that have ordinal responses as the primary endpoint. This article extends the recently developed robust semiparametric ordinal regression model to estimate regional treatment effects, in which the regression coefficients and regional effects are modeled parametrically for ease of interpretation, and the regression link function is specified nonparametrically for robustness. The model parameters are estimated by semiparametric maximum likelihood estimation, and the null hypothesis of no regional effect is tested by the Wald test. Simulation studies are conducted to evaluate the performance of the proposed method and compare it with the commonly used parametric model. The results of the former show an improved overall performance over the latter. In particular, the model yields much higher precision in estimation and prediction than the fixed-link model. This result is especially appealing since our interest is to estimate the treatment effect more efficiently and the estimand is of particular interest in multiregional clinical trials. We then apply the method by analyzing real multiregional clinical trials with ordinal responses as their primary endpoint.

Nonparametric and semiparametric estimation with sequentially truncated survival data.

In observational cohort studies with complex sampling schemes, truncation arises when the time to event of interest is observed only when it falls below or exceeds another random time, that is, the truncation time. In more complex settings, observation may require a particular ordering of event times; we refer to this as sequential truncation. Estimators of the event time distribution have been developed for simple left-truncated or right-truncated data. However, these estimators may be inconsistent under sequential truncation. We propose nonparametric and semiparametric maximum likelihood estimators for the distribution of the event time of interest in the presence of sequential truncation, under two truncation models. We show the equivalence of an inverse probability weighted estimator and a product limit estimator under one of these models. We study the large sample properties of the proposed estimators and derive their asymptotic variance estimators. We evaluate the proposed methods through simulation studies and apply the methods to an Alzheimer's disease study. We have developed an R package, seqTrun, for implementation of ourmethod.

Consistency of semi-parametric maximum likelihood estimator under identifiability conditions for the linear regression model with type I right censoring data

The consistency of the semi-parametric maximum likelihood estimator (SMLE) under the semi-parametric linear regression model with right-censoring data (SPLRRC model) has not been studied under the necessary and sufficient condition for the identifiability of the parameters. In this paper, we discuss the necessary and sufficient condition for the consistency of SMLE under type I right censoring.

Efficient semiparametric estimation of time‐censored intensity‐reduction models for repairable systems

AbstractThe rate reduction models have been widely used to model the recurrent failure data for their capabilities in quantifying the repair effects. Despite the widespread popularity, there have been limited studies on statistical inference of most failure rate reduction models. In view of this fact, this study proposes a semiparametric estimation framework for a general class of such models, called extended geometric failure rate reduction (EGFRR) models. Covariates are considered in our analysis and their effects are modeled as a log‐linear factor on the baseline failure rate. Unlike the existing inference methods for the EGFRR models that assume the failure data are censored at a fixed number of failures, our study considers covariates and time‐censoring, which are more common in practice. The semiparametric maximum likelihood (ML) estimators are obtained by carefully constructing the likelihood function. Asymptotic properties including consistency and weak convergence of the ML estimators are established by using the properties of the martingale process. In addition, we show that the semiparametric estimators are asymptotically efficient. A real example from the automobile industry illustrates the usefulness of the proposed framework and extensive simulations show its outstanding performance when comparing with the existing methods.

ESTIMATES OF DERIVATIVES OF (LOG) DENSITIES AND RELATED OBJECTS

We estimate the density and its derivatives using a local polynomial approximation to the logarithm of an unknown density function f. The estimator is guaranteed to be non-negative and achieves the same optimal rate of convergence in the interior as on the boundary of the support of f. The estimator is therefore well-suited to applications in which non-negative density estimates are required, such as in semiparametric maximum likelihood estimation. In addition, we show that our estimator compares favorably with other kernel-based methods, both in terms of asymptotic performance and computational ease. Simulation results confirm that our method can perform similarly or better in finite samples compared to these alternative methods when they are used with optimal inputs, that is, an Epanechnikov kernel and optimally chosen bandwidth sequence. We provide code in several languages.

Logistic regression analysis of two-phase studies using generalized method of moments.

Two-phase designs can reduce the cost of epidemiological studies by limiting the ascertainment of expensive covariates or/and exposures to an efficiently selected subset (phase-II) of a larger (phase-I) study. Efficient analysis of the resulting data set combining disparate information from phase-I and phase-II, however, can be complex. Most of the existing methods, including semiparametric maximum-likelihood estimator, require the information in phase-I to be summarized into a fixed number of strata. In this paper, we describe a novel method for the analysis of two-phase studies where information from phase-I is summarized by parameters associated with a reduced logistic regression model of the disease outcome on available covariates. We then setup estimating equations for parameters associated with the desired extended logistic regression model, based on information on the reduced model parameters from phase-I and complete data available at phase-II after accounting for nonrandom sampling design. We use generalized method of moments to solve overly identified estimating equations and develop the resulting asymptotic theory for the proposed estimator. Simulation studies show that the use of reduced parametric models, as opposed to summarizing data into strata, can lead to more efficient utilization of phase-I data. An application of the proposed method is illustrated using the data from the U.S. National Wilms TumorStudy.

Combining multiple imputation with raking of weights: Anefficient and robust approach in the setting of nearly true models.

Multiple imputation (MI) provides us with efficient estimators in model-based methods for handling missing data under the true model. It is also well-understood that design-based estimators are robust methods that do not require accurately modeling the missing data; however, they can be inefficient. In any applied setting, it is difficult to know whether a missing data model may be good enough to win the bias-efficiency trade-off. Raking of weights is one approach that relies on constructing an auxiliary variable from data observed on the full cohort, which is then used to adjust the weights for the usual Horvitz-Thompson estimator. Computing the optimally efficient raking estimator requires evaluating the expectation of the efficient score given the full cohort data, which is generally infeasible. We demonstrate MI as a practical method to compute a raking estimator that will be optimal. We compare this estimator to common parametric and semi-parametric estimators, including standard MI. We show that while estimators, such as the semi-parametric maximum likelihood and MI estimator, obtain optimal performance under the true model, the proposed raking estimator utilizing MI maintains a better robustness-efficiency trade-off even under mild model misspecification. We also show that the standard raking estimator, without MI, is often competitive with the optimal raking estimator. We demonstrate these properties through several numerical examples and provide a theoretical discussion of conditions for asymptotically superior relative efficiency of the proposed raking estimator.

Estimation of a binary dependent variable sample selection model with endogeneity under a nonparametric selection mechanism

In this paper, we consider an estimation of a sample selection model with a binary dependent variable and possible endogenous regressors. Under the semiparametric framework, we impose neither the parametric specification of the error distribution nor the functional form of the selection equation, which largely reduces the risk of model misspecification. Throughout this paper, we present the identification 条件 and propose a two-step semiparametric maximum likelihood estimator with the first step being a nonparametric regression for the selection variable. A control function approach is used to control for the possible endogeneity. The proposed estimator is shown to be consistent and asymptotically normal. In the simulation studies, we compare the finite sample properties of our estimator with those of existing alternatives, demonstrating the significant advantages of our approach especially in robustness to the model misspecification. Finally, an economic application on labor economics is carried out to illustrate the usefulness of our estimator.

Consistency of the Semi-parametric MLE under the Cox Model with Right-Censored Data

Objective: We studied the consistency of the semi-parametric maximum likelihood estimator (SMLE) under the Cox regression model with right-censored (RC) data. Methods: Consistency proofs of the MLE are often based on the Shannon-Kolmogorov inequality, which requires finite E(lnL), where L is the likelihood function. Results: The results of this study show that one property of the semi-parametric MLE (SMLE) is established. Conclusion: Under the Cox model with RC data, E(lnL) may not exist. We used the Kullback-Leibler information inequality in our proof.

Targeted design for adaptive clinical trials via semiparametric model

Precision medicine approach that assigns treatment according to an individual's personal (including molecular) profile is revolutionizing health care. Existing statistical methods for clinical trial design typically assume a known model to estimate characteristics of treatment outcomes, which may yield biased results if the true model deviates far from the assumed one. This article aims to achieve model robustness in a phase II multi-stage adaptive clinical trial design. We propose and study a semiparametric regression mixture model in which the mixing proportions are specified according to the subjects' profiles, and each sub-group distribution is only assumed to be unimodal for robustness. The regression parameters and the error density functions are estimated by semiparametric maximum likelihood and isotonic regression estimators. The asymptotic properties of the estimates are studied. Simulation studies are conducted to evaluate the performance of the method after a real data analysis.

A gamma-frailty model for interval-censored data with dependent examination times: a computationally efficient approach

Interval-censored survival data arise naturally in many fields such as medical follow-up studies, in which the event or failure is not observed exactly but only known to occur within a time interval. Most existing approaches for analyzing interval-censored failure time data assume that the examination times and the failure time are independent or conditionally independent given covariates. While this assumption offers considerable simplification, it is not plausible in some situations, e.g., the visiting rate can be positively or negatively correlated with the risk of failure due to unobservable health status even after adjusting for observable covariates. In this article, we consider dependent interval-censored data, where there exists dependence between the failure time and the entire visiting process. A shared frailty is used to characterize the dependence of hazard function of failure time and intensity function of visiting process. Moreover, the joint model could describe the possible none, positive or negative association between failure time and visiting process. We propose the semiparametric maximum likelihood estimators and develop an EM algorithm based on a Poisson data augmentation. The performance of the proposed method is examined through extensive simulation studies and an application to a bladder cancer dataset is presented.

On computation of semiparametric maximum likelihood estimators with shape constraints.

Large sample theory of semiparametric models based on maximum likelihood estimation (MLE) with shape constraint on the nonparametric component is well studied. Relatively less attention has been paid to the computational aspect of semiparametric MLE. The computation of semiparametric MLE based on existing approaches such as the expectation-maximization (EM) algorithm can be computationally prohibitive when the missing rate is high. In this paper, we propose a computational framework for semiparametric MLE based on an inexact block coordinate ascent (BCA) algorithm. We show theoretically that the proposed algorithm converges. This computational framework can be applied to a wide range of data with different structures, such as panel count data, interval-censored data, and degradation data, among others. Simulation studies demonstrate favorable performance compared with existing algorithms in terms of accuracy and speed. Two data sets are used to illustrate the proposed computational method. We further implement the proposed computational method in R package BCA1SG, available atCRAN.