Nonparametric Likelihood Estimator Research Articles

Time to First Onset of Chest Binding-Related Symptoms in Transgender Youth.

Most transgender individuals assigned female at birth use chest binding (ie, wearing a tight garment to flatten chest tissue for the purpose of gender expression), often beginning in adolescence, to explore their gender identity. Although binding is often critical for mental health, negative physical side effects, ranging from chronic pain to rib fractures, are common. Time to first onset of symptoms is unknown. A community-engaged, online, cross-sectional survey ("The Binding Health Project") enrolled 1800 assigned female at birth or intersex individuals who had ever used chest binding. Lifetime prevalence of 27 pain, musculoskeletal, neurologic, gastrointestinal, generalized, respiratory, and skin or soft tissue symptoms related to binding was assessed. Nonparametric likelihood estimation methods were used to estimate survival curves. More than one-half (56%) of participants had begun binding by age 21, and 30% had begun by age 18. In 18 of 27 symptoms, the majority of people who go on to experience the event will do so within the first binding-year, but several skin-related and rare but serious outcomes (eg, rib fracture) took longer to occur. Pain presents rapidly but continues to rise in intensity over time, peaking at >5 years of binding. Although many symptoms emerge quickly, others can take years to develop. Individuals and their clinicians can use this information to make informed decisions on how to structure binding practices and top surgery timing while meeting goals related to gender expression and mental health. Access to puberty blockers may delay initiation of binding, preventing binding-related symptoms in youth.

Two-step combined nonparametric likelihood estimation of misspecified semiparametric models

This paper proposes to estimate possibly misspecified semiparametric estimating equations models using a two-step combined nonparametric likelihood method. The method uses in the first step the plug in principle and replaces the infinite dimensional parameter with a consistent estimator. In the second step an estimator for the finite dimensional parameter is obtained by combining exponential tilting with a another member of the empirical Cressie-Read discrepancy. The resulting class of semiparametric estimators are robust to misspecification and have the same asymptotic variance as that of the efficient semiparametric generalised method of moment estimator under correct specification. It is also shown that the asymptotic distributions of the proposed estimators can be consistently estimated by a multiplier bootstrap procedure. The results of the paper are illustrated with a quadratic inference function model and an instrumental variable partially linear additive model. Monte Carlo evidence suggests that the proposed estimators have competitive finite sample properties.

Semi‐supervised logistic learning based on exponential tilt mixture models

Consider semi‐supervised learning for classification, where both labelled and unlabelled data are available for training. The goal is to exploit both datasets to achieve higher prediction accuracy than just using labelled data alone. We develop a semi‐supervised logistic learning method based on exponential tilt mixture models by extending a statistical equivalence between logistic regression and exponential tilt modelling. We study maximum nonparametric likelihood estimation and derive novel objective functions that are shown to be Fisher probability consistent. We also propose regularized estimation and construct simple and highly interpretable expectation–maximization (EM) algorithms. Finally, we present numerical results that demonstrate the advantage of the proposed methods compared with existing methods.

A COMPARISON OF NONPARAMETRIC ESTIMATORS FOR LENGTH DISTRIBUTION IN LINE SEGMENT PROCESSES

We study nonparametric estimation of the length distribution for stationary line segment processes in the d-dimensional Euclidean space. Several methods have been proposed in the literature. We review different approaches (Horvitz-Thompson type estimator, reduced-sample estimator, Kaplan-Meier estimator, nonparametric maximum likelihood estimator, stochastic restoration estimation) and compare the finite sample behaviour by means of a simulation study for stationary line segment processes in 2D and 3D. Several data generating processes (Poisson point process, Matérn cluster process and Matérn hard-core process II) are considered with both independent and dependent segments. Our finite sample comparison reveals that the nonparametric likelihood estimator provides the most preferable method which works reasonably also if its assumptions are not satisfied.

Dimension reduction for integrating data series in Bayesian inversion of geostatistical models

This study explores methods with which multidimensional data, e.g. time series, can be effectively incorporated into a Bayesian framework for inferring geostatistical parameters. Such series are difficult to use directly in the likelihood estimation procedure due to their high dimensionality; thus, a dimension reduction approach is taken to utilize these measurements in the inference. Two synthetic scenarios from hydrology are explored in which pumping drawdown and concentration breakthrough curves are used to infer the global mean of a log-normally distributed hydraulic conductivity field. Both cases pursue the use of a parametric model to represent the shape of the observed time series with physically-interpretable parameters (e.g. the time and magnitude of a concentration peak), which is compared to subsets of the observations with similar dimensionality. The results from both scenarios highlight the effectiveness for the shape-matching models to reduce dimensionality from 100+ dimensions down to less than five. The models outperform the alternative subset method, especially when the observations are noisy. This approach to incorporating time series observations in the Bayesian framework for inferring geostatistical parameters allows for high-dimensional observations to be faithfully represented in lower-dimensional space for the non-parametric likelihood estimation procedure, which increases the applicability of the framework to more observation types. Although the scenarios are both from hydrogeology, the methodology is general in that no assumptions are made about the subject domain. Any application that requires the inference of geostatistical parameters using series in either time of space can use the approach described in this paper.

Smoothed nonparametric estimation for current status competing risks data

We study the nonparametric estimation of the cumulative incidence function and the cause-specific hazard function for current status data with competing risks via kernel smoothing. A smoothed naive nonparametric maximum likelihood estimator and a smoothed full nonparametric maximum likelihood estimator are shown to have pointwise asymptotic normality and faster convergence rates than the corresponding unsmoothed nonparametric likelihood estimators. Using the smoothed estimators and the plug-in principle, we can estimate the cause-specific hazard function, which has not been studied previously. We also propose semi-smoothed estimators of the cause-specific hazard as an alternative to the smoothed estimator and demonstrate that neither is uniformly more efficient than the other. Numerical studies show that a smoothed bootstrap method works well for selecting the bandwidths in the smoothed nonparametric estimation. The use of the estimators is exemplified by an application to cumulative incidence and hazard of subtype-specific HIV infection from a sero-prevalence study in injecting drug users in Thailand. Copyright 2013, Oxford University Press.

Nuisance parameter elimination for proportional likelihood ratio models with nonignorable missingness and random truncation

We show that the proportional likelihood ratio model proposed recently by Luo & Tsai (2012) enjoys model-invariant properties under certain forms of nonignorable missing mechanisms and randomly double-truncated data, so that target parameters in the population can be estimated consistently from those biased samples. We also construct an alternative estimator for the target parameters by maximizing a pseudo-likelihood that eliminates a functional nuisance parameter in the model. The corresponding estimating equation has a U-statistic structure. As an added advantage of the proposed method, a simple score-type test is developed to test a null hypothesis on the regression coefficients. Simulations show that the proposed estimator has a small-sample efficiency similar to that of the nonparametric likelihood estimator and performs well for certain nonignorable missing data problems.

Constrained nonparametric maximum likelihood estimation of stochastically ordered survivor functions

AbstractThis paper considers estimators of survivor functions subject to a stochastic ordering constraint based on right censored data. We present the constrained nonparametric maximum likelihood estimator (C‐NPMLE) of the survivor functions in one‐and two‐sample settings where the survivor distributions could be discrete or continuous and discuss the non‐uniqueness of the estimators. We also present a computationally efficient algorithm to obtain the C‐NPMLE. To address the possibility of non‐uniqueness of the C‐NPMLE of $S_1(t)$ when $S_1(t)\le S_2(t)$, we consider the maximum C‐NPMLE (MC‐NPMLE) of $S_1(t)$. In the one‐sample case with arbitrary upper bound survivor function $S_2(t)$, we present a novel and efficient algorithm for finding the MC‐NPMLE of $S_1(t)$. Dykstra (1982) also considered constrained nonparametric maximum likelihood estimation for such problems, however, as we show, Dykstra's method has an error and does not always give the C‐NPMLE. We corrected this error and simulation shows improvement in efficiency compared to Dykstra's estimator. Confidence intervals based on bootstrap methods are proposed and consistency of the estimators is proved. Data from a study on larynx cancer are analysed to illustrate the method. The Canadian Journal of Statistics 40: 22–39; 2012 © 2012 Statistical Society of Canada

On estimation of conditional density models with two-phase sampling

Suppose that the conditional density of a response variable given a vector of explanatory variables is parametrically modelled, and that data are collected by a two-phase sampling design. First, a simple random sample is drawn from the population. The stratum membership in a finite number of strata of the response and explanatory variables is recorded for each unit. Second, a subsample is drawn from the phase-one sample such that the selection probability is determined by the stratum membership. The response and explanatory variables are fully measured at this phase. We synthesize existing results on nonparametric likelihood estimation and present a streamlined approach for the computation and the large sample theory of profile likelihood in four different situations. The amount of information in terms of data and assumptions varies depending on whether the phase-one data are retained, the selection probabilities are known, and/or the stratum probabilities are known. We establish and illustrate numerically the order of efficiency among the maximum likelihood estimators, according to the amount of information utilized, in the four situations.

Asymptotics for non-parametric likelihood estimation with doubly censored multivariate failure times

This paper considers non-parametric estimation of a multivariate failure time distribution function when only doubly censored data are available, which occurs in many situations such as epidemiological studies. In these situations, each of multivariate failure times of interest is defined as the elapsed time between an initial event and a subsequent event and the observations on both events can suffer censoring. As a consequence, the estimation of multivariate distribution is much more complicated than that for multivariate right- or interval-censored failure time data both theoretically and practically. For the problem, although several procedures have been proposed, they are only ad-hoc approaches as the asymptotic properties of the resulting estimates are basically unknown. We investigate both the consistency and the convergence rate of a commonly used non-parametric estimate and show that as the dimension of multivariate failure time increases or the number of censoring intervals of multivariate failure time decreases, the convergence rate for non-parametric estimate decreases, and is slower than that with multivariate singly right-censored or interval-censored data.

ASYMPTOTIC DISTRIBUTIONS OF SEMIPARAMETRIC MAXIMUM LIKELIHOOD ESTIMATORS WITH ESTIMATING EQUATIONS FOR GROUP-CENSORED DATA

Summary Semiparametric maximum likelihood estimation with estimating equations (SMLE) is more flexible than traditional methods; it has fewer restrictions on distributions and regression models. The required information about distribution and regression structures is incorporated in estimating equations of the SMLE to improve the estimation quality of non-parametric methods. The likelihood of SMLE for censored data involves complicated implicit functions without closed-form expressions, and the first derivatives of the log-profile-likelihood cannot be expressed as summations of independent and identically distributed random variables; it is challenging to derive asymptotic properties of the SMLE for censored data. For group-censored data, the paper shows that all the implicit functions are well defined and obtains the asymptotic distributions of the SMLE for model parameters and lifetime distributions. With several examples the paper compares the SMLE, the regular non-parametric likelihood estimation method and the parametric MLEs in terms of their asymptotic efficiencies, and illustrates application of SMLE. Various asymptotic distributions of the likelihood ratio statistics are derived for testing the adequacy of estimating equations and a partial set of parameters equal to some known values.

Nonparametric estimation of a distribution with type I bias with applications to competing risks

A random variable X has a symmetric distribution about a if and only if X − a and −X + a are identically distributed. By considering various types of partial orderings between the distributions of X − a and −X + a, one obtains various types of partial skewness or one-sided bias. For example, F has Type I bias about a if F¯(a + x) ≥ F((a − x)−) for all x > 0; here F¯ = 1 − F. In this article we assume that a = 0, and propose a nonparametric estimator of a continuous distribution function F under the restriction that it has Type I bias. We derive the weak convergence of the resulting process which is used to test for symmetry against that type of bias. The new estimator is then compared with the nonparametric likelihood estimator (NPMLE), [Fcirc] n , of F in terms of mean squared error. A simulation study seems to indicate that the new estimator outperforms the NPMLE uniformly at all the quantiles of the distributions that we have investigated. It turns out that the results developed here could be used to compare two risks in a competing risks problem. We show how this can be done and illustrate the theory with a real life example.