Parameter identifiability in a class of random graph mixture models

We prove identifiability of parameters for a broad class of random graph mixture models. These models are characterized by a partition of the set of graph nodes into latent (unobservable) groups. The connectivities between nodes are independent random variables when conditioned on the groups of the nodes being connected. In the binary random graph case, in which edges are either present or absent, these models are known as stochastic blockmodels and have been widely used in the social sciences and, more recently, in biology. Their generalizations to weighted random graphs, either in parametric or non-parametric form, are also of interest. Despite these many applications, the parameter identifiability issue for such models has only been touched upon in the literature. We give here a thorough investigation of this problem. Our work also has consequences for parameter estimation. In particular, the estimation procedure proposed by Frank and Harary for binary affiliation models is revisited in this article.     

Locally efficient semiparametric estimators for a class of Poisson models with measurement error

The presence of measurement error may cause bias in parameter estimation and can lead to incorrect conclusions in data analyses. Despite a large body of literature on general measurement error problems, relatively few works exist to handle Poisson models. In this article we thoroughly study Poisson models with errors in covariates and propose consistent and locally efficient semiparametric estimators. We assess the finite sample performance of the estimators through extensive simulation studies and illustrate the proposed methodologies by analyzing data from the Stroke Recovery in Underserved Populations Study. The Canadian Journal of Statistics 47: 157–181; 2019 © 2019 Statistical Society of Canada     

Semiparametric Mixtures of Symmetric Distributions

We consider in this paper the semiparametric mixture of two unknown distributions equal up to a location parameter. The model is said to be semiparametric in the sense that the mixed distribution is not supposed to belong to a parametric family. To insure the identifiability of the model, it is assumed that the mixed distribution is zero symmetric, the model being then defined by the mixing proportion, two location parameters and the probability density function of the mixed distribution. We propose a new class of M‐estimators of these parameters based on a Fourier approach and prove that they are ‐consistent under mild regularity conditions. Their finite sample properties are illustrated by a Monte Carlo study, and a benchmark real dataset is also studied with our method.     

Books on probability ideas are reviewed

In this paper, semiparametric methods are applied to estimate multivariate volatility functions, using a residual approach as in [J. Fan and Q. Yao, Efficient estimation of conditional variance functions in stochastic regression, Biometrika 85 (1998), pp. 645–660; F.A. Ziegelmann, Nonparametric estimation of volatility functions: The local exponential estimator, Econometric Theory 18 (2002), pp. 985–991; F.A. Ziegelmann, A local linear least-absolute-deviations estimator of volatility, Comm. Statist. Simulation Comput. 37 (2008), pp. 1543–1564], among others. Our main goal here is two-fold: (1) describe and implement a number of semiparametric models, such as additive, single-index and (adaptive) functional-coefficient, in volatility estimation, all motivated as alternatives to deal with the curse of dimensionality present in fully nonparametric models; and (2) propose the use of a variation of the traditional cross-validation method to deal with model choice in the class of adaptive functional-coefficient models, choosing simultaneously the bandwidth, the number of covariates in the model and also the single-index smoothing variable. The modified cross-validation algorithm is able to tackle the computational burden caused by the model complexity, providing an important tool in semiparametric volatility estimation. We briefly discuss model identifiability when estimating volatility as well as nonnegativity of the resulting estimators. Furthermore, Monte Carlo simulations for several underlying generating models are implemented and applications to real data are provided.     

The Identifiability of Dependent Competing Risks Models Induced by Bivariate Frailty Models

In this paper, we propose to use a special class of bivariate frailty models to study dependent censored data. The proposed models are closely linked to Archimedean copula models. We give sufficient conditions for the identifiability of this type of competing risks models. The proposed conditions are derived based on a property shared by Archimedean copula models and satisfied by several well‐known bivariate frailty models. Compared with the models studied by Heckman and Honoré and Abbring and van den Berg, our models are more restrictive but can be identified with a discrete (even finite) covariate. Under our identifiability conditions, expectation–maximization (EM) algorithm provides us with consistent estimates of the unknown parameters. Simulation studies have shown that our estimation procedure works quite well. We fit a dependent censored leukaemia data set using the Clayton copula model and end our paper with some discussions. © 2014 Board of the Foundation of the Scandinavian Journal of Statistics     

Estimation of the mixtures of GLMs with covariate-dependent mixing proportions

ABSTRACT

In this article, we study the estimation for a class of semiparametric mixtures of generalized linear models where mixing proportions depend on a covariate non parametrically. We investigate a backfitting estimation procedure and show the asymptotic normality of the proposed estimators under mild conditions. We conduct simulation to show the good performance of our methodology and give a real data analysis as an illustration.     

Finite Mixture of Generalized Semiparametric Models: Variable Selection via Penalized Estimation

Selection of the important variables is one of the most important model selection problems in statistical applications. In this article, we address variable selection in finite mixture of generalized semiparametric models. To overcome computational burden, we introduce a class of variable selection procedures for finite mixture of generalized semiparametric models using penalized approach for variable selection. Estimation of nonparametric component will be done via multivariate kernel regression. It is shown that the new method is consistent for variable selection and the performance of proposed method will be assessed via simulation.     

Optimum mixture designs for the log-logistic dose–response model with mixture of two similar compounds

The article studies the log-logistic class of dose–response bioassay models in the binomial set-up. The dose is identified by the potency adjusted mixing proportions of two similar compounds. Models for both absence and presence of interaction between the compounds have been considered. The aim is to investigate the D- and D_s-optimal mixture designs for the estimation of the full set of parameters or for the estimation of potency for a best guess of the parameter values. We also indicate how to find the optimal design to estimate the mixing proportions at which the probability of success attains a given value in the absence of the interaction effect.     

Semiparametric Estimation of a Two-component Mixture Model where One Component is known

Abstract.  We consider a two-component mixture model where one component distribution is known while the mixing proportion and the other component distribution are unknown. These kinds of models were first introduced in biology to study the differences in expression between genes. The various estimation methods proposed till now have all assumed that the unknown distribution belongs to a parametric family. In this paper, we show how this assumption can be relaxed. First, we note that generally the above model is not identifiable, but we show that under moment and symmetry conditions some 'almost everywhere' identifiability results can be obtained. Where such identifiability conditions are fulfilled we propose an estimation method for the unknown parameters which is shown to be strongly consistent under mild conditions. We discuss applications of our method to microarray data analysis and to the training data problem. We compare our method to the parametric approach using simulated data and, finally, we apply our method to real data from microarray experiments.     

Semiparametric Regression with Kernel Error Model

Abstract.  We propose and study a class of regression models, in which the mean function is specified parametrically as in the existing regression methods, but the residual distribution is modelled non-parametrically by a kernel estimator, without imposing any assumption on its distribution. This specification is different from the existing semiparametric regression models. The asymptotic properties of such likelihood and the maximum likelihood estimate (MLE) under this semiparametric model are studied. We show that under some regularity conditions, the MLE under this model is consistent (when compared with the possibly pseudo-consistency of the parameter estimation under the existing parametric regression model), is asymptotically normal with rate and efficient. The non-parametric pseudo-likelihood ratio has the Wilks property as the true likelihood ratio does. Simulated examples are presented to evaluate the accuracy of the proposed semiparametric MLE method.     

Semiparametric score test for varying copula parameter in Markov time series

This article examines a semiparametric test for checking the constancy of serial dependence via copula models for Markov time series. A semiparametric score test is proposed for testing the constancy of the copula parameter against stochastically varying copula parameter. The asymptotic null distribution of the test is established. A semiparametric bootstrap procedure is employed for the estimation of the variance of the proposed score test. Illustrations are given based on simulated series and historic interest rate data.     

Semiparametric median residual life model and inference

For randomly censored data, the authors propose a general class of semiparametric median residual life models. They incorporate covariates in a generalized linear form while leaving the baseline median residual life function completely unspecified. Despite the non‐identifiability of the survival function for a given median residual life function, a simple and natural procedure is proposed to estimate the regression parameters and the baseline median residual life function. The authors derive the asymptotic properties for the estimators, and demonstrate the numerical performance of the proposed method through simulation studies. The median residual life model can be easily generalized to model other quantiles, and the estimation method can also be applied to the mean residual life model. The Canadian Journal of Statistics 38: 665–679; 2010 © 2010 Statistical Society of Canada     

A class of semiparametric transformation models for survival data with a cured proportion

We propose a new class of semiparametric regression models based on a multiplicative frailty assumption with a discrete frailty, which may account for cured subgroup in population. The cure model framework is then recast as a problem with a transformation model. The proposed models can explain a broad range of nonproportional hazards structures along with a cured proportion. An efficient and simple algorithm based on the martingale process is developed to locate the nonparametric maximum likelihood estimator. Unlike existing expectation-maximization based methods, our approach directly maximizes a nonparametric likelihood function, and the calculation of consistent variance estimates is immediate. The proposed method is useful for resolving identifiability features embedded in semiparametric cure models. Simulation studies are presented to demonstrate the finite sample properties of the proposed method. A case study of stage III soft-tissue sarcoma is given as an illustration.     

A (Semi)Parametric Functional Coefficient Logarithmic Autoregressive Conditional Duration Model

In this article, we propose a class of logarithmic autoregressive conditional duration (ACD)-type models that accommodates overdispersion, intermittent dynamics, multiple regimes, and asymmetries in financial durations. In particular, our functional coefficient logarithmic autoregressive conditional duration (FC-LACD) model relies on a smooth-transition autoregressive specification. The motivation lies on the fact that the latter yields a universal approximation if one lets the number of regimes grows without bound. After establishing sufficient conditions for strict stationarity, we address model identifiability as well as the asymptotic properties of the quasi-maximum likelihood (QML) estimator for the FC-LACD model with a fixed number of regimes. In addition, we also discuss how to consistently estimate a semiparametric variant of the FC-LACD model that takes the number of regimes to infinity. An empirical illustration indicates that our functional coefficient model is flexible enough to model IBM price durations.     

Marginal Regression Model with Time-Varying Coefficients for Panel Data

In this article, we formulate a class of semiparametric marginal means models with a mixture of time-varying and time-independent parameters for analyzing panel data. For inference about the regression parameters, an estimation procedure is developed and asymptotic properties of the proposed estimators are established. In addition, some tests are presented for investigating whether or not covariate effects vary with time. The finite-sample behavior of the proposed methods is examined in simulation studies, and the data from an AIDS clinical trial study are used to illustrate the methodology.     

Statistical Inference and Applications of Mixture of Varying Coefficient Models

In this paper, we consider a new mixture of varying coefficient models, in which each mixture component follows a varying coefficient model and the mixing proportions and dispersion parameters are also allowed to be unknown smooth functions. We systematically study the identifiability, estimation and inference for the new mixture model. The proposed new mixture model is rather general, encompassing many mixture models as its special cases such as mixtures of linear regression models, mixtures of generalized linear models, mixtures of partially linear models and mixtures of generalized additive models, some of which are new mixture models by themselves and have not been investigated before. The new mixture of varying coefficient model is shown to be identifiable under mild conditions. We develop a local likelihood procedure and a modified expectation–maximization algorithm for the estimation of the unknown non‐parametric functions. Asymptotic normality is established for the proposed estimator. A generalized likelihood ratio test is further developed for testing whether some of the unknown functions are constants. We derive the asymptotic distribution of the proposed generalized likelihood ratio test statistics and prove that the Wilks phenomenon holds. The proposed methodology is illustrated by Monte Carlo simulations and an analysis of a CO₂‐GDP data set.     

Gamma failure‐time mixture models: yet another way to establish efficacy

Using a Yamaguchi‐type generalized gamma failure‐time mixture model, we analyse the data from a study of autologous and allogeneic bone marrow transplantation in the treatment of high‐risk refractory acute lymphoblastic leukaemia, focusing on the time to recurrence of disease. We develop maximum likelihood techniques for the joint estimation of the surviving fractions and the survivor functions. This includes an approximation to the derivative of the survivor function with respect to the shape parameter. We obtain the maximum likelihood estimates of the model parameters. We also compute the variance‐covariance matrix of the parameter estimators. The extended family of generalized gamma failure‐time mixture models is flexible enough to include many commonly used failure‐time distributions as special cases. Yet these models are not used in practice because of computational difficulties. We claim that we have overcome this problem. The proposed approximation to the derivative of the survivor function with respect to the shape parameter can be used in any statistical package. We also address the issue of lack of identifiability. We point out that there can be a substantial advantage to using the gamma failure‐time mixture models over nonparametric methods. Copyright © 2003 John Wiley & Sons, Ltd.     

Root-n-consistent Estimation in Partial Linear Models with Long-memory Errors

We consider estimation of β in the semiparametric regression model y ( i ) - x ^T( i )β + f ( i / n ) + ε( i ) where x ( i ) = g ( i )/ n ) + e ( i , f and g are unknown smooth functions and the processes ε( i ) and e ( i ) are stationary with short- or long-range dependence. For the case of i.i.d. errors, Speckman (1988) proposed a √ n –consistent estimator of β. In this paper it is shown that, under suitable regularity conditions, this estimator is asymptotically unbiased and √ n –consistent even if the errors exhibit long-range dependence. The orders of the finite sample bias and of the required bandwidth depend on the long-memory parameters. Simulations and a data example illustrate the method     

A Class of Weighted Estimating Equations for Semiparametric Transformation Models with Missing Covariates

In survival analysis, covariate measurements often contain missing observations; ignoring this feature can lead to invalid inference. We propose a class of weighted estimating equations for right‐censored data with missing covariates under semiparametric transformation models. Time‐specific and subject‐specific weights are accommodated in the formulation of the weighted estimating equations. We establish unified results for estimating missingness probabilities that cover both parametric and non‐parametric modelling schemes. To improve estimation efficiency, the weighted estimating equations are augmented by a new set of unbiased estimating equations. The resultant estimator has the so‐called ‘double robustness’ property and is optimal within a class of consistent estimators.     

An asymptotic theory for semiparametric generalized least squares estimation in partially linear regression models

Consider a partially linear regression model with an unknown vector parameter β, an unknown functiong(·), and unknown heteroscedastic error variances. In this paper we develop an asymptotic semiparametric generalized least squares estimation theory under some weak moment conditions. These moment conditions are satisfied by many of the error distributions encountered in practice, and our theory does not require the number of replications to go to infinity.