Finite-dimensional Parameter Research Articles

On semiparametric transformation model with LTRC data: pseudo likelihood approach

When the distribution of the truncation time is known up to a finite-dimensional parameter vector, many researches have been conducted with the objective to improve the efficiency of estimation for nonparametric or semiparametric model with left-truncated and right-censored (LTRC) data. When the distribution of truncation times is unspecified, one approach is to use the conditional maximum likelihood estimators (cMLE) (Chen and Shen in Lifetime Data Anal https://doi.org/10.1007/s10985-016-9385-9, 2017). Although the cMLE has nice asymptotic properties, it is not efficient since the conditional likelihood function does not incorporate information on the distribution of truncation time. In this article, we aim to develop a more efficient estimator by considering the full likelihood function. Following Turnbull (J R Stat Soc B 38:290–295, 1976) and Qin et al. (J Am Stat Assoc 106:1434–1449, 2011), we treat the unobserved (left-truncated) subpopulation as missing data and propose a two-stage approach for obtaining the pseudo maximum likelihood estimators (PMLE) of regression parameters. In the first stage, the distribution of left truncation time is estimated by the inverse-probability-weighted (IPW) estimator (Wang in J Am Stat Assoc 86:130–143, 1991). In the second stage, we obtain the pseudo complete-data likelihood function by replacing the distribution of truncation time with the IPW estimator in the full likelihood. We propose an expectation–maximization algorithm for obtaining the PMLE and establish the consistency of the PMLE. Simulation results show that the PMLE outperforms the cMLE in terms of mean squared error. The PMLE can also be used to analyze the length-biased data, where the truncation time is uniformly distributed. We demonstrate that the PMLE works more robust against the support assumption of truncation time for length-biased data compared with the MLE proposed by Qin et al. (2011). We apply our proposed method to the channing house data. While the PMLE is quite appealing under specific cases with independent censoring and time-invariant covariates, its applicability, as shown in simulation study, can be rather restricted for more general settings.

The semi-parametric Bernstein-von Mises theorem for regression models with symmetric errors

In a smooth semi-parametric model, the marginal posterior distribution for a finite dimensional parameter of interest is expected to be asymptotically equivalent to the sampling distribution of any efficient point-estimator. The assertion leads to asymptotic equivalence of credible and confidence sets for the parameter of interest and is known as the semi-parametric Bernstein-von Mises theorem. In recent years, it has received much attention and has been applied in many examples. We consider models in which errors with symmetric densities play a role; more specifically, it is shown that the marginal posterior distributions of regression coefficients in the linear regression and linear mixed effect models satisfy the semi-parametric Bernstein-von Mises assertion. As a consequence, Bayes estimators in these models achieve frequentist inferential optimality, as expressed e.g. through Hajek's convolution and asymptotic minimax theorems. Conditions for the prior on the space of error densities are relatively mild and well-known constructions like the Dirichlet process mixture of normal densities and random series priors constitute valid choices. Particularly, the result provides an efficient estimate of regression coefficients in the linear mixed effect model, for which no other efficient point-estimator was known previously.

FPCA-based estimation for generalized functional partially linear models

In real data analysis, practitioners frequently come across the case that a discrete response will be related to both a function-valued random variable and a vector-value random variable as the predictor variables. In this paper, we consider the generalized functional partially linear models (GFPLM). The infinite slope function in the GFPLM is estimated by the principal component basis function approximations. Then, we consider the theoretical properties of the estimator obtained by maximizing the quasi likelihood function. The asymptotic normality of the estimator of the finite dimensional parameter and the rate of convergence of the estimator of the infinite dimensional slope function are established, respectively. We investigate the finite sample properties of the estimation procedure via Monte Carlo simulation studies and a real data analysis.

Three-stage semi-parametric inference: Control variables and differentiability

We show the usefulness of the path-derivative calculations that were introduced in econometrics by Newey (1994) for multi-step semi-parametric estimators. These estimators estimate a finite-dimensional parameter using moment conditions that depend on nonparametric regressions on observed and estimated regressors that are estimated in the second and first step of the estimation procedure, respectively. Our earlier paper showed that Newey’s calculations can be extended to three-step estimators. In the current paper we consider the control variable (CV) estimator and related statistics in semi-parametric econometric models with non-separable errors and regressors that are correlated with these errors. Non-separable econometric models with endogenous regressors are often identified by average moment restrictions that average over control variables, and these control variables are estimated in a first stage by (non)parametric regression. We study aspects of inference for such estimators where we focus on a finite-dimensional parameter vector or statistic. The asymptotic distribution and a closed-form expression for the asymptotic variance of the CV estimator were not available until now. Our path derivative calculations are much simpler than the derivation of the asymptotic distribution by a stochastic expansion that is particularly complicated for multi-step semi-parametric estimators. We also consider just- and overidentification of the parameters and we propose a diagnostic test for overidentifying restrictions in models with non-separable errors and endogenous regressors. Finally, the path-derivative calculation breaks down if the moment condition is not differentiable. In an example we show that non-differentiability is associated with irregular behavior of the estimator.

Full families of generalized interval exchange transformations

We consider generalized interval exchange transformations, or briefly GIETs, that is bijections of the interval which are piecewise increasing homeomorphisms with finitely many branches. When all continuous branches are translations, such maps are classical interval exchange transformations, or briefly IETs. The well-known Rauzy renormalization procedure extends to a given GIET and a Rauzy renormalization path is defined, provided that the map is infinitely renormalizable. We define full families of GIETs, that is optimal finite dimensional parameter families of GIETs such that any prescribed Rauzy renormalization path is realized by some map in the family. In particular, a GIET and a IET with the same Rauzy renormalization path are semi-conjugated. This extends a classical result of Poincaré relating circle homeomorphisms and irrational rotations.

PROPERTIES OF DOUBLY ROBUST ESTIMATORS WHEN NUISANCE FUNCTIONS ARE ESTIMATED NONPARAMETRICALLY

An estimator of a finite-dimensional parameter is said to be doubly robust (DR) if it imposes parametric specifications on two unknown nuisance functions, but only requires that one of these two specifications is correct in order for the estimator to be consistent for the object of interest. In this article, we study versions of such estimators that use local polynomial smoothing for estimating the nuisance functions. We show that such semiparametric two-step (STS) versions of DR estimators have favorable theoretical and practical properties relative to other commonly used STS estimators. We also show that these gains are not generated by the DR property alone. Instead, it needs to be combined with an orthogonality condition on the estimation residuals from the nonparametric first stage, which we show to be satisfied in a wide range of models.

Current status linear regression

We construct $\sqrt{n}$-consistent and asymptotically normal estimates for the finite dimensional regression parameter in the current status linear regression model, which do not require any smoothing device and are based on maximum likelihood estimates (MLEs) of the infinite dimensional parameter. We also construct estimates, again only based on these MLEs, which are arbitrarily close to efficient estimates, if the generalized Fisher information is finite. This type of efficiency is also derived under minimal conditions for estimates based on smooth nonmonotone plug-in estimates of the distribution function. Algorithms for computing the estimates and for selecting the bandwidth of the smooth estimates with a bootstrap method are provided. The connection with results in the econometric literature is also pointed out.

Method of Moments Estimators and Multi–step MLE for Poisson Processes

We introduce two types of estimators of the finite–dimensional parameters in the case of observations of inhomogeneous Poisson processes. These are the estimators of the method of moments and Multi–step MLE. It is shown that the estimators of the method of moments are consistent and asymptotically normal and the Multi–step MLE are consistent and asymptotically efficient. The construction of Multi–step MLE–process is done in two steps. First we construct a consistent estimator by the observations on some learning interval and then this estimator is used for construction of One–step and Two–step MLEs. The main advantage of the proposed approach is its computational simplicity.

Representation of Hashin–Shtrikman Bounds in Terms of Texture Coefficients for Arbitrarily Anisotropic Polycrystalline Materials

The present work generalizes the results of Bohlke and Lobos (Acta Mater. 67:324–334, 2014) by giving an explicit representation of the Hashin–Shtrikman (HS) bounds of linear elastic properties in terms of tensorial Fourier texture coefficients not only for cubic materials but for arbitrarily anisotropic linear elastic polycrystalline materials. Based on the HS bounds as given by Walpole (J. Mech. Phys. Solids 14(3):151–162, 1966) and tensor functions for the representation of the crystallite orientation distribution function, it is shown that the HS bounds are represented in terms of the exact same second- and fourth-order texture coefficients which appear in the consideration of the Voigt and Reuss bounds. The derived representations in terms of tensorial texture coefficients are valid for all symmetry classes in elasticity and present expressions highly attractive for the description of physical quantities in terms of tensorial variables. In order to make these results also accessible for the community of quantitative texture analysis, transformation relations between experimentally obtained Bunge’s or Roe’s coefficients and the tensorial texture coefficients are given. The representation of the present work offers a finite and low dimensional parametrization of the fully anisotropic Hashin–Shtrikman bounds, which can be used in inverse materials design problems in order to explore the set of possible materials properties or for the determination of optimal microstructural influence with respect to prescribed material properties. Examples for orthotropic polycrystals of cubic materials and transversely isotropic polycrystals of hexagonal materials (showing the connection and applicability of the results also to fiber orientation distributions) are discussed. Finally, an implementation in Mathematica® 11 of the HS bounds for arbitrarily anisotropic materials is offered, such that readers can reproduce all the results of this work and use them for their own purposes.

On the choice of test statistic for conditional moment inequalities

This paper derives asymptotic approximations to the power of Cramer–von Mises (CvM) style tests for inference on a finite dimensional parameter defined by conditional moment inequalities in the case where the parameter is set identified. Combined with power results for Kolmogorov–Smirnov (KS) tests, these results can be used to choose the optimal test statistic, weighting function and, for tests based on kernel estimates, kernel bandwidth. The results show that, in the setting considered here, KS tests are preferred to CvM tests, and that a truncated variance weighting is preferred to bounded weightings.

On penalized estimation for dynamical systems with small noise

We consider a dynamical system with small noise for which the drift is parametrized by a finite dimensional parameter. For this model, we consider minimum distance estimation from continuous time observations under $l^{p}$-penalty imposed on the parameters in the spirit of the Lasso approach, with the aim of simultaneous estimation and model selection. We study the consistency and the asymptotic distribution of these Lasso-type estimators for different values of $p$. For $p=1,$ we also consider the adaptive version of the Lasso estimator and establish its oracle properties.

Superharmonic priors for autoregressive models

Tanaka and Komaki (Sankhya Ser A Indian Stat Inst 73-A:162–184, 2011) proposed superharmonic priors in Bayesian time series analysis as alternative to the famous Jeffreys prior. By definition the existence of superharmonic priors on a specific time series model with finite-dimensional parameter is equivalent to that of positive nonconstant superharmonic functions on the corresponding Riemannian manifold endowed with the Fisher metric. In the autoregressive models, whose Fisher metric and its inverse have quite messy forms, we obtain superharmonic priors in an explicit manner. To derive this result, we developed a systematic way of dealing with symmetric polynomials, which are related to Schur functions.

FLCRM: Functional linear cox regression model.

We consider a functional linear Cox regression model for characterizing the association between time-to-event data and a set of functional and scalar predictors. The functional linear Cox regression model incorporates a functional principal component analysis for modeling the functional predictors and a high-dimensional Cox regression model to characterize the joint effects of both functional and scalar predictors on the time-to-event data. We develop an algorithm to calculate the maximum approximate partial likelihood estimates of unknown finite and infinite dimensional parameters. We also systematically investigate the rate of convergence of the maximum approximate partial likelihood estimates and a score test statistic for testing the nullity of the slope function associated with the functional predictors. We demonstrate our estimation and testing procedures by using simulations and the analysis of the Alzheimer's Disease Neuroimaging Initiative (ADNI) data. Our real data analyses show that high-dimensional hippocampus surface data may be an important marker for predicting time to conversion to Alzheimer's disease. Data used in the preparation of this article were obtained from the ADNI database (adni.loni.usc.edu).

Maximum likelihood estimation for semiparametric regression models with multivariate interval-censored data.

SummaryInterval-censored multivariate failure time data arise when there are multiple types of failure or there is clustering of study subjects and each failure time is known only to lie in a certain interval. We investigate the effects of possibly time-dependent covariates on multivariate failure times by considering a broad class of semiparametric transformation models with random effects, and we study nonparametric maximum likelihood estimation under general interval-censoring schemes. We show that the proposed estimators for the finite-dimensional parameters are consistent and asymptotically normal, with a limiting covariance matrix that attains the semiparametric efficiency bound and can be consistently estimated through profile likelihood. In addition, we develop an EM algorithm that converges stably for arbitrary datasets. Finally, we assess the performance of the proposed methods in extensive simulation studies and illustrate their application using data derived from the Atherosclerosis Risk in Communities Study.

Multiple robustness in factorized likelihood models.

SummaryWe consider inference under a nonparametric or semiparametric model with likelihood that factorizes as the product of two or more variation-independent factors. We are interested in a finite-dimensional parameter that depends on only one of the likelihood factors and whose estimation requires the auxiliary estimation of one or several nuisance functions. We investigate general structures conducive to the construction of so-called multiply robust estimating functions, whose computation requires postulating several dimension-reducing models but which have mean zero at the true parameter value provided one of these models is correct.

A GENERAL CLASS OF NON-NESTED TEST STATISTICS FOR MODELS DEFINED THROUGH MOMENT RESTRICTIONS

In this article, we introduce a new class of Cox’s non-nested test statistics for models defined through overidentifying moment restrictions that depend on a finite dimensional parameter vector. In addition to showing that the GEL test statistics proposed in Smith (1997,Economic Journal107, 503–519) and Ramalho and Smith (2002,Journal of Econometrics107, 99–125) are members of this class, we reveal that further members can be constructed using the artificial compound model approach, which was originally applied by Atkinson (1970,Journal of the Royal Statistical Society, Series B32, 323–335) in the parametric setting. We investigate the asymptotic properties of the statistics and propose tests based on modified versions of these statistics that have correct asymptotic size in a uniform sense, a requirement not satisfied by existing Cox’s non-nested tests. A Monte Carlo study examines the performance of the proposed tests.

Interdependent Durations with Incomplete Information

In this paper, we propose an interdependent durations model with incomplete information structure for multiple players. Our model generates a system of simultaneous equations of the conditional marginal distributions for duration outcomes, and implies that the corresponding copula function is the copula of the unobserved heterogeneity terms. We establish the uniqueness of the Bayesian Nash Equilibrium for the model resorting to a moderate social influence condition. We obtain identification of the finite dimensional parameters capturing the interaction effects and all nonparametric model primitives from the simultaneous equations of conditional marginal distributions and the dependence among unobserved heterogeneity through the conditional joint distribution of duration outcomes. Thus, the direct strategic interaction effects and the selection effects due to unobserved heterogeneity are separately identified.

SEMIPARAMETRIC EFFICIENCY FOR CENSORED LINEAR REGRESSION MODELS WITH HETEROSKEDASTIC ERRORS

Using a simplified approach developed by Severini and Tripathi (2001), we calculate the semiparametric efficiency bound for the finite-dimensional parameters of censored linear regression models with heteroskedastic errors. Under an additional identification at infinity type assumption, we propose an efficient estimator based on a novel result from Lewbel and Linton (2002). An extension to censored partially linear single-index models is also presented.

Efficient estimation in models with independence restrictions

Unconditional and conditional independence restrictions are used in many econometric models to identify their parameters. However, there are few results about efficient estimation procedures for finite-dimensional parameters under these independence restrictions. This paper computes the efficiency bound for finite-dimensional parameters under independence restrictions, and proposes an estimator that is consistent, asymptotically normal and which achieves the efficiency bound. The estimator is based on a growing number of zero-covariance conditions that are asymptotically equivalent to the independence restriction. The results are illustrated with examples, including an instrumental variables regression model and partially linear regression models. A small Monte Carlo study is performed to investigate the estimator’s small sample properties and to quantify the efficiency gains that can be made by using the proposed efficient estimator.

FEM-based discretization-invariant MCMC methods for PDE-constrained Bayesian inverse problems

We present a systematic construction of FEM-based dimension-independent (discretization-invariant) Markov chain Monte Carlo (MCMC) approaches to explore PDE-constrained Bayesian inverse problems in infinite dimensional parameter spaces. In particular, we consider two frameworks to achieve this goal: Metropolize-then-discretize and discretize-then-Metropolize. The former refers to the method of discretizing function-space MCMC methods. The latter, on the other hand, first discretizes the Bayesian inverse problem and then proposes MCMC methods for the resulting discretized posterior probability density. In general, these two frameworks do not commute, that is, the resulting finite dimensional MCMC algorithms are not identical. The discretization step of the former may not be trivial since it involves both numerical analysis and probability theory, while the latter, perhaps ``easier'', may not be discretization-invariant using traditional approaches. This paper constructively develops finite element (FEM) discretization schemes for both frameworks and shows that both commutativity and discretization-invariance are attained. In particular, it shows how to construct discretize-then-Metropolize approaches for both Metropolis-adjusted Langevin algorithm and the hybrid Monte Carlo method that commute with their Metropolize-then-discretize counterparts. The key that enables this achievement is a proper FEM discretization of the prior, the likelihood, and the Bayes' formula, together with a correct definition of quantities such as the gradient and the covariance matrix in discretized finite dimensional parameter spaces. The implication is that practitioners can take advantage of the developments in this paper to straightforwardly construct discretization-invariant discretize-then-Metropolize MCMC for large-scale inverse problems. Numerical results for one- and two-dimensional elliptic inverse problems with up to $17899$ parameters are presented to support the proposed developments.