Showing papers on "Semiparametric model published in 1996"

Efficient estimation for the proportional hazards model with interval censoring

Abstract: The maximum likelihood estimator (MLE) for the proportional hazards model with "case 1" interval censored data is studied. It is shown that the MLE for the regression parameter is asymptotically normal with $\sqrt{n}$ convergence rate and achieves the information bound, even though the MLE for the baseline cumulative hazard function only converges at $n^{1/3}$ rate. Estimation of the asymptotic variance matrix for the MLE of the regression parameter is also considered. To prove our main results, we also establish a general theorem showing that the MLE of the finite-dimensional parameter in a class of semiparametric models is asymptotically efficient even though the MLE of the infinite-dimensional parameter converges at a rate slower than $\sqrt{n}$. The results are illustrated by applying them to a data set from a tumorigenicity study.

Nonparametric and semiparametric estimation of the receiver operating characteristic curve

Abstract: curve (ODC) given by F(G-'(t)). First we consider nonparametric estimators based on empirical distribution functions and derive asymptotic properties. Next we consider the so-called semiparametric "binormal" model, in which it is assumed that the distributions F and G are normal after some unknown monotonic transformation of the measurement scale. For this model, we propose a generalized least squares procedure and compare it with the estimation algorithm of Dorfman and Alf, which is based on grouped data. Asymptotic results are obtained; small sample properties are examined via a simulation study. Finally, we describe a minimum distance estimator for the ROC curve, which does not require grouping the data.

Locally parametric nonparametric density estimation

Abstract: This paper develops a nonparametric density estimator with parametric overtones. Suppose f(x, θ) is some family of densities, indexed by a vector of parameters θ. We define a local kernel-smoothed likelihood function which, for each x, can be used to estimate the best local parametric approximant to the true density. This leads to a new density estimator of the form f(x, o(x)), thus inserting the best local parameter estimate for each new value of x. When the bandwidth used is large, this amounts to ordinary full likelihood parametric density estimation, while for moderate and small bandwidths the method is essentially nonparametric, using only local properties of data and the model. Alternative ways more general than via the local likelihood are also described. The methods can be seen as ways of nonparametrically smoothing the parameter within a parametric class. Properties of this new semiparametric estimator are investigated. Our preferred version has approximately the same variance as the ordinary kernel method but potentially a smaller bias. The new method is seen to perform better than the traditional kernel method in a broad nonparametric vicinity of the parametric model employed, while at the same time being capable of not losing much in precision to full likelihood methods when the model is correct. Other versions of the method are approximately equivalent to using particular higher order kernels in a semiparametric framework. The methodology we develop can be seen as the density estimation parallel to local likelihood and local weighted least squares theory in nonparametric regression.

Semiparametric estimation of a hedonic price function

Abstract: SUMMARY Previous work on the preferred specification of hedonic price models usually recommended a Box-Cox model. In this paper we note that any parametric model involves implicit restrictions and they can be reduced by using a semiparametric model. We estimate a benchmark parametric model which passes several common specification tests, before showing that a semiparametric model outperforms it significantly. In addition to estimating the model, we compare the predictions of the models by deriving the distribution of the predicted log(price) and then calculating the associated prediction intervals. Our data show that the semiparametric model provides more accurate mean predictions than the benchmark parametric model.

Bandwidth Choice in Gaussian Semiparametric Estimation of Long Range Dependence

Abstract: We consider covariance stationary processes with spectral density which behaves according to a power law around zero frequency, where it can be infinite (long range dependence), finite and positive (short range dependence), or zero (antipersistence). This behaviour is governed by a self-similarity parameter which can be estimated semiparametrically by one of several methods, all of which require a choice of bandwidth. We consider a Gaussian estimate which seems likely to have good efficiency, and whose asymptotic distributional properties have already been determined. The minimum mean squared error optimal bandwidth is heuristically derived and feasible approximations to it are proposed, these being assessed in Monte Carlo experiments and applied to financial and Nile river data.

A Semiparametric Proportional Odds Regression Model for the Analysis of Current Status Data

Abstract: An approach to covariate analysis of current status data is described The method is applicable when the logit of the conditional probability of survival given the covariates is some increasing function of time plus a linear combination of the covariates The approach is based on approximating the infinite-dimensional nuisance parameter, the baseline log-odds of failure, with a step function, and carrying out a maximum likelihood procedure The resulting finite dimensional parameter estimates for the regression parameters are shown to be asymptotically normal and semiparametric efficient Numerical studies with small and moderate samples are discussed

Parametric and semi-parametric modelling of vacation expenditures

Abstract: We analyse several limited dependent variable models explaining the budget share that Dutch families spend on vacations. To take account of the substantial number of zero shares, two types of models are used. The first is the single-equation censored regression model. We estimate and test several parametric and semi-parametric extensions of the Tobit model. Second, we consider two-equation models, in which the participation decision and the decision on the amount to spend are treated separately. The first decision is modelled as a binary choice model; the second as a conditional regression. We estimate and test parametric and semi-parametric specifications.

A semi-parametric additive model for variance heterogeneity

Abstract: This paper presents a flexible model for variance heterogeneity in a normal error model. Specifically, both the mean and variance are modelled using semi-parametric additive models. We call this model a ‘Mean And Dispersion Additive Model’ (MADAM). A successive relaxation algorithm for fitting the model is described and justified as maximizing a penalized likelihood function with penalties for lack of smoothness in the additive non-parametric functions in both mean and variance models. The algorithm is implemented in GLIM4, allowing flexible and interactive modelling of variance heterogeneity. Two data sets are used for demonstration.

Bayesian Inference for Semiparametric Binary Regression

Abstract: We propose a regression model for binary response data that places no structural restrictions on the link function except monotonicity and known location and scale. Predictors enter linearly. We demonstrate Bayesian inference calculations in this model. By modifying the Dirichlet process, we obtain a natural prior measure over this semiparametric model, and we use Polya sequence theory to formulate this measure in terms of a finite number of unobserved variables. We design a Markov chain Monte Carlo algorithm for posterior simulation and apply the methodology to data on radiotherapy treatments for cancer.

Nonparametric Methods for Quantitative Analysis

Abstract: (1999). Nonparametric Methods for Quantitative Analysis. Technometrics: Vol. 41, No. 1, pp. 75-75.

Parametric and semi‐parametric estimation of the binary response model of labour market participation

Abstract: This paper compares the familiar probit model with three semiparametric estimators of binary response models in an application to labour market participation of married women. This exercise is performed using two different cross-section data sets from Switzerland and Germany. For the Swiss data the probit specification cannot be rejected and the models yield similar results. In the German case the probit model is rejected, but the coefficient estimates do not vary substantially across the models. The predicted choice probabilities, however, differ systematically for a subset of the sample. The results of this paper indicate that more work is necessary on specification tests of semiparametric models and on simulations using these models.

Asymptotically uniformly most powerful tests in parametric and semiparametric models

Abstract: Tests of hypotheses about finite-dimensional parameters in a semiparametric model are studied from Pitman's moving alternative (or local) approach using Le Cam's local asymptotic normality concept. For the case of a real parameter being tested, asymptotically uniformly most powerful (AUMP) tests are characterized for one-sided hypotheses, and AUMP unbiased tests for two-sided ones. An asymptotic invariance principle is introduced for multidimensional hypotheses, and AUMP invariant tests are characterized. These provide optimality for Wald, Rao (score), Neyman-Rao (effective score) and likelihood ratio tests in parametric models, and for Neyman-Rao tests in semiparametric models when constructions are feasible. Inversions lead to asymptotically uniformly most accurate confidence sets. Examples include one-, two- and k-sample problems, a linear regression model with unknown error distribution and a proportional hazards regression model with arbitrary baseline hazards. Results are presented in a format that facilitates application in strictly parametric models.

Efficient maximum likelihood estimation in semiparametric mixture models

Abstract: We consider maximum likelihood estimation in several examples of semiparametric mixture models, including the exponential frailty model and the errors-in-variables model. The observations consist of a sample of size n from the mixture density $\int p_{\theta}(x|z) d \eta(z)$. The mixing distribution is completely unknown. We show that the first component $\hat{\theta}_n$ of the joint maximum likelihood estimator , $(\hat{\theta}_n \hat{\eta}_n)$ is asymptotically normal and asymptotically efficient in the semiparametric sense.

A semiparametric extension of the Mann-Whitney test for randomly truncated data.

Abstract: In many applications, statistical data are frequently observed subject to a retrospective sampling criterion resulting in pure right-truncated data. In classical testing problems, the Mann-Whitney test is used for testing the equality of two distributions. A semiparametric extension of this test is developed for the case when truncation is present. We consider a model in which the truncation distribution is parameterized, while the lifetime distribution is left as a nonparametric component. The method is seen to be applicable to many patterns of truncation including left truncation, right truncation, and doubly truncated data for which no other nonparametric or semiparametric test is currently available. Applications of the semiparametric method are given. Simulation results indicate that for pure right-truncated data the semiparametric test is more powerful than a recent nonparametric test.

NONPARAMETRIC ESTIMATION OF Low-FLOW FREQUENCIES

Abstract: A new nonparametric method is proposed for low-flow frequency analysis. Low-flow series often contain years with zero values. The occurrence of zero and nonzero events are treated by first estimating the probability of zero flows using a plotting formula, and then fitting the nonzero events by nonparametric method. A Monte Carlo–simulation experiment is performed to compare the proposed nonparametric method with two parametric distributions, namely, log-Pearson Type III and Weibull. Results show that the nonparametric method gives more accurate results than currently used parametric methods. It is also concluded that the nonparametric method is uniform because it does not require a priori specification of any parametric distribution.

Locally Efficient Median Regression with Random Censoring and Surrogate Markers

Abstract: Robins and Rotnitzky (1992) proved a general representation theorem for (1) the efficient score and (2) the set of influence functions for regular asymptotically linear (RAL) estimators in arbitrary semiparametric models with (i) the data missing or coarsened at random, and (ii) the probability of observing complete data bounded away from zero. We use this representation theorem to construct locally efficient estimators (at a parametric submodel) in a censored median regression model where the hazard of censoring at u (i) may depend on both the regressors and on the history up to u of a surrogate process of prognostic factors, but (ii) does not further depend on the possibly unobserved failure time. Our model incorporates both the Ying et al. (1994) random censoring model and the Newey and Powell (1990) observed potential censoring time model as special cases.

Edgeworth Approximation for MINPIN Estimators in Semiparametric Regression Models

Abstract: We examine the higher order asymptotic properties of semiparametric regression estimators that were obtained by the general MINPIN method described in Andrews (1989, Semiparametric Econometric Models: I. Estimation, Discussion paper 908, Cowles Foundation). We derive an order n−1 stochastic expansion and give a theorem justifying order n−1 distributional approximation of the Edgeworth type.

Nonparametric Statistics: Methods for Analyzing Data Not Meeting Assumptions Required for the Application of Parametric Tests

Abstract: This article discusses the basic characteristics of nonparametric statistical tests, contrasting them with the characteristics of parametric statistical tests. Examples for performing nonparametric statistical tests on practitioners' own data also are included.Nonparametric tests can be used

Regression rank scores estimation in ANOCOVA

Abstract: In semiparametric ANOCOVA (mixed-effects) models, the role of regression rank scores in robust estimation of fixed-effects parameters as well as covariate regression functionals is critically appraised, and the relevant asymptotic theory is presented.

Uniform rates of estimation in the semiparametric Weibull mixture model

Abstract: This paper presents a uniform estimator for a finite-dimensional parameter in the semiparametric Weibull mixture model. The rates achieved by the estimator hold uniformly over shrinking sequences of models much more general than traditional sequences that are required to satisfy a Hellinger differentiable property. We show that these rates are optimal in a class of identified models constrained by a moment condition on the nonparametric mixing distribution.

Testing Parametric versus Semiparametric Modelling in Generalized Linear Models

Abstract: We consider a generalized partially linear model E(Y|X,T) = G{X'b + m(T)} where G is a known function, b is an unknown parameter vector, and m is an unknown function. The paper introduces a test statistic which allows to decide between a parametric and a semiparametric model: (i) m is linear, i.e.m(t) = t'g for a parameter vector g, (ii) m is a smooth (nonlinear) function. Under linearity (i) it is shown that the test statistic is asymptotically normal. Moreover, for the case of binary responses, it is proved that the bootstrap works asymptotically. Simulations suggest that (in small samples) bootstrap outperforms the calculation of critical values from the normal approximation. The practical performance of the test is shown in applications to data on East--West German migration and credit scoring.