Comparison of renewal function estimators for censored data

The computation of the renewal function when the distribution function is completely known has received much attention in the literature. However, in many cases the form of the distribution function is unknown and has to be estimated nonparametrically. A nonparametric estimator for the renewal function for complete data was suggested by Frees (1986). In many cases, however, censoring of the lifetime might occur. We shall present parametric and nonparametric estimators of the renewal function based on censored data. In a simulation study we compare the nonparametric estimators with parametric estimators for the Weibull and lognormal distribution. The study suggests that the nonparametric estimator is a viable alternative to the parametric estimators when the lifetime distribution is unknown. Also, the nonparametric estimator is computationally simpler than the parametric estimator.     

Statistic inference for a single-index ARCH-M model

For a single-index autoregressive conditional heteroscedastic (ARCH-M) model, estimators of the parametric and non parametric components are proposed by the profile likelihood method. The research results had shown that all the estimators have consistency and the parametric estimators have asymptotic normality. We extend this line of research by deriving the asymptotic normality of the non parametric estimator. Based on the asymptotic properties, we propose Wald statistic and generalized likelihood ratio statistic to investigate the testing problems for ARCH effect and goodness of fit, respectively. A simulation study is conducted to evaluate the finite-sample performance of the proposed estimation methodology and testing procedure.     

Consistency of non parametric estimators of the area under the ROC curve

ABSTRACT

The area under the receiver operating characteristic (ROC) curve is a popular summary index that measures the accuracy of a continuous-scale diagnostic test to measure its accuracy. Under certain conditions on estimators of distribution functions, we prove a theorem on strong consistency of the non parametric “plugin” estimators of the area under the ROC curve. Based on this theorem, we construct some new “plugin” consistent estimators. The performance of the non parametric estimators considered is illustrated numerically and the estimators are compared in terms of bias, variance, and mean square error.     

Finite Sample Comparison of Parametric,Semiparametric, and Wavelet Estimators of Fractional Integration

ABSTRACT

In this paper we compare through Monte Carlo simulations the finite sample properties of estimators of the fractional differencing parameter, d. This involves frequency domain, time domain, and wavelet based approaches, and we consider both parametric and semiparametric estimation methods. The estimators are briefly introduced and compared, and the criteria adopted for measuring finite sample performance are bias and root mean squared error. Most importantly, the simulations reveal that (1) the frequency domain maximum likelihood procedure is superior to the time domain parametric methods, (2) all the estimators are fairly robust to conditionally heteroscedastic errors, (3) the local polynomial Whittle and bias-reduced log-periodogram regression estimators are shown to be more robust to short-run dynamics than other semiparametric (frequency domain and wavelet) estimators and in some cases even outperform the time domain parametric methods, and (4) without sufficient trimming of scales the wavelet-based estimators are heavily biased.     

Generalised quasi‐likelihood inference in a semi‐parametric binary dynamic mixed logit model

There exists a recent study where dynamic mixed‐effects regression models for count data have been extended to a semi‐parametric context. However, when one deals with other discrete data such as binary responses, the results based on count data models are not directly applicable. In this paper, we therefore begin with existing binary dynamic mixed models and generalise them to the semi‐parametric context. For inference, we use a new semi‐parametric conditional quasi‐likelihood (SCQL) approach for the estimation of the non‐parametric function involved in the semi‐parametric model, and a semi‐parametric generalised quasi‐likelihood (SGQL) approach for the estimation of the main regression, dynamic dependence and random effects variance parameters. A semi‐parametric maximum likelihood (SML) approach is also used as a comparison to the SGQL approach. The properties of the estimators are examined both asymptotically and empirically. More specifically, the consistency of the estimators is established and finite sample performances of the estimators are examined through an intensive simulation study.     

Difference-based estimation and model identification for panel data semiparametric models with cross-section dependence

Abstract

In this article, we consider a panel data partially linear regression model with fixed effect and non parametric time trend function. The data can be dependent cross individuals through linear regressor and error components. Unlike the methods using non parametric smoothing technique, a difference-based method is proposed to estimate linear regression coefficients of the model to avoid bandwidth selection. Here the difference technique is employed to eliminate the non parametric function effect, not the fixed effects, on linear regressor coefficient estimation totally. Therefore, a more efficient estimator for parametric part is anticipated, which is shown to be true by the simulation results. For the non parametric component, the polynomial spline technique is implemented. The asymptotic properties of estimators for parametric and non parametric parts are presented. We also show how to select informative ones from a number of covariates in the linear part by using smoothly clipped absolute deviation-penalized estimators on a difference-based least-squares objective function, and the resulting estimators perform asymptotically as well as the oracle procedure in terms of selecting the correct model.     

Finite Sample Comparison of Parametric, Semiparametric, and Wavelet Estimators of Fractional Integration

In this paper we compare through Monte Carlo simulations the finite sample properties of estimators of the fractional differencing parameter, d. This involves frequency domain, time domain, and wavelet based approaches, and we consider both parametric and semiparametric estimation methods. The estimators are briefly introduced and compared, and the criteria adopted for measuring finite sample performance are bias and root mean squared error. Most importantly, the simulations reveal that (1) the frequency domain maximum likelihood procedure is superior to the time domain parametric methods, (2) all the estimators are fairly robust to conditionally heteroscedastic errors, (3) the local polynomial Whittle and bias-reduced log-periodogram regression estimators are shown to be more robust to short-run dynamics than other semiparametric (frequency domain and wavelet) estimators and in some cases even outperform the time domain parametric methods, and (4) without sufficient trimming of scales the wavelet-based estimators are heavily biased.     

Non‐parametric Copula Estimation Under Bivariate Censoring

In this paper, we consider non‐parametric copula inference under bivariate censoring. Based on an estimator of the joint cumulative distribution function, we define a discrete and two smooth estimators of the copula. The construction that we propose is valid for a large range of estimators of the distribution function and therefore for a large range of bivariate censoring frameworks. Under some conditions on the tails of the distributions, the weak convergence of the corresponding copula processes is obtained in l^∞([0,1]²). We derive the uniform convergence rates of the copula density estimators deduced from our smooth copula estimators. Investigation of the practical behaviour of these estimators is performed through a simulation study and two real data applications, corresponding to different censoring settings. We use our non‐parametric estimators to define a goodness‐of‐fit procedure for parametric copula models. A new bootstrap scheme is proposed to compute the critical values.     

Semiparametric multiple kernel estimators and model diagnostics for count regression functions

Abstract

This study concerns semiparametric approaches to estimate discrete multivariate count regression functions. The semiparametric approaches investigated consist of combining discrete multivariate nonparametric kernel and parametric estimations such that (i) a prior knowledge of the conditional distribution of model response may be incorporated and (ii) the bias of the traditional nonparametric kernel regression estimator of Nadaraya-Watson may be reduced. We are precisely interested in combination of the two estimations approaches with some asymptotic properties of the resulting estimators. Asymptotic normality results were showed for nonparametric correction terms of parametric start function of the estimators. The performance of discrete semiparametric multivariate kernel estimators studied is illustrated using simulations and real count data. In addition, diagnostic checks are performed to test the adequacy of the parametric start model to the true discrete regression model. Finally, using discrete semiparametric multivariate kernel estimators provides a bias reduction when the parametric multivariate regression model used as start regression function belongs to a neighborhood of the true regression model.     

Estimation of the Entropy Rate of a Countable Markov Chain

We consider here ergodic homogeneous Markov chains with countable state spaces. The entropy rate of the chain is an explicit function of its transition and stationary distributions. We construct estimators for this entropy rate and for the entropy of the stationary distribution of the chain, in the parametric and nonparametric cases. We study estimation from one sample with long length and from many independent samples with given length. In the parametric case, the estimators are deduced by plug-in from the maximum likelihood estimator of the parameter. In the nonparametric case, the estimators are deduced by plug-in from the empirical estimators of the transition and stationary distributions. They are proven to have good asymptotic properties.     

Estimation of the Moment Generating Function

A number of statistical problems use the moment generating function (mgf) for purposes other than determining the moments of a distribution. If the distribution is not completely specified, then the mgf must be estimated from available data. The empirical mgf makes no assumptions concerning the underlying distribution except for the existence of the mgf. In contrast to the nonparametric approach provided by the empirical mgf, alternative estimators can be formed based on an assumed parametric model. Comparison of these approaches is considered for two parametric models; the normal and a one parameter gamma. Comparison criteria are efficiency and empirical confidence interval coverage. In general the parametric estimators outperform the empirical mgf when the model is correct. The comparisons are extended to underlying models which are two component mixtures from the distributional family assumed by the parametric estimators. Under the mixture models the superiority of the parametric estimator depends upon the model, value of the argument of the mgf, and the comparison criterion. The empirical mgf is the better estimator in some cases.     

Statistical properties of parametric estimators for Markov chain vectors based on copula models

To estimate and measure risks, two key classes of dependence relationship must be identified: temporal dependence and contemporaneous dependence. In this paper, we propose a parametric estimation model that uses a three-stage pseudo maximum likelihood estimation (3SPMLE), and we investigate the consistency and asymptotic normality of parametric estimators. The proposed model combines the concept of a copula and the methods of parametric estimators of two-stage pseudo maximum likelihood estimation (2SPMLE). The selection of a copula model that best captures the dependence structure is a critical problem. To solve this problem, we propose a model selection method that is based on the parametric pseudo-likelihood ratio under the 3SPMLE for stationary Markov vector-type models.     

On Minimum Variance Unbiased Estimators

The problem of finding minimum variance unbiased estimators of various parameters for parametric distributions is an important one in statistics. This article gives analytical formulas for the minimum variance unbiased estimators of parametric functions, which are usually used in a classroom, for two types of densities. The first type is the one-parameter regular exponential family, and the second is a two-parameter family of a continuous random variable whose range depends on the unknown parameters.     

A class of nonparametric bivariate survival function estimators for randomly censored and truncated data

This paper proposes a class of nonparametric estimators for the bivariate survival function estimation under both random truncation and random censoring. In practice, the pair of random variables under consideration may have certain parametric relationship. The proposed class of nonparametric estimators uses such parametric information via a data transformation approach and thus provides more accurate estimates than existing methods without using such information. The large sample properties of the new class of estimators and a general guidance of how to find a good data transformation are given. The proposed method is also justified via a simulation study and an application on an economic data set.     

A comparison between two competing fixed parameter constrained general linear models with new regressors

Assume that two competing general linear models with fixed coefficients and identical parameter restrictions are given by adding new or deleting existing unknown parameters. In this situation, estimators of the parametric functions in the contexts of the two competing models are not necessarily the same. We consider the relationships between the best linear unbiased estimators (BLUEs) of parametric functions under these two competing constricted general linear models, and derive necessary and sufficient conditions for their BLUEs to be equal.     

The Double Gaussian Approximation for High Frequency Data

Abstract High frequency data have become an important feature of many areas of research. They permit the creation of estimators in highly non‐parametric classes of continuous‐time models. In the context of continuous semi‐martingale models, we here provide a locally parametric ‘double Gaussian’ approximation, to facilitate the analysis of estimators. As in Mykland and Zhang (Econometrica, 77, 2009, p. 1403), the error in the approximation can be offset with a postasymptotic likelihood correction. The current approximation is valid in large neighbourhoods, permitting a sharp analysis of estimators that use local behaviour over asymptotically increasing numbers of observations.     

Dual divergences estimation for censored survival data

This paper is devoted to robust estimation based on dual divergences estimators for parametric models in the framework of right censored data. We give limit laws of the proposed estimators and examine their asymptotic properties through a simulation study.     

Parametrically Guided Non-parametric Regression

We present a new approach to regression function estimation in which a non-parametric regression estimator is guided by a parametric pilot estimate with the aim of reducing the bias. New classes of parametrically guided kernel weighted local polynomial estimators are introduced and formulae for asymptotic expectation and variance, hence approximated mean squared error and mean integrated squared error, are derived. It is shown that the new classes of estimators have the very same large sample variance as the estimators in the standard non-parametric setting, while there is substantial room for reducing the bias if the chosen parametric pilot function belongs to a wide neighbourhood around the true regression line. Bias reduction is discussed in light of examples and simulations.     

Local Box–Cox transformation on time-varying parametric models for smoothing estimation of conditional CDF with longitudinal data

Nonparametric estimation and inferences of conditional distribution functions with longitudinal data have important applications in biomedical studies, such as epidemiological studies and longitudinal clinical trials. Estimation approaches without any structural assumptions may lead to inadequate and numerically unstable estimators in practice. We propose in this paper a nonparametric approach based on time-varying parametric models for estimating the conditional distribution functions with a longitudinal sample. Our model assumes that the conditional distribution of the outcome variable at each given time point can be approximated by a parametric model after local Box–Cox transformation. Our estimation is based on a two-step smoothing method, in which we first obtain the raw estimators of the conditional distribution functions at a set of disjoint time points, and then compute the final estimators at any time by smoothing the raw estimators. Applications of our two-step estimation method have been demonstrated through a large epidemiological study of childhood growth and blood pressure. Finite sample properties of our procedures are investigated through a simulation study. Application and simulation results show that smoothing estimation from time-variant parametric models outperforms the existing kernel smoothing estimator by producing narrower pointwise bootstrap confidence band and smaller root mean squared error.     

Estimation on semi-functional linear errors-in-variables models

Abstract

Semi-functional linear regression models are important in practice. In this paper, their estimation is discussed when function-valued and real-valued random variables are all measured with additive error. By means of functional principal component analysis and kernel smoothing techniques, the estimators of the slope function and the non parametric component are obtained. To account for errors in variables, deconvolution is involved in the construction of a new class of kernel estimators. The convergence rates of the estimators of the unknown slope function and non parametric component are established under suitable norm and conditions. Simulation studies are conducted to illustrate the finite sample performance of our method.