Kernel Method Starting with Half-Normal Detection Function for Line Transect Density Estimation

In this article, we introduce the nonparametric kernel method starting with half-normal detection function using line transect sampling. The new method improves bias from O(h ²), as the smoothing parameter h → 0, to O(h ³) and in some cases to O(h ⁴). Properties of the proposed estimator are derived and an expression for the asymptotic mean square error (AMSE) of the estimator is given. Minimization of the AMSE leads to an explicit formula for an optimal choice of the smoothing parameter. Small-sample properties of the estimator are investigated and compared with the traditional kernel estimator by using simulation technique. A numerical results show that improvements over the traditional kernel estimator often can be realized even when the true detection function is far from the half-normal detection function.     

Asymptotic Unbiased Kernel Estimator for Line Transect Sampling

In this article, we propose a new estimator for the density of objects using line transect data. The proposed estimator combines the nonparametric kernel estimator with parametric detection function: the exponential or the half normal detection function to estimate the density of objects. The selection of the detection function depends on the testing of the shoulder condition assumption. If the shoulder condition is true then the half-normal detection function is introduced together with the kernel estimator. Otherwise, the negative exponential is combined with the kernel estimator. Under these assumptions, the proposed estimator is asymptotically unbiased and it is strongly consistent estimator for the density of objects using line transect data. The simulation results indicate that the proposed estimator is very successful in taking the advantage of the parametric detection function available.     

Histogram Model with Plausible Parametric Detection Function for Line Transect Data

This article develops a new model that combines between the histogram and plausible parametric detection function to estimate the population density (abundance) by using line transects technique. A parametric detection function is introduced to improve the properties of the classical histogram estimator. Asymptotic properties of the resulting estimator are derived and an expression for the asymptotic mean square error (AMSE) is given. A general formula for the optimal choice of the histogram bin width based on AMSE is derived. Moreover, other possible alternative procedures to select the bin width are suggested and studied via simulation technique. The results show the superiority of the proposed estimators over both the classical histogram and the usual kernel estimators in most reasonable cases. In addition, the simulation results indicate that the choice of a plausible detection function is less sensitive than the choice of a bin width on the performance of the proposed estimator.     

Kernel density estimation for heavy-tailed distributions using the champernowne transformation

When estimating loss distributions in insurance, large and small losses are usually split because it is difficult to find a simple parametric model that fits all claim sizes. This approach involves determining the threshold level between large and small losses. In this article, a unified approach to the estimation of loss distributions is presented. We propose an estimator obtained by transforming the data set with a modification of the Champernowne cdf and then estimating the density of the transformed data by use of the classical kernel density estimator. We investigate the asymptotic bias and variance of the proposed estimator. In a simulation study, the proposed method shows a good performance. We also present two applications dealing with claims costs in insurance.     

Semiparametric Density Deconvolution

Abstract.  A new semiparametric method for density deconvolution is proposed, based on a model in which only the ratio of the unconvoluted to convoluted densities is specified parametrically. Deconvolution results from reweighting the terms in a standard kernel density estimator, where the weights are defined by the parametric density ratio. We propose that in practice, the density ratio be modelled on the log-scale as a cubic spline with a fixed number of knots. Parameter estimation is based on maximization of a type of semiparametric likelihood. The resulting asymptotic properties for our deconvolution estimator mirror the convergence rates in standard density estimation without measurement error when attention is restricted to our semiparametric class of densities. Furthermore, numerical studies indicate that for practical sample sizes our weighted kernel estimator can provide better results than the classical non-parametric kernel estimator for a range of densities outside the specified semiparametric class.     

Discrete Nonparametric Kernel and Parametric Methods for the Modeling of Pavement Deterioration

This article is concerned with one discrete nonparametric kernel and two parametric regression approaches for providing the evolution law of pavement deterioration. The first parametric approach is a survival data analysis method; and the second is a nonlinear mixed-effects model. The nonparametric approach consists of a regression estimator using the discrete associated kernels. Some asymptotic properties of the discrete nonparametric kernel estimator are shown as, in particular, its almost sure consistency. Moreover, two data-driven bandwidth selection methods are also given, with a new theoretical explicit expression of optimal bandwidth provided for this nonparametric estimator. A comparative simulation study is realized with an application of bootstrap methods to a measure of statistical accuracy.     

Fourth-order kernel method for simple linear degradation model

Degradation analysis is a useful technique when life tests result in few or even no failures. The degradation measurements are recorded over time and the estimation of time-to-failure distribution plays a vital role in degradation analysis. The parametric method to estimate the time-to-failure distribution assumed a specific parametric model with known shape for the random effects parameter. To avoid any assumption about the model shape, a nonparametric method can be used. In this paper, we suggest to use the nonparametric fourth-order kernel method to estimate the time-to-failure distribution and its percentiles for the simple linear degradation model. The performances of the proposed method are investigated and compared with the classical kernel; maximum likelihood and ordinary least squares methods via simulation technique. The numerical results show the good performance of the fourth-order kernel method and demonstrate its superiority over the parametric method when there is no information about the shape of the random effect parameter distribution.     

Robust estimation of complicated profiles using wavelets

Some quality characteristics are well defined when treated as the response variables and their relationships are identified to some independent variables. This relationship is called a profile. The parametric models, such as linear models, may be used to model the profiles. However, due to the complexity of many processes in practical applications, it is inappropriate to model the process using parametric models. In these cases non parametric methods are used to model the processes. One of the most applicable non parametric methods used to model complicated profiles is the wavelet. Many authors considered the use of the wavelet transformation only for monitoring the processes in phase II. The problem of estimating the in-control profile in phase I using wavelet transformation is not deeply addressed. Usually classical estimators are used in phase I to estimate the in-control profiles, even when the wavelet transformation is used. These estimators are suitable if the data do not contain outliers. However, when the outliers exist, these estimators cannot estimate the in-control profile properly. In this research, a robust method of estimating the in-control profiles is proposed, which is insensitive to the presence of outliers and could be applied when the wavelet transformation is used. The proposed estimator is the combination of the robust clustering and the S-estimator. This estimator is compared with the classical estimator of the in-control profile in the presence of outliers. The results from a large simulation study show that using the proposed method, one can estimate the in-control profile precisely when the data are contaminated either locally or globally.     

Nonparametric kernel density estimation for general grouped data

Interval-grouped data are defined, in general, when the event of interest cannot be directly observed and it is only known to have been occurred within an interval. In this framework, a nonparametric kernel density estimator is proposed and studied. The approach is based on the classical Parzen–Rosenblatt estimator and on the generalisation of the binned kernel density estimator. The asymptotic bias and variance of the proposed estimator are derived under usual assumptions, and the effect of using non-equally spaced grouped data is analysed. Additionally, a plug-in bandwidth selector is proposed. Through a comprehensive simulation study, the behaviour of both the estimator and the plug-in bandwidth selector considering different scenarios of data grouping is shown. An application to real data confirms the simulation results, revealing the good performance of the estimator whenever data are not heavily grouped.     

Estimating Percentiles of Time-to-Failure Distribution Obtained from a Linear Degradation Model Using Kernel Density Method

In this article, we propose a nonparametric estimator for percentiles of the time-to-failure distribution obtained from a linear degradation model using the kernel density method. The properties of the proposed kernel estimator are investigated and compared with well-known maximum likelihood and ordinary least squares estimators via a simulation technique. The mean squared error and the length of the bootstrap confidence interval are used as the basis criteria of the comparisons. The simulation study shows that the performance of the kernel estimator is acceptable as a general estimator. When the distribution of the data is assumed to be known, the maximum likelihood and ordinary least squares estimators perform better than the kernel estimator, while the kernel estimator is superior when the assumption of our knowledge of the data distribution is violated. A comparison among different estimators is achieved using a real data set.     

Statistical inference under imputation for proportional hazard model with missing covariates

Missing covariate data are common in biomedical studies. In this article, by using the non parametric kernel regression technique, a new imputation approach is developed for the Cox-proportional hazard regression model with missing covariates. This method achieves the same efficiency as the fully augmented weighted estimators (Qi et al. 2005. Journal of the American Statistical Association, 100:1250) and has a simpler form. The asymptotic properties of the proposed estimator are derived and analyzed. The comparisons between the proposed imputation method and several other existing methods are conducted via a number of simulation studies and a mouse leukemia data.     

Outlier Resistant Estimation in Difference-Based Semiparametric Partially Linear Models

Partially linear models are extensions of linear models that include a nonparametric function of some covariate allowing an adequate and more flexible handling of explanatory variables than in linear models. The difference-based estimation in partially linear models is an approach designed to estimate parametric component by using the ordinary least squares estimator after removing the nonparametric component from the model by differencing. However, it is known that least squares estimates do not provide useful information for the majority of data when the error distribution is not normal, particularly when the errors are heavy-tailed and when outliers are present in the dataset. This paper aims to find an outlier-resistant fit that represents the information in the majority of the data by robustly estimating the parametric and the nonparametric components of the partially linear model. Simulations and a real data example are used to illustrate the feasibility of the proposed methodology and to compare it with the classical difference-based estimator when outliers exist.     

Kernel‐based Generalized Cross‐validation in Non‐parametric Mixed‐effect Models

Abstract. Although generalized cross‐validation (GCV) has been frequently applied to select bandwidth when kernel methods are used to estimate non‐parametric mixed‐effect models in which non‐parametric mean functions are used to model covariate effects, and additive random effects are applied to account for overdispersion and correlation, the optimality of the GCV has not yet been explored. In this article, we construct a kernel estimator of the non‐parametric mean function. An equivalence between the kernel estimator and a weighted least square type estimator is provided, and the optimality of the GCV‐based bandwidth is investigated. The theoretical derivations also show that kernel‐based and spline‐based GCV give very similar asymptotic results. This provides us with a solid base to use kernel estimation for mixed‐effect models. Simulation studies are undertaken to investigate the empirical performance of the GCV. A real data example is analysed for illustration.     

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.     

Efficient estimation in partially linear single‐index models for longitudinal data

In this paper, we consider the estimation of both the parameters and the nonparametric link function in partially linear single‐index models for longitudinal data that may be unbalanced. In particular, a new three‐stage approach is proposed to estimate the nonparametric link function using marginal kernel regression and the parametric components with generalized estimating equations. The resulting estimators properly account for the within‐subject correlation. We show that the parameter estimators are asymptotically semiparametrically efficient. We also show that the asymptotic variance of the link function estimator is minimized when the working error covariance matrices are correctly specified. The new estimators are more efficient than estimators in the existing literature. These asymptotic results are obtained without assuming normality. The finite‐sample performance of the proposed method is demonstrated by simulation studies. In addition, two real‐data examples are analyzed to illustrate the methodology.     

Guided Censored Regression

Parametrically guided non‐parametric regression is an appealing method that can reduce the bias of a non‐parametric regression function estimator without increasing the variance. In this paper, we adapt this method to the censored data case using an unbiased transformation of the data and a local linear fit. The asymptotic properties of the proposed estimator are established, and its performance is evaluated via finite sample simulations.     

Model-based confidence bands for survival functions

This paper focuses on a novel method of developing one-sample confidence bands for survival functions from right censored data. The approach is model-based, relying on a parametric model for the conditional expectation of the censoring indicator given the observed minimum, and derives its strength from easy access to a good-fitting model among a plethora of choices available for binary response data. The substantive methodological contribution is in exploiting a semiparametric estimator of the survival function to produce improved simultaneous confidence bands. To obtain critical values for computing the confidence bands, a two-stage bootstrap approach that combines the classical bootstrap with the more recent model-based regeneration of censoring indicators is proposed and a justification of its asymptotic validity is also provided. Several different confidence bands are studied using the proposed approach. Numerical studies, including robustness of the proposed bands to misspecification, are carried out to check efficacy. The method is illustrated using two lung cancer data sets.     

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.     

On kernel density estimation based on different stratified sampling with optimal allocation

Kernel density estimation is probably the most widely used non parametric statistical method for estimating probability densities. In this paper, we investigate the performance of kernel density estimator based on stratified simple and ranked set sampling. Some asymptotic properties of kernel estimator are established under both sampling schemes. Simulation studies are designed to examine the performance of the proposed estimators under varying distributional assumptions. These findings are also illustrated with the help of a dataset on bilirubin levels in babies in a neonatal intensive care unit.     

Estimation of regression parameters in a semiparametric transformation model

A semiparametric approach to model skewed/heteroscedastic regression data is discussed. We work with a semiparametric transform-both-sides regression model, which contains a parametric regression function and a nonparametric transformation. This model is adequate when the relationship between the median response and the explanatory variable has been specified by a theoretical result or a previous empirical study. The transform-both-sides model with a parametric transformation has been studied extensively and applied successfully to a number data sets. Allowing a nonparametric transformation function increases the flexibility of the model. In this article, we estimate the nonparametric transformation function by the conditional kernel density approach developed by Wang and Ruppert (1995), and then use a pseudo-maximum likelihood estimator to estimate the regression parameters. This estimate of the regression parameters has not been studied previously. In this article, the asymptotic distribution of this pseudo-MLE is derived. We also show that when σ, the standard deviation of the error, goes to zero (small σ asymptotics), this estimator is adaptive. Adaptive means that the regression parameters are estimated as precisely as when the transformation is known exactly. A similar result holds in the parametric approaches of Carroll and Ruppert (1984) and Ruppert and Aldershof (1989). Simulated and real examples are provided to illustrate the performance of the proposed estimator for finite sample size.