A bias-corrected histogram estimator for line transect sampling

The classical histogram method has already been applied in line transect sampling to estimate the parameter f(0), which in turns is used to estimate the population abundance D or the population size N. It is well know that the bias convergence rate for histogram estimator of f(0) is o(h²) as h → 0, under the shoulder condition assumption. If the shoulder condition is not true, then the bias convergence rate is only o(h). This paper proposed two new estimators for f(0), which can be considered as modifications of the classical histogram estimator. The first estimator is derived when the shoulder condition is assumed to be valid and it reduces the bias convergence rate from o(h²) to o(h³). The other one is constructed without using the shoulder condition assumption and it reduces the bias convergence rate from o(h) to o(h²). The asymptotic properties of the proposed estimators are derived and formulas for bin width are also given. The finite properties based on a real data set and an extensive simulation study demonstrated the potential practical use of the proposed estimators.     

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.     

A Semiparametric Model for Line Transect Sampling

This paper introduces an appealing semiparametric model for estimating wildlife abundance based on line transect data. The proposed method requires the existence of a parametric model and then improves the estimator using a kernel method. Properties of the resultant 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 proposed estimator using the parametric half-normal model are investigated and compared with the classical kernel estimator using both simulations and real data. Numerical results show that improvements over the classical kernel estimator often can be realized even when the true density is far from the half-normal model.     

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.     

A Pointwise Estimator for the k-Fold Convolution of a Distribution Function

ABSTRACT

This article is concerned with some parametric and nonparametric estimators for the k-fold convolution of a distribution function. An alternative estimator is proposed and its unbiasedness, asymptotic unbiasedness, and consistency properties are investigated. The asymptotic normality of this estimator is established. Some applications of the estimator are given in renewal processes. Finally, the computational procedures are described and the relative performance of these estimators for small sample sizes is investigated by a simulation study.     

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.     

Seemingly Unrelated Ridge Regression in Semiparametric Models

This article is concerned with the problem of multicollinearity in the linear part of a seemingly unrelated semiparametric (SUS) model. It is also suspected that some additional non stochastic linear constraints hold on the whole parameter space. In the sequel, we propose semiparametric ridge and non ridge type estimators combining the restricted least squares methods in the model under study. For practical aspects, it is assumed that the covariance matrix of error terms is unknown and thus feasible estimators are proposed and their asymptotic distributional properties are derived. Also, necessary and sufficient conditions for the superiority of the ridge-type estimator over the non ridge type estimator for selecting the ridge parameter K are derived. Lastly, a Monte Carlo simulation study is conducted to estimate the parametric and nonparametric parts. In this regard, kernel smoothing and cross validation methods for estimating the nonparametric function are used.     

Uniform strong consistency of histogram density estimation for dependent process

Histogram density estimator is very intuitive and easy to compute and has been widely adopted. Especially in today's big data environment, people pay more attention to the computational cost and are more willing to choose estimators with less to compute. And so, many scholars have been interested in the various estimates based on the histogram technique. Under strong mixing process, this article studies the uniform strong consistency of histogram density estimator and the convergence rate. Our conditions on the mixing coefficient and the bin width are very mild.     

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.     

Bias reduction in kernel density estimation

In this paper, we propose two kernel density estimators based on a bias reduction technique. We study the properties of these estimators and compare them with Parzen–Rosenblatt's density estimator and Mokkadem, A., Pelletier, M., and Slaoui, Y. (2009, ‘The stochastic approximation method for the estimation of a multivariate probability density’, J. Statist. Plann. Inference, 139, 2459–2478) is density estimators. It turns out that, with an adequate choice of the parameters of the two proposed estimators, the rate of convergence of two estimators will be faster than the two classical estimators and the asymptotic MISE (Mean Integrated Squared Error) will be smaller than the two classical estimators. We corroborate these theoretical results through simulations.     

Shape Constrained Non‐parametric Estimators of the Baseline Distribution in Cox Proportional Hazards Model

Abstract. We investigate non‐parametric estimation of a monotone baseline hazard and a decreasing baseline density within the Cox model. Two estimators of a non‐decreasing baseline hazard function are proposed. We derive the non‐parametric maximum likelihood estimator and consider a Grenander type estimator, defined as the left‐hand slope of the greatest convex minorant of the Breslow estimator. We demonstrate that the two estimators are strongly consistent and asymptotically equivalent and derive their common limit distribution at a fixed point. Both estimators of a non‐increasing baseline hazard and their asymptotic properties are obtained in a similar manner. Furthermore, we introduce a Grenander type estimator for a non‐increasing baseline density, defined as the left‐hand slope of the least concave majorant of an estimator of the baseline cumulative distribution function, derived from the Breslow estimator. We show that this estimator is strongly consistent and derive its asymptotic distribution at a fixed point.     

A comparison of dependence function estimators in multivariate extremes

Various nonparametric and parametric estimators of extremal dependence have been proposed in the literature. Nonparametric methods commonly suffer from the curse of dimensionality and have been mostly implemented in extreme-value studies up to three dimensions, whereas parametric models can tackle higher-dimensional settings. In this paper, we assess, through a vast and systematic simulation study, the performance of classical and recently proposed estimators in multivariate settings. In particular, we first investigate the performance of nonparametric methods and then compare them with classical parametric approaches under symmetric and asymmetric dependence structures within the commonly used logistic family. We also explore two different ways to make nonparametric estimators satisfy the necessary dependence function shape constraints, finding a general improvement in estimator performance either (i) by substituting the estimator with its greatest convex minorant, developing a computational tool to implement this method for dimensions \(D\ge 2\) or (ii) by projecting the estimator onto a subspace of dependence functions satisfying such constraints and taking advantage of Bernstein–Bézier polynomials. Implementing the convex minorant method leads to better estimator performance as the dimensionality increases.     

Liu-type estimator in semiparametric partially linear additive models

Partially linear additive model is useful in statistical modelling as a multivariate nonparametric fitting technique. This paper considers statistical inference for the semiparametric model in the presence of multicollinearity. Based on the profile least-squares (PL) approach and Liu estimation method, we propose a PL Liu estimator for the parametric component. When some additional linear restrictions on the parametric component are available, the corresponding restricted Liu estimator for the parametric component is constructed. The properties of the proposed estimators are derived. Some simulations are conducted to assess the performance of the proposed procedures and the results are satisfactory. Finally, a real data example is analysed.     

ESTIMATION OF MULTIVARIATE SHANNON ENTROPY USING MOMENTS

Three new entropy estimators of multivariate distributions are introduced. The two cases considered here concern when the distribution is supported by a unit sphere and by a unit cube. In the former case, the consistency and the upper bound of the absolute error for the proposed entropy estimator are established. In the latter one, under the assumption that only the moments of the underlying distribution are available, a non‐traditional estimator of the entropy is suggested. We also study the practical performances of the constructed estimators through simulation studies and compare the estimators based on the moment‐recovered approaches with their counterparts derived by using the histogram and k th nearest neighbour constructions. In addition, one worked example is briefly discussed.     

A consistent simulation-based estimator in generalized linear mixed models

We propose a strongly root-n consistent simulation-based estimator for the generalized linear mixed models. This estimator is constructed based on the first two marginal moments of the response variables, and it allows the random effects to have any parametric distribution (not necessarily normal). Consistency and asymptotic normality for the proposed estimator are derived under fairly general regularity conditions. We also demonstrate that this estimator has a bounded influence function and that it is robust against data outliers. A bias correction technique is proposed to reduce the finite sample bias in the estimation of variance components. The methodology is illustrated through an application to the famed seizure count data and some simulation studies.     

Semiparametric Regression Analysis of Longitudinal Skewed Data

In this paper, we develop a semiparametric regression model for longitudinal skewed data. In the new model, we allow the transformation function and the baseline function to be unknown. The proposed model can provide a much broader class of models than the existing additive and multiplicative models. Our estimators for regression parameters, transformation function and baseline function are asymptotically normal. Particularly, the estimator for the transformation function converges to its true value at the rate n ^{? 1 ∕ 2}, the convergence rate that one could expect for a parametric model. In simulation studies, we demonstrate that the proposed semiparametric method is robust with little loss of efficiency. Finally, we apply the new method to a study on longitudinal health care costs.     

Product-limit Estimators and Cox Regression with Missing Censoring Information

The Kaplan–Meier estimator of a survival function requires that the censoring indicator is always observed. A method of survival function estimation is developed when the censoring indicators are missing completely at random (MCAR). The resulting estimator is a smooth functional of the Nelson–Aalen estimators of certain cumulative transition intensities. The asymptotic properties of this estimator are derived. A simulation study shows that the proposed estimator has greater efficiency than competing MCAR-based estimators. The approach is extended to the Cox model setting for the estimation of a conditional survival function given a covariate.     

Combining poisson means

The problem of estimating the Poisson mean is considered based on the two samples in the presence of uncertain prior information (not in the form of distribution) that two independent random samples taken from two possibly identical Poisson populations. The parameter of interest is λ₁ from population I. Three estimators, i.e. the unrestricted estimator, restricted estimator and preliminary test estimator are proposed. Their asymptotic mean squared errors are derived and compared; parameter regions have been found for which restricted and preliminary test estimators are always asymptotically more efficient than the classical estimator. The relative dominance picture of the estimators is presented. Maximum and minimum asymptotic efficiencies of the estimators relative to the classical estimator are tabulated. A max-min rule for the size of the preliminary test is also discussed. A Monte Carlo study is presented to compare the performance of the estimator with that of Kale and Bancroft (1967).     

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.     

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.