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.     

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 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.     

Kernel estimation in transect sampiing withoyt the shoulder condition

We consider the estiinution of wildlife population density based on line transect data. Nonparametric kernel method is employed, without the usual assumption that the detection curve has a shoulder at distance zero, with the help of a special class of kernels called boundary kernels. Asymptotic distribution results are included. It is pointed out that the boundery kernel of Zhang and Karunamuni (1998) (see also Müller and Wang (1994)) performs better (for asmyptotic mean square error consideration) than that of the boundary kernel of M¨ller (1991). But both of these kernels are clearly superior to the half-nonnal and one-sided Epanechnikov kernel when the shoulder condition fails to hold. In practice, however, for small to moderate sample sizes, caution should be exercised in using bounrlary kernels in that the density estimate might become negative. A Monte Carlo study is also presented, comparing the performance of four kernels applied to detection data, with and without the shoulder condition, Two bundary kernels for deriatives are also included for the point transect case.     

Uniformly most powerful unbiased test for shoulder condition in point transect sampling

Point transect sampling is a well-known methodology for estimating wildlife population density. In this context, the usual approach is to assume a model for the detection function. Thus, the estimate depends on the shape of the detection function. In particular, the estimation is influenced by the so-called shoulder condition, which guarantees that detection is nearly certain at small distances from the observer. For instance, the half-normal model satisfies this condition, whereas the negative exponential model does not. Testing whether the shoulder condition is consistent with data is a crucial issue. In this paper we propose the uniformly most powerful unbiased test for the shoulder condition in the exponential mixture model of the half-normal and the negative exponential. Critical values of the proposed test are calculated for large samples by means of asymptotic distribution theory and for small samples via Monte Carlo simulations. Finally, a case study is presented.     

Kernel density estimation under negative superadditive dependence and its application for real data

In this paper, the kernel density estimator for negatively superadditive dependent random variables is studied. The exponential inequalities and the exponential rate for the kernel estimator of density function with a uniform version, over compact sets are investigated. Also, the optimal bandwidth rate of the estimator is obtained using mean integrated squared error. The results are generalized and used to improve the ones obtained for the case of associated sequences. As an application, FGM sequences that fulfil our assumptions are investigated. Also, the convergence rate of the kernel density estimator is illustrated via a simulation study. Moreover, a real data analysis is presented.     

Robust importance sampling for some typical types of utility-based shortfall risk measures using exponential twisting and kernel density techniques

A robust algorithm for utility-based shortfall risk (UBSR) measures is developed by combining the kernel density estimation with importance sampling (IS) using exponential twisting techniques. The optimal bandwidth of the kernel density is obtained by minimizing the mean square error of the estimators. Variance is reduced by IS where exponential twisting is applied to determine the optimal IS distribution. Conditions for the best distribution parameters are derived based on the piecewise polynomial loss function and the exponential loss function. The proposed method not only solves the problem of sampling from the kernel density but also reduces the variance of the UBSR estimator.     

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.     

Estimation of a probability density function using interval aggregated data

ABSTRACT

In economics and government statistics, aggregated data instead of individual level data are usually reported for data confidentiality and for simplicity. In this paper we develop a method of flexibly estimating the probability density function of the population using aggregated data obtained as group averages when individual level data are grouped according to quantile limits. The kernel density estimator has been commonly applied to such data without taking into account the data aggregation process and has been shown to perform poorly. Our method models the quantile function as an integral of the exponential of a spline function and deduces the density function from the quantile function. We match the aggregated data to their theoretical counterpart using least squares, and regularize the estimation by using the squared second derivatives of the density function as the penalty function. A computational algorithm is developed to implement the method. Application to simulated data and US household income survey data show that our penalized spline estimator can accurately recover the density function of the underlying population while the common use of kernel density estimation is severely biased. The method is applied to study the dynamic of China's urban income distribution using published interval aggregated data of 1985–2010.     

Variable kernel density estimation

This paper considers the problem of selecting optimal bandwidths for variable (sample‐point adaptive) kernel density estimation. A data‐driven variable bandwidth selector is proposed, based on the idea of approximating the log‐bandwidth function by a cubic spline. This cubic spline is optimized with respect to a cross‐validation criterion. The proposed method can be interpreted as a selector for either integrated squared error (ISE) or mean integrated squared error (MISE) optimal bandwidths. This leads to reflection upon some of the differences between ISE and MISE as error criteria for variable kernel estimation. Results from simulation studies indicate that the proposed method outperforms a fixed kernel estimator (in terms of ISE) when the target density has a combination of sharp modes and regions of smooth undulation. Moreover, some detailed data analyses suggest that the gains in ISE may understate the improvements in visual appeal obtained using the proposed variable kernel estimator. These numerical studies also show that the proposed estimator outperforms existing variable kernel density estimators implemented using piecewise constant bandwidth functions.     

Kernel Estimation of the Spectral Density of Stationary Random Closed Sets

A non‐parametric kernel estimator of the spectral density of stationary random closed sets is studied. Conditions are derived under which this estimator is asymptotically unbiased and mean‐square consistent. For the planar Boolean model with isotropic compact and convex grains, an averaged version of the kernel estimator is compared with the theoretical spectral density.     

Composite Estimation for Single‐Index Models with Responses Subject to Detection Limits

We propose a semiparametric estimator for single‐index models with censored responses due to detection limits. In the presence of left censoring, the mean function cannot be identified without any parametric distributional assumptions, but the quantile function is still identifiable at upper quantile levels. To avoid parametric distributional assumption, we propose to fit censored quantile regression and combine information across quantile levels to estimate the unknown smooth link function and the index parameter. Under some regularity conditions, we show that the estimated link function achieves the non‐parametric optimal convergence rate, and the estimated index parameter is asymptotically normal. The simulation study shows that the proposed estimator is competitive with the omniscient least squares estimator based on the latent uncensored responses for data with normal errors but much more efficient for heavy‐tailed data under light and moderate censoring. The practical value of the proposed method is demonstrated through the analysis of a human immunodeficiency virus antibody data set.     

Strong consistency of non parametric kernel regression estimator for strong mixing samples

For α-mixing samples, we study Priestley–Chao kernel estimator for non parametric regression model. By using the moment inequality and the exponential inequality, the strong consistency and the uniformly strong consistency of the estimator are obtained for some weak conditions.     

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.     

Robust and efficient parameter estimation based on censored data with stochastic covariates

Analysis of random censored life-time data along with some related stochastic covariables is of great importance in many applied sciences. The parametric estimation technique commonly used under this set-up is based on the efficient but non-robust likelihood approach. In this paper, we propose a robust parametric estimator for censored data with stochastic covariates based on the minimum density power divergence approach. The resulting estimator also has competitive efficiency with respect to the maximum likelihood estimator under pure data. The strong robustness property of the proposed estimator with respect to the presence of outliers is examined and illustrated through an appropriate real data example and simulation studies. Further, the theoretical asymptotic properties of the proposed estimator are also derived in terms of a general class of M-estimators based on the estimating equation.     

Nonparametric density deconvolution by weighted kernel estimators

Nonparametric density estimation in the presence of measurement error is considered. The usual kernel deconvolution estimator seeks to account for the contamination in the data by employing a modified kernel. In this paper a new approach based on a weighted kernel density estimator is proposed. Theoretical motivation is provided by the existence of a weight vector that perfectly counteracts the bias in density estimation without generating an excessive increase in variance. In practice a data driven method of weight selection is required. Our strategy is to minimize the discrepancy between a standard kernel estimate from the contaminated data on the one hand, and the convolution of the weighted deconvolution estimate with the measurement error density on the other hand. We consider a direct implementation of this approach, in which the weights are optimized subject to sum and non-negativity constraints, and a regularized version in which the objective function includes a ridge-type penalty. Numerical tests suggest that the weighted kernel estimation can lead to tangible improvements in performance over the usual kernel deconvolution estimator. Furthermore, weighted kernel estimates are free from the problem of negative estimation in the tails that can occur when using modified kernels. The weighted kernel approach generalizes to the case of multivariate deconvolution density estimation in a very straightforward manner.     

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.     

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.     

Moment density estimation for positive random variables

An unknown moment-determinate cumulative distribution function or its density function can be recovered from corresponding moments and estimated from the empirical moments. This method of estimating an unknown density is natural in certain inverse estimation models like multiplicative censoring or biased sampling when the moments of unobserved distribution can be estimated via the transformed moments of the observed distribution. In this paper, we introduce a new nonparametric estimator of a probability density function defined on the positive real line, motivated by the above. Some fundamental properties of proposed estimator are studied. The comparison with traditional kernel density estimator is discussed.     

Nonparametric density estimation based on beta prime kernel

Abstract

In this work, we propose beta prime kernel estimator for estimation of a probability density functions defined with nonnegative support. For the proposed estimator, beta prime probability density function used as a kernel. It is free of boundary bias and nonnegative with a natural varying shape. We obtained the optimal rate of convergence for the mean squared error (MSE) and the mean integrated squared error (MISE). Also, we use adaptive Bayesian bandwidth selection method with Lindley approximation for heavy tailed distributions and compare its performance with the global least squares cross-validation bandwidth selection method. Simulation studies are performed to evaluate the average integrated squared error (ISE) of the proposed kernel estimator against some asymmetric competitors using Monte Carlo simulations. Moreover, real data sets are presented to illustrate the findings.