Omar Eidous

On Improving Kernel Estimators Using Line Transect Sampling

ABSTRACT We investigate some possibilities for improvements of kernel density estimates using line transect sampling. A bias correction approach is used which produces a new estimator for density of object D. The asymptotic bias of kernel estimator is subtracted from the kernel estimator to form the proposed estimator. It reduced the bias from O(h 2) to O(h 3) as h → 0 under the shoulder condition assumption and in some cases the bias reduced to O(h 4). The asymptotic theoretical properties of the proposed estimator are presented and the optimal formula for the bandwidth parameter h is given based on minimizing the asymptotic mean square error (AMSE). Moreover, in the absence of our knowledge whether or not the shoulder condition is valid another estimator is suggested, small-sample properties of the proposed estimators are studied and compared with some existing estimators by simulation technique. The results show the superiority of the proposed estimators among the other estimators for most cases considered.

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.

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.

Additive histogram frequency estimator for wildlife abundance using line transect data without the shoulder condition

SummaryThis paper proposes an additive histogram estimator for the wildlife population density from the line transect data when the shoulder condition is not satisfied. The proposed estimator is a simple and interpretable as the usual histogram estimator while holding theoretical and practical advantages. The new estimator is compared with the boundary kernel method and the results show the good properties of the proposed estimator from both asymptotic theoretical and numerical results.

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.

Nonparametric Estimation of f(0) Applying Line Transect Data with and without the Shoulder Condition

Abstract The problem of estimating f (0); the probability density function of observed distances at the left boundary x = 0 using line transect data is considered. A general nonparametric histogram estimator (0) for f (0) is proposed and investigated. The proposed estimator is formulated as linear of indicator functions with corresponding weights. These weights can be determined in a variety of different ways to ensure desirable properties for the proposed estimator and to deal with the presence or absence of the shoulder condition assumption. Theoretical motivation is provided for the existence of weights that prove to be effective in reducing the bias without increasing variance. We consider a direct implementation of the proposed estimator in which the weights are optimized subject to some constraints. The optimization problem converts into the problem of solving a system of linear equations. Some comparison performances are obtained, which demonstrate the good results of the proposed estimator at least for adequately large samples.

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.

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.

Combination of parametric and nonparametric estimators for population abundance using line transect sampling

Abstract This paper introduced a modification of the classical kernel estimator for population abundance using line transect sampling. The procedure allows the incorporation of a small set of parametric models with the usual kernel estimator. Akaike Information Criterion (AIC) is used to choose the most appropriate parametric detection function. The resultant estimators of f(0); the probability density function at perpendicular distance x = 0, are compared with the existing usual kernel estimator by using the simulation technique and a set of real data. Numerical results demonstrate the superiority of these estimators over the usual kernel estimator for almost all considered cases.

A bias-corrected histogram estimator for line transect sampling

ABSTRACT 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.