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This paper deals with the problem of estimating the overlapping (OVL) Weitzman measure (Δ) when two independent random variables 𝑿 and 𝒀 are given by two-parameter Weibull distribution. The measure Δ has been studied in the literature in the case of two Weibull distributions under the assumption that the two shape parameters are equal. In this work, a general expression for the Weitzmans measure is provided under the Weibull distribution without using any assumptions about the distribution parameters. Some new estimators for Δ are developed depending on three numerical integration rules known as trapezoidal, Simpson 1/3 and Simpson 3/8 rules. The performance of the proposed estimators were investigated and compared with some existing estimators via simulation technique and real data. The results demonstrated the superiority of the proposed estimators over the existing one in almost all considered cases.

Most studies of estimating overlapping coefficients assume two specific parametric models for population densities. The methods that used such of this assumption are called parametric methods, which work well when these model assumptions are valid. Violation of parametric model assumption often leads to a poor estimation. The alternative nonparametric methods are more flexible than the parametric one and they required no assumptions about the parametric structure of population densities. Recently, some efforts were introduced to estimate the overlapping coefficients non-parametrically. In this paper, we introduced the nonparametric kernel method to estimate the three overlapping coefficients that known in the literature as Matusita's measure ρ , Morisita's measure λ and Weitzman's measure Δ. As the simulation results shown and based on the relative root mean square error (RRMSE) measure, if the two population distributions are assumed to be known then the parametric method performs better than the nonparametric kernel method. If there is no information about the structural form of the population distributions then the performance of the kernel method is very acceptable and it performs better than the parametric one. [ABSTRACT FROM AUTHOR]

In this study, we proposed the variable scale kernel estimator for analyzing the degradation data. The properties of the proposed method are investigated and compared with the classical method such as; maximum likelihood and ordinary least square methods via simulation technique. The criteria bias and MSE are used for comparison. Simulation results showed that the performance of the variable scale kernel estimator is acceptable as a general estimator. It is nearly the best estimator when the assumption of the distribution is invalid. Application to real data set is also given. [ABSTRACT FROM AUTHOR]

The non-parametric kernel density estimation is used in practice to estimate population abundance using the line transect sampling method. Kernel estimator of f(0) usually produces an underestimated value. Assuming the kernel method, this article applies shifted logarithmic transformation to line transect data that violating shoulder condition. Mathematically, the proposed log-transform estimator was shown to be more efficient than the classical kernel estimator. The simulation results present the good properties of the proposed estimator compared to the performance of the classical kernel estimators. [ABSTRACT FROM AUTHOR]

This article proposes nine new approximations for the standard normal cumulative distribution function Φ z. In addition, it collects most of the approximations existing in the literature. The accuracy of the proposed approximations is evaluated by using the maximum absolute error and the mean absolute error. The maximum absolute errors fall between 0.0422613 × 10−6 and 950.55 × 10−6, which indicates high accuracy for some of them. A comparison study between the different existing approximations and the proposed approximations is also accomplished. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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 onAMSEis 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. [ABSTRACT FROM AUTHOR]

The variable location kernel (VLK) method provides a nonparametric estimator for a probability density function. This article proposes the VLK method to fit line transect data in order to estimate the density of a biological population. The method produces two promising estimators for the density of objects which improve upon the performance of the classical kernel estimator. Although the two proposed estimators share a common form, they exhibit rather different performances. To compute the bias and variance of the proposed estimators, the bootstrap technique is proposed. For a wide range of possible models for line transect data, a comparison of the two estimators and the classical kernel estimator is carried out by simulation. The results show the practical potential of the proposed estimators over the classical kernel estimator for almost all cases considered. Two previously published data sets are also analyzed and the results confirm the good performances of the proposed estimators. Copyright © 2010 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]