16 research outputs found

    Improvements of Polya Upper Bound for Cumulative Standard Normal Distribution and Related Functions

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    Although there is an extensive literature on the upper bound for cumulative standard normal distribution, there are relatively not sharp for all values of the interested argument x. The aim of this paper is to establish a sharp upper bound for standard normal distribution function, in the sense that its maximum absolute difference from phi(x) is less than for all values of x. The established bound improves the well-known Polya upper bound and it can be used as an approximation for Phi(x) itself with a very satisfactory accuracy. Numerical comparisons between the proposed upper bound and some other existing upper bounds have been achieved, which show that the proposed bound is tighter than alternative bounds found in the literature.Comment: 11 pages, 3 figure

    Some Improvements in Kernel Estimation Using Line Transect Sampling

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    Kernel estimation provides a nonparametric estimate of the probability density function from which a set of data is drawn. This article proposes a method to choose a reference density in bandwidth calculation for kernel estimator using line transect sampling. The method based on testing the shoulder condition, if the shoulder condition seems to be valid using as reference the half normal density, while if the shoulder condition does not seem to be valid, we will use exponential reference density. Accordingly, the performances of the resultant estimator are studied under a wide range of underlying models using simulation techniques. The results demonstrate the improvements that can be obtained by applying this technique

    Kernel-Based Estimation of P(X less than Y)With Paired Data

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    A point estimation of P(X \u3c Y) was considered. A nonparametric estimator for P(X \u3c Y) was developed using the kernel density estimator of the joint distribution of X and Y, may be dependent. The resulting estimator was found to be similar to the estimator based on the sign statistic, however it assigns smooth continuous scores to each pair of the observations rather than the zero or one scores of the sign statistic. The asymptotic equivalence of the sign statistic and the proposed estimator is shown and a simulation study is conducted to investigate the performance of the proposed estimator. Results indicate that the estimator has a good overall performance

    A Weighted Exponential Detection Function Model for Line Transect Data

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    A new parametric model is proposed for modeling the density function of perpendicular distances in line transects sampling. The model can be considered a weighted exponential model in the sense that it combines two exponential models with different weights. The proposed model is appealing because it is monotone decreasing with distance from transect line; in contrast to the classical exponential model, it satisfies the shoulder condition at the origin. Simulation results for a wide range of target densities show reasonable and good performances of the weighted exponential model in most considered cases compared to the classical exponential and the half-normal models

    Akaike Information Criterion to Select the Parametric Detection Function for Kernel Estimator Using Line Transect Data

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    Among different candidate parametric detection functions, it is suggested to use Akaike Information Criterion (AIC) to select the most appropriate one of them to fit line transect data. Four different detection functions are considered in this paper. Two of them are taken to satisfy the shoulder condition assumption and the other two estimators do not satisfy this condition. Once the appropriate detection function is determined, it also can be used to select the smoothing parameter of the nonparametric kernel estimator. For a wide range of target densities, a simulation results show the reasonable and good performances of the resulting estimators comparing with some existing estimator, particularly the usual kernel estimator when the half normal model is use as a reference to select the smoothing parameter

    Akaike Information Criterion to Select the Parametric Detection Function for Kernel Estimator Using Line Transect Data

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    Among different candidate parametric detection functions, it is suggested to use Akaike Information Criterion (AIC) to select the most appropriate one of them to fit line transect data. Four different detection functions are considered in this paper. Two of them are taken to satisfy the shoulder condition assumption and the other two estimators do not satisfy this condition. Once the appropriate detection function is determined, it also can be used to select the smoothing parameter of the nonparametric kernel estimator. For a wide range of target densities, a simulation results show the reasonable and good performances of the resulting estimators comparing with some existing estimator, particularly the usual kernel estimator when the half normal model is use as a reference to select the smoothing parameter

    Semi-Parametric Method to Estimate the Time-to-Failure Distribution and its Percentiles for Simple Linear Degradation Model

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    Most reliability studies obtained reliability information by using degradation measurements over time, which contains useful data about the product reliability. Parametric methods like the maximum likelihood (ML) estimator and the ordinary least square (OLS) estimator are used widely to estimate the time-to-failure distribution and its percentiles. In this article, we estimate the time-to-failure distribution and its percentiles by using a semi-parametric estimator that assumes the parametric function to have a half- normal distribution or an exponential distribution. The performance of the semi-parametric estimator is compared via simulation study with the ML and OLS estimators by using the mean square error and length of the 95% bootstrap confidence interval as the basis criteria of the comparison. An application to real data is given. In general, if there are assumptions on the random effect parameter, the ML estimator is the best; otherwise the kernel semi- parametric estimator with half-normal distribution is the best

    A Comparative Study for Bandwidth Selection in Kernel Density Estimation

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    Nonparametric kernel density estimation method does not make any assumptions regarding the functional form of curves of interest; hence it allows flexible modeling of data. A crucial problem in kernel density estimation method is how to determine the bandwidth (smoothing) parameter. This article examines the most important bandwidth selection methods, in particular, least squares cross-validation, biased crossvalidation, direct plug-in, solve-the-equation rules and contrast methods. Methods are described and expressions are presented. The main practical contribution is a comparative simulation study that aims to isolate the most promising methods. The performance of each method is evaluated on the basis of the mean integrated squared error for small-to-moderate sample size. Simulation results show that the contrast method is the most promising methods based on the simulated families considered

    A Comparative Study for Bandwidth Selection in Kernel Density Estimation

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    Nonparametric kernel density estimation method makes no assumptions on the functional form of the curves of interest and hence allows flexible modeling of the data. Many authors pointed out that the crucial problem in kernel density estimation method is how to determine the bandwidth (smoothing) parameter. In this paper, we introduce the most important bandwidth selection methods. In particular, least squares cross-validation, biased cross-validation, direct plug-in, solve-the-equation rules and contrast methods are considered. These methods are described and their expressions are presented. Our main practical contribution is a comparative simulation study that aims to isolate the most promising methods. The performance of each method is evaluated on the basis of the mean integrated squared error and for small-to-moderate sample size. The simulation results showed that the contrast method is the most promising methods based on the simulated families that are considered

    Variable Scale Kernel Density Estimation for Simple Linear Degradation Model

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    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
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