Variable bandwidth Kernel estimator of the mode

In this paper, the problem of estimating the mode of a probability density function has been studied.  Parzen (1962) proposed a kernel estimator of the mode depends on a single bandwidth. In this paper, the Parzen estimator has been improved by proposing a kernel estimator with variable bandwidth for the mode of the density function. Proceeding as in Parzen (1962), the consistency and asymptotic normality of the proposed estimator are shown. Moreover, the good  performance of the proposed estimator is tested via simulation study and it is shown that the proposed estimator is more efficient than the Parzen estimator

Adaptive Kernel Estimation of The Hazard Rate Function

In this paper, we generalized the constant bandwidth kernel es-timator of the hazard rate function from Watson and Leadbetter (1964), which depends on a single bandwidth to the adaptive ker-nel estimator, which depends on different bandwidths. We derive the asymptotic normality of the adaptive kernel estimator. Also we illustrate the use of the adaptive kernel hazard rate estima-tor in both simulation and real life data and compared it to the constant bandwidth kernel estimator. In our study, we show that the adaptive estimator has no boundary effects as the constant bandwidth kernel estimator, and has smaller bias. Keywords: Hazard rate function, adaptive kernel estimation, constant bandwidth kernel estimation, density estimation

تقدير دالة معدل المخاطرة باستخدام نواة دالة جاوس العكسية

In this paper, we consider the nonparametric estimation of the hazard rate function for independent identically distributed (iid) data using kernel estimation techniques. Since survival times are positive with potentially a high concentration at zero, one has to take into account the bias problem when the hazard rate function is estimated in the boundary region. To overcome the boundary bias problem, we use the Inverse Gaussian (IG) kernel, since it has a positive support. The asymptotic mean squared error (AMSE) and the asymptotic normality of the proposed estimator are investigated. Also, the selection of an optimal bandwidth is discussed since it plays an important role in the kernel estimation. Keywords: Inverse Gaussian kernel, hazard rate function, kernel estimation, asymptotic mean square error, boundary bias. : . . . . .في هذا البحث ندرس التقدير اللامعلمي لدالة المخاطرة باستخدام طرق تقدير النواة. ولأن تحليل البقاء يعتمد على بيانات ايجابية متركزة بالقرب من الصفر فيجب أن يؤخذ بعين الاعتبار مشكلة التحيز عند تقدير دالة المخاطرة في منطقة الحدود بالقرب من الصفر. وللتغلب على هذه المشكلة فإننا نستخدم تقدير معكوس جاوس والذي يتميز بأنه يمتلك دعم ايجابيا. كما تم في هذا البحث دراسة تقارب الخطأ التربيعي للتقدير المقترح وتقاربه لتوزيع طبيعي. بالإضافة لذلك تم مناقشة كيفية اختيار اتساع النافذ حيث إنها تلعب دورا مهما في تقدير النواة

Multivariate time series modeling of monthly rainfall amounts

This paper discusses the tting of suitable models to rainfall observations.Daily rainfall amounts were aggregated to monthly data using the Thiessenpolygons method and multivariate seasonal vector integrated autoregressivemoving average models (sVARIMA) were tted to the monthly cumulative rainfall volume. The data were obtained from the 12 Palestinian meteorological gauge stations located across the 5 governorates of the Gaza Strip and incorporated 42 years (from 1973 to 2014) of irregular daily precipitation. It can be concluded that the use of sVARIMAmodels in the environmental science provide a useful method to forecast rainfall data as a preliminary guideline for short and long-term sustainable water resources management

تقريب دالة الكثافة الاحتمالية باستخدام الموجات الصغيرة

Wavelets are a new family of basis functions that can be used to express and approximate other functions. In this thesis, we investigate the use of wavelets in Probability Theory. This thesis consists of two parts. In the first part, we present the continuous wavelet transform and multiresolution analysis as ways of introducing wavelets. Then we study some powerful properties of wavelets such as orthogonality and compact support. Then we study the use of wavelets in data compression. In the second part, we study the use of wavelets in Probability Theory and we show how we can use wavelets to approximate Random Density Functions.تقريب دالة الكثافة الاحتمالية باستخدام الموجات الصغير

التقدير المشترك للممئينات المشروطة لعملية متوقفة قطعاً

In this paper, the kernel estimation of the conditional quantiles for a strictly stationary stochastic process satisfying the strong mixing condition, which was proposed by Abberger (1997) is studied. Under some mild conditions, the joint asymptotic normality of the kernel estimation of several conditional quantiles estimated at the same conditional point and the joint asymptotic normality of the kernel estimation of the same conditional quantile estimated at different conditional points are established. The performance of the estimations is tested by an application for a real life data .في هذا البحث درسنا تقدير النواة للمئينات المشروطة لعملية متوقفة قطعاً تحقق شرط تبعية ضعيف والذي تم اقتراحه منAbberger (1997) . تحت شروط معينة أثبتنا أن تقدير النواة المشترك لعدة مئينات مشروطة مقدرة عند نقطة شرط محددة وتقدير النواة المشترك لمئين مشروط واحد مقدراّ عند عدد من نقاط الشرط المختلفة تتقارب لتوزيع طبيعي متعدد. ولقد تم اختبار كفاءة التقديرات من خلال تطبيقها على بيانات من الواقع الحياتي. الكلمات المفتاحية: تقدير النواة، المئين المشروط، تقارب لتوزيع طبيعي، شروط تبعية

Kernel estimation of the regression mode for fixed design model

In this paper, we study the problem of estimating nonparametrically the regression mode for fixed design model. We suppose the error random variables are independent. The joint asymptotic normality of the regression mode estimator at different fixed design points is established under some regularity conditions. The performance of the proposed estimator is tested via a simulation study.

تقدير دالة الكثافة ودالة معدل المخاطرة باستخدام كرنال معكوس جاوس

In this paper, we use the Reciprocal Inverse Gaussian (RIG) kernel to estimate nonparametrically the probability density function (pdf) and the hazard rate function for independent and identically distributed (iid) data. The estimator uses adaptive weights depending on the points at which we estimate the functions. We derive the strong consistency, the asymptotic normality and the asymptotic mean squared error (AMSE) of the proposed estimator. Also, the selection of the optimal bandwidth is investigated. The performance of the proposed estimator is compared to that of the Gaussian kernel.تقدير دالة الكثافة ودالة معدل المخاطرة باستخدام كرنال معكوس جاو

The joint asymptotic normality of the conditional quantiles

  Abstract: Let (X,Y)  be a two dimensional random variable with a joint distribution function F(X,Y). This paper studies the kernel estimation of the conditional quantiles of for a given value of  based on a random sample from the above distribution, which was proposed by [12]. In this paper, the joint asymptotic normality of the conditional quantile estimated at a finite number of distinct points is established under some regularity conditions. Moreover, the performance of the conditional quantile estimation in constructing prediction intervals is tested through two applications. The first application deals with simulated data set and the second deals with real life data set

دراسة مقارنة بين مقدر ندارايا- واتسن ومقدر ندارايا- واتسن المرجح لدالة متوسط الانحدار

Nonparametric kernel estimators are widely used in regression estimation. One of the most important kernel estimators of the regression mean function is the Nadaraya-Watson estimator. Another estimator, which is called the Reweighted Nadaraya-Watson estimator has been proposed to improve the performance of the Nadaraya- Watson estimator. In this paper, we have compared theoretically between the two estimators by looking at their asymptotic bias, variance and the mean squared error. The results of this comparison indicated that the bias of the Reweighted Nadaraya-Watson estimator is better than that of the Nadaraya-Watson estimator. Also, a comparison of the practical performance of the two estimators based on simulated and real data has been given. The results of this comparison was consistent with the results of the theoretical comparison and indicated that, the Reweighted Nadaraya-Watson estimator has modified the weakness of the Nadaraya-Watson estimator.تستخدم مقد ا رت النواة اللامعلمية بكثرة في تقدير الانحدار. من أهم مقد ا رت النواة لدالة متوسط الانحدار هو مقدر ندارايا – واتسن. مقدر أخر يسمى مقدر ندارايا- واتسن المرجح قد تم اقتراحه لتحسين أداء مقدر ندارايا – واتس . في هذا البحث نقارن بين المقدرين نظريا بمقارنة تقارب كل من تحيزهم، تباينهم و متوسط مربعات الخطأ. نتائج هذه المقارنة أشارت إلى أن تحيز مقدر ندارايا – واتسن المرجح كان الأقل. بالإضافة لذلك فإنه تم مقارنة الأداء العملي للمقدرين باستخدام بيانات محاكاة وأخرى حقيقية و النتائج كانت متسقة مع نتائج المقارنة النظرية وأشارت إلى أن مقدر ندارايا – واتسن المرجح قد عدل جوانب الضعف في مقدر ندارايا – واتسن