The weighted Hellinger distance in the multivariate kernel density estimation

The kernel multivariate density estimation is an important technique to estimate the multivariate density function. In this investigation we will use Hellinger Distance as a measure of error to evaluate the estimator, we will derive the mean weighted Hellinger distance for the estimator, and we obtain the optimal bandwidth based on Hellinger distance. Also, we propose and study a new technique to select the matrix of bandwidths based on Hellinger distance, and compare the new technique with the plug-in and the least squares techniques

Application of Fractional Moments for Comparing Random Variables with Varying Probability Distributions

New methods are being presented for statistical treatment of different random variables with unknown probability distributions. These include analysis based on the probability circles, probability ellipses, generalized mean values, generalized Pearson correlation coefficient and the beta-function analysis. Unlike other conventional statistical procedures, the main distinctive feature of these new methods is that no assumptions are made about the nature of the probability distribution of the random series being evaluated. Furthermore, the suggested procedures do not introduce uncontrollable errors during their application. The effectiveness of these methods is demonstrated on simulated data with extended and reduced sample sizes having different probability distributions

Statistical properties of a localization-delocalization transition induced by correlated disorder

The exact probability distributions of the resistance, the conductance and the transmission are calculated for the one-dimensional Anderson model with long-range correlated off-diagonal disorder at E=0. It is proved that despite of the Anderson transition in 3D, the functional form of the resistance (and its related variables) distribution function does not change when there exists a Metal-Insulator transition induced by correlation between disorders. Furthermore, we derive analytically all statistical moments of the resistance, the transmission and the Lyapunov Exponent. The growth rate of the average and typical resistance decreases when the Hurst exponent $H$ tends to its critical value ($H_{cr}=1/2$) from the insulating regime. In the metallic regime $H\geq1/2$, the distributions become independent of size. Therefore, the resistance and the transmission fluctuations do not diverge with system size in the thermodynamic limit

A Bandwidth Selection For Kernel Density Estimation Of Functions Of Random Variables

In this investigation, the problem of estimating the probability density function of a function of m independent identically distributed random variables, g(X1,X2,...,Xm) is considered. The choice of the bandwidth in the kernel density estimation is very important. Several approaches are known for the choice of bandwidth in the kernel smoothing methods for the case m=1 and g is the identity. In this study we will derive the bandwidth using the least square cross validation and the contrast methods. We will compare between the two methods using Monte Carlo simulation and using an example from the real life. © 2003 Elsevier B.V. All rights reserved

Moment Inequalities Derived From Comparing Life With Its Equilibrium Form

A life is characterized by a nonnegative random variable. Associated with life, two notions are interesting in applications such as biomedical research, engineering and statistics. These are the notions of random remaining life at a certain age and its equilibrium (stationary) limit as the age tends to infinity. In the current investigation, comparisons between these three notions both stochastically and in expectation are discussed where moments inequalities based on these comparisons are presented. These inequalities are then used to develop testing the hypothesis of exponentiality against orderings based on these comparisons. © 2004 Elsevier B.V. All rights reserved

Weighted Hellinger Distance As An Error Criterion For Bandwidth Selection In Kernel Estimation

Ever since the pioneering work of Parzen [Parzen, E., 1962, On estimation of a probability density function and mode. Annales of Mathematics and Statistics , 33, 1065-1076.], the mean-square error (MSE) and its integrated form (MISE) have been used as the criteria of error in choosing the window size in kernel density estimation. More recently, however, other criteria have been advocated as competitors to the MISE, such as the mean absolute deviation or the Kullback-Leibler loss. In this note, we define a weighted version of the Hellinger distance and show that it has an asymptotic form, which is one-fourth the asymptotic MISE under a slightly more stringent smoothness conditions on the density f . In addition, the proposed criteria give rise to a new way for data-dependent bandwidth selection, which is more stable in the sense of having smaller MSE than the usual least-squares cross-validation, biased cross-validation or the plug-in methodologies when estimating f . Analogous results for the kernel distribution function estimate are also presented

Testing Normality Using Kernel Methods

Testing normality is one of the most studied areas in inference. Many methodologies have been proposed. Some are based on characterization of the normal variate, while most others are based on weaker properties of the normal. In this investigation, we propose a new procedure, which is based on the well-known characterization; if X1 and X2 are two independent copies of a variable with distribution F, then X1 and X2 are normal if and only if X1 - X2 and X1 + X2 are independent. If X1, ..., Xn is a random sample from F, we test that F is normal by testing nonparametrically that uii* = Xi - Xi* and vii* = Xi + Xi* are independent, i ≠ i* = 1, 2, ..., n. This procedure has several major advantages; it applies equally to one-dimensional or multi-dimensional cases, it does not require estimation of parameters, it does not require transformation to uniformity, it does not require use of special tables of coefficients, and it does have very good power requiring much less number of iterations to reach stable results

Further Moments Inequalities Of Life Distributions With Hypothesis Testing Applications: The Ifra, Nbuc And Dmrl Classes

In a recent article, Ahmad (J. Statist. Plann. Inference 92 (2001) 121-132), one of the authors presented moments inequalities for classes of life distributions including increasing failure rate, new better than used, new better than used in expectation and harmonic new better than used in expectation. He then uses these inequalities to devise new testing procedures for exponentiality against an alternative among the four classes indicated above. Three other classes, the increasing failure rate average, the new is better than used in convex ordering and the decreasing mean residual lifetime, were left untreated since the techniques employed in Ahmad (2001) did not apply to them. Thus, new techniques are developed here to address these three classes completing the study of the perennial group of life distributions. The tests proposed here are simpler than those known in the literature with very good efficiencies. © 2002 Elsevier B.V. All rights reserved

Analysis Of Kernel Density Estimation Of Functions Of Random Variables

In the current investigation, the problem of estimating the probability density of a function of m independent identically distributed random variables, g(X1..... Xm) is considered. Defining the integrated square contrast (ISC)and its mean (MISC), we study the central limit theorem of (ISO-MISC) as well as the second order approximation of both ISC and MISC. Via simulation and also using real data, we address some of the practical aspects of choosing the optimal bandwidth which minimizes the asymptotic MISC and its data based analog which minimizes ISC

Bounds Of Moment Generating Functions Of Some Life Distributions

In this article we show that if a life has new better than used in expectation (NBUE) ageing property and if the mean life is finite then the moment generating function exists and is finite. In fact, the moment generating function is shown to be bounded above by that of the exponential distribution with the same mean. Analogous results are also proven for two much bigger families of life distribution, namely, the new better than renewal used in expectation (NBRUE) and the renewal new is better than used in expectation (RNBUE) and the renewal new better than renewal used in expectation (RNBRUE), provided that the life has finite two moments. Further, stronger results are also obtained for the smaller new better than used version of the above classes. © 2005 Springer-Verlag