Yogendra P. Chaubey

Application of Bernstein Polynomials for smooth estimation of a distribution and density function

The empirical distribution function is known to have optimum properties as an estimator of the underlying distribution function. However, it may not be appropriate for estimating continuous distributions because of its jump discontinuities. In this paper, we consider the application of Bernstein polynomials for approximating a bounded and continuous function and show that it can be naturally adapted for smooth estimation of a distribution function concentrated on the interval [0,1] by a continuous approximation of the empirical distribution function. The smoothness of the approximating polynomial is further used in deriving a smooth estimator of the corresponding density. The asymptotic properties of the resulting estimators are investigated. Specifically, we obtain strong consistency and asymptotic normality under appropriate choice of the degree of the polynomial. The case of distributions with other compact and non-compact support can be dealt through transformations. Thus, this paper gives a general method for non-parametric density estimation as an alternative to the current estimators. A small numerical investigation shows that the estimator proposed here may be preferable to the popular kernel-density estimator.

ON SMOOTH ESTIMATION OF SURVIVAL AND DENSITY FUNCTIONS

Incorporating the Hille theorem, smooth estimators of survival and density func- tions are considered , and their (asymptotic) properties are studied in an unified manner. A comparative picture of the so called kernel and nearest neighbor methods and the proposed one is presented with due emphasis on the (asymptotic) bias and mean square error criteria.

On smooth estimation of mean residual life

Abstract The methodology developed in Chaubey and Sen (1996) , ( Statistics and Decision , 14 , 1–22) is adopted here for smooth estimation of mean residual life. It is seen that Hille (1948) , ( Functional Analysis and Semigroups , AMS, New York) theorem, which has been vital in the development of smooth estimators of the distribution, density, hazard and cumulative hazard functions, does not work well in the current context. For this reason a modified weighting scheme is proposed for estimation of the mean residual life. Asymptotic properties of the resulting estimator is investigated along with its aging aspects.

On the computation of aggregate claims distributions: some new approximations

Abstract This paper proposes a new approximation to the aggregate claims distribution based on the inverse Gaussian (IG) distribution. It is compared to several other approximations in the literature. The IG approximation compares favorably to the well-known gamma approximation. We also propose an IG-gamma mixture that approximates the true distribution extremely accurately, in a large variety of situations.

Moments of a ratio of two positive quadratic forms in normal variables

In this note we have derived the expression for the ath moment of the ratio , where represent independent chi-square random variables with r degrees of freedom, in terms of a finite series. Some particular cases of importance are also discussed.

Smooth Isotonic Estimation of Density, Hazard and MRL Functions

Incorporating the classical smoothing lemma of Hille (1948) , as adapted in Chaubey and Sen (1996), smooth isotonic estimators of density, hazard, mean residual life, and other functionals of (survival) distribution functions are proposed here. Along with the computational aspects, their (asymptotic) properties are discussed. Some extensions to the case of random censoring are also considered.

Noise Reduction of cDNA Microarray Images Using Complex Wavelets

Noise reduction is an essential step of cDNA microarray image analysis for obtaining better-quality gene expression measurements. Wavelet-based denoising methods have shown significant success in traditional image processing. The complex wavelet transform (CWT) is preferred to the classical discrete wavelet transform for denoising of microarray images due to its improved directional selectivity for better representation of the circular edges of spots and near shift-invariance property. Existing CWT-based denoising methods are not efficient for microarray image processing because they fail to take into account the signal as well as noise correlations that exist between red and green channel images. In this paper, two bivariate estimators are developed for the CWT-based denoising of microarray images using the standard maximum a posteriori and linear minimum mean squared error estimation criteria. The proposed denoising methods are capable of taking into account both the interchannel signal and noise correlations. Significance of the proposed denoising methods is assessed by examining the effect of noise reduction on the estimation of the log-intensity ratio. Extensive experimentations are carried out to show that the proposed methods provide better noise reduction of microarray images leading to more accurate estimation of the log-intensity ratios as compared to the other CWT-based denoising methods.

Wavelet-based estimation of regression function for dependent biased data under a given random design

In this article, we consider the estimation of the regression function in a dependent biased model. It is assumed that the observations form a stationary α-mixing sequence. We introduce a new estimator based on a wavelet basis. We explore its asymptotic performances via the supremum norm error and the mean integrated squared error. Fast rates of convergence are established.

On some improved ridge estimators

SummaryIn this paper two variants of ridge estimators are proposed. One is a linear function of the generalized ridge estimator (GRE) of Hoerl and Kennard and the other is a convex combination of OLSE and GRE. These seem to have some desirable properties. Operational versions are studied using simulation.

Wavelet Based Estimation of the Derivatives of a Density for a Negatively Associated Process

Here we adopt the method of estimation for the derivatives of a probability density function based on wavelets discussed in Prakasa Rao (1996) to the case of negatively associated random variables. An upper bound on Lp-loss for the resulting estimator is given which extends such a result for the integrated mean square error (IMSE) given in Prakasa Rao (1996). Also, considering the case of derivative of order zero, the results given by Kerkyacharian and Picard (1992), Tribouley (1995) and Leblanc (1996) are obtained as special cases.