A Pointwise Estimator for the k-Fold Convolution of a Distribution Function

ABSTRACT This article is concerned with some parametric and nonparametric estimators for the k-fold convolution of a distribution function. An alternative estimator is proposed and its unbiasedness, asymptotic unbiasedness, and consistency properties are investigated. The asymptotic normality of this estimator is established. Some applications of the estimator are given in renewal processes. Finally, the computational procedures are described and the relative performance of these estimators for small sample sizes is investigated by a simulation study.

[1]  P. L. Novi Inverardi,et al.  MSE Comparison of Some Different Estimators of Entropy , 2003 .

[2]  D. J. McConalogue,et al.  Numerical treatment of convolution integrals involving distributions with densities having singularities at the origin , 1981 .

[3]  Lirong Cui,et al.  Some Normal Approximations for Renewal Function of Large Weibull Shape Parameter , 2003 .

[4]  Min Xie,et al.  On the solution of renewal-type integral equations , 1989 .

[5]  D. J. McConalogue,et al.  A Numerical Algorithm for Recursively-Defined Convolution Integrals Involving Distribution Functions , 1976 .

[6]  Ali Esmaili,et al.  Probability and Random Processes , 2005, Technometrics.

[7]  Ing Rj Ser.  Approximation Theorems of Mathematical Statistics , 1980 .

[8]  M. Pagano,et al.  On Obtaining Permutation Distributions in Polynomial Time , 1983 .

[9]  F. M. Gryna.  Quality planning and analysis , 1970 .

[10]  Kenny S. Crump,et al.  Numerical Inversion of Laplace Transforms Using a Fourier Series Approximation , 1976, J. ACM.

[11]  J. Bert Keats,et al.  Quality Planning and Analysis , 1981 .

[12]  Edward W. Frees,et al.  Warranty analysis and renewal function estimation , 1986 .

[13]  Nonparametric estimation of some quantities of interest from renewal theory , 2003 .

[14]  Helmut Schneider,et al.  Comparison of Nonparametric Estimators for the Renewal Function , 1990 .

[15]  Herbert Solomon,et al.  Monte Carlo simulation of the renewal function , 1981 .

[16]  I. Gertsbakh,et al.  Renewal Function and Interval Availability: A Numerical Monte-Carlo Study , 2004 .

[17]  L. Baxter.  Some remarks on numerical convolution , 1981 .