Estimation of the cumulative by fourier series methods and application to the insertion problem

In a paper published in last year's A.C.M. Proceedings a new method for estimating the population density, i.e., f, was described1 The procedure was based upon the use of sample trigonometric moments as unbiased estimates of density f's Fourier coefficients. It was noted, but not further discussed, that the cumulative distribution function F associated with density f could be approximated by an estimator Fˆm associated with density estimator Fˆm, and that Fˆm possessed the following properties: The coefficients Bˆk are unbiased estimators of the Fourier coefficients of the function F(x)-[(x-a)/(b-a)] and therefore simple M.I.S.E. (Mean Integrated Square Error) expressions could be obtained for estimator Fˆm. Also as m→@@, Fˆm→F* where F* represents the sample cumulative (step function). In this paper the estimator Fˆm will be more fully discussed and an application to the problem of address calculation insertion will be described.