The purpose of this dissertation is to examine the problem of estimation of a univariate probability density function. Let Y₁, Y₂, …, Y_n be a sample of n independent observations, each distributed according to an unknown continuous density function f(y). Given this sequence of observations, how can one estimate f(y)? Chapter I presents the historical background and a literature review.of existing methods for the estimation of a probability. density function. In addition, a section is devoted to the application of density estimation. In particular, we consider the practical application to reliability analysis. The estimator of the unknown density function developed in Chapter II is similar to one proposed by Rosenblatt (1956) and by Parzen (1962). However, the kernel we consider is a function of the rank of each observation. We use, as our kernel, the asymptotic distribution of the order statistics of a sample. We refer to this estimator as the normal rank kernel estimator. In order to test the performance of our estimator, we have performed an experimental analysis by monte carlo studies. Chapter III uses the estimator of Chapter II as the foundation for the development of a. recursive estimation procedure. The technique employed is the method of successive substitution in which the solution at each iteration is used to generate the next solution until convergence is achieved. Consequently, we call this estimator the iterative estimator. Again, we have performed a simulation to compare the estimator of Chapter III with that of Chapter II. In Chapter IV, a sequential procedure is developed for estimation of a probability density function. Initially, a normal distribution with mean and variance y̅and s², respectively, is fitted to the data and a goodness of fit test is performed. This hypothesis rejected, a sequential procedure employing the concept of spline functions is used. Several examples are given in Chapter V which illustrate the various methods of density estimation introduced.in the preceding chapters. The examples use data that are both simulated and real. Also, an example estimates both the reliability and hazard functions. Finally, relevant computer programs (Fortran) and descriptions of their utilization appear in the appendices. The program contained in Appendix A has a dual purpose in that by proper choice of an input parameter, the program will be executed for either the rank kernel estimator or for the iterative estimator, while Appendix B contains the computer program for Chapter IV. / Ph. D.