| The raw moments of a random variable, or of a distribution, are the expectations of the powers of the random variable with a the given distribution. The moments of a density function play an important role in theoretical and applied statistics. In fact, in some cases, if the moments are known the density can be determined. A function called the moment generating function (MGF) gives a representation of all moments without reference to the random variable. When it exists, the MGF completely determines the distribution of a random variable.;A major drawback of the MGF is that it does not exist for every distribution. To remedy this drawback, the characteristic function (CF), which exists for every distribution, is often used.;For estimating the empirical distribution function is somewhat smooth and is a consistently strong estimator of the distribution function, although further smoothing may be advantageous. In this dissertation, a nonparametric estimator was developed. This estimator smoothes the estimate of the characteristic function of the underlying distribution obtained using the estimated CF of the kernel method. This dissertation proposes a method to estimate the underlying distribution via the estimated CF. This estimator is based on the kernel estimator of the PDF. One marked difference, however, is that the ECF is used to approximate the optimal kernel function. Further, for this estimator the bandwidth depends on t, the argument of the CF. Minimizing the ℓ2 norm of the MSE results in an estimator that does not depend directly on a user-supplied kernel function, or bandwidth. Various tests of this new technique indicated the proposed method performed better or at least comparably to the kernel method. |
