A Semiparametric Approach to Density Estimation

One method of fitting a parametric density function f(x, θ) is first to estimate θ by maximum likelihood, say, and then to estimate f(x, θ) by . On the other hand, when the parametric model does not hold, the true density f(x) may be estimated nonparametrically, as in the case of a kernel estimate . The key idea proposed is to fit a combination of the parametric and nonparametric estimates, namely where π (0 ≤ π ≤ 1) is unknown. The parameter π is estimated from the data, and its estimate is then used in as the proposed density estimate. The main point is that we expect to be close to unity when the parametric model prevails, and close to zero when it does not hold. We show that, under certain conditions, converges to the parametric density when the parametric model holds and approaches the true f(x) when the parametric model does not hold. The procedure was applied to a number of actual data sets. In each case the maximum likelihood estimate was readily obtained and the semiparametric density estimate provided a better understanding of which model prevailed. Other compromise estimates may also be used. One such is the geometric mean of cumulative distribution functions: where is a parametric estimate of the parametric distribution function F(x, θ) and is a nonparametric estimate of F(x).