A nonparametric approach to analyzing large human populations.

ABSTRACT

There are many statistical techniques that require the assumption that the population being studied is normally distributed--regression analysis, multivariate analysis, time series analysis, and so on. Unfortunately, as the development of survey sampling has long acknowledged, large human populations are usually stratified into several different subpopulations. Since the boundaries between the strata are somewhat blurred, they are not independent, so the overall distribution of the population tends to be multimodal rather than normal. In this paper, a technique is developed to find these multimodal techniques using nonparametric density estimation. Its effectiveness is demonstrated by means of an example.

ABSTRACT

PIP The nonparametric kernal density estimation technique based on the survey sampling technique is explained, and its effectiveness is demonstrated by a simulation and by an actual example in order to show its usefulness in capturing multimodel distribution. The universe under investigation is usually stratified into several subpopulations. Incorrect assumptions are made when unimodel techniques are used on multimodel populations. Multimodel populations can also be dealt with by using only large samples with estimates made only on the sample mean, but this approach is not always possible. Another option is to study only one of the subpopulations, which can result in bias or oversmoothing the data. Other methods of multimodel analysis involve use of multiple regression, which does not consider overall distribution in the splintering of the population. Mixing distributions, which attempts to find a linear combination of usually normal distributions is another multimodel method which may result in data which have a nonnormal distribution. The method described in this article estimates the density function on each subpopulation and then assigns weights based on knowledge of the subpopulations before the densities are combined. In density estimation, calculation of window size is based on additional random variables, and then a determination is made of how many modes are required. The ramifications of a theorem which considers that several density estimates of the density functions are continuous and bounded and K(x) is a known density function that is bounded and symmetric and has compact support are described. The simulation example generates sample means and variances on 3 subpopulations P1, P2, and P3. P1 is normal with a mean of 5.01 and variance of .09 squared, and P3 is normal with a mean of 6.23 and variance of .11 squared. Means and variances are generated with and without stratification, and a comparison is made by computing the integrated mean square error. The result is a reduction in error by a factor of 10 using stratification. The example is based on a clinical trial of 100 people to compare safety and efficacy of antidepressants.