Comparison Density Representation of Traditional Test Statistics for the Equality of Two Population Proportions

Let <TEX>$p_1$</TEX> and <TEX>$p_2$</TEX> be the proportions of two populations. To test the hypothesis <TEX>$H_0 : p_1 = p_2$</TEX>, we usually use the <TEX>$x^2$</TEX> statistic, the large sample binomial statistic Z, and the Generalized Likelihood Ratio statistic-2log <TEX>$lambda$</TEX>developed based on different mathematical rationale, respectively. Since testing the above hypothesis is equivalent to testing whether two populations follow the common Bernoulli distribution, one may also test the hypothesis by comparing 1 with the ratio of each density estimate and the hypothesized common density estimate, called comparison density, which was devised by Parzen(1988). We show that the above traditional test statistics ate actually estimating the measure of distance between the true densities and the common density under <TEX>$H_0$</TEX> by representing them with the comparison density.