Mixtures of Mixtures Based on Multiresolution Analysis Theory

In this article, the approaches for exploiting mixtures of mixtures are expanded by using the Multiresolution family of probability density functions (MR pdf). The flexibility and the properties of local analysis of the MR pdf facilitate the location of subpopulations into a given population. In order to do this, two algorithms are provided.

The MR model is more flexible in adapting to the different subpopulations than the traditional mixtures. In addition, the problems of identification of mixtures distributions and the label-switching do not appear in the MR pdf context.     

A note on approximate confidence limits for the negative binomial model

Four methods of approximating confidence limits for the single negative binomial parameter, P, are outlined and an empirical study is presented. Some remarks on prediction intervals are also included.     

Irregularities in X(Y) from Y(X) in linear calibration

Let X be an input measurement and Y the output reading of a calibrated instrument, with Y(X) as the calibration curve. Solving X(Y) projects an instrumental reading back onto the scale of measurements as an object of pivotal interest. The arrays of instrumental readings are projected in this manner in practice, yielding arrays of calibrated measurements, typically subject to errors of calibration. The effects of calibration errors on the properties of calibrated measurements are examined here under linear calibration. Irregularities arise as induced dependencies, inflated variances, non-standard distributions, inconsistent sample means, the underestimation of measurement variance, and other unintended consequences. On the other hand, conventional properties are seen to remain largely in place in the use of selected regression diagnostics and in one-way comparative experiments using calibrated data.     

Trimmed and Outer Means and Their Variances

Some properties of trimmed and outer means for the normal situation are considered; in particular, the mean of the outer half of the sample has the same variance as the mean of the inner half. An inequality involving variances of unbiased estimators of location and their complements is derived and some of its consequences are examined.     

Use of the Correlation Coefficient with Normal Probability Plots

The use of the correlation coefficient is suggested as a technique for summarizing and objectively evaluating the information contained in probability plots. Goodness-of-fit tests are constructed using this technique for several commonly used plotting positions for the normal distribution. Empirical sampling methods are used to construct the null distribution for these tests, which are then compared on the basis of power against certain nonnormal alternatives. Commonly used regression tests of fit are also included in the comparisons. The results indicate that use of the plotting position p_i = (i - .375)/(n + .25) yields a competitive regression test of fit for normality.     

Aggregation of Variables in Least-Squares Regression

I consider the properties of the estimator when the true model is y = β₁ x ₁ + β₂ x ₂ + u, but the restriction β₁ = β₂ = β is incorrectly imposed. I show that the probability limit of is a weighted sum of β₁ and β₂; the weights sum to 1 but do not necessarily lie in the unit interval, so plim need not be bounded by β₁ and β₂. Sufficient conditions for such bounding are derived. Certain changes in the moments of x ₁ and x ₂ have “perverse” effects on the weights. I illustrate the consequences of inappropriate aggregation of variables with an empirical example of the effect of research and development investment on productivity.     

Editor's Report

There are two common methods for statistical inference on 2 × 2 contingency tables. One is the widely taught Pearson chi-square test, which uses the well-known χ²statistic. The chi-square test is appropriate for large sample inference, and it is equivalent to the Z-test that uses the difference between the two sample proportions for the 2 × 2 case. Another method is Fisher’s exact test, which evaluates the likelihood of each table with the same marginal totals. This article mathematically justifies that these two methods for determining extreme do not completely agree with each other. Our analysis obtains one-sided and two-sided conditions under which a disagreement in determining extreme between the two tests could occur. We also address the question whether or not their discrepancy in determining extreme would make them draw different conclusions when testing homogeneity or independence. Our examination of the two tests casts light on which test should be trusted when the two tests draw different conclusions.