Mean-variance Relationship Research Articles

Dispersion statistics in overdispersed mixture models

Samples are often found to be too heterogeneous to be explained by a one-parameter family of models in the sense that the implicit mean-variance relationship in such a family is violated by the data. This phenomenon is often called over-dispersion. The most frequently used method in dealing with over-dispersion is to mix a one-parameter family creating a two parameter marginal mixture family for the data. In this paper, we investigate performance of estimators such as maximum likelihood estiamtor, method of moment estimator, and maximum quasi-likelihood estimator in negative binomial and beta-binomial distribution. Numerical results are obtained for various mean parameter and dispersion parameter in both distributions, and we conclude that the moment estimators are very superior in the sense of bias and asymptotic relative efficiency. Two real data sets are used for applications.

Sample Sizes and Distributions of Japanese Beetles (Coleoptera: Scarabaeidae) Captured in Lure Traps

Sampling statistics were determined for Japanese beetles, Popillia japonica Newman, caught in traps baited with a 3:7:3 mixture of phenethyl propionate/eugenol/geraniol, or with ( R, Z )-5-(1-decenyl)dihydro-2(3 H )-furanone (Japonilure), or with a combination of both lures. Iwao’s patchiness regression and a quadratic model were inappropriate for modeling the mean-variance relationships. Taylor’s power law was used to model these data, and relationships were developed for beetle numbers captured by traps baited with phenethyl propionate-eugenol-geraniol alone or in combination with Japonilure and for males and females captured by traps baited with Japonilure alone. Relationships to determine sample sizes for fixed levels of precision were also developed. The I n ( X + 1) transformation did not standardize the mean-variance relationship for any data set, but the Healy & Taylor transformation was effective on all data sets.

Some approximations for the moments of a process used in diffusion of new products

We consider approximations for the moments of a well known birth process commonly used in a variety of applications involving diffusion of new products and services. This process also has applications in Epidemiology and Social Sciences. The available approximation for the mean function involves the use of a well known population dynamics model. Direct substitution of this formula in the mean variance relationship leads to an identically zero variance function. We propose an improvement for approximating the mean function. Further, we propose two nontrivial approximations for the variance function. Both of the approximations are evaluated in a limited simulation study. As a extension, the problem of approximating the correlation function is also considered.

Paradox or At Least Variance Found: A Comment on “Mean-Variance Approaches to Risk-Return Relationships in Strategy: Paradox Lost”

In general, the problem is that the computed mean-variance relationship for a period of time cannot be identified in distinction to the effects of shifts in the relationship over time—without additional information or assumptions. Thus, using a mean-variance approach to risk-return relationships means that statements about the nature of the mean-variance association cannot be confirmed in a nontrivial fashion within the empirical system nor generalized to any other time period—including subperiods. (Ruefli 1990) (emphasis in original)

Regression-based tests for overdispersion in the Poisson model

A property of the Poisson regression model is mean-variance equality, conditional on explanatory variables. ‘Regression-based’ tests for this property are proposed in a very general setting. Unlike classical statistical tests, these tests require specification of only the mean-variance relationship under the alternative, rather than the complete distribution whose choice is usually arbitrary. The optimal regression-based test is easily computed as the t-test from an auxiliary regression. If a distribution under the alternative hypothesis is in fact specified and is in the Katz system of distributions or is Cox's local approximation to the Poisson, the score test for the Poisson distribution is equivalent to the optimal regression-based test.

Sampling Distributions and Sequential Sampling Plans for Perkinsiella saccharicida Kirkaldy (Hemiptera: Delphacidae) and Tytthus spp. (Hemiptera: Miridae) on Sugarcane

Sampling statistics of eggs, nymphs, and adults of Perkinsiella saccharicida Kirkaldy and its predators, Tytthus spp., were determined on six varieties of sugarcane ( Saccharum spp.) in southern Queensland, Australia. Iwao’s patchiness regression and Taylor’s power law were used to model the relationship between mean and variance of counts. Taylor’s power law gave better fits to all data except for counts of eggs per egg mass. Sample sizes for fixed levels of precision and fixed-precision-level stop lines for sequential sampling were developed for all stages of P. saccharicida and nymphs and adults of Tytthus spp. The In(x + 1) transformation standardized the mean-variance relationship for P. saccharicida nymphs, adults, and eggs per egg mass and Tytthus spp. adults, but not for P. saccharlctda egg masses or Tytthus spp. nymphs. The Healy and Taylor transformation was not effective on P. saccharicida data but standardized Tytthus spp. nymph and adult counts.

Heterogeneity of variance in field experiments: some causes and practical implications

SUMMARYData from 62 field experiments on various crops in 1973–85 at Rothamsted Experimental Station were used to obtain knowledge about the heterogeneity of variance of yield. The results suggest that explaining heterogeneity of variance in terms of a mean–variance relationship is inadequate; better understanding of the patterns of changing variance can be gained by investigating the influence of the experimental factors on the variance. The effects of the factor ‘amount of nitrogen’ on mean yield and on the variance were positively correlated; factors which tend to affect variability but not the mean were identified. Suggestions are given for controlling variance but the need for further experimental evidence is emphasized.