The Distribution of Family Sizes Under a Time-Homogeneous Birth and Death Process

The number of extant individuals within a lineage, as exemplified by counts of species numbers across genera in a higher taxonomic category, is known to be a highly skewed distribution. Because the sublineages (such as genera in a clade) themselves follow a random birth process, deriving the distribution of lineage sizes involves averaging the solutions to a birth and death process over the distribution of time intervals separating the origin of the lineages. In this article, we show that the resulting distributions can be represented by hypergeometric functions of the second kind. We also provide approximations of these distributions up to the second order, and compare these results to the asymptotic distributions and numerical approximations used in previous studies. For two limiting cases, one with a relatively high rate of lineage origin, one with a low rate, the cumulative probability densities and percentiles are compared to show that the approximations are robust over a wide range of parameters. It is proposed that the probability distributions of lineage size may have a number of relevant applications to biological problems such as the coalescence of genetic lineages and in predicting the number of species in living and extinct higher taxa, as these systems are special instances of the underlying process analyzed in this article.     

Comparisons of certain distribution functions

Certain common distributions, including the exponential, gamma, Gumbel. Weibull and normal are compared by means of partial orderings, extending results given in LISEK (1978).     

Testing optimality of experimental designs for a regression model with random variables

Tsukanov (Theor. Probab. Appl. 26 (1981) 173–177) considers the regression model $\text{E(}\text{y}\text{|}\text{Z}\text{)=}\text{Fp}\text{+}\text{Zq}$, $\text{D(}\text{y}\text{|}\text{Z}\text{)=σ}{}^2\text{I}{}_{\mathrm{n}}$, where $\text{y}\text{(n×1)}$ is a vector of measured values,$\text{F}\text{(n×k)}$ contains the control variables, $\text{Z}\text{(n×l)}$ contains the observed values, and $\text{p}\text{(k×1)}$ and $\text{q}\text{(l×1)}$ are being estimated. Assuming that $\text{Z}\text{=}\text{FL}\text{+}\text{R}$, where $\text{L}\text{(k×l)}$ is non-random, and the rows of $\text{R}\text{(n×l)}$ are i.i.d. $\text{N(}\text{0}\text{,Σ)}$, we extend Tsukanov's results by (i) computing E(detH_p), where H_p is the covariance matrix of p?, the l.s.e. of p, (ii) considering ‘optimality in the mean’ for the largest root criterion, (iii) discussing these equations when the matrix R has a left-spherical distribution.     

An introduction to hypergeometric functions for economists

Hypergeometric functions are a generalization of exponential functions. They are explicit, computable functions that can also be manipulated analytically. The functions and series we use in quantitative economics are all special cases of them. In this paper, a unified approach to hypergeometric functions is given. As a result, some potentially useful general applications emerge in a number of areas such as in econometrics and economic theory. The greatest benefit from using these functions stems from the fact that they provide parsimonious explicit (and interpretable) solutions to a wide range of general problems.     

A gamma-mixture class of distributions with Bayesian application

In this article, a subjective Bayesian approach is followed to derive estimators for the parameters of the normal model by assuming a gamma-mixture class of prior distributions, which includes the gamma and the noncentral gamma as special cases. An innovative approach is proposed to find the analytical expression of the posterior density function when a complicated prior structure is ensued. The simulation studies and a real dataset illustrate the modeling advantages of this proposed prior and support some of the findings.     

A stratified randomized response model for sensitive characteristics using the negative hypergeometric distribution

Taking clue from the pioneer work of Singh and Sedory (2013 Singh, S., Sedory, S.A. (2013). A new randomized response device for sensitive characteristics: an application of the negative hypergeometric distribution. Metron 71:3–8.[Crossref] , [Google Scholar]), we have suggested a new stratified randomized response (RR) model. The properties of the suggested stratified RR model have been studied under proportional and “Neyman” allocations. The study has been also carried out in the presence of crude prior estimates. Numerical illustrations are also given in support of the present study.     

不同总体量和样本量时如何计算比例的置信区间

在总体或者总体子集不大情况下的抽样调查中,往往不易得出合理的关于比例的区间估计。这一类问题在抽样调查实践中已经严重到非说不可的地步。文章讨论了在样本量不大或者(和)在总体不大时估计比例的置信区间时往往忽略的问题,并给出了在不同情况下如何计算置信区间的方法。     

Bounds for the Bayes Error in Classification: A Bayesian Approach Using Discriminant Analysis

We study two of the classical bounds for the Bayes error P _e , Lissack and Fu’s separability bounds and Bhattacharyya’s bounds, in the classification of an observation into one of the two determined distributions, under the hypothesis that the prior probability χ itself has a probability distribution. The effectiveness of this distribution can be measured in terms of the ratio of two mean values. On the other hand, a discriminant analysis-based optimal classification rule allows us to derive the posterior distribution of χ, together with the related posterior bounds of P _e . Research partially supported by NSERC grant A 9249 (Canada). The authors wish to thank two referees, for their very pertinent comments and suggestions, that have helped to improve the quality and the presentation of the paper, and we have, whenever possible, addressed their concerns.