A Pathway Idea for Model Building.

Models, mathematical or stochastic, which move from one functional form to another through pathway parameters, so that in between stages can be captured, are examined in this article. Models which move from generalized type-1 beta family to type-2 beta family, to generalized gamma family to generalized Mittag-Leffler family to Lévy distributions are examined here. It is known that one can likely find an approximate model for the data at hand whether the data are coming from biological, physical, engineering, social sciences or other areas. Different families of functions are connected through the pathway parameters and hence one will find a suitable member from within one of the families or in between stages of two families. Graphs are provided to show the movement of the different models showing thicker tails, thinner tails, right tail cut off etc.

Two-way model with random cell sizes.

We consider inference for row effects in the presence of possible interactions in a two-way fixed effects model when the numbers of observations are themselves random variables. Let Nij be the number of observations in the (i, j) cell, πij be the probability that a particular observation is in that cell and µij be the expected value of an observation in that cell. We assume that the {Nij } have a joint multinomial distribution with parameters n and {πij }. Then µÌ„i . = Σ jπijµij /Σ jπij is the expected value of a randomly chosen observation in the ith row. Hence, we consider testing that the µÌ„i . are equal. With the {πij } unknown, there is no obvious sum of squares and F-ratio computed by the widely available statistical packages for testing this hypothesis. Let Ȳi .. be the sample mean of the observations in the ith row. We show that Ȳi .. is an MLE of µÌ„i ., is consistent and is conditionally unbiased. We then find the asymptotic joint distribution of the Ȳi .. and use it to construct a sensible asymptotic size α test of the equality of the µÌ„i . and asymptotic simultaneous (1 - α) confidence intervals for contrasts in the µÌ„i ..

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.