Abstract:
Pathway idea is a switching mechanism by which one can go from one functional
form to another, and to yet another. It is shown that through a parameter α, called the
pathway parameter, one can connect generalized type-1 beta family of densities, generalized
type-2 beta family of densities, and generalized gamma family of densities, in the scalar
as well as the matrix cases, also in the real and complex domains. It is shown that when
the model is applied to physical situations then the current hot topics of Tsallis statistics
and superstatistics in statistical mechanics become special cases of the pathway model, and
the model is capable of capturing many stable situations as well as the unstable or chaotic
neighborhoods of the stable situations and transitional stages. The pathway model is shown
to be connected to generalized information measures or entropies, power law, likelihood ratio
criterion or λ−criterion in multivariate statistical analysis, generalized Dirichlet densities,
fractional calculus, Mittag-Leffler stochastic process, Krätzel integral in applied analysis,
and many other topics in different disciplines. The pathway model enables one to extend
the current results on quadratic and bilinear forms, when the samples come from Gaussian
populations, to wider classes of populations.
