Abstract
In this article we study and obtain some general results on the quantile functions for the generalized beta family and the family of beta generated distributions. Having described the standardization rule, we have derived the quantile function of the 5 parameter generalized beta family of distributions [21, 22]. Further, quantile rules for distributional model building have been applied to generate quantile functions of several known and unknown distributions. Attempts have been made to obtain and study the quantile functions of size biased generalized beta distributions and generalized beta generated distributions. Finally, we have applied the proposed results to simulated as well as real datasets.
Highlights
A quantile is the value that corresponds to a specified proportion of a sample or population
SOME GENERAL RESULTS ON quantile function (QF) FOR THE generalized beta distribution (GB) FAMILY
The QF defined in (20) is in the form, λ + ηS(p, α), where λ = 0, η = b and α = (a, c, α, β). This QF includes the quantile functions of the generalized beta I (GBI) and generalized beta II (GBII) distribution corresponding to c = 0 and c = 1, which in turn includes the QF of a numerous distributions as special cases, few of which are shown in table 2
Summary
A quantile is the value that corresponds to a specified proportion of a sample or population. If F (Y ) is the distribution function of Y such that F (Y ) = p Y = F −1(p) = Q(p) is the corresponding quantile function. Lemma 1 (The standardization rule) The quantile function of any distribution can be written in the general form. Q(p) = λ + ηS(p; α) where, λ is the location parameter, η is the scale parameter and α is a vector of shape parameters and S(p, α) is the quantile function of the standard distribution. The standard distribution F0(Y ) is free from the location and scale parameter, but it may contain parameter(s) that control the shape of the distribution, so that the standard quantile function, Q0(p) may contain shape parameter(s) as well.
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