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International Biometric Society
THE KUMMER BETA UNIFORM DISTRIBUTION AND ITS APPLICATIONS
Thiago Gentil Ramires1, Ana Julia Righetto1, Rodrigo Rossetto Pescim1, Edwin Moises
Marcos Ortega1
Programa de Pós-Graduação em Estatística e Experimentação Agronômica –
Departamento de Ciências Exatas, ESALQ – Universidade de São Paulo, Brasil.
1
Recently, attempts have been made to define new families of probability distributions that
provide great flexibility in modeling skewed data in practice. Some recent distributions with
positive support are developed such as the beta generalized half-normal geometric
(BGHNG) distribution (Ramires et al. 2013), the beta Rayleigh generalized (BRG)
distribution (Cordeiro et al. 2013a), the Burr XII geometric (BXIIG) distribution (Lanjoni et al.
2013), the beta exponentiated Weibull (BEW) distribution (Cordeiro et al. 2013b). Pescim et
al. (2013) defined a new class of distributions so-called the Kummer beta generalized (KBG) which adds three shape parameters, a>0 and b>0 for introducing skewness and thereby
promoting weight variation of the tails and the parameter c ϵ R that “squeezes” the
probability density function (pdf) to the left or to right. For modeling response variables with
limited support, for example data score, it is necessary to truncate the distribution so that the
estimated values do not exceed their natural limits. Another alternative for modeling some
data set is to use distribution whose support is limited such as the uniform distribution. This
distribution is widely uses for generate random values from another distribution, but its use
becomes limited because it has constant probability throughout of the range. An alternative
distribution to model data sets with limited support is proposed in this work. We, introduce a
new distribution based on the KB-G family of distributions called the Kummer beta
generalized uniform (KBU) distribution taking the cumulative distribution function (cdf) of the
uniform distribution as the cdf baseline. Let Y be a uniform random variable with cdf G(y)=(xl)/(u-l) for l ≤ y ≤ u, the random variable X, say X~KBU(a,b,c,l,u), has five parameters. We
note that for l=0 and u=1 the Kummer-beta (KB) distribution is obtained as particular case.
We also provide the general expansions for the moments, generating function, mean
absolute deviations, Rényi entropy and order statistics for the new distribution. We adopt the
maximum likelihood method to fit the distribution and illustrate its potentiality with an
application to a real data set which describes a random sample (n=372) of the Brazilian
schoolchildren of 8 to10 years old which was examined. They were separate in two sample
groups: 1) 186 schoolchildren in group denominated “dental treatment group”, who
presented cavitated dental caries lesions and dental treatment needs; 2) 186 schoolchildren
in the group denominated the “group without caries”. To evaluate impact of dental treatment
on schoolchildren’s Oral Health Related Quality of Life (OHRQoL), the Child Perceptions
Questionnaires (CPQ) instrument was administered in the begining of the study and in a
four weeks follow-up time in both groups. The CPQ consists of 25 items covering 4 domains:
oral symptoms (OS), functional limitation (FL), emotional well-being (EWB), and social wellbeing (SWB). Each item is scored on a 5-point Likert scale to rate the impact of oral health
status on an aspect of quality of life (described by the item), with responses ranging from
“none of the time” (score 0) to “every day or almost every day” (score 4). The overall CPQ
(range 0-100) is sum of the results of all questions, and the higher the value of this sum, the
worse the impact of oral health on quality of life is considered to be. Each domain is
characterized: OS had 5 questions and value range 0-20; LF had 5 questions and value
range 0-20; EWB had 5 questions and value range 0-20; SWB had 10 questions and value
range 0-40.
International Biometric Conference, Florence, ITALY, 6 – 11 July 2014