Abstract
The Kumaraswamy distribution is useful for modeling variables whose support is the standard unit interval, i.e., (0, 1). It is not uncommon, however, for the data to contain zeros and/or ones. When that happens, the interest shifts to modeling variables that assume values in [0, 1), (0, 1] or [0, 1]. Our goal in this paper is to introduce inflated Kumaraswamy distributions that can be used to that end. We consider inflation at one of the extremes of the standard unit interval and also the more challenging case in which inflation takes place at both interval endpoints. We introduce inflated Kumaraswamy distributions, discuss their main properties, show how to estimate their parameters (point and interval estimation) and explain how testing inferences can be performed. We also present Monte Carlo evidence on the finite sample performances of point estimation, confidence intervals and hypothesis tests. An empirical application is presented and discussed.
Highlights
Oftentimes practitioners need to model variables that assume values in the standard unit interval, (0, 1), such as rates, proportions and concentration indices
In this paper we develop alternative laws: we introduce the class of inflated Kumaraswamy distributions
Even though the null hypothesis is not rejected for both distributions, the fact that the test statistic is smaller for the inflated Kumaraswamy law indicates there is more evidence in favor of the inflated Kumaraswamy distribution relative to the alternative law
Summary
Oftentimes practitioners need to model variables that assume values in the standard unit interval, (0, 1), such as rates, proportions and concentration indices. We say that the random variable Y is Kumaraswamy-distributed with shape parameters α > 0 and β > 0, denoted by Y ∼ Kum(α, β), if its probability density function (pdf) is given by g(y; α, β) = αβyα–1(1 – yα)β–1, y ∈ (0, 1), (1) The zero or one inflated Kumaraswamy log-likelihood function is given by (θ; y) = 1(λ; y) + 2(α, β; y), where n n
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