The Kumaraswamy distribution: median-dispersion re-parameterizations for regression modeling and simulation-based estimation

Mitnik, Pablo A.; Baek, Sunyoung

doi:10.1007/s00362-011-0417-y

Cited by 92 publications

(65 citation statements)

References 42 publications

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“…Therefore, we consider the median-based re-parameterization proposed by (Mitnik and Baek, 2013) aiming to facilitate its use in regression models. In order to develop the proposed model, we will follow the generalized linear models (McCullagh and Nelder, 1989) and Beta regression (Ferrari and Cribari-Neto, 2004) methodologies,; although as stated before, we will be modeling the median instead of the mean.…”

Section: Pakistan Journal Of Statistics and Operation Researchmentioning

confidence: 99%

Kumaraswamy regression modeling for Bounded Outcome Scores

Hamedi−Shahraki

Rasekhi

Yekaninejad³

et al. 2021

Pak.j.stat.oper.res.

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In this paper, we use a regression model for modeling bounded outcome scores (BOS), where the outcome is Kumaraswamy distributed. Similar to the Beta distribution, this distribution can take a variety of shapes while being computationally easier to use. Thus, it is deemed as a suitable alternative distribution to the Beta in modeling bounded random processes. In the proposed model, the median of a bounded response is modeled by the linear predictors which is defined through regression parameters and explanatory variables. We obtained the maximum likelihood estimates (MLE) of the parameters, provided closed-form expressions for the score functions and Fisher information matrix, and presented some diagnostic measures. We conducted Monte Carlo simulations to investigate the finite-sample performance of the MLEs of the parameters. Finally, two practical applications of this model to the real data sets are presented and discussed.

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Section: Pakistan Journal Of Statistics and Operation Researchmentioning

confidence: 99%

Kumaraswamy regression modeling for Bounded Outcome Scores

Hamedi−Shahraki

Rasekhi

Yekaninejad³

et al. 2021

Pak.j.stat.oper.res.

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“…Using (), an expression for the median of the distribution can be obtained, 11 which is given by

md (Y_{K}) = {\tilde{μ}}_{K} = {(1 - 0 . 5^{1 / γ_{K}})}^{1 / δ_{K}} .

Thus, making ϕ K = δ K (dispersion parameter), the parameter γ K can be expressed by

γ_{K} = \frac{\ln (0.5)}{\ln (1 - {\tilde{μ}}_{K}^{ϕ_{K}})} .

For the Kumaraswamy distribution, we adopted the following notation

Y_{K} \sim Kuma false({\overset{μ}{true}}_{K}, ϕ_{K} false)

. Considering this reparameterization, the probability density, cumulative distribution, and quantile functions can be re‐expressed as

f (y_{K}; {\tilde{μ}}_{K}, ϕ_{K}) = \frac{ϕ_{K} \ln (0.5)}{\ln (1 - {\tilde{μ}}_{K}^{ϕ_{K}})} y_{K}^{ϕ_{K} - 1} {false(1 - y_{K}^{ϕ_{K}} false)}^{\frac{normalln false(0.5 false)}{normalln false(1 - {\overset{μ}{true}}_{K}^{ϕ_{K}} false)} - 1},

F false(y_{K}; tr...

…”

Section: Distributions To Model Double‐bounded Processesmentioning

confidence: 99%

Median control charts for monitoring asymmetric quality characteristics double bounded

Lima-Filho

Bourguignon

et al. 2020

Quality & Reliability Eng

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In many practical situations, the quality characteristics of interest assume values in the range (0,1), like rates and proportions (but they are not results from Bernoulli experiments). Most control charts built for these quality characteristics rely on monitoring parameters of their probability distribution functions or on their averages after some reparameterization of their density probability function. However, for highly asymmetric distributions, the median is a more appropriate location parameter than the average. In this paper, we propose Shewhart-type control charts for monitoring the median of observations taken from quality characteristics double bounded after reparameterization of two probability density functions: Kumaraswamy and unit Weibull. The performance of the control charts is evaluated and compared in terms of run length (RL) analysis considering three estimators for the median. Finally, we also carry out two applications to demonstrate the applicability of these control charts.

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“…We now turn to considering the conditions under which σ is a dispersion parameter. Following Mitnik and Baek (, pp. 181–182), we show that σ is a dispersion parameter via an argument based on the ‘quantile spread order’ from Townsend and Colonius ().…”

Section: Basic Propertiesmentioning

confidence: 99%

CDF‐quantile distributions for modelling random variables on the unit interval

Smithson

Shou

2017

Brit J Math & Statis

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This paper introduces a two-parameter family of distributions for modelling random variables on the (0,1) interval by applying the cumulative distribution function of one 'parent' distribution to the quantile function of another. Family members have explicit probability density functions, cumulative distribution functions and quantiles in a location parameter and a dispersion parameter. They capture a wide variety of shapes that the beta and Kumaraswamy distributions cannot. They are amenable to likelihood inference, and enable a wide variety of quantile regression models, with predictors for both the location and dispersion parameters. We demonstrate their applicability to psychological research problems and their utility in modelling real data.

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The Kumaraswamy distribution: median-dispersion re-parameterizations for regression modeling and simulation-based estimation

Cited by 92 publications

References 42 publications

Kumaraswamy regression modeling for Bounded Outcome Scores

Kumaraswamy regression modeling for Bounded Outcome Scores

Median control charts for monitoring asymmetric quality characteristics double bounded

CDF‐quantile distributions for modelling random variables on the unit interval

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