Abstract
Statistical distributions play a prominent role in applied sciences, particularly in biomedical sciences. The medical data sets are generally skewed to the right, and skewed distributions can be used quite effectively to model such data sets. In the present study, therefore, we propose a new family of distributions to model right skewed medical data sets. The proposed family may be named as a flexible reduced logarithmic-X family. The proposed family can be obtained via reparameterizing the exponentiated Kumaraswamy G-logarithmic family and the alpha logarithmic family of distributions. A special submodel of the proposed family called, a flexible reduced logarithmic-Weibull distribution, is discussed in detail. Some mathematical properties of the proposed family and certain related characterization results are presented. The maximum likelihood estimators of the model parameters are obtained. A brief Monte Carlo simulation study is done to evaluate the performance of these estimators. Finally, for the illustrative purposes, three applications from biomedical sciences are analyzed and the goodness of fit of the proposed distribution is compared to some well-known competitors.
Highlights
We propose a new family of distributions to model right skewed medical data sets. e proposed family may be named as a flexible reduced logarithmic-X family. e proposed family can be obtained via reparameterizing the exponentiated Kumaraswamy G-logarithmic family and the alpha logarithmic family of distributions
A number of parametric continuous distributions for modeling lifetime data sets have been proposed in literature including exponential, Rayleigh, gamma, lognormal, and Weibull, among others. e exponential, Rayleigh, and Weibull distributions are more popular than the gamma and lognormal distributions since the survival functions of the gamma and the lognormal distributions cannot be expressed in closed forms and both require numerical integration to arrive at the mathematical properties. e exponential and Rayleigh distributions are commonly used in lifetime analysis. ese distributions, are not flexible enough to counter complex forms of the data
The exponential distribution is capable of modeling data with constant failure rate function, whereas the Rayleigh distribution offers data modeling with only increasing failure rate function. e Weibull distribution, known as the super exponential distribution, is more flexible than the aforementioned distributions. e Weibull distribution offers the characteristics of both the exponential and Rayleigh distributions and is capable of modeling data with monotonic hazard rate function
Summary
Some statistical properties of the FRL-X family are derived. σ where u ∈ (0, 1). From the expression (11), it is clear that the FRL-X family has a closed form solution of its quantile function which makes it easier to generate random numbers. The reverse residual life of the FRL-X random variable, denoted by R(t), is. Let X: Ω ⟶ H be a continuous random variable with the distribution function G and let q1. Where the function s is a solution of the differential equation s/(u) η/q1/ηq1 − q2 and C is the normalization constant, such that HdF 1. E random variable X has pdf (5) if and only if there exist functions q2(x) and η(x) defined in eorem 1 satisfying the following differential equation: s/(x) η/(x)q1(x) f(x; ξ) , x ∈ R. We like to point out that one set of functions satisfying the above differential equation is given in Proposition 1 with D 1/2. What we try to do with MLE’s is to maximize ni 1 f(xi) or, equivalently, minimize ni 1 − log(f(xi)) [16] proposed to generalize this to the minimization of ospfnite ec1niρa(lbxecia)d,soewnhoeef breMy ρ-deisiffsteimsroeanmttoiearstf.iunnMgcitρnioiamnn.idzMinsLogElvsinanig r1eρt(nih xe1ir)φe(fcoxarine) where φ(x) zρ(x)/zx; for further detail, we refer the interested readers to [17, 18]
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