Statistical Inference of the Half-Logistic Inverse Rayleigh Distribution

Abstract

The inverse Rayleigh distribution finds applications in many lifetime studies, but has not enough overall flexibility to model lifetime phenomena where moderately right-skewed or near symmetrical data are observed. This paper proposes a solution by introducing a new two-parameter extension of this distribution through the use of the half-logistic transformation. The first contribution is theoretical: we provide a comprehensive account of its mathematical properties, specifically stochastic ordering results, a general linear representation for the exponentiated probability density function, raw/inverted moments, incomplete moments, skewness, kurtosis, and entropy measures. Evidences show that the related model can accommodate the treatment of lifetime data with different right-skewed features, so far beyond the possibility of the former inverse Rayleigh model. We illustrate this aspect by exploring the statistical inference of the new model. Five classical different methods for the estimation of the model parameters are employed, with a simulation study comparing the numerical behavior of the different estimates. The estimation of entropy measures is also discussed numerically. Finally, two practical data sets are used as application to attest of the usefulness of the new model, with favorable goodness-of-fit results in comparison to three recent extended inverse Rayleigh models.Entropy 2020, 22, 449 2 of 24 respectively, where α is a scale parameter. As notable features, the IR distribution has tractable and simple probability functions, is unimodal and right-skewed, and possesses a hazard rate function with a singular curvature: it increases at a certain value, then decreases until attain a kind of stabilization. The pioneer studies are [2] which presents some properties of the maximum likelihood estimator of α, and [3] which provides closed-form expressions for the (standard) mean, harmonic mean, geometric mean, mode and median of the IR distribution. Also, among the amount of works investigating the statistical aspects of the IR distribution, the reader can be referred to [4][5][6][7][8][9][10][11][12].In the recent years, several extensions of the IR distribution were developed, using different mathematical techniques, often at the basis of general families of distributions. Among them, there are the beta IR (BIR) distribution by [13], transmuted IR (TIR) distribution by [14], modified IR (MIR) distribution by [15], transmuted modified IR (TMIR) distribution by [16], transmuted exponentiated IR (TEIR) distribution by [17], Kumaraswamy exponentiated IR (KEIR) distribution by [18], weighted IR (WIR) distribution by [19], odd Fréchet IR (OFIR) distribution by [20], type II Topp-Leone IR (TIITLIR) distribution by [21], type II Topp-Leone generalized IR (TIITLGIR) distribution by [22] and exponentiated IR (EIR) distribution by [23].However, to the best of our knowledge, the use of the half-logistic transformation to extend the IR distribution remains unexplored, despite recent success in this regard. This half-log...

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