Multivariate Inverted Kumaraswamy Distribution: Derivation and Estimation

Abstract

Industrial revolution leads to the manufacturing of multicomponent products; to guarantee the sufficiency of the product and consumer satisfaction, the producer has to study the lifetime of the products. This leads to the use of bivariate and multivariate lifetime distributions in reliability engineering. The most popular and applicable is Marshall–Olkin family of distributions. In this paper, a new bivariate lifetime distribution which is the bivariate inverted Kumaraswamy (BIK) distribution is found and its properties are illustrated. Estimation using both maximum likelihood and Bayesian approaches is accomplished. Using different selection criteria, it is found that BIK provides the best performance compared with other bivariate distributions like bivariate exponential and bivariate inverse Weibull distributions. As a generalization, the multivariate inverted Kumaraswamy (MIK) distribution is derived. Few studies have been conducted on the multivariate Marshall–Olkin lifetime distributions. To the best of our knowledge, none of them handle estimation process. In this paper, we developed an algorithm to show how to estimate the unknown parameters of MIK using both maximum likelihood and Bayesian approaches. This algorithm could be applied in estimating other Marshall–Olkin multivariate lifetime distributions.

Discrete Inverted Kumaraswamy Distribution: Properties and Estimation

“…where 𝑝(𝑥) and 𝑆(𝑥) are given, respectively, by ( 5) and (7). The 𝑥 (𝑖) ′s are ordered times for 𝑖 = 1,2, … 𝑟.…”

“…Depending on the invariance property, the ML estimators of 𝑆(𝑥), ℎ(𝑥) and 𝑎ℎ(𝑥) can be obtained by replacing 𝛼 and 𝛽 with their corresponding ML estimators 𝛼 ̂ and 𝛽 ̂, respectively, in (7), ( 8) and ( 9), as given below…”

Alpha power inverted Kumaraswamy distribution: Definition, different estimation methods, and application

“…are obtained by taking partial derivatives of the ln L function given in (6) with respect to the parameters α, β and λ, and equating them zero. The ML estimates of the parameters α, β and λ are obtained by solving the likelihood equations ( 7) -( 9), simultaneously.…”

Inference for dependent complementary competing risks model from an inverted Kumaraswamy distribution under ranked set sampling

“…The IK distribution has been discussed by many authors, see, for example, some recent works of Abu-Moussa and El-Din, 41 Pasha-Zanoosi and Pourdarvish, 42 Aly and Abuelamayem, 43 Iqbal et al, 44 and Muhammed 45 among others. For illustration, Figure 1 displays the density and HRF plots of the IK distribution to demonstrate its flexibility.…”

“…Consequently, the survival function (SF) and hazard rate function (HRF) of the IK distribution at mission time $$t$$ can be expressed as $$$$\begin{align} S_{IK}(t;\alpha,\beta)=1-[1-(1+t)^{-\alpha}]^{\beta}\nobreakspace \nobreakspace \text{and}\nobreakspace \nobreakspace h_{IK}(t;\alpha,\beta)=\frac{\alpha \beta (1+t)^{-\alpha -1}[1-(1+t)^{-\alpha}]^{\beta -1}}{1-[1-(1+t)^{-\alpha}]^{\beta}}. \end{align}$$$$The IK distribution has been discussed by many authors, see, for example, some recent works of Abu‐Moussa and El‐Din, 41 Pasha‐Zanoosi and Pourdarvish, 42 Aly and Abuelamayem, 43 Iqbal et al., 44 and Muhammed 45 among others. For illustration, Figure 1 displays the density and HRF plots of the IK distribution to demonstrate its flexibility.…”