Modified beta modified-Weibull distribution

Abstract

We introduce a flexible modified beta modified-Weibull model, which can accommodate both monotonic and non-monotonic hazard rates such as a useful long bathtub shaped hazard rate in the middle. Several distributions can be obtained as special cases of the new model. We demonstrate that the new density function is a linear combination of modified-Weibull densities. We obtain the ordinary and central moments, generating function, conditional moments and mean deviations, residual life functions, reliability measures and mean and variance (reversed) residual life. The method of maximum likelihood and a Bayesian procedure are used for estimating the model parameters. We compare the fits of the new distribution and other competitive models to two real data sets. We prove empirically that the new distribution gives the best fit among these distributions based on several goodness-of-fit statistics.

Appendices

Appendix A. The unified Fox–Wright generalized hypergeometric function

Here,

$$\begin{aligned} {}_p\varPsi _q^*\Big [ \begin{array}{c} (a, A)_p\\ (b, B)_q \end{array} \Big |\, z\,\Big ] = \sum _{n=0}^\infty \frac{\prod _{j=1}^p (a_j)_{A_jn}}{\prod _{j=1}^q (b_j)_{B_jn}}\, \frac{z^n}{n!} \end{aligned}$$

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stands for the unified variant of the Fox–Wright generalized hypergeometric function with p upper and q lower parameters; $(a,A)_p$ denotes the parameter p-tuple $(a_1, A_1), \ldots , (a_p, A_p)$ and $a_j \in \mathbb C$, $b_i \in \mathbb C\setminus \mathbb Z_0^-$, $A_i, B_j>0$ for all $j=\overline{1,p}, i=\overline{1,q}$. The power series converges for suitably bounded values of |z| when

$$\begin{aligned} \varDelta _{p,q} = 1 - \sum _{j=1}^pA_j + \sum _{j=1}^qB_j >0\, . \end{aligned}$$

In the case $\varDelta = 0$, the convergence holds in the open disc $|z|<\beta = \prod _{j=1}^qB_j^{B_j}\cdot \prod _{j=1}^pA_j^{-A_j}$.

The function ${}_1\varPsi _0^*$ is called confluent. The convergence condition $\varDelta _{1,0} = 1-A_1>0$ is of special interest for us.

We point out that the original definition of the Fox–Wright function ${}_p\varPsi _q[z]$ (consult formula collection (Erdélyi et al. 1995) and the monographs (Kilbas et al. 2006; Mathai and Saxena 1978)) contains gamma functions instead of the generalized Pochhammer symbols used here. However, these two functions differ only up to constant multiplying factor, that is

$$\begin{aligned} {}_p\varPsi _q \Big [ \begin{array}{c} (a, A)_p\\ (b, B)_q \end{array} \Big |\, z\,\Big ] = \frac{\prod _{j=1}^p \varGamma (a_j)}{\prod _{j=1}^q\varGamma (b_j)}\, {}_p\varPsi _q^* \Big [ \begin{array}{c} (a, A)_p\\ (b, B)_q \end{array} \Big |\, z\,\Big ]\, . \end{aligned}$$

The unification’s motivation is clear - for $A_1 = \cdots = A_p = B_1 = \cdots = B_q = 1$, the fucntion ${}_p\varPsi _q^*[z]$ reduces exactly to the well-known generalized hypergeometric function ${}_pF_q[z]$.

Appendix B. Meijer G-function

The symbol $G_{p,q}^{m,n}( \cdot |\, \cdot )$ denotes Meijer’s G-function (Meijer 1946) defined in terms of the Mellin–Barnes integral as

$$\begin{aligned}&G_{p,q}^{m,n}\Big ( z \,\Big | \begin{array}{c} a_1, \ldots , a_p\\ b_1, \ldots , b_q \end{array}\Big ) = \frac{1}{2\pi {\mathrm i}}\oint _{\mathfrak C}\frac{\prod _{j=1}^{m}\varGamma (b_j-s) \prod _{j=1}^{n}\varGamma ( 1-a_j + s)}{\prod _{j=m+1}^q \varGamma ( 1-b_j + s) \prod _{j=n+1}^{p}\varGamma (a_j - s)} z^s \mathrm d s, \end{aligned}$$

where $0\le m\le q,\, 0\le n\le p$ and the poles $a_j, b_j$ are such that no pole of $\varGamma (b_j - s), j=\overline{1,m}$ coincides with any pole of $\varGamma (1-a_j+s), j=\overline{1,n}$; i.e. $a_k-b_j \not \in \mathbb N$, while $z \ne 0$. $\mathfrak C$ is a suitable integration contour, see [p.143] Luke (1969) and Meijer (1946) for more details.

The G function’s Mathematica code reads$\texttt {MeijerG[}\{\{a_1,\ldots ,a_n\}, \{a_{n+1},\ldots ,a_p\}\}, \{\{b_1,\ldots ,b_m\}, \{b_{m+1},\ldots ,b_q\}\}, z\texttt {]}.$