The Weibull Burr XII distribution in lifetime and income analysis

Abstract

We study a five-parameter model called the Weibull Burr XII (WBXII) distribution, which extends several models, including new ones. This model is quite flexible in terms of the hazard function, which exhibits increasing, decreasing, upside-down bathtub, and bathtub shapes. Its density function allows different forms such as left-skewed, right-skewed, reversed-J, and bimodal. We aim to provide some general mathematical quantities for the proposed distribution, which can be useful to real data analysis. We develop a shiny application to provide interactive illustrations of the WBXII density and hazard functions. We estimate the model parameters using maximum likelihood and derive a profile log-likelihood for all members of the Weibull-G family. The survival analysis application reveals that the WBXII model is suitable to accommodate left-skewed tails, which are very common when the variable of interest is the time to failure of a product. The income application is related to player salaries within a professional sports league and it is peculiar because the mean of the player's salaries is much higher than for most professions. Both applications illustrate that the new distribution provides much better fits than other models with the same and less number of parameters.

Another unit Burr XII quantile regression model based on the different reparameterization applied to dropout in Brazilian undergraduate courses

“…In each Monte Carlo replication, the inversion method is used to generate n occurrences of a random variable Y i * UBXII (q i , c). By assuming the regression structure defined in Eq (19), it follows that…”

“…The regression structure for the Kw, UW, and beta distributions is analogous to (19). The main differences are the assumptions under the random components and modeled location parameters.…”

“…The main differences are the assumptions under the random components and modeled location parameters. To get the Kw regression, q must be replaced by the median (ω) in Eq (19) and supposed that Y i * Kw(ω i , d p ). The UW regression is obtained by considering the structure (19) and assuming that Y i * UW(q i , γ).…”

“…To get the Kw regression, q must be replaced by the median (ω) in Eq (19) and supposed that Y i * Kw(ω i , d p ). The UW regression is obtained by considering the structure (19) and assuming that Y i * UW(q i , γ). In the beta regression, the location parameter is the mean (μ).…”

“…We use an approach based on the unit Burr XII (UBXII) distribution, which was pioneered [ 8 ] by applying the transformation method in a Burr XII (BXII) random variable. Their choice was based on the versatility of the baseline, which has been applied in reliability analysis [ 17 ], regression modeling [ 18 ], generalized distributions [ 19 , 20 ], and several other disciplines. Let Y be a unit random variable having the UBXII distribution.…”