## Brazilian Journal of Probability and Statistics

- Braz. J. Probab. Stat.
- Volume 29, Number 3 (2015), 565-589.

### The Burr XII power series distributions: A new compounding family

Rodrigo B. Silva and Gauss M. Cordeiro

#### Abstract

Generalizing lifetime distributions is always precious for applied statisticians. In this paper, we introduce a new family of distributions by compounding the Burr XII and power series distributions. The compounding procedure follows the key idea by Adamidis and Loukas (
*Statist. Probab. Lett.*
**39** (1998) 35–42) or, more generally, by Chahkandi and Ganjali (
*Comput. Statist. Data Anal.*
**53** (2009) 4433–4440) and Morais and Barreto-Souza (
*Comput. Statist. Data Anal.*
**55** (2011) 1410–1425). The proposed family includes as a basic exemplar the Burr XII distribution. We provide some mathematical properties including moments, quantile and generating functions, order statistics and their moments, Kullback–Leibler divergence and Shannon entropy. The estimation of the model parameters is performed by maximum likelihood and the inference under large sample. Two special models of the new family are investigated in details. We illustrate the potential of the new family by means of two applications to real data. It provides better fits to these data than other important lifetime models available in the literature.

#### Article information

**Source**

Braz. J. Probab. Stat., Volume 29, Number 3 (2015), 565-589.

**Dates**

Received: June 2013

Accepted: November 2013

First available in Project Euclid: 11 June 2015

**Permanent link to this document**

https://projecteuclid.org/euclid.bjps/1433983066

**Digital Object Identifier**

doi:10.1214/13-BJPS234

**Mathematical Reviews number (MathSciNet)**

MR3355748

**Zentralblatt MATH identifier**

1326.62031

**Keywords**

Burr XII distribution information matrix Kullback–Leibler divergence order statistic power series distribution

#### Citation

Silva, Rodrigo B.; Cordeiro, Gauss M. The Burr XII power series distributions: A new compounding family. Braz. J. Probab. Stat. 29 (2015), no. 3, 565--589. doi:10.1214/13-BJPS234. https://projecteuclid.org/euclid.bjps/1433983066