A new method for adding a parameter to a family of distributions with application to the exponential and Weibull families. (English) Zbl 0888.62012

Marshall, Albert W.; Olkin, Ingram

Biometrika 84, No. 3, 641-652 (1997), correction ibid. 92, No. 2, 505 (2005).

Summary: A new way of introducing a parameter to expand a family of distributions is introduced and applied to yield a new two-parameter extension of the exponential distribution which may serve as a competitor to such commonly-used two-parameter families of life distributions as the Weibull, gamma and lognormal distributions. In addition, the general method is applied to yield a new three-parameter Weibull distribution. Families expanded using the method introduced here have the property that the minimum of a geometric number of independent random variables with common distribution in the family has a distribution again in the family. Bivariate versions are also considered.
A mistake in equ. 3.1 is corrected. (Correction 2005).

MSC:

62E10	Characterization and structure theory of statistical distributions
62P10	Applications of statistics to biology and medical sciences; meta analysis

Keywords:

bivariate geometric distribution; copula; geometric extreme stability; parametric family; life distributions