Independent non-identical five-parameter gamma-Weibull variates and their sums

Abstract Gamma-Weibull variates with five parameters are defined by multiplication of gamma and Weibull densities and renormalising. Sums of independent such variates are distributed as combinations of products of gammas and confluent hypergeometric functions and are explicitly determined. Sums of independent non-identical Weibulls arise as a special case. These variates can be used to model moderately extreme scenarios between gamma and Weibull that occur in many natural applications. All results are exact.

[1]  N. Balakrishnan,et al.  Order statistics from the type I generalized logistic distribution , 1988 .

[2]  H. A. David,et al.  Order Statistics (2nd ed). , 1981 .

[3]  Waloddi Weibull,et al.  Probabilistic methods in the mechanics of solids and structures : symposium Stockholm, Sweden June 19-21, 1984 : to the memory of Waloddi Weibull , 1985 .

[4]  M. Fiorentino,et al.  Reply [to Comment on Two-Component Extreme Value Distribution for Flood Frequency Analysis by Fab , 1986 .

[5]  J. Hosking,et al.  Comment on “Two‐Component Extreme Value Distribution for Flood Frequency Analysis” by Fabio Rossi, Mauro Florentino, and Pasquale Versace , 1986 .

[6]  J. D. T. Oliveira,et al.  The Asymptotic Theory of Extreme Order Statistics , 1979 .

[7]  W. Weibull,et al.  The phenomenon of rupture in solids , 1939 .

[8]  G. A. Mack,et al.  Order Statistics (2nd Ed.) , 1983 .

[9]  E. J. Gumbel,et al.  Statistics of Extremes. , 1960 .

[10]  M. Fréchet.  Sur la loi de probabilité de l'écart maximum , 1928 .

[11]  Benjamin Epstein,et al.  Applications of Extreme Value Theory to Problems of Material Behavior , 1985 .

[12]  A. M. Freudenthal,et al.  The statistical aspect of fatigue of materials , 1946, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[13]  W. Weibull.  A statistical theory of the strength of materials , 1939 .

[14]  B. Gnedenko.  Sur La Distribution Limite Du Terme Maximum D'Une Serie Aleatoire , 1943 .

[15]  I. S. Gradshteyn,et al.  Table of Integrals, Series, and Products , 1976 .

[16]  Mauro Fiorentino,et al.  Two‐Component Extreme Value Distribution for Flood Frequency Analysis , 1984 .

[17]  R. Fisher,et al.  Limiting forms of the frequency distribution of the largest or smallest member of a sample , 1928, Mathematical Proceedings of the Cambridge Philosophical Society.

[18]  A. A. Griffith.  The Phenomena of Rupture and Flow in Solids , 1921 .

[19]  E. Dodd.  The greatest and the least variate under general laws of error , 1923 .

[20]  M. R. Leadbetter,et al.  Extremes and Related Properties of Random Sequences and Processes: Springer Series in Statistics , 1983 .