Introduction to Morris (1982) Natural Exponential Families with Quadratic Variance Functions

1997 ◽  
pp. 369-394 ◽  
Author(s):  
G. G. Letac
Keyword(s):  
Exponential Families ◽  
Natural Exponential Families ◽  
Variance Functions

scholarly journals Increasing Hazard Rate of Mixtures for Natural Exponential Families

2012 ◽  
Vol 44 (02) ◽  
pp. 373-390 ◽  
Author(s):  
Shaul K. Bar-Lev ◽  
Gérard Letac
Keyword(s):  
Hazard Rate ◽  
Exponential Families ◽  
Sufficient Condition ◽  
Natural Exponential Families ◽  
Mixtures Of Distributions ◽  
Variance Functions ◽  
Hyperbolic Cosine ◽  
Actuarial Studies ◽  
Two Populations

Hazard rates play an important role in various areas, e.g. reliability theory, survival analysis, biostatistics, queueing theory, and actuarial studies. Mixtures of distributions are also of great preeminence in such areas as most populations of components are indeed heterogeneous. In this study we present a sufficient condition for mixtures of two elements of the same natural exponential family (NEF) to have an increasing hazard rate. We then apply this condition to some classical NEFs having either quadratic or cubic variance functions (VFs) and others as well. Particular attention is paid to the hyperbolic cosine NEF having a quadratic VF, the Ressel NEF having a cubic VF, and the NEF generated by Kummer distributions of type 2. The application of such a sufficient condition is quite intricate and cumbersome, in particular when applied to the latter three NEFs. Various lemmas and propositions are needed to verify this condition for such NEFs. It should be pointed out, however, that our results are mainly applied to a mixture of two populations.

scholarly journals Increasing Hazard Rate of Mixtures for Natural Exponential Families

2012 ◽  
Vol 44 (2) ◽  
pp. 373-390 ◽  
Author(s):  
Shaul K. Bar-Lev ◽  
Gérard Letac
Keyword(s):  
Hazard Rate ◽  
Exponential Families ◽  
Sufficient Condition ◽  
Natural Exponential Families ◽  
Mixtures Of Distributions ◽  
Variance Functions ◽  
Hyperbolic Cosine ◽  
Actuarial Studies ◽  
Two Populations

Hazard rates play an important role in various areas, e.g. reliability theory, survival analysis, biostatistics, queueing theory, and actuarial studies. Mixtures of distributions are also of great preeminence in such areas as most populations of components are indeed heterogeneous. In this study we present a sufficient condition for mixtures of two elements of the same natural exponential family (NEF) to have an increasing hazard rate. We then apply this condition to some classical NEFs having either quadratic or cubic variance functions (VFs) and others as well. Particular attention is paid to the hyperbolic cosine NEF having a quadratic VF, the Ressel NEF having a cubic VF, and the NEF generated by Kummer distributions of type 2. The application of such a sufficient condition is quite intricate and cumbersome, in particular when applied to the latter three NEFs. Various lemmas and propositions are needed to verify this condition for such NEFs. It should be pointed out, however, that our results are mainly applied to a mixture of two populations.

scholarly journals Independent, Tough Identical Results: The Class of Tweedie on Power Variance Functions and the Class of Bar-Lev and Enis on Reproducible Natural Exponential Families

2019 ◽  
Vol 9 (1) ◽  
pp. 30 ◽  
Author(s):  
Shaul K. Bar-Lev
Keyword(s):  
Exponential Families ◽  
Natural Exponential Families ◽  
Self Confidence ◽  
Variance Functions ◽  
Complete Study ◽  
Exponential Dispersion Models ◽  
Power Variance ◽  
Parameter Case ◽  
First Time ◽  
Fundamental Paper

The Rao-Blackwell theorem has had a fundamental role in statistical theory. However, as opposed to what seems natural, Rao and Blackwell did not investigate and write the theorem jointly. In fact, they both published the same result independently, two years apart. Indeed, as C.R. Rao writes in Wikipedia: ”the result on one parameter case was published by Rao (1945) in the Bulletin of the Calcutta Mathematical Society and by Blackwell (1947) in The Annals of Mathematical Statistics. Only Lehmann and Sche ´e (1950) called the result as Rao-Blackwell theorem”. Forty years later, a situation very similar to the previous one seems to have happened. Tweedie (1984) in a paper published in a proceedings to a conference held in Calcutta and Bar-Lev and Enis (1986) in a paper published in The Annals of Statistics both presented for the first time, albeit two years apart, independently and in di erent contexts, the class of natural exponential families having power variance functions (NEF-PVFs). Tweedie’s results were then mentioned by Jorgensen (1987) in his fundamental paper on exponential dispersion models published in the Journal of the Royal Statistical Society, Series B. Jorgensen, however, mentioned also other researchers, including Bar-Lev and Enis, as dealt with the same problem. Nonetheless, Jorgensen (1987) stated in his paper that ”The most complete study” of NEF-PVFs was given by Tweedie (1984), a statement which has led to naming the class of NEF-PVFs as the Tweedie class. This statement of Jorgensen is entirely and utterly incorrect. Accordingly, one of the goals of this note is to 'prove' such incorrectness. Based on this 'proof' it will be evident, so I trust, that both Bar-Lev and Enis should have received the appropriate credit by re-naming the class of NEF-PVFs via the exploitation of the names of Tweedie, Bar-Lev and Enis. This would resemble the dignified and elegant manner Lehmann and Sche ´e acted on the Rao-Blackwell Theorem. Notwithstanding, the main aim of the note is to encourage young researchers to present their results with self-confidence and to get the credit they deserve.

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