|
Organizers |
Exchangeability and applications in statistics and system reliability
by
Fabio Spizzichino
Department of Mathematics, University "La Sapienza", Rome
1. Exchangeability and the problem of statistical prediction.
The original work of Bruno de Finetti (1928-1938) in introducing and developing the theory of exchangeability was essentially motivated by the need of giving a sound foundation to the study of statistical inference and prediction, in the frame of a subjective interpretation of probability.
We aim to illustrate these motivations and explain de Finetti's solutions of the posed problems in terms of the concept of exchangeability.
We will then present a discussion on the theme of inference, showing the related role of the concept of statistical sufficiency and of the de Finetti's-type theorems.
This path will give us the opportunity to revisit some basic examples and probabilistic properties of exchangeable sequences of events and random variables.
2. Exchangeability and symmetrization of multivariate distributions; uses in system reliability.
The topic of exchangeability is strictly related with the study of order statistics and, in turn, order statistics have a relevant role in the field of system reliability (see [1]).
The concept of signature of a system, introduced in [2], can provide a new insight in this special direction (see [3]).
In the frame of a discussion about these topics, we focus new attention on the analysis of symmetries of a reliability system and of the probability distribution of its components' lifetimes.
In more detail, our discussion will develop along the following lines.
We shall be dealing with the longitudinal observation of vectors of non-negative random variables X1, X2, ..., Xn, where X1, X2, ..., Xn can be interpreted as the lifetimes of components in a system.
We will generally assume absolute continuity and compare the different, possible descriptions of the joint law of ( X1, X2, ..., Xn) ; one such description is based on the stochastic intensity of the pure-death process {Nt}t ≥ 0, where Nt:=∑k=1n1{Xk ≤ t}.
This description is strictly related with the so-called multivariate conditional hazard rate functions (see e.g. [4], [5]).
Based on the above, we first compare different characterizations of the condition of exchangeability for non-negative random variables (see [6]); then we discuss the theme of symmetrization of the law of X = ( X1, X2, ..., Xn) , by analyzing different ways to describe the joint law of an exchangeable vector T = (T1, T2, ..., Tn) such that the corresponding vectors of order statistics ( X(1), X(2), ..., X(n)) and ( T(1), T(2), ..., T(n)) share the same joint law.
A second part of this lecture will more specifically concern the field of system reliability. A central problem in this field is to analyze how the reliability function depends, on the one hand, on the structure of the system and on the joint law of the lifetimes X1, X2, ..., Xn of the system's components, on the other hand.
Commonly, both these two objects present relevant properties of symmetry that can have a direct impact in the reliability analysis.
The form of the reliability function for a coherent system (see e.g. [7]) is particularly simple when X1, X2, ..., Xn are themselves exchangeable (i.e. X=T). In this case, in fact, it only depends on the signature of the system and on the marginal laws of X(1), X(2), ..., X(n).
For the case when X1, X2, ..., Xn are not exchangeable, we point out some relations among the structure of system's signature, the amount of symmetry in the system, and the distance between the reliability functions obtained for X and for the symmetrized vector of lifetimes T respectively.
References
[1] Galambos, J. (1978). The Asymptotic Theory of Extreme Order Statistics. J. Wiley & Sons, New York - Chichester - Brisbane -Toronto.
[2] Samaniego, F. J. (1985). On the closure of the IFR class under formation of coherent systems. IEEE Trans., R-34, 69-72.
[3] Kochar, S.; Mukerjee, H.; Samaniego, F. J. (1999). The ßignature" of a coherent system and its application to comparisons among systems. Naval Res. Logist., 46 , no. 5, 507-523.
[4] Arjas, E. (1989). Survival models and martingale dynamics. Scand. J. Statist. 16, 177-225.
[5] Shaked, M.; Shanthikumar, J. G. (1987). Multivariate hazard rates and stochastic ordering. Adv. in Appl. Probab., 19, 123-137.
[6] Spizzichino F. (2001). Subjective probability models for lifetimes. Chapman and Hall/CRC. Boca Raton.
[7] Barlow, R. E.; Proschan, F. (1975). Statistical theory of reliability and life testing. Probability models. Holt, Rinehart and Winston, Inc., New York-Montreal, Que.-London.
Date received: April 7, 2006
Copyright © 2006 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Mathematical Conference Abstracts. Document # cart-09.