Subjective Probability Models for Lifetimes
Proportional hazards, conditionally i. Whence, using 2. Continuing Example 2. Consider the case of Example 2. A fundamental property of conditional probability specializes in the following formula. Common mode failures. Consider two units which are simi- lar and work independently one of the other; they are, however, sensible to the same destructive shock. We assume that, in the absence of a shock, the two lifetimes would be independent with the same survival function G and that the waiting time until the shock is W , where the survival function of W is H.
Bivariate model of Marshall-Olkin. Notice that the one-dimensional marginal distribution is exponential. Remark 2.
In the reliability practice, data are often generated according to a scheme as follows. We then consider a duration experiment where the units U1 ; ; Un are new at time 0 and are simultaneously put under test, progressively recording all the subsequent failure times. U in U in A dynamic approach is appropriate to deal with such situations. This means in particular that the form of interdependence existing among lifetimes is more conveniently described in terms of the notion of multivariate conditional hazard rates, which will be introduced in the next section.
Survival data typically come along with the presence of some rule for stopping the life-data observation. In some cases the stopping rule can itself be informative, i.
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Barlow and Proschan, In the derivation of Proposition 2. Parametric models described by 2. In the reliability practice, for instance, this is the case when, for a component in a position of a system, we plan a replacement policy by installing at the instant of failure or at a xed age other similar components: the lifetimes of new components, which are subsequently put into operation, form a conceivably in nite population. Conditional independence in 2.
Actually, this is, in particular, formalized by the formula 2. In recent statistical literature, more and more interest has been directed toward the study of interdependence among lifetimes arising from time-varying environmental conditions. A simple model of this type is provided by a straightforward generalization of the proportional hazard model of Example 2. Apart from speci c probabilistic models, the mathematical tools to deal with general problems involving time-varying environmental conditions is to be found in the framework of the theory of point processes and of stochastic ltering Bremaud, ; Arjas, Assuming a common dynamic environment presupposes the consideration of units which can work simultaneously, as for instance happens in the situations described in the previous Remark 2.
However this is in general a rather complicated situation to deal with and one usually restricts attention to the case where dynamic histories are observed of the type in 2.
This is the case which will be considered in detail in the next section when introducing the concept of multivariate conditional hazard rate functions. Such a concept is in particular helpful to describe cases when, even conditionally on the knowledge of the environment process, the lifetimes are not necessarily independent. The following two examples show special models obtained by specifying arguments contained in Singpurwalla and Yougren to the case of exchangeable lifetimes.
Cox and Isham, , p. Furthermore assume that X1 ; X2 ; are independent of T 1 ; T 2 ; This situation leads, in a natural way, to the de nition of the concept of multivariate conditional hazard rate m. Also in formulating the de nition of m.
Subjective Probability Models for Lifetimes (Monographs on Statistics and Applied Probability 91)
More generally, this concept was considered e. We shall brie y report the general de nition in the next Chapter. This is related to the more general concept of stochastic intensity, introduced in the framework of the theory of point processes see Arjas, ; Bremaud, If not otherwise speci ed, the individuals U1 ; ; Un will be thought of as n pieces of an industrial equipment, since such interpretation actually provides a more exible basis for the language that we shall use.
De nition 2. Note that, due to exchangeability, the above limit does not depend on the choice of the index j 2 fj1 ; ; jn;h g. Thus we can consider that, in place of 2. Proposition 2. In the case when T1; ; Tn are i. Equation 2. Some of the examples show the special form taken by the m. From De nition 2. In the case of two individuals in Example 2. Consider the situation of a Schur-constant joint density already analyzed in Example 1. Moreover, by 2.
Think of two similar individuals, between whom there is a strong rivalry, so that each individual undergoes a stress which is proportional to the total age cumulated by the other individual. We model this by assuming that their lifetimes T1 and T2 are exchangeable random quantities with a joint distribution characterized by m. Note that the condition 2. Let the joint density function be of the form 2.
Replace f n in 2. Now contrast 2. Proportional hazard models. The condition 2. This will be rendered in more precise terms in Chapter 3 Subsection 3. It is easy to understand that cases of positive dependence are also those described in the above Example 2. Lemma 2. We start by proving 2. First we need to extend Equation 2. From Equation 2. By Lemma 2. In such cases the equations in 2. In order to obtain such formulae a setting of stochastic processes and appropriate general assumptions are needed.
However it is still possible to see directly, by means of a heuristic reasoning, what in tractable cases the appropriate generalization of 2.
Here we shall not consider such a generalization. Rigorous results can be obtained as particular cases of a general theorem from the theory of point processes see e. This is the case, for instance, in Example 2. This means that we completely ignore the structure function of the system. For the case when environmental factors are constant in time, this is better explained by means of Theorem 2.
In this section we focus attention on some further aspects of interest related to the notion of multivariate conditional hazard rate.
Subjective probability models for lifetimes - کتابخانه دیجیتال جندی شاپور اهواز
For this purpose, we keep the same setting and same notation as in Section 2. T1 ; ; Tn are i. More generally, 2. In the same cases, on the contrary, it can take a while to gure out what the corresponding joint density or the joint survival function should be. This fact allows us to focus one reason of interest in Proposition 2. One aspect of the above is that, in particular, no simple equation relates m. The following de nition is also of interest in the setting of longitudinal observations of failure data.
The inverse of 2. Barlow and Proschan, , p. T1; ; Tn are independent, identically distributed, standard exponential variables if and only if the corresponding variables C1 ; ; Cn are also independent, identically distributed, standard exponential variables. By combining this result with the relation 2. V1; ; Vn is distributed as the vector of the order statistics of n independent, identically distributed, standard exponential variables if and only if R1 ; ; Rn are independent, identically distributed, standard exponential variables.
The above Lemma can be used in the derivation of the following result, concerning the vectors W1 ; ; Wn and 1 ; ; n , respectively, de ned in 2. Assume that 2. Then W1 ; ; Wn are also independent, identically distributed, standard exponential variables and 1 ; ; n is distributed as the vector of the order statistics of n independent, identically distributed, standard exponential variables. From Lemma 2. In order to conclude the proof we only have to take into account 2. A remarkable fact is that the property of W1 ; ; Wn and 1 ; ; n , proved in Proposition 2.
To this purpose it is helpful to keep in mind the following elementary result. Notice that now we are not requiring, for f n , the condition 2. As shown by Equation 2. Then W1 has a standard exponential distribution according to Lemma 2. Now we aim to obtain the conditional distribution of W2 given W1 : Since w1 t1 is a one-to-one mapping, we can equivalently compute the conditional distribution of W2 given T 1 : By 2. In order to prove b we simply have to recall Equation 2. A presentation of the result above in a much more general setting has been given in Arjas The proposition 2.
Such a construction goes along as follows: i x a vector of i.