COMPOSITE RELIABILITY MODELS FOR SYSTEMS WITH TWO DISTINCT KINDS OF STOCHASTIC DEPENDENCES BETWEEN THEIR COMPONENTS LIFE TIMES

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1 COMPOSITE RELIABILITY MODELS FOR SYSTEMS WITH TWO DISTINCT KINDS OF STOCHASTIC DEPENDENCES BETWEEN THEIR COMPONENTS LIFE TIMES Jerzy Filus Department of Mathematics and Computer Science, Oakton Community College, Des Plaines, IL 60016, USA Lidia Filus Department of Mathematics, Northeastern Illinois University Chicago, IL 60625, USA, Key words: reliability of multi-component systems, multivariate probability densities as models, fusion of two system models, bivariate Gumbel pdf in a reliability setting. Abstract Multivariate, especially bivariate, probability densities are constructed and applied as stochastic models for the reliability of multi-component parallel systems. We consider two classes of models; each of them describing different stochastic failure mechanisms, and consequently a different kind of component life times stochastic dependence. One of the classes contains (exclusively) the first type of Gumbel bivariate exponential pdfs. In one of our earlier works, the Gumble pdfs gained (perhaps for the first time) an interpretation as reliability models. In this paper, these models describe one of two types of system components failure. The other class of pdfs contains the bivariate pseudoexponential ( ex-exponential ) generalization of the univariate exponentials. After some elaboration of the models we construct the fusion of the considered two classes, such that the resulting composite Gumbel - pseudoexponential densities become, at least potentially, models for systems subjected to two distinct physical processes. Each of these processes, separately, may add to any of the two component s failure. As it was pointed out, the fusion of the two failure models has an analytical reflection by a nice non-trivial factorization of the composite description. 1. Introduction Since about 1960 many stochastic models for reliability of multi-component systems appeared in the literature, see e.g. [2]. (By models we mean the bi- and multi-variate probability distributions of system components life times, given a reliability structure that is parallel throughout this text). One has, unfortunately, to admit that a majority of the constructed models are only potential with respect to their reliability application. Some of them found their reliability applications through time. For example, this is likely the case of the (considered in this paper) first Gumbel bivariate exponential density (see [10], [11]

2 and specifically [6], [8] ), while many other models still remain only mathematical entities, some of significant theoretical importance with promising analytical properties, that may indicate possibilities for future valuable applications. Few models describe explicitly some realistic physical situations, or, more generally, types of situation. In some cases, physics of the reliability problem motivated the stochastic description. Such classical constructions were established first of all by Freund [9], Marshal and Olkin [14], [12], and others, as well as by a number of their followers. For the latter, see for example [3] and [13] ). This paper is devoted to the investigation of two different reliability models of the latter type. The first is the bivariate exponential Gumbel probability distribution of the first kind. This mathematically very nice model (perhaps the first important bivariate probability distribution, that is not the bivariate normal, published in 1960), until recent days lacked any reliability interpretation. Probably, the first such interpretation of the Gumbel distribution was a (very) special case of a larger class of the reliability models constructed and investigated in [8]. As the second class of the models we will consider one of our earlier obtained class of bivariate densities, i.e., the so called, exexponential ( ex like extended, but better known under the name pseudoexponential ) pdfs (see, for example [4], [5], [7] ). Both the Gumbelian and the pseudoexponentials may be viewed as two independent stochastic descriptions, each of one out of two distinct kinds of physical phenomena, respectively. The task we perform here is to build the joint stochastic model in a situation where the physical system is subjected to two independent failure mechanisms: one described by the bivariate Gumbelian (in [8] ), and the other by the pseudoexponential probability density. This fusion of the two models is derived in section 5 with the final result given by (7). The nice model s factorizations, included in formulas (8) and (9), stand, in a sense, for an analytical reflection of the composite nature of the obtained fusion. The theory of the exexponential models is not understood well enough by a majority of reliability theory people. For that reason, before the actual composite models are introduced, some more detailed preliminary work on the construction and analysis of the exexponential models alone is presented in sections 2, 3, and 4. Occasionally, in these three sections we also included some information about other similar models (for example, the pseudoweibullian and the pseudonormal), that are based on the same pattern of the construction as the exexponentials. For the reliability framework of the Gumbelian class of the pdfs, and also of an associated wider class of reliability models, we refer to [8]. In particular, in section 2 we deal with the core of the theory, that is, with an analytical description (definition) of a relatively new kind of stochastic dependences. In that section the description has rather a general mathematical character, and comprises both the main bivariate case as well as its multivariate extensions. Parallel derivation of the same probability densities, this time based on physical reliability motivated analysis, is the subject of section 3. In both sections, the so called by us method of parameter replacement is employed for the pdfs construction. The alternative method of the construction i.e., the triangular (here, with its special pseudoaffine version) transformation method is presented in section 4. The

3 advantage of the transformation method over the first method, is the ability to use the model s defining transformations for both purposes: easy sampling from the models and for facilitating an underlying statistical analysis of unknown pdfs parameters. On the other hand, the first method is simpler and somehow more straightforward. The actual results are formulated in section Stochastic Dependences Description Via the Method of Parameter Replacement As already mentioned, the two methods of construction of the exexponential (and other, for example the pseudonormal ) multivariate probability distributions, here presented, are relatively new. The first procedure of the construction, in the bivariate case, can be described as follows: Consider a class of continuous (i.e., possessing a joint pdf ) random vectors (X 1, X 2 ). It is assumed that the following is either known or can be imposed: A) a conditional pdf g 2 (x 2 x 1 ), while, in general, the other conditional pdf g 1 (x 1 x 2 ) may not be given. B) a marginal pdf g 1 (x 1 ) of X 1 is known, while the marginal g 2 (x 2 ) of X 2 is usually unknown. Given, the two pdfs g 1 (x 1 ), and g 2 (x 2 x 1 ) the joint pdf g(x 1,x 2 ) of the random vector (X 1, X 2 ) is simply equal to the arithmetic product: g 1 (x 1 ) g 2 (x 2 x 1 ) = g(x 1, x 2 ). In other words, the random variable X 1 may be thought of as an explanatory variable for X 2. In the general multivariate case, the formal pattern of construction of joint pdfs g(x 1,,x n ) (n=2,3,..) of r. vectors (X 1,,X n ) obeys the following restrictions: 1) For j = 2, 3,,n the random variables X 1,,X j-1 (or just X j-1, in 2) Markovian cases ) are explanatory variables for X j. 3) The marginal g 1 (x 1 ) of X 1 is assumed to be known. 4) For all j = 2,,n the X j has the conditional pdf g j (x j x 1,,x j-1 ) to be determined. (Remark: The above pattern also applies to the situation when {X j } is a stochastic process, j = 2, 3, with n = ). Return to the n = 2 case. Consider the product g 1 (x 1 ) g 2 (x 2 x 1 ). In general, the factors may belong to different pdfs classes. For simplicity let s assume, tentatively, that both factors belong to the same class of pdfs. The first method of construction of bivariate pdfs is the Parameter Replacement Method or the Method of Partial Conditioning (transformation of probability densities). Let f 1 (x 1 ;θ 1 ), f 2 (x 2 ;θ 2 ) be pdfs of some independent random variables T 1, T 2. Transform f 1 (x 1 ;θ 1 ), f 2 (x 2 ;θ 2 ) into other (related) pdfs from some class of distributions by imposing that that either θ 1 = θ 1 (x 2 ) or θ 2 = θ 2 (x 1 ) admitting only one of the two possibilities: the second (in the case here investigated). Define f 2 (x 2 ; θ 2 (x 1 ) ) = g 2 (x 2 x 1 ), while g 1 (x 1 ; θ 1 ) = f 1 (x 1 ; θ 1 ). In this way a class of new bivariate pdfs of (X 1, X 2 ) is obtained as:

4 g(x 1,x 2 ) = g 1 (x 1 ) g 2 (x 2 x 1 ). Anticipating the context of next section, it is possible that the same pdfs g(x 1, x 2 ) may, alternatively, be obtained by a suitable triangular transformations: (T 1,T 2 ) (X 1,X 2 ), where the random variables T 1, T 2 either are independent or the joint pdf of (T 1, T 2 ) is considered to be a well known pdf, such as bivariate normal or the first exponential Gumbel, for example). An Example: Suppose the random variables T 1, T 2 are independent and f k (t k ) = (1/θ k ) exp[ - t k /θ k ] is a pdf of T k (k = 1,2). Letting θ 2 = θ 2 (x 1 ) in the exponential f 2 (x 2 ;θ 2 ), given above, one obtains a bivariate exexponential pdf: g(x 1, x 2 ) = g 1 ( x 1 ) g 2 (x 2 x 1 ) = (θ 1 ) -1 exp [-x 1 /(θ 1 )](θ 2 (x 1 )) -1 exp[-x 2 /(θ 2 (x 1 ))] where, in particular, one may specify: θ 2 (x 1 ) = θ 2 (1 + Ax 1 r ), where A and r are positive reals. Other, analytically interesting model is: θ 2 (x 1 ) = θ 2 exp[ax 1 r ], with A arbitrary. Note that both factors g 1 (x 1 ), g 2 (x 2 x 1 ) of g(x 1, x 2 ), given above, are exponentials, so it is called exexponential (or pseudoexponential). In a similar way one can also obtain a class of bivariate exnormal (former name pseudonormal ) pdfs h(x 1, x 2 ), given independent univariate normal pdfs: f 1 (x 1 ) = N(μ 1, σ 1 ), f 2 (x 2 ) = N(μ 2, σ 2 ), and new stochastic dependencies are imposed by the following analytical relationships: μ 2 = μ 2 (x 1 ), σ 2 = σ 2 (x 1 ), while keeping f 1 (x 1 ) invariant. In such a way one obtains the bivariate exnormal pdfs as the products h(x 1,x 2 ) = h 1 (x 1 ) h 2 (x 2 x 1 ), where h 1 (x 1 ) = f 1 (x 1 ) = N(μ 1, σ 1 ), and h 2 (x 2 x 1 ) = N( μ 2 (x 1 ), σ 2 (x 1 ) ) are normal in x 1, and x 2 respectively. Finally one obtains: h(x 1, x 2 ) = [σ 1 (2π)] -1 exp[-(x 1 - μ 1 ) 2 /2σ 1 2 ] [σ 2 (x 1 ) (2π)] -1 exp[-(x 2 μ 2 (x 1 )) 2 / 2[σ 2 (x 1 )] 2 where μ 2 (x 1 ) = E[X 2 x 1 ] is (possibly nonlinear) continuous regression function, and the conditional variance: [σ 2 (x 1 )] 2 = Var [X 2 x 1 ] may be chosen to be arbitrary continuous function in x 1. In particular, one may consider the (exnormal) regression function in the form: E[X 2 x 1 ] = μ 2 + a (x 1 - μ 1 ) + A (x 1 - μ 1 ) n, with μ 2 (this time) being constant a, A arbitrary real, and n = 2, 3, For the coefficient A, small in comparison to the a, the term A(x 1 - μ 1 ) n may be considered as a nonlinear correction to the Gaussian (linear) regression. The cases n = 2, and n = 3 i.e., quadratic (asymmetric), and cubic (symmetric) are of special interest. Parameter n also may be considered an arbitrary real (in particular negative). 3. The Stochastic Dependencies in Reliability Context The origin of the ideas presented in this work takes roots in the following system

5 reliability problem: Given a parallel system composed of the components e 1, e 2,,e n. Suppose that initially the life time of each of the n system components is tested in separation of others in idealized laboratory conditions. Therefore, at this primary stage, the component life times, here denoted by T 1,,T n, are considered to be stochastically independent. As a result of a testing procedure one obtains estimated pdfs f 1 (t 1 ; θ 1 ),,f n (t n ; θ n ) of the life times T 1,,T n respectively, where for every j = 1,,n, θ j denotes a scalar or a vector parameter of the j-th pdf. In the next stage new components e 1 *,,e n * are installed into the real system. They are assumed to be statistically identical to the components e 1,,e n before the laboratory conditions testing procedure started. For that reason we still denote them by the same symbols e 1,,e n as before. The pdfs of the component life times, once they are installed into the system, change a bit because some physical interactions between the working components occur. The life times of the components, now working in the system, differ from those T 1,,T n in the laboratory conditions. The new life times, which in general remain in some associations with the original ones, will be denoted by X 1,,X n throughout, and are considered to be stochastically dependent. The physical interactions between the components e 1, e 2,, e n, when in the system, are assumed to be continuous in the sense that they take place at any time instant when the components work together. The physical, and the resulting stochastic, dependencies differ essentially from those considered, in [9] and [14]. In the case, here considered, the changes in the component life times are actually not supposed to be caused directly by failing of some other components. The interactions presently considered are caused by the components mutual cooperation, when they work, while in the approach developed by Freund in [9] the disturbing factor is lack of other components in the system. Here we explore, in a sense, a reverse idea by considering the absence of some components as the factor making the work conditions of the surviving ones closer to the normal, i. e., to the laboratory conditions. Also, regardless of some similarities in defining the conditional distributions, the method of construction of the joint probability densities we develop, essentially differs from that described, for example in [1]. One of the obvious reasons for that difference is that the obtained classes of the models, ours and those cited in [1], are essentially disjoint. Suppose that in the system the following phenomena are observed or can be deduced: [For better clarity: The symbol (s) and (ies), present in what follows, means the plural. For example, by component(s) we mean one component or, for a more general case, components ; also e j (s) = one component e j or, perhaps several components, all depending on a degree of generality we chose.] One (or more than one) component(s), say, e i (s) by its (their) activity(ies) change(s) the environment or work conditions of some other component(s) e j (s). The influence of e i (s) on e j (s) may cause either improvement or deterioration of the component(s) e j (s) functioning (from its (their) reliability viewpoint). These changes cause the life time(s) X j (s) of e j (s), in the system, to become statistically longer or shorter than their original life times T j (s) under laboratory conditions. In order to grasp somewhat fugitive physical interactions between the system components

6 in a stochastic reliability model, (note that the system component life times X 1,,X n are now stochastically dependent ) the following will be assumed: 1) The magnitude or the amount of an influence (from the reliability point of view) of, say, e i on e j ( i j) depends explicitly on the length of time X i of the components mutual cooperation, provided the random event X i < X j occurs. This dependence can be described by a suitable continuous function of X i in the following manor. 2) If the component e i fails before the failure of e j, the primary laboratory conditions pdf f j (t j θ j ) of the life time T j of e j are affected as a result of the components cooperation during the random time X i. 3) For a simple and stochastically meaningful measure of an influence on e j s life-time during that cooperation a change in the parameter(s) θ j of the original e j -th life time T j pdf f(t j, θ j ) has been chosen. 4) In order to grasp a magnitude of this measure (i.e., measure of dependence of the j-th parameter on the time X i ) a suitable continuous function θ j * = θ j * (θ j, x i ) should be chosen, while it will be assumed that the random events X i = x i and x i < X j occurred. Additionally, the condition θ j *(0) = θ j should be imposed whenever the reliability modeling is under consideration. In other possible applications this condition may not be mandatory. [The failure model, so far described, is proposed to be named continuous (micro)shocks (micro)damages model, with the micro-shocks and resulting microdamages mathematically can be described by infinitesimal quantities.] As a part of the proposed stochastic model one obtains the conditional pdfs g j (x j x i ) of the life time X j, given that X i = x i (with i < j, and x i < x j ) which are defined by g j (x j x i ) = f j(x j ; θ j *( θ j, x i )). Generalizing the pattern, the influence of all the components, say, e 1, e 2,,e j-1 on e j s life time length can possibly be described by a suitably chosen ( where the choice should depend on physical properties of a particular real system ) function θ j * = θ j *(θ j, x 1,,x j-1 ), continuous, at least with respect to each argument x k separately, k = 1,, j-1. Furthermore, in order to achieve more simplicity in case of the reliability modeling, the two following assumptions are to be adopted: (S1) for j = 1,,n-1 the components e j+1, e j+2, e n have no physical influence on the component e j, regardless of the obvious presence of stochastic dependences of the r. variable X j on the possible values of the variables X j+1, X j+2,,x n. Consequently it must be assumed that no component has a physical influence on e 1. (S2) The probability of the random event (X 1 < X 2 < < X n ) is sufficiently close to one. { An idea (to be verified in a future) of how to overcome this or a similar difficulty i.e., when the condition S2 is not satisfied (with a significant probability) is as follows: Suppose n = 2, and instead of X 1 < X 2 the opposite event had happened. Now, one may try to apply the old models and then go with them to a limit as x 1 x 2, whenever possible. On the other hand, the

7 order assumption S2 is not always mandatory.} If no additional stochastic dependencies between the life times X 1,,X n are assumed to occur (or other possible dependencies are negligible) one obtains the system reliability model, in the form of a joint pdf g(x 1,,x n ) of the random vector (X 1,,X n ), in the following two stages: First. The following common representation of n-variate pdfs, i.e, the product form: g(x 1,,x n ) = g 1 (x 1 ) g 2 (x 2 x 1 ) g 3 (x 3 x 1, x 2 ) g n (x n x 1,,x n-1 ) (1) is chosen for the further constructions. Second. The factors of the product (1) can be defined in the following way: 1) In accordance with the assumption (S1) we set X 1 = T 1, in the sense that g 1 (x 1 ) = f 1 (x 1 ) as x 1 = t 1. 2) The remaining n-1 factors in (1) are given by all the equalities: g j (x j x 1,,x j-1 )= f j (x j ;θ j *(θ j, x 1,,x j-1 ), j = 2,3,,n, where the functions θ j * (θ j, x 1,,x j-1 ) (with their reliability meaning described above), are only assumed to be arbitrary continuous (or piecewise continuous) at least with respect to each of the arguments x 1, x 2,,x j-1 separately, with the condition θ j *(θ j, 0,,0) = θ j holds. As an example of a parameter function one may consider the function: θ j *( θ j ; x 1,,x j-1 ) = θ j cosh( c 1 j x 1 ά + + c j-1 j x j-1 ά ). 4. The (Triangular) Transformations Method Consider transformations of random vectors: (T 1,, T n ) (X 1,,X n ), where the distribution of (T 1,,T n ) is usually a well known one. All the transformations here considered are assumed to have a triangular Jacobi matrix. One of the simplest cases is the following class of the Pseudoaffine Transformations (easily invertible): X 1 = ϕ 0 T 1 + θ 0 X 2 = ϕ 1 (X 1 )T 2 + θ 1 (X 1 ) (2)... X n= ϕ n-1 (X 1,,X n-1 )T n + θ n-1 (X 1,,X n-1 ) (notice that, for n : we obtain a transition to stochastic processes ) The continuous functions ϕ 0, ϕ 1 ( ),, ϕ n-1 ( ), and θ 0, θ 1 ( ),, θ n-1 ( ) are called parameter functions. In the case when θ 0 = θ 1 ( ) = = θ n-1 ( ) = 0 the transformations in (2) will be named diagonal pseudolinears with respect to the r. variables T 1,,T n. If ϕ 0 = ϕ 1 ( ) = = ϕ n-1 ( ) = 1 they will be named pseudotranslations. The inverse to the foregoing pseudoaffine transformation is: T 1 = (X 1 - θ 0 ) / ϕ 0 T 2 = (X 2 - θ 1 (X 1 )) / ϕ 1 (X 1 ) (3) T n = (X n - θ n-1 (X 1,,X n-1 )) / ϕ n-1 (X 1,,X n-1 ) The jacobian of the inverse transformation is in a simple form of a product: [ϕ 0 ϕ 1 (x 1 ) ϕ 2 (x 1, x 2 ) ϕ n-1 (x 1,,x n-1 ) ] -1. The r. variables T 1,,T n are assumed all to have one of the following pdfs: 1) the gaussian (normal, arbitrary: no independence assumed),

8 2) the two or the three parameter Weibullian, 3) the gamma, the generalized gamma and, 4) as particular case, the one or the two parameters exponential. Accordingly, the resulting joint pdfs g(x 1,,x n ) of the r. vectors (X 1,,X n ) are: exnormal, exweibullian, exgamma, exgeneralized gamma, and exexponential. All the joint pdfs are given in the common product form (1), where g 1 (x 1 ), and g j (x j x 1,...,x j-1) are either all normal or all Weibullian, or all gamma, or all generalized gamma or all exponential pdfs in x 1 and x j respectively. When T 1,,T n are independent normal then the pdf g(x 1,,x n) of (X 1,,X n ) is given by the initial marginal g 1 (x 1 )=N(μ 1,σ 1 ), and for each j=2, 3,, n g j (x j x 1,...,x j-1) = (σ j φ j-1 ( x 1,...,x j-1 ) 2π ) -1 exp[- (1/2σ j 2 (φ j-1 (x 1,..., x j-1)) 2 ) (x j - θ j-1 (x 1,...,x j-1)) 2 ] therefore the pseudonormal g(x 1,,x n) is uniquely represented by (1). 5. An Extension of the First Gumbel Bivariate Exponential Densities Toward the First Gumbel Bivariate exexponential Models Recall that the first Gumbel bivariate pdf family of random vectors, say (Y 1, Y 2 ) ), in its standard form, (see, for example [11] ) obeys the following scheme: h (y 1, y 2 ) = exp [- y 1 y 2 ] exp [ - η y 1 y 2 ] {(1 + η y 1 ) (1 + η y 2 ) - η} (4) with y 1, y 2 0. For our future needs we will write the above formula in the form: h (y 1, y 2 ) = exp [- y 1 y 2 ] Gumb(y 1, y 2 ; η). (Notice that Gumb(y 1, y 2 ; 0) = 1, so that the condition η = 0 makes the r. variables Y 1, Y 2 independent (standard) exponential.) Parallelly, consider class of the bivariate exexponential pdfs, say g (x 1, x 2 ) of random vectors (X 1, X 2 ), that also are obtained from the product exp [- t 1 t 2 ] of the standard exponential pdfs of some independent r. variables, say, T 1, T 2 but now by use of the corresponding class of pseudolinear transformations : X 1 = ϕ 0 T 1 X 2 = ϕ 1 (X 1 ) T 2. (5) Each transformation satisfying the pattern (5) results with the (standard) exexponential joint pdf of the random vector (X 1, X 2 ) that is given by: g (x 1, x 2 ) = (ϕ 0 ) -1 exp[- x 1 /ϕ 0 ] { (ϕ 1 (x 1 ) -1 exp[ -x 2 / ϕ 1 (x 1 ) ] } (6) for any constant ϕ 0 and an arbitrary continuous function ϕ 1 (x 1 ). In order to construct the composite model we apply transformations (5) to the Gumbelian random vectors (Y 1, Y 2 ), defined above. We then consider the formula (5) with the input (Y 1, Y 2 ) instead of the input T 1, T 2. That is: Z 1 = ϕ 0 Y 1 Z 2 = ϕ 1 (Z 1 ) Y 2. (5*) Recall that the joint (Gumbel) pdf of (Y 1, Y 2 ) is

9 h (y 1, y 2 ) = exp [- y 1 y 2 ] exp [ - η y 1 y 2 ] {(1 + η y 1 ) (1 + η y 2 ) - η}. After performing usual calculations a new composite 1 th Gumbelian pdf p (g, h) (z 1, z 2 ) is obtained in the form: p (g, h) (z 1, z 2 ) = ({(1/ϕ 0 ) exp[ -z 1 /ϕ 0 ] } {(1/ϕ 1 (z 1 ) ) exp[ - z 2 /ϕ 1 (z 1 ) ] }) { [{(1 + η z 1 /ϕ 0 ) (1 + η z 2 / ϕ 1 (z 1 ) ) - η)] exp[ - η z 1 z 2 / ϕ 0 ϕ 1 (z 1 ) ] }. (7) This probability density is a result of the «composition» of two different stochastic models of failure mechanisms: one that, when alone, results with the exexponential pdf g (x 1, x 2 ), and the other (when alone) expresses, say, Gumbel s failure rule h (y 1, y 2 ) of the system component lifetimes dependence. Notice, that each of the two stochastic rules may reflect different physical phenomena or mechanisms. Our hypothesis is, that the composition of these physical phenomena results with the joint stochastic model (7) that describes the net efect of the two phenomena on the system reliability. It is nice that the fusion of the two models i.e., the junction of the densities g(, ), and h(, ) finds its analytical expression through the following factorization: p (g, h) (z 1, z 2 ) = g (z 1, z 2 ) Gumb(η) (8) where Gumb(η) = { [{(1 + η z 1 /ϕ 0 ) (1 + η z 2 / ϕ 1 (z 1 ) ) - η)] exp[ - η z 1 z 2 / ϕ 0 ϕ 1 (z 1 ) ] } is uniquelly determined by the pdf h(, ). Notice too, that Gumb(0) = 1 so that here the parameter η may serve as a measure of an influence of a ( physical) «Gumbel mechanism» or just the «Gumbel part» of the stochastic dependence. The factorization (8) allows to consider the density p (g, h) (z 1, z 2 ) as a kind of «product» of the densities g( ) and h( ). This «multiplication» is, in its sense, «comutative». The analitic form of the density p (g, h) (z 1, z 2 ) also exhibits possibility of other meaningfull factorization of the composite model. Firstly, it can be shown that (if ϕ 0 = 1) then the marginal pdfs of X 1 and Z 1 are the same, i.e., both having the same (in particular standard ) exponential pdfs g 1 (z 1 ) = h 1 (z 1 ), when setting x 1 = y 1 = z 1. As g (z 1, z 2 ) = g 1 (z 1 ) g 2 (z 2 z 1 ), one obtains that p (g, h) (z 1, z 2 ) = g 1 (z 1 ) h 2 (z 2 z 1 ), and consequently one obtains a following, slightly different, factorization for the conditionals: h 2 (z 2 z 1 ) = g 2 (z 2 z 1 ) Gumb(η). (9) References [1] B. C. Arnold, E. Castillo and J. M. Sarabia, Conditionally Specified Distributions, Lecture Notes in Statistics - 73, New York: Springer- Verlag, [2] R.E. Barlow and F. Proschan, Statistical Theory of Reliability and Life Testing, Holt, Rinehart and Winston, New York, [3] J. K. Filus, On a Type of Dependencies between Weibull Life times of System Components, Reliability Engineering and

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