RELIABILITY FOR SOME BIVARIATE EXPONENTIAL DISTRIBUTIONS

Transcription

1 RELIABILITY FOR SOME BIVARIATE EXPONENTIAL DISTRIBUTIONS SARALEES NADARAJAH AND SAMUEL KOTZ Received 8 January 5; Revised 7 March 5; Accepted June 5 In the area of stress-strength models, there has been a large amount of work as regards estimation of the reliability R = PrX <Y). The algebraic form for R = PrX <Y)has been worked out for the vast majority of the well-known distributions when X and Y are independent random variables belonging to the same univariate family. In this paper, forms of R are considered when X,Y) follow bivariate distributions with dependence between X and Y. In particular, eplicit epressions for R are derived when the joint distribution is bivariate eponential. The calculations involve the use of special functions. An application of the results is also provided. Copyright 6 S. Nadarajah and S. Kotz. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.. Introduction Without a doubt, eponential distributions are the most popular and the most applied life time models in many areas, including life testing and telecommunications. In the contet of reliability, the stress-strength model describes the life of a component which has a random strength Y and is subjected to random stress X. The component fails at the instant that the stress applied to it eceeds the strength, and the component will function satisfactorily whenever Y>X.Thus,R = PrX <Y) is a measure of component reliability. It has many applications especially in engineering concepts such as structures, deterioration of rocket motors, static fatigue of ceramic components, fatigue failure of aircraft structures, and the aging of concrete pressure vessels. Some eamples are as follows: i) if X represents the maimum chamber pressure generated by ignition of a solid propellant and Y represents the strength of the rocket chamber, then R is the probability of successful firing of the engine; ii) if X represents the diameter of a shaft and Y represents the diameter of a bearing that is to be mounted on the shaft, then R is the probability that the bearing fits without interference; Mathematical Problems in Engineering Volume 6, Article ID 465, Pages 4 DOI.55/MPE/6/465

2 Reliability for some bivariate eponential distributions iii) let Y and X be the remission times of two chemicals when they are administered to two kinds of mechanical systems. Inferences about R present a comparison of the effectiveness of the two chemicals; iv) if X and Y are future observations on the stability of an engineering design, then R would be predictive probability that X is less than Y. Similarly, if X and Y represent lifetimes of two electronic devices, then R is the probability that one fails before the other; v) if Y represents the distance of a pyrotechnic igniter from its adjacent pellet and X represents its ignition distance,then R is the probability that the igniter succeeds to bridge the gap in the pyrotechnic chain; vi) a certain unit be it a receptor in a human eye, ear, or any other organ including seual) operates only if it is stimulated by source of random magnitude Y and the stimulus eceeds a lower threshold X specific for that unit. In this case, R is the probability that the unit functions; vii) in military warfare, R could be interpreted as the probability that a given round of ammunition will penetrate its target; viii) in quality control, the use of process capability indices is motivated by a desire to have an inde related to the probability that an attribute Z) ofacomponent size, density, elastic strength, etc.) falls within fied specification limits. However, in some circumstances it may be desirable to have an inde allowing for possibly varying limits T L or T U, say, for lower and upper limits, respectively. One is then interested in PrT L <Z<T U ). If only one of the limits is finite, then this probability reduces to calculations of R = PrX <Y)type. Because of these applications, the calculation and the estimation of R = PrX <Y)isimportant for the class of bivariate eponential distributions. The calculation of R has been etensively investigated in the literature when X and Y are independent random variables belonging to the same univariate family of distributions. The algebraic form for R has been worked out for the majority of the well-known distributions, including normal, uniform, eponential, gamma, beta, etreme value, Weibull, Laplace, logistic, and the Pareto distributions Nadarajah, [ 6]; Nadarajah and Kotz [7]). In this paper, we calculate R when X and Y are dependent random variables from si fleible families of bivariate eponential distributions Sections to 7). We also provide an application of the results to receiver operating characteristic curves Section 8). We will assume throughout this paper that X,Y) has a bivariate eponential distribution with joint probability density function pdf) f and joint survivor function F. One can write R = f, y)dyd..) Our calculations of R make use of a number of special functions. They are the complementary incomplete gamma function defined by Γa,) = t a ep t)dt,.)

3 S. Nadarajah and S. Kotz 3 the eponential integral function defined by Ei) = ep t) dt,.3) t and the modified Bessel function of the first kind of order zero defined by I z) = k= z k 4 k k!)..4) The properties of these special functions being used can be found in Prudnikov et al. [9 ] and Gradshteyn and Ryzhik [5].. Gumbel s bivariate eponential distribution Gumbel s [6] bivariate eponential distribution has the joint survivor function and joint pdf specified by F, y) = ep { α + βy+ θαβy) }, f, y) = { θ)αβ + θα β + θαβ y + θ α β y } F, y),.) respectively, for >, y>, α, β, and <θ<. This is the earliest and the simplest known bivariate eponential distribution. It has received applications in many areas, including competing risks, etreme values, failure times, regional analyses of precipitation, and reliability. The marginal distributions of X and Y are eponential with parameters α and β, respectively; so, in particular, EX) = /α and EY) = /β. The correlation coefficient ρ = CorrX,Y)isgivenby ρ = θ ep θ ) ) Ei..) θ The correlation is, of course, zero when θ = the case of independence between X and Y) and it decreases to.4365 as θ increases to. The reliability in.) can be epressed as R = { θ)αβ + θα β } F, y)dyd + { θαβ + θ α β } y F, y)dyd θ)α + θα = F,)d + +θα = αi + θαβi, θα+ θ α + θα) F,)d.3)

4 4 Reliability for some bivariate eponential distributions where I = I = F,)d, F,)d. Application of Lemma A. shows that one can reduce { } π α + β) I = θαβ ep Φ α + β, 4θαβ θαβ I = { } πα + β) α + β) θαβ ep Φ α + β, θαβ) 3/ 4θαβ θαβ.4).5) where Φ ) denotes the cdf of the standard normal distribution. Thus, it follows using.3) that the form of R for Gumbel s bivariate eponential distribution is given by R = { } πα β) α + β) + θαβ ep Φ α + β..6) 4θαβ θαβ Note that if α<β,thenr</; and if α>β,thenr>/. Moreover, if α = β, thenr = /. 3. Hougaard s bivariate eponential distribution Hougaard s [7] bivariate eponential distribution has the joint survivor function and joint pdf specified by [ { ) r ) y r } /r ] F, y) = ep +, θ φ { ) f, y) = y)r r ) y r } /r [ { ) r ) y r } /r ] 3.) θφ) r + r + + F, y), θ φ θ φ respectively, for >, y>, θ, φ, and r>. This distribution has been quite popular as a frailty model. The marginal distributions of X and Y are eponential with parameters /θ and /φ, respectively; so, in particular EX) = θ and EY) = φ. Thecorrelation coefficient ρ = CorrX,Y)isgivenby ρ = Γ /r). 3.) rγ/r) Note that if one transforms U,V) = X/θ,Y/φ), then the joint survivor function and joint pdf of U,V) take the simpler forms Fu,v) = ep { u r + v r) /r }, f u,v) = uv) r u r + v r) /r { r + u r + v r) /r } Fu,v). 3.3)

5 Thus, the form of R corresponding to 3.) can be computed as S. Nadarajah and S. Kotz 5 R = PrθU/φ < V) = = u r θu/φ u r = r )I + I, v r u r + v r) /r { r + u r + v r) /r } Fu,v)dvdu θ r +φ r ) /r u/φ z r r +z)ep z)dzdu where the transformation z = u r + v r ) /r has been applied, and θ I = u r r Γ r, I = u r Γ r, ) /r ) φ r + u du, θ r ) /r ) φ r + u du. 3.4) 3.5) Application of Lemma A. shows that one can reduce φ r I = r θ r + φ r), φ I r 3.6) = r θ r + φ r). Thus, it follows from 3.4) that the form of.) for Houggard s bivariate eponential distribution takes the simple form Note that if θ = φ,thenr = /. R = φr θ r + φ r. 3.7) 4. Downton s bivariate eponential distribution Downton s [3] bivariate eponential distribution has the joint pdf specified by f, y) = μ μ ep μ ) + μ y I { ρμ μ y } 4.) for >, y>, μ >, μ >, and ρ<. This distribution arises from shocks causing various types of failure to components which have geometric distributions. It has received applications in queueing systems and hydrology and has been used a model for Wold s Markov dependent processes, intensity and duration of rainfall, and height of water waves. The marginal distributions of X and Y are eponential with parameters μ and μ, respectively; so, in particular, EX) = /μ and EY) = /μ.theparameterρ is the correlation coefficient between X and Y with independence corresponding to ρ =.

6 6 Reliability for some bivariate eponential distributions Using the definition of I ) giveninsection, the corresponding form of R can be calculated as R = μ ) μ k ρμ μ ) k= k k!) = μ ) μ k ρμ μ ) k= k k!) ) k ρμ = μ ) k= k k! ) k ρμ k = μ ) k= k k! l! l= ) k ρμ k = μ ) k k! l! k= l= ρ = ) k μ k! μ + μ k= { k ep μ k y k ep k ep μ μ ) + μ y dyd ) ) } μ + μ k l! l= ) l { ) k+l μ + μ ep ) l ) μ k+l+ k + l)! μ + μ ) k+ k l= ) k + l)! l μ. l! μ + μ y k ep μ ) y dyd ) μ l d } d In the particular case μ = μ, one can show that the above reduces to R = /. 5. Arnold and Strauss bivariate eponential distribution Arnold and Strauss [] bivariate eponential distribution has the joint pdf specified by 4.) f, y) = K ep { a + by+ cy) } 5.) for >, y >, a >, b >, and c >, where K denotes the normalizing constant given by ) ) ab ab K = c ep Ei. 5.) c c Note that the marginal pdfs of X and Y are not eponential and are given by f X ) = ep a) b + c, f Y y) = ep by), a + cy respectively, with the epectations EX) = b ) K c ab, EY) = a ) K c ab. 5.3) 5.4)

7 S. Nadarajah and S. Kotz 7 However, the conditional distributions of Y X = and X Y = y are eponential with parameters b + c and a + cy, respectively. Arnold and Strauss [] motivated the use of 5.) by observing that it often happens that a researcher has better insight into the forms of conditional distributions rather than the joint distribution. The distribution has been used to model such variables as blood counts and survival times of patients. The correlation coefficient ρ = CorrX,Y)isgivenby ρ = abc + abk K Kab + c K). 5.5) One can show that the correlation is always negative and is bounded from below by the value.3. The form of R can be derived as R = K ep a) ep { b + c)y } dyd ep { a + b + c) } = K d b + c = K { a + b) c ep } ep z /c ) 4c z +b a)/ dz a+b)/ = K { a + b) c ep } 4c a+b)/ = K { } a + b) c ep 4c = K c ep { a + b) 4c = K c ep { a + b) 4c ep z /c ) ) b a k ) k z k+) dz k= ) b a k ) k z k+) ep z /c ) dz k= a+b)/ } ) b a k ) k w k/+) ep w)dw c k= a+b) /4c) } ) b a k ) Γ k k ) a + b),, c 4c k= 5.6) where the transformations z = c +a + b)/ andw = z /c have been applied, and the series epansion z + d = ) k d k z k+) 5.7) k= used. Thus, one has an epression for R which is an infinite sum of incomplete gamma functions. In the particular case a = b, one can show that R = /. 6. Freund s bivariate eponential distribution Freund s [4] bivariateeponentialdistributionhasthejointpdf specifiedby α β ep { β y ) } α + α β, if<y, f, y) = α β ep { β ) } 6.) α + α β y, if y<

8 8 Reliability for some bivariate eponential distributions for >, y>, α >, α >, β >, and β >. This distribution arises in the following setting: X and Y are the lifetimes of two components assumed to be independent eponentials with parameters α and α, respectively; but, a dependence between X and Y is introduced by taking a failure of either component to change the parameter of the life distribution of the other component; if component fails, the parameter for Y changed to β ; and if component fails, the parameter for X changed to β. As in the previous section, the marginal pdfs of X and Y are not eponential. They take the form of eponential mitures given by [ ) ) { ) } f X ) = α β α + α ep α + α + β α ep β )], α + α β [ ) ) { ) } f Y y) = α β α + α ep α + α y + β α ep β y )], α + α β 6.) respectively, with the epectations EX) = β + α β α + α ), EY) = β + α β α + α ). 6.3) The correlation coefficient ρ = CorrX,Y)is given by β β α α ρ =. 6.4) β +α α + α β +α α + α One can show that /3 <ρ<. The form of R is given by R = α β ep { β y α + α β ) } dyd = α ep { ) } α + α d 6.5) = α α + α. This epression is independent of β and β because of the contet described above: since X and Y are independent eponentials with parameters α and α, one must have X<Y with probability α /α + α ). 7. Marshall and Olkin s bivariate eponential distribution Marshall and Olkin s [9, ] bivariateeponentialdistribution has the joint pdf specified by ) { θ θ + θ 3 ep θ ) } θ + θ 3 y, if<y, ) { f, y) = θ θ + θ 3 ep θ y ) } θ + θ 3, if y<, θ 3 ep { 7.) ) } θ + θ + θ 3 y, if = y

9 S. Nadarajah and S. Kotz 9 for >, y>, θ >, θ >, and θ 3 >. This distribution arises in the following contet: X and Y are the lifetimes of two components subjected to three kinds of shocks; these shocks are assumed to be governed by independent Poisson processes with parameters θ, θ,andθ 3, according as the shock applies to component only, component only, or both components. The distribution has received wide applicability in nuclear reactor safety, competing risks, reliability and in quantal response contets. The marginal pdfs of X and Y are eponential with parameters θ + θ 3 and θ + θ 3, respectively; so, in particular, EX) =, θ + θ 3 EY) =. θ + θ 3 7.) The correlation coefficient ρ = CorrX,Y)is given by ρ = θ 3 θ + θ + θ ) Solving the three preceding equations, one can epress θ, θ,andθ 3 as θ = EX) θ 3, θ = EY) θ 3, { } EX)+EY) ρ θ 3 = EX)EY) + ρ). 7.4) It is easily seen that R can be epressed as R = θ θ + θ 3 ) = θ ep { ) } θ + θ + θ 3 d ep { θ θ + θ 3 ) y } dyd 7.5) 8. Application = θ θ + θ + θ 3. For a good part of the th century, the assumption of independent random samples from continuous distributions dominated applications of statistical methodology. From the middle of the eighties of the th century, we are beginning to observe deviations from this setup, mainly because real-world sources of data were not conforming to the i.i.d. continuous model. In fact, a substantial amount of categorized data plays an important role, especially in medical-orientated applications. One of the developments of this type is the analysis of receiver operating characteristic ROC) curves. This topic was a real hit in the last decade with a large number of publications appearing.

10 Reliability for some bivariate eponential distributions ROC curve is a particular type of an ordinal dominance OD) graph. Consider random variables X and Y, andforeveryrealnumberc plot a point Tc) in a Cartesian coordinate system with the coordinates PrX c),pry c)). The collection of the points Tc) form a ROC graph. Note that the coordinates of Tc) lie between and, so that the ROC graph is always located within the unit square {, y):, y }. Moreover, by letting c =±, we conclude that ROC graph always starts at,) and ends at,). The relation between OD graphs and R = PrX < Y) was originally pointed out by Bamber [] and brought a variety of methods developed for an inference about PrX <Y) into analysis of ROC curves. Bamber [] observed that the area above the OD graph for continuous X and Y is equal to AX,Y) = PrX )d PrY c) = PrX ) f Y c)dc = PrX Y) = R. 8.) In view of this relation, the area AX,Y) can be utilized as a measure of the size difference between two populations with AX,Y) = if and only if the distribution of X lies entirely below the distribution of Y. On the other hand, if X and Y are identically distributed, AX,Y) = /. ROC curve analysis has been used in various fields of medical imaging, radiology, psychiatry, nondestructive, and manufacturing inspection systems see, e.g., Hsiao et al. [8], Metz [], Nockermann et al. [8], Reiser [], Swets [3], and Swets and Pickett [4]). As an eample, consider the yes-no signal detection eperiment. In this eperiment, the observer is told to respond yes if he/she thinks that the signal was presented on the trial, and to respond no otherwise. It is assumed that the observer performs this task as follows. First, he/she adopts often subjectively) an impression strength criterion, say c. Then, on each trial if the impression strength reaches or eceeds the criterion, he/she responds yes, and responds no otherwise. Let I s and I n be continuous random variables denoting the strengths of sensory impressions aroused by signal and noise events, respectively. Then, Pryes signal) = PrI s c) and Pryes noise) = PrI n c). ROC curve is then a collection of points PrI n c),pri s c)) in a unit square. If numerical data on I n, I s, and yes-no responses is available, one can construct a sample ROC curve by estimating probabilities PrI n c)andpri s c) under certain parametric assumptions. Figure 8. illustrates the variation of AX,Y) = PrX <Y) = R with respect to ρ, EX), and EY) for the si bivariate eponential distributions discussed in the previous sections. The general pattern of variation is R>/ and is an increasing function of ρ when EX) >EY); R</ and is a decreasing function of ρ when EX) <EY); and R = / when EX) = EY). The variations can be grouped into four classes as follows: ) Arnold and Strauss and Freund into the first class showing the least sensitivity to ρ. For these two models, one can compute R without much error by assuming that X and Y are independent. Of these two, Freund is the simpler since the

11 S. Nadarajah and S. Kotz.8.8 R.6.4 R ρ.5.5 ρ a) b).8.8 R.6.4 R ρ.3.. ρ c) d).8.8 R.6.4 R ρ ρ e) f) Figure 8.. Variation of R versus ρ for the si bivariate eponential distributions and for selected values of EX) and EY): Gumbel a); Hougaard b); Downton c); Arnold and Strauss d); Freund e); Marshall and Olkin f); the four curves in each plot from a) to f) correspond to EX),EY)) = 5,),,),,), and,5). epressions for it are elementary and do not involve special functions. Furthermore, Freund ehibits a wider range of values for ρ.3 <ρ< for Arnold and Strauss, and /3 <ρ< for Freund) and attains larger values for R.

12 Reliability for some bivariate eponential distributions ) Gumbel and Downton into the second class showing moderate sensitivity to ρ.of these two, although Gumbel is the earliest and simplest known model, one might choose Downton because it has a wider range of values for ρ.4365 <ρ< for Gumbel, and ρ< for Downton) and attains larger values for R.Also,the limits of R are attained as ρ in the case of Downton. 3) Hougaard into the third class which shows the highest sensitivity to ρ. Hougaard ehibits the widest range of values for ρ <ρ<) and attains the limits of R as ρ. 4) The case of Marshall and Olkin stands out from the rest because of the presence of singularity along the ais = y none of the other models have this). Here, R is a decreasing function of ρ for all values of EX) andey), so the system is most reliable when X and Y are independent. The amount of sensitivity to ρ is comparable to that of Hougaard. Based on the above discussion, the best model to choose would be that due to Hougaard since it gives the widest range of values for both ρ and R. However, if one is not interested in the dependence, then the model due to Freund might give as good a result. When selecting a model, one should also take into account the physical contets described in Sections 4, 6,and7. 9. Conclusions We have calculated the forms of R = PrX <Y) for si fleible families of bivariate eponential distributions and discussed their utility to ROC curve analysis. It would be of interest to emulate this work for other continuous bivariate distributions, including bivariate beta distributions, bivariate gamma distributions, and bivariate Pareto distributions. It would also be of interest to etend this work for continuous multivariate distributions. We hope to address some of these issues in a future paper. Appendi Some technical lemmas required for the calculations above are noted below. Lemma A.Prudnikovetal.[9,.3.5.7)]). For p>, n ep { p + q )} d = ) n π p n { q ) q n ep Φ q )}, A.) 4p p whereφ ) denotes the cdf of the standard normal distribution. Lemma A.Prudnikovetal.[9,...)]). For α>, α Γν,c)d = Γα + ν) αc α. A.)

13 Acknowledgments S. Nadarajah and S. Kotz 3 The first author is grateful to Professor Samuel Kotz George Washington University, USA) and to Professor Marianna Pensky University of Central Florida, USA) for introducing him to this area of research. Both authors would like to thank the referee and the editors for carefully reading the paper and for their great help in improving the paper. References [] B. C. Arnold and D. Strauss, Bivariate distributions with eponential conditionals,journalofthe American Statistical Association ), no. 4, [] D. Bamber, The area above the ordinal dominance graph and the area below the receiver operating characteristic graph, Mathematical Psychology 975), no. 4, [3] F. Downton, Bivariate eponential distributions in reliability theory, the Royal Statistical Society. Series B. 3 97), [4] J. E. Freund, A bivariate etension of the eponential distribution, the American Statistical Association 56 96), [5] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 6th ed., Academic Press, California,. [6] E. J. Gumbel, Bivariate eponential distributions, the American Statistical Association 55 96), [7] P. Hougaard, A class of multivariate failure time distributions, Biometrika ), no. 3, [8] J. K. Hsiao, J. J. Bartko, and W. Z. Potter, Diagnosing diagnoses, Archives of General Psychiatry ), no. 7, [9] A. W. Marshall and I. Olkin, A generalized bivariate eponential distribution, Applied Probability 4 967), 9 3. [], A multivariate eponential distribution, the American Statistical Association 6 967), [] C. E. Metz, Some practical issues of eperimental design and data analysis in radiological ROC studies,investigative Radiology 4 989), no. 3, [] S. Nadarajah, Reliability for beta models, Serdica. Mathematical Journal 8 ), no. 3, [3], Reliability for etreme value distributions, Mathematical and Computer Modelling 37 3), no. 9-, [4], Reliability for lifetime distributions, Mathematical and Computer Modelling 37 3), no. 7-8, [5], Reliability for Laplace distributions, Mathematical Problems in Engineering 4 4), no., [6], Reliability for Logistic distributions, Engineering Simulation 6 4), [7] S. Nadarajah and S. Kotz, Reliability for Pareto models, Metron6 3), no., 9 4. [8] C. Nockermann, H. Heidt, and N. Thomsen, Reliability in NTD: ROC study of radiographic weld inspections,ndt &EInternatioal 4 99), no. 5, [9] A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series. Vol., Gordon& Breach, New York, 986. [], Integrals and Series. Vol., Gordon & Breach, New York, 986. [], Integrals and Series. Vol. 3, Gordon & Breach, New York, 986. [] B. Reiser, Measuring the effectiveness of diagnostic markers in the presence of measurement error through the use of ROC curves,statistics in Medicine 9 ), no. 6, 5 9.

14 4 Reliability for some bivariate eponential distributions [3] J. A. Swets, Signal Detection Theory and ROC Analysis in Psychology and Diagnostics: Collected Papers, Lawrence Erlbaum Associates, New Jersey, 996. [4] J. A. Swets and R. M. Pickett, Evaluation of Diagnostic Systems: Methods from Signal Detection Theory, Academic Press, New York, 98. Saralees Nadarajah: Department of Statistics, University of Nebraska, Lincoln, NE 68583, USA address: Samuel Kotz: Department of Engineering Management and Systems Engineering, The George Washington University, Washington, DC 5, USA address:

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