Extreme shock models: an alternative perspective
aa r X i v : . [ s t a t . O T ] S e p Extreme shock models:an alternative perspective
Pasquale Cirillo and J¨urg H¨usler
Institute of Mathematical Statistics and Actuarial Sciences, University of Bern
Abstract
Extreme shock models have been introduced in Gut and H¨usler (1999) to studysystems that at random times are subject to shock of random magnitude. Thesesystems break down when some shock overcomes a given resistance level.In this paper we propose an alternative approach to extreme shock models us-ing reinforced urn processes. As a consequence of this we are able to look at thesame problem under a Bayesian nonparametric perspective, providing the predictivedistribution of systems’ defaults.
Key words: extreme shock models, urn models, default probability
Shock models are a particular class of models in which a system is randomlysubject to different shocks of random magnitude that can make it fail.In the literature (see [8] for a good survey) there are essentially two distincttypes of shock models: cumulative shock models, in which the failure of thesystem is due to a cumulative effect; and extreme shock models, whose defaultis caused by one single extreme shock. In this paper we focus our attentiononly on the second type, referring to [6] for cumulative shock models.Consider a family { Z i , U i } i ≥ of nonnegative i.i.d. two-dimensional randomvectors, such that Z i represents the intensity of the i -th shock and U i is thetime between the ( i − i -th shock. Set Z = U = T = 0 and, for n ≥ T n = P ni =1 U i is the amount of time that has elapsed after n shocks.In extreme shock models, one is interested in the behavior of the stoppingtime τ ( t ) = min { n : Z n > t } , i.e. the minimum time at which a large extreme Email addresses: [email protected] (Pasquale Cirillo), [email protected] (J¨urg H¨usler).
Preprint submitted to Elsevier October 29, 2018 hock occurs causing the default of the system at T τ t . We refer to [7] for furtherdetails and results.More recently, in [8], extreme shock models have been generalized by assum-ing that large but not fatal shocks may effect system’s tolerance to subsequentshocks. To be more exact, for a fixed t , a shock Z i can damage the systemif it is larger than a certain boundary value β t < t . As long as Z i < t thesystem does not fail. The crucial hypothesis is the following: if a first nonfatalshock comes with values in [ β t , t ] the maximum load limit of the system is nomore t , but decreases to α t (1) ∈ [ β t , t ], since the system has been damaged.At this point, if another large but not too strong shock occurs in [ β t , α t (1)],the new crucial threshold is lowered again to α t (2) ∈ [ β t , α t (1)] and so onuntil the system fails. We could call all this “risky threshold mechanism”.The relevant stopping time is now τ ( t ) = min { n : Z n ≥ α t ( L t ( n − } with L t ( n ) = P ni =1 { Z i ≥ β t } and L t (0) = 0. All the results can be found in [8].In this work we want to present an alternative perspective to extreme shockmodel by using a particular class of reinforced urn processes, as introducedby [12]. The aim is to develop a Bayesian nonparametric approach to shockmodels, in the wake of some recent works like [1]. As we will see the choice ofusing urn schemes for modeling is strictly related to their ability of reproduc-ing the Bayesian paradigm of information update in a rather intuitive way.In the literature there are very few papers that combine shock models and urnprocesses. In [11] a Polya urn is used to model a system subject to non-i.i.d.shocks, while in [1] a first urn-based approach to generalized extreme shockmodels is discussed.Urn processes form a very large family of probabilistic models in which theprobability of certain events is represented in terms of sampling, replacing andadding balls in one or more urns or boxes. A good introduction to urn modelsand their properties is represented by [9] and [10].In reality it is not difficult to derive simple shock models using urns, as weonly need basic tools as Bernoulli and Poisson trials. Think for example of asimple urn containing b black balls and w white balls, which is sampled withreplacement. If black balls are associated to the event “extreme fatal shock”and white balls represent the case in which there is no dangerous shock, thenthe waiting time related to the extraction of the first black ball from the urnis somehow related to the τ ( t ) in extreme shock models.Naturally, this is just a trivial example. A more interesting urn scheme hasbeen proposed in [1] to model generalized extreme shock models, where a tri-angular reinforcement matrix is used to mymic the risky threshold mechanism.This model is called UbGesm (urn-based generalized extreme shock model)and, in the next sections, we will show how to obtain it as a special case ofour new approach.Very briefly, having in mind the mechanism of generalized extreme shock mod-els, the idea is to create three different risk areas for the system subject toshocks - no risk or safe, risky and default, to link every area to a particularcolor and to work with the probability for the process to enter each area.2f every time the process enters the risky area the probability of failing in-creases, and this can be obtained with a triangular reinforcement matrix, wecan consider such a modeling a sort of intuitive approach to generalized ex-treme shock models, getting around the definition of the moving threshold.In some sense, reinforcing the probability for the system to fail is like makingthe risky threshold move down and vice-versa.Consider an urn containing balls of three different colors: x , y , and w . The x -balls are related to the safe state, y -balls to the risky state and w -ballsembody the default state. The process evolves as follows: at time n a ball issampled from the urn, the probability of sampling a particular ball dependingon the urn composition after time n −
1; then, according to the color of thesampled ball, the process enters (or remains in) one of the three states of risk(e.g. if the sampled ball is of type x , the process is in a safe state, while it failsif the chosen ball is w ); finally the urn is reinforced according to its balancedreinforcement matrix, as shown in (1). It means that if the sampled ball is oftype x , then the ball is returned and s > x -balls are added to the urn,if the sampled ball is of type y , then it is replaced together with r > y -ballsand p > w -balls, and if the sampled ball is of type w , then s extra w -ballsare added. RM = xyw s r p s , where r = s − p (1)The distributions of the different colors and the main properties of the urnprocess can be described analytically through the analysis of its generatingfunction, as discussed in [1], where all the connections with standard shockmodels are also analyzed. The basic ingredient of our construction is the Polya urn, introduced by [5].The behavior of this urn model is very simple yet ingenious. In its simplestversion, we have an urn containing balls of two different colors. Every timewe sample the urn we look at the color of the chosen ball and then put itback into the urn together with another ball of the same color. In this way,the more a given color has been sampled in the past, the more likely it willbe sampled in the future. It is easy to understand how this naive replacementrule is able to reproduce a lot of self-reinforcing and contagious phenomena.More formally, at time t = 0, consider an urn U with initial composition C = ( w , b ), where w ≥ b ≥ t ≥
1, urn U is sampled and the extracted ball isreplaced into it together with s > X t be arandom variable equal to 1 if the sampled ball at time t ≥ X ∼ Bernoulli ( Z ), where Z = b / ( w + b ) is the proportion of black balls at time t = 0, and, for t ≥ X t +1 ∼ Bernoulli ( Z t ), where Z t = b t / ( w t + b t ) is now a random variable. It isnot difficult to demonstrate that the sequence { X t } t ≥ is exchangeable in thesense of de Finetti [2].Regarding the evolution of balls in the urn, we simply have( w t +1 , b t +1 ) = ( w t , b t + s ) if X t = 1( w t + s, b t ) if X t = 0 . (2)Thanks to its reinforcement mechanism, Polya urn is widely used in Bayesiananalysis (see for example [9] and [12]): the initial composition of the urn canbe considered as the prior knowledge about the occurrence of an event; thesampling of the balls represents the outcome of the statistical experiment;and the reinforcement of balls can be seen as the Bayesian updating, whosestrength is given by s , of the prior knowledge.Polya urns have a lot of interesting properties. For example, thanks to ex-changeability and de Finetti’s representation theorem, conditionally on a ran-dom variable Θ, the variables X t are i.i.d Bernoulli (Θ). Moreover, it can beshown that Θ ∼ Beta ( w /s, b /s ) and Z t → a.s. Θ. These are the essentialproperties for our approach to extreme shock models. We refer to [9] and [10]for a much more complete treatment.Recently, in [12], a generalization of the Polya urn scheme has been introducedunder the name of reinforced urn process (RUP). Here we directly introduceour special case, referring to the next section for a more general setting.Imagine to have a discrete set of states v < v < v < ... and associate to eachof them a Polya urn U ( v i ), i ≥
0. Every urn has its own initial composition C ( v i ) = ( w ( v i ) , b ( v i )), for i ≥ { X n } on { v , v , ... } is defined iteratively as follows:set X = v and, for n ≥
1, if X n − = v i sample urn U ( v i ). If the sampledball is white, then Polya reinforce urn U ( v i ) with s > X n = v i +1 . Otherwise, whether the ball is black, add s black balls and set X t = v . In other words, starting from urn U ( v ) we sequentially sample fromurns in v , v , v , ... until a black ball is chosen, then we start again the se-quential sampling.Now, let black balls represent the event “extreme shock”, while white ballsindicate the event “no shock or weak shock”. Moreover, imagine for simplicitythat the states v < v < v < ... are just time instants, such that v i = i for i ≥ v i can be of every kind).Repeating the iterative sampling several times, we obtain a sequence like this4s realization: { X n } n ≥ = { v , v , v , v , v , v , v , v , v , v , ..., v , v , v , .... } = { , , , , , , , , , , ..., , , , .... } (3)In other words, every time a black ball is sampled, a so called 0-block iscreated, i.e. a sequence of states between two different 0’s.Now, assume we have several different systems, say k ≥
1, that are subjectsto shocks as in the standard extreme shock model setting. Let us hypothesizethat these systems are exchangeable, i.e. it is not relevant, for our analysis, theorder in which we consider them. If the life of every system is represented byone of the 0-blocks, we have that the process { X n } is partially exchangeable inthe sense of Diaconis and Freedman [3]. In other words { X n } can be expressedas a mixture of Markov chains. In fact, it suffices to notice that by assumptionthe 0-blocks are exchangeable (every block represents the life of a system) andevery block is a Markov chain, since, within each block, the probability ofmoving from one state to the other only depends on the last visited state.Let ξ i be the last coordinate of the i -th 0-block, for i = 1 , ..., k , of the process { X n } . Since every ξ i is a function of the corresponding 0-block i , and theblocks are exchangeable, then the sequence { ξ i } is exchangeable as well.Now, in order to guarantee that the process { X n } is recurrent, i.e. P [ X n =0 for infinitely many n ] = 1, let us assume thatlim n →∞ n Y i =1 b ( v i ) w ( v i ) + b ( v i ) = 0 . (4)In this way we are sure that infinitely many 0-blocks can be created andwe can thus define the sequence { τ n } of stopping times, with τ = 0 and τ n = inf { n > τ n − : X n = 0 } . Naturally, we have that ξ n = X τ n − .Let us finally enter in the Bayesian interpretation of extreme shock modelsusing reinforced urn processes. Consider the first system subject to shocks.All the information we have about the probability of an extreme shock instate v = 0 is just represented by the a priori information about the initialcomposition of urn U ( v ) (remember that an extreme shock is represented bythe extraction of a black ball). Hence we have that P ( ξ = 0) = b (0) w (0) + b (0) and P ( ξ >
0) = w (0) w (0) + b (0) . (5)If a black ball is sampled and the system fails, then urn U (0) is reinforcedwith s extra black balls. Otherwise we add s extra white balls. For clarity wecall the new reinforced urn U (0). From now on U i ( v ) and C i ( v ) represent thePolya urn and its composition in state v after the i -th sampling.If the system does not fail in v , since we have extracted a white ball, wemove to urn U ( v ), and so on until we pick a black ball. View that we areconsidering the first system and we do not have any information about past5xperiments, we simply obtain P ( ξ = v r ) = b ( v r ) w ( v r ) + b ( v r ) r − Y j =0 w ( v j ) w ( v j ) + b ( v j ) . (6)Now imagine that the first system fails in v r , i.e. at time instant r , then we havethat C ( v r ) = ( w ( v r ) , b ( v r ) + s ), C ( v j ) = ( w ( v j ) + s, b ( v j )) for j = 0 , ..., r − C ( v j ) = C ( v j ) for j > r .At this point we start considering the second system. We begin from theurn in state v , that now has been updated according to the history of thefirst system, being U ( v ). Up to state v r we will use the information we haveacquired about system one. From v r +1 on we will come back to the initial priorinformation, since no history has been observed. These iterative procedure isrepeated for all the systems considered. We can then state the following result. Proposition 1
Given a reinforced urn process { X n } as described above, afterobserving m systems, we have that the probability that the ( m + 1) -th systemfails in v r is equal to P ( ξ m +1 = v r | ξ , ..., ξ m ) = b ( v r ) + sd r w ( v r ) + sf r + b ( v r ) + sd r r − Y j =0 w ( v j ) + sf j w ( v j ) + sf j + b ( v j ) + sd j , (7) while P ( ξ m +1 > v r | ξ , ..., ξ m ) = r Y j =0 w ( v j ) + sf j w ( v j ) + sf j + b ( v j ) + sd j , (8) where f l = P mp =1 { v l <ξ p } and d l = P mp =1 { v l = ξ p } . Proof.
The proof is simply given by applying the iterative construction ofthe process as for equation (6) and by counting the number of defaults in thedifferent states at every sampling.As far as inference is concerned, from equations (7) and (8), we derive thatˆ ξ m +1 = E [ ξ m +1 | ξ , ..., ξ m ] = X r v r P ( ξ m +1 = v r | ξ = a , ..., ξ m = a m ) . (9)Anyway, the most interesting aspect of the reinforced urn process constructionfor extreme shock models is represented by the following proposition, that givesuseful information about the distribution of default times. Proposition 2 If { X n } is a recurrent reinforced urn process, the sequence { ξ n } is exchangeable, i.e. there exists a random distribution function F suchthat, conditionally on F , the random variables of the sequence { ξ n } are i.i.d.with distribution F , whose law is that of a beta-Stacy process. In other words,for v i +1 ∈ V , the random mass assigned by F to the subset { v , v , ..., v i + 1 } s equal to − i +1 Y j =1 (1 − Y j ) , (10) where Y i , i = 1 , , ... , are independent random variables such that Y i is dis-tributed as Beta ( w ( v i ) /s, b ( v i ) /s ) . Proof.
The proposition is an application of Theorem 3.26 in [12], to whichwe refer for further details. The fact that Y i , i = 1 , , ... , are independent Beta ( w ( v i ) /s, b ( v i ) /s ) random variables is a direct consequence of the use ofPolya urns.The beta-Stacy process has been introduced in [15] as a special case of neu-tral to the right processes (see [4]). An interesting property of beta-Stacyprocesses is conjugacy, that is to say that, in our construction, the poste-rior distribution for F , after observing the history of the different systems, isstill a discrete time beta-Stacy process, with jumps in { v r } and parameters( w ( v r ) + sf r , b ( v r ) + sd r ). Example
We now show a possible simple application of the urn-based shock model inthe field of material science and engineering.Imagine we have several equal bars of the same metal and we want to testwhich is the maximum load that they can bear before rupture. A similar situ-ation can be efficiently modeled using the reinforced urn process we have justdescribed.Set v < v < v < ... to be different loadings in increasing order. Then equipevery state v i with an urn U ( v i ) as seen before.Start with the first bar and apply loading v . If it cracks then read this as theextraction of a black ball from urn U ( v ), reinforce the urn with s black ballsand pass to the second bar. If the loading is not sufficient for the bar to break,we have sampled a white ball, we reinforce the urn and then move to loading v > v . Finally we go on until the metal bar shatters.Repeating the procedure for the different bars we are able to make Bayesianinference about the loading capacity of the different bars (remember equations(7) and (8)). Calibrating the magnitude s of reinforcement, we can weight theinformation given to the experiment with respect to our prior knowledge, asexpressed by the initial compositions of the different urns.This kind of model can be considered a special case of extreme shock model,in which the magnitude of shocks (loading) is not random, but it increasesstate after state. 7 The general construction and generalized extreme shock models
The model we have introduced for extreme shook models is just one of thepossible application of RUP to shock models. For example, if we move to amore general definition of RUP, in which every urn needs not to be a Polyaurn, we can obtain several interesting results. For example, we can reproducethe urn-based generalized extreme shock model of [1].A generalized reinforced urn process, i.e. a generalization of the constructionin [12], can be obtained using the following ingredients:(1) A countable state space V .(2) A finite set E of colors, whose cardinality is at least 1.(3) For every v ∈ V there exists an urn U ( v ) characterized by an initialcomposition C ( v ) and a reinforcement matrix M ( v ). In particular C ( v ) = { n v ( c ) : c ∈ E } where n v ( c ) ≥ c in U ( v ); and M ( v ) describes how the urn behaves when it is sampled,i.e. how balls are replaced or reinforced.(4) A function m : V × E → V that indicates how the process moves fromstate to state, that is from one urn to the other.Imagine we are in v ∈ V . We sample the urn U ( v ) and look at the color c of theextracted ball, whose probability of being sampled depends on the composition C ( v ) of the urn. At this point the matrix M ( v ) says how we have to behave(see [1] and [10] for more details on reinforcement matrices). For example if M ( v ) is the null matrix, it could represent sampling without replacement, sothat the sampled ball is thrown away. If M ( v ) is the identity matrix, then weconsider sampling with replacement. A Polya urn is represented by a diagonalmatrix having 1 + s , with s >
0, on the main diagonal. And so on for all theother possible schemes. At this point the law of motion m says how we moveform state v ∈ V to say state v ∈ V , i.e. m ( v , c ) = v .To clarify the exposition, let us consider the simple model we have introducedbefore. Set { X t } to be our process defined on V . Assume V = { , , , ... } , E = { b, w } where b means black and w stands for white, C ( v ) = [ w ( v ) , b ( v )]and M ( v ) = s
00 1 + s , s >
0, for every v ∈ V . Moreover the law ofmotion m is such that m ( v, b ) = 0 and m ( v, w ) = v + 1.For what concerns generalized extreme shock models, as we have seen in [1]the authors propose a generalized triangular Polya-like urn, in order to modelthe risky threshold mechanism introduced in [8].It is not difficult to reproduce the results in [1] using the generalized reinforcedurn process. In fact it suffices to mimic the behavior of the original triangularurn using our sequence of urns. The trick is to introduce a recurrence relationamong the urns in order to imitate the evolution of the urn-based generalized8xtreme shock model.Set V = { , , , ... } and E = { w, r, b } . For every v ∈ V define the balancedmatrix M = M ( v ) = s r p s . (11)Assume that the vector representing the initial composition of urn U (0) is C (0) = ( a , b , c ) and, for every v >
0, set C ( v ) = C ( v −
1) + e v − M, (12)where e v is a random vector equal to (cid:20) (cid:21) if the ball extracted in U ( v ) iswhite, (cid:20) (cid:21) if red and (cid:20) (cid:21) if black. Furthermore, for the law of motion,simply set m ( v, w ) = m ( v, r ) = v + 1 and m ( v, b ) = 0.It is effortless to verify that the so-defined urn chain exactly reproduces theurn model introduced in [1], even if in a less intuitive way. As a consequenceof this, all the results there stated still hold. We have seen that extreme shock models are models in which a system issubject to random shocks of random magnitude, which make it fail as soonas they overcome a certain resistance threshold. Extreme shock models canhave several applications in material science (as in the simple example we haveshown) and engineering in general, but also in economics and finance, thinkfor example about firms’ defaults.In this paper we have proposed an alternative modeling based on a special ver-sion of the reinforced urn processes introduced by [12]. The main novelty ofthis approach is given by the possibility of performing a Bayesian nonparamet-ric analysis of extreme shocks. Given a set of systems subjects to shocks, andassuming that they are exchangeable, we have shown how to exploit the infor-mation about their history for making predictions concerning default times.Moreover, the use of Polya urns for the construction of the model allows theresearcher to introduce his/her prior knowledge of the phenomena, by modi-fying the initial composition of the different urns.Possible extensions of the present work could be its development in continuoustime, on the basis of the main results in [13], and its applications to practicalproblems, such as for example fatigue analysis [14]. Moreover, it could be alsoworth of investigation to work with the general definition of reinforced urnprocesses, introducing for example time-variant reinforcement matrices, in or-9er to model other shock models as discussed in [8].
Acknowledgements:
The present work has been supported by the SwissNational Science Foundation.
References [1] Cirillo, P., and H¨usler, J., 2009. An urn-based approach to generalized extremeshock models. Statistics and Probability Letters 79, 969-976.[2] de Finetti, B., 1975. Theory of Probability II. Wiley, New York.[3] Diaconis, P., and Freedman, D., 1980. de Finetti’s theorem for Markov chains.Annals of Probability 8, 115-130.[4] Doksum, K., 1974. Tailfree and neutral random probabilities and their posteriordistributions, Annals of Statistics 2, 183-201.[5] Eggenberger, F., and Polya, G., 1923. ¨Uber die Statistik verketteter Vorg¨ange.Zeitschrift f¨ur Angerwandte Mathematik and Mechanik 1, 279-289.[6] Gut, A., 1990. Cumulative shock models. Advances in Applied Probability 22,504-507.[7] Gut, A., and H¨usler, J., 1999. Extreme shock models. Extremes 2, 293-305.[8] Gut, A., and H¨usler, J., 2005. Realistic variation of shock models. Statistics &Probability Letters 74, 187-204.[9] Johnson, N. L., and Kotz, S., 1977. Urn models and their applications. Wiley,New York.[10] Mahmoud, H.M., 2009. Polya urn models. CRC Press, New York.[11] Marshall, A. W., and Olkin, I., 1993. Bivariate life distributions from Polya’surn model for contagion. Journal of Applied Probability 30, 497-508.[12] Muliere. P., Secchi, P., and Walker S. G., 2000. Urn schemes and reinforcedrandom walks. Stochastic Processes and their Applications 88, 59-78.[13] Muliere P., Secchi P., and Walker S. G., 2003. Reinforced random processes incontinuous time. Stochastic Processes and their Applications 104, 117-130.[14] Stephens, R.I., Fatemi, A., and Fuchs, H.O., 2000. Metal Fatigue in Engineering.Wiley, New York.[15] Walker, S., and Muliere, P., 1997. Beta-Stacy processes and a generalization ofthe Polya urn scheme. Annals of Statistics 25, 1762-1780.[1] Cirillo, P., and H¨usler, J., 2009. An urn-based approach to generalized extremeshock models. Statistics and Probability Letters 79, 969-976.[2] de Finetti, B., 1975. Theory of Probability II. Wiley, New York.[3] Diaconis, P., and Freedman, D., 1980. de Finetti’s theorem for Markov chains.Annals of Probability 8, 115-130.[4] Doksum, K., 1974. Tailfree and neutral random probabilities and their posteriordistributions, Annals of Statistics 2, 183-201.[5] Eggenberger, F., and Polya, G., 1923. ¨Uber die Statistik verketteter Vorg¨ange.Zeitschrift f¨ur Angerwandte Mathematik and Mechanik 1, 279-289.[6] Gut, A., 1990. Cumulative shock models. Advances in Applied Probability 22,504-507.[7] Gut, A., and H¨usler, J., 1999. Extreme shock models. Extremes 2, 293-305.[8] Gut, A., and H¨usler, J., 2005. Realistic variation of shock models. Statistics &Probability Letters 74, 187-204.[9] Johnson, N. L., and Kotz, S., 1977. Urn models and their applications. Wiley,New York.[10] Mahmoud, H.M., 2009. Polya urn models. CRC Press, New York.[11] Marshall, A. W., and Olkin, I., 1993. Bivariate life distributions from Polya’surn model for contagion. Journal of Applied Probability 30, 497-508.[12] Muliere. P., Secchi, P., and Walker S. G., 2000. Urn schemes and reinforcedrandom walks. Stochastic Processes and their Applications 88, 59-78.[13] Muliere P., Secchi P., and Walker S. G., 2003. Reinforced random processes incontinuous time. Stochastic Processes and their Applications 104, 117-130.[14] Stephens, R.I., Fatemi, A., and Fuchs, H.O., 2000. Metal Fatigue in Engineering.Wiley, New York.[15] Walker, S., and Muliere, P., 1997. Beta-Stacy processes and a generalization ofthe Polya urn scheme. Annals of Statistics 25, 1762-1780.