TESTING A HOMOGENEITY OF STOCHASTIC PROCESSES

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1 K Y B E R N E I K A V O L U M E , N U M B E R 4, P A G E S ESING A HOMOGENEIY OF SOCHASIC PROCESSES Jaromír Antoch and Daniela Jarušová he paper concentrates on modeling the data that can be described by a homogeneous or non-homogeneous Poisson process. he goal is to decide whether the intensity of the process is constant or not. In technical practice, e.g., it means to decide whether the reliability of the system remains the same or if it is improving or deteriorating. We assume two situations. First, when only the counts of events are nown and, second, when the times between the events are available. Several statistical tests for a detection of a change in an intensity of the Poisson process are described and illustrated by an example. We cover both the case when the time of the change is assumed to be nown or unnown. Keywords: homogeneous and non-homogeneous Poisson process, counting process, change point detection AMS Subject Classification: 6F03, 6E0, 60K99. INRODUCION he paper concentrates on modeling data that describe occurrences of certain events in time. he reliability data where the events correspond to a failures of a product is a typical example. In the scope of mathematical statistics a counting process, often a homogeneous or non-homogeneous Poisson process, serves as a successful model. We consider two situations. In the first one all we now are numbers of observed events { N i }, i =,..., q, in consecutive non-overlapping intervals of the length { ti }, i =,..., q, t + + t q =. In the second one, a sequence of times of event occurrences, say 0 < τ < τ <..., is observed. It is clear that in such a case we also now lengths of intervals between events {Y i }. he basic question that arises when trying to find a good model is the following: Is the observed process stationary or does its intensity change? In the case of reliability data changes in the intensity may be caused by improvement or deterioration of the system. For detection of changes the hypotheses testing may be applied, where the null hypothesis claims that the process is stationary. According to an alternative under which a certain type of change is considered a test statistic may be developed. o decide whether the null or the alternative hypothesis holds the distribution of the suggested test statistic under the null hypothesis has to be derived. As the exact distribution is often complex an asymptotic distribution for a large or a large

2 46 J. ANOCH AND D. JARUŠKOVÁ number of observations n may be applied. In our paper several test statistics are proposed and their distribution under the null distribution is studied. After the problem of stationarity is solved and an appropriate parametric model is chosen, an estimation of parameters concludes the statistical inference. herefore, we show how to estimate parameters of several simple models usually applied to data described above.. COUNING PROCESS We suppose that we observe a stream of events, failures, e. g., that occurred in times 0 < τ < τ <... Such a sequence of variables {τ i } is called a simple point process. At any time t we can count the number of events Nt that occurred before or at the time t. he process {Nt, t 0} is called a counting process. Between the point process {τ i } and its counting process the following relation holds, i. e., P Nt n = P τ n t. A counting process { Nt, t 0 } is a renewal process if:. N 0 = 0.. he variable Y = τ the time to the occurrence of the first event from t = 0 and the variables Y j = τ j τ j, j, Y j being time between the j st and jth event, form a sequence of i.i.d. random variables with a distribution function F x. 3. Nt = sup { n : S n t }, where S 0 = 0, S n = n i= Y i, n. If times between events {Y i } are i.i.d. with an exponential distribution then the process is called a homogeneous Poisson process. Renewal processes do not allow to model improving and deteriorating systems. For modeling non-stationary systems trend renewal processes may be applied. For more detailed information about trend renewal processes see [4]. An example of a trend renewal process is a non-homogeneous Poisson process. Non-homogeneous as well as homogeneous Poisson processes with intensity {λs > 0, s 0} may be defined by the following set of properties:. For every n and every 0 < t < t < < t n the variables Nt, Nt Nt,..., Nt n Nt are independent.. For every 0 s < t the variable Nt Ns has a Poisson distribution with the parameter t λz dz. Denoting {Λt, t 0} a cumulative intensity s Λt = t λz dz, it holds 0 Nt P Ns = = exp Λt Λs Λt Λs, =0,,....!

3 esting a Homogeneity of Stochastic Processes 47 Suppose that the interval [0, ] is divided into smaller time intervals of the lengths t, t,..., t q such that t + t + + t q = and N denotes the number of events in the interval [0, t ], N the number of events in the interval t, t + t ] etc. Denote by N the total number of events in the time interval [0, ]. Clearly, N = N + + N q. he variables {N i, i =,..., q} are independent and each has a Poisson distribution with the parameter Λ i, where Λ = Λt ; Λ i = Λt + + t i Λt + + t i, i =,..., q. If the intensity of the underlying Poisson process does not change in time then the variable N i follows a Poisson distribution with parameter λ t i, i =,..., q, and the distribution of N i, given the number of all observations N = n, is binomial with parameters n and p i = Λ i /Λ. For a homogeneous Poisson process the distribution does not depend on the intensity λ and is binomial with parameters n and p i = t i /, i. e., P N i = x N = n = n x ti x t i n x. For large N an approximation of binomial distribution by a normal distribution yields L N i N = n N n t i, n t i t i or N i N L ti N N = n N0,, t i t i where Nµ, σ denotes a normal distribution with a mean µ and a variance σ. Similarly, the distribution of the vector N,..., N q, given N = n, is multinomial with parameters p,..., p q and n. For a homogeneous Poisson process we have P N = x,..., N q = x q = n! x!... x q! t x tq xq q... if x i = n, x i 0, i= = 0 otherwise. 3. SAISICAL ANALYSIS OF REND IN INENSIY For the proper statistical analysis of a system it is important to detect possible changes in the length of intervals between events. For example, reliability growth corresponds to times between failures becoming longer as time goes on improving system, whereas aging effects often lead to decreasing inter-failure times deteriorating system. he intensity may change in many different ways. he change may be sudden or gradual. For the decision whether the intensity is constant one may use statistical tests. We describe here statistical tests suitable for testing that the process is stationary against alternatives depending on the ind of trend to detect.

4 48 J. ANOCH AND D. JARUŠKOVÁ 3.. Detection of a change in an intensity of a Poisson process 3... Inspections at several discrete time points Suppose that the interval [0, ] is divided into smaller intervals of the lengths t i, i =,..., q, and that all we now are the corresponding numbers of events N i, i =,..., q, in the respective intervals. he decision whether a change in an intensity occurred may be based on hypotheses testing. he null hypothesis claims that N i, i =,..., q, are independent random variables distributed according to a Poisson distribution Poλ t i. he alternative claims that there exists an index such that N is not distributed according to the Poisson distribution with the parameter λ t. First, we deal with a simple situation when the interval [0, ] is divided into two intervals [0, t ] and t, ], where N denotes a number of events in the interval [0, t ] and N = N N a number of events in the interval t, ]. We decide that the intensity changed, i. e., we reject the null hypothesis of no change at the significance level α, either if or if N i=0 N i=n N i N i t t i t N i α i t N i α. For N large the rule for rejecting the null hypothesis of no change has the form N N t t t N > u α/ where u α/ is a α/00 % quantile of N0,. In the case that all we now are numbers of observed events { N i }, i=,..., q, in consecutive non-overlapping intervals of the length { t i }, i=,..., q, t + +t q =, and if the number of all events N is large, we decide that the intensity changed if q i= Ni t i N t i N > χ αq, where χ αq is a α00 % quantile of χ distribution with q degrees of freedom Decision based on all events In the preceding paragraph we supposed that we observed a counting process of a underlying Poisson process in several discrete time points t, t + t, t + t +

5 esting a Homogeneity of Stochastic Processes 49 t 3,..., t +t + +t q =. herefore, our decision whether the intensity of the process is constant was based only on the values of the counting process {Nt, Nt + t,..., N } or on the values {N,..., N q }, where Nt + +t i = N + +N i. However, it can happen that we follow the process continuously so that we now all time points 0 < τ < τ < < τ n in which the events occurred. As we have already explained, if the intensity is constant then for every fixed t [0, ] the distribution of Nt, given N = n, is binomial Bi n, t/. he process has a non-constant intensity if at least for one 0 < t < the variable Nt N t N t t is large. herefore, we may base our decision on the statistic Nt N t CP = sup N 0<t< t t. If the intensity is constant and the number of observations N large, one may replace t/ in the denominator of by Nt/N and apply the statistic Nt N t CP = sup N 0<t< Nt Nt N N. 3 For large it holds under the null hypothesis that Nt N P t sup N 0<t< t t x + b a and P sup 0<t< Nt N t Nt Nt N N N x + b a exp e x 4 exp e x, 5 where denotes that the distribution of the LHS can be for large values of approximated by the RHS, a = log log and b = log log + log log log log π. he approximations 4 and 5 provide us with approximate critical values. For more details see [6]. Another interesting approach is described in [7]. 3.. Detection of a change in mean time between events We suppose again that we register a sequence of times of events 0 < τ < τ <... Until now we were interested in a counting process {Nt}, where Nt denotes the number of occurrences of an event in the interval [0, t]. However, a different

6 40 J. ANOCH AND D. JARUŠKOVÁ approach for decision whether a change in the process occurred may be based on times between events {Y i = τ i τ i }. If studied Poisson process is homogeneous then all the variables { Y i } are i.i.d. with an exponential distribution. We will discuss this situation more profoundly in the following paragraphs esting for a change with a nown change point It can happen that after having observed th event the conditions that rule the process changed and we as whether this change affected the mean time between events. In the case of a constant intensity the times between events are exponentially distributed so that the problem is to decide whether all variables Y,..., Y n have the same mean δ or whether the mean δ of the variables Y,..., Y differs from the mean δ of the variables Y +,..., Y n. If we formulate the problem within the theory of hypotheses testing, we are to test the null hypothesis H 0 against the alternative A: H 0 : Y i Exp δ, i =,..., n, A : Y i Exp δ, i =,...,, Y i Exp δ, i = +,..., n, where δ > 0, δ > 0 and δ > 0 are unnown. It depends on our prior nowledge about the sign of δ δ whether we consider a one-sided or a two-sided alternative. We consider a two-sided alternative δ δ if we do not now a priori whether the change causes an increase or decrease of the mean time between events. We consider a one-sided alternative δ < δ resp. δ > δ if we expect that the change might increase resp. decrease the mean time between events. he problem is so called two samples problem for exponentially distributed random variables. It is often used if we compare the life times of two sets of products. he most frequently applied test statistics are equivalent to the log-lielihood ratio Denoting sup δ,δ i= fy i; δ n i=+ fy i; δ n sup δ i= fy. i; δ S = Y i, S 0 = n Y i, Y = S / and Y 0 = S/n 0, i= i=+ the maximum lielihood estimates of δ, δ and δ are δ = Y n, δ = Y and δ = Y 0, and the logarithm of the lielihood ratio is equal to 0 Y n Z = log Y n log Y Y n n = log S n log S n n = log S /S 0 + S /S 0 n n S0 S n n log n n + S /S

7 esting a Homogeneity of Stochastic Processes 4 Clearly, we reject H 0 against the two-sided alternative A with δ δ if the value of Z is larger than a value C which corresponds to a chosen significance level. Moreover, P Z < C = P a C < S /S n < b C, where a C and b C are solutions of the equation n n log x n log n x = C. 8 Further, note that where P a C < S /S n < b C = P d C < Y /Y 0 < h C, d C = a C n a C and h C = b C b C n. Under H 0 the statistic S /S n has a beta distribution Be, n with the density f Be y;, n = y y n, B, n while Y /Y 0 has a F distribution with and n degrees of freedom. hus P < Y F α/ n, Y 0 < F α/, n = α, where F β p, q is a β 00 % quantile of the F distribution with p and q degrees of freedom. herefore, we reject the null hypothesis H 0 against the twosided alternative δ δ at the significance level α if either Y /Y 0 is larger than F α/, n or if it is smaller than /F α/ n,. Analogously we reject H 0 against the one-sided alternative δ > δ if Y /Y 0 is larger than F α, n, and similarly against the one-sided alternative δ < δ if Y /Y 0 is smaller than /F α n, esting for a change with an unnown change point maximum type test statistics In the preceding paragraph we suspected that after we observed the th event the change in the mean time between events might occur. he problem was to decide whether it really happened. he situation is more complicated if we would lie to now whether the mean time between events remains the same all the time or whether it changed but we have no idea where the change might occur. In the scope of hypotheses testing the problem may be specified as follows: H 0 : Y i Exp δ, i =,..., n, A : {,..., n } such that Y i Exp δ, i =,...,, Y i Exp δ, i = +,..., n, 9

8 4 J. ANOCH AND D. JARUŠKOVÁ where δ > 0, δ > 0 and δ > 0. We reject the null hypothesis against the two-sided alternative with δ δ if at least one of the statistics { Z, =,..., n } given by 7 is large or, more precisely, if max {Z } is larger than an appropriate critical value. Unfortunately, the distribution of max {Z } under H 0 is so complex that it is not possible to use it for finding critical values and this is why some approximations are needed. he simple approximation is obtained by the Bonferroni inequality = P max { } Z > C = P P a C < S < b C = P S n { Z > C } P Z > C 0 d C < Y Y 0 < h C where a C, b C, d C and h C were introduced in the previous section. If we find a constant C such that PZ > C α for all, then it is obvious that the exact α00 % critical value is smaller than C or, in other words, C is a conservative estimate of the α00 % critical value. he inequality 0 may also serve for finding an upper bound of the p-value. Despite the distribution of statistics { { } Z} being of the same type, Z are not identically distributed parameters depend on, and that is why the smallest bound C that fulfills P Z > C α for all is unnecessarily large. As the contributions of different terms of the sum 0 are for fixed C different, it is better to loo for such a C that P a C < S < b C = P d C < Y S n Y 0 < h C = α. As a result we obtain a less conservative estimate of the α00 % critical value. o get an idea about the magnitude of the critical values, following table presents the critical values calculated using simulations and the Bonferroni inequality. We recommend to apply the Bonferroni inequality for smaller samples with a number of observations less than 70. able. Critical values for the maximum type statistics calculated using simulations, the Bonferroni inequality and asymptotic approximation for standard exponential distribution Exp. α = 0. α = 0.05 α = 0.0 n Simul. Bonf. Asymp. Simul. Bonf. Asymp. Simul. Bonf. Asymp ,

9 esting a Homogeneity of Stochastic Processes 43 If the number{ of } observations n is large, we can use the asymptotic distribution of max Z to find approximate critical values. Under H0, the aylor expansion yields that { } max Z is equivalent to max n n S n S n S n /n. From the theory of extremes of random processes, cf. [6], we have P max { Z } > x + b n exp e x, a n where a n = log log n and b n = log log n + log log log n log π. { } For one-sided alternative δ >δ we may use the test statistic max S /S n { or equivalently max Y /Y 0 }. o find approximate critical values we recommend again to use the Bonferroni inequality for small n, while for large n we recommend to use the asymptotic distribution P max { n n S n S } n S n /n > x + b n a n exp e x. Another results based on the strong invariance principle can be found in [8] Sum type test statistics A different approach how to derive test statistics for testing problem 6 is a so called pseudo Bayesian approach. his approach was applied for the first time by [4] and [0] for testing a sudden change in a mean of normally distributed random variables. he test statistic is again an analogue to the log-lielihood ratio under H 0 provided a prior distribution of a change point is nown and the amount of change i. e. δ δ tends to zero. For the problem 9 with the one-sided alternative δ > δ resp. δ < δ and under the assumption of a uniform prior distribution of over {,..., n }, the statistic = n n n j=+ Yj Y n = n S n Y n n S n is obtained. he exact distribution of under H 0 is again complex but it is possible to calculate explicitly all its moments since P Y Y Yi,..., Pn Yi has a Dirichlet distribution with parameters,,...,. Using the central limit theorem it may be shown that the statistic has asymptotically a standard normal distribution

10 44 J. ANOCH AND D. JARUŠKOVÁ N0,. Hence, appropriate quantiles of a normal distribution may serve as approximate critical values. For a better approximation of critical values we may use the Edgeworth expansion. It is interesting that the statistic is asymptotically equivalent for large values of n to the Laplace test statistic L = i= τ i τ n. τ n More precisely, it holds that = n L. he Laplace test statistic was suggested to test the null hypothesis that a process is a homogeneous Poisson process against the alternative that it is a non-homogeneous Poisson process with an intensity λt = e α+β t, for details see Cox and Lewis [5]. It is also worth to notice that the test statistic L is asymptotically equivalent to a standardized least squares estimator of a slope parameter b in a linear regression E Y i = a + b i, i =,..., n, where the standard deviation is estimated by Y n = n n i= Y i = τ n /n. If b = 0 then the variables {Y i } have the same exponential distribution with E Y i = std Y i = δ and Y n is a good estimate of the standard deviation. However, if the variables {Y i } are distributed according to another distribution with smaller variability then it is reasonable to replace the estimator of the standard deviation by another estimator. If we estimate the variance by the standard sample variance s LR = n n we get the Lewis Robinson test statistic i= Yi Y n, LR = Y n i= τ i s LR τ n τ n Clearly, the statistic s LR is a good estimator of the variance under the null hypothesis claiming that there is no trend. In the case where there is a trend then s LR overestimates the true variance. herefore some authors recommend to estimate the variance by s LI = Yi+ Y i n and to use the test statistic LR = Y n s LI i= i= τ i τ n τ n..

11 esting a Homogeneity of Stochastic Processes 45 For a two-sided alternative with δ δ we get from the pseudo Bayesian approach the statistic = n n n j=+ Yj = Y n n Y n S n S n. Under the null hypothesis the statistic converges, as the number of observations tends to infinity, in distribution to 0 B s ds, where { Bt, t 0 } is a Brownian bridge. he distribution of 0 B s ds was studied by Kiefer in []. he 5 % asymptotic critical value is equal to 0.46, the.5 % asymptotic critical value to and the % asymptotic critical value to Instead of the statistic some authors recommend to use the statistic 3 = n n n n j=+ Yj Y n = Y n S. n n n S he statistic 3 is a so called Anderson Darling statistic. Under H 0 its distribution tends as the number of observations n increases to the distribution of B s 0 s s ds. he 5 % asymptotic critical value is equal to.49, the.5 % asymptotic critical value to 3.08 and the % asymptotic critical value to In comparison with the statistic 3 has a larger power when a change occurs in the beginning or the end of our observations. On the contrary the statistic has a larger power when a change occurs in the middle of the sample. Similarly as above, if we are not sure whether the observations {Y i } are distributed according to an exponential distribution we estimate the true variance by the estimator s LI, instead of Y n, and we get a generalized Anderson Darling statistic as suggested by Kvaløy in [4]. If nothing is nown about the distribution of the variables {Y i = τ i+ τ i }, the Mann test may be applied. he Mann test is a ran test for testing the null hypothesis that all variables {Y i } are independent identically distributed i.e a renewal process is observed against an alternative of a monotone trend. he test statistic M is computed by counting the number of reverse arrangements among the interarrival times {Y i }. We spea about a reverse arrangement of Y i and Y j if Y i < Y j for i < j. hus M = n i= j=i+ I Y i < Y j, where I is an indicator function. Under the null hypothesis M is approximately normally distributed with the expectation nn /4 and the variance n 3 + 3n 5n/7. If the variables {Y i } are stochastically increasing the intensity of the underlying processes decreases, then the statistic M attains large values. On the other hand, if the variables {Y i } are stochastically decreasing the intensity increases the statistic M attains small values. For more details see, e. g., classical monograph [9].

12 46 J. ANOCH AND D. JARUŠKOVÁ 4. ESIMAION OF INENSIY OF A POISSON PROCESS For a homogeneous Poisson process on the interval [0, ], the maximum lielihood estimate of λ is of the form λ = N. Notice that all the information about the intensity λ is contained in the value of the counting process N so that it is not necessary to now times when events failures occurred. For large we have N λ and N λ N0,. λ he maximum lielihood estimate of the mean time between events δ = /λ is δ = N. he exact α 00 % confidence interval for λ has the form χ α/ N +, χ α/ N +. 3 For large a normal approximation may be used, giving the interval N + u α/ u α/ u α/ 4 +N, N + u α/ +u α/ u α/ 4 +N. 4 Now we will turn to the estimation of a trend in intensity. We start with a situation that the times 0 < τ < < τ n t when the events occurred are nown. Let Y = τ and Y i = τ i τ i denote the times between events. We are especially interested in models with a monotone intensity to model the reliability growth or decrease. he goal of statistical inference is to estimate the unnown parameters. A class of models which have been widely used because of its simplicity is the non-homogeneous Poisson process described earlier. We focus here on two models. In the first one the intensity is given by λt = α β t β and the cumulative intensity function is thus Λt = α t β. he model is called the Weibull process or the power-law process. It describes either a reliability growth when β > or a reliability decrease when β <. When β =, it reduces to a homogeneous Poisson process. he maximum lielihood estimators of the parameters based on the observation of the n event times τ,..., τ n, have a very simple closed form α = n n β τ b and β = n n i=. τn ln τ i

13 esting a Homogeneity of Stochastic Processes 47 In the second model, so called exponential-law model, the intensity is given by λt = e α+β t and the cumulative intensity is thus Λt = eα e βt. It describes a reliability growth if β > 0 or a reliability decrease if β < 0. If β < 0 then Λ+ < and it means that with a positive probability the process dies out and we do not get more information about the intensity if we observe the process for a longer time. o obtain the maximum lielihood estimators of the parameters based on the observation of the n event times τ,..., τ n, we have to solve the equation for β n β nτ n e + n τ i = 0. βτn After obtaining the estimator β as the solution of the preceding equation the estimator α of α has a simple form α = log n β βτ log e b n. Finally we consider the situation that all we now are numbers of events {N i, i =,..., q} in consecutive intervals {t i, i =,..., q}. As E Nt = Λt a regression analysis can be done directly in terms of the {N i }. Supposing that {N i } have a Poisson distribution, their variance increases with their mean. herefore, [5] suggested to apply one of the following transformations either logn i + / or N i + /. If Λt = αt β, then log Λt = log α + β log t. Considering log t to be independent and log Nt dependent variable, log α and β may be obtained by the least squares method. As the values {log Nt i } are neither independent nor identically distributed the regression technique has to be applied cautiously. i= 5. EXAMPLE Sigma Re insurance company published in [6] a list of most costly insurance losses in due to the hurricanes, tornados, earthquaes, floods, etc., worth of US $. hese data were republished and analyzed from the point of view of large deviations by [7]. Let us loo on these data from the point of view of the theory described in this paper. he times of events, in days starting January st, 970, are following: 5, 0, 55, 7, 9, 353, 3698, 4976, 5098, 676, 689, 797, 79, 735, 739, 7338, 736, 7899, 7939, 796, 87, 889, 8469, 8646, 8699, 878, 947, 95, 956, Moreover, the end of observations, i. e. December 3st, 995, corresponds to the value It follows that times between the catastrophes in days are 995, 34, 69, 47, 340, 66, 78,, 663, 68, 368, 3, 6, 94, 9, 3, 538, 40, 3, 309, 8, 80, 77, 53, 83, 365, 4, 05, 5. Figure presents lengths of intervals between the events in days versus index. Figure presents the counting process Nt versus time t. Looing at these figures it seems that there is a change around the tenth observation, i. e., around the 7 000th day, when the frequency of events started to increase. Let us tae a loo whether this suspicion can be confirmed by the theory described above.

14 48 J. ANOCH AND D. JARUŠKOVÁ Fig.. Lengths of intervals between events Fig.. Corresponding counting process. We have calculated values of the test statistics presented in this paper. All of them were statistically significant of the level 5 %. he following table summarizes the values of them. est statistic CP CP L LR LR 3 Value % asympt. crit.value In Figure 3 we present the values of statistics forming CP and CP, for details see and 3, while in Figure 4 we present the values of statistics n n S n S n S n /n corresponding to the statistic of the max-type considered in Subsection 3... We can see that their maximum is larger than both asymptotical critical value as well as the critical value obtained using the Bonferroni approach, so that we can reject the null hypothesis of no change in the intensity of appearances of the catastrophes. 6 5 CP CP 5 4 Asymp. crit. value Bonfer. crit. value Fig. 3. Statistics forming CP and CP. Fig. 4. Statistics forming.

15 esting a Homogeneity of Stochastic Processes 49 ACKNOWLEDGEMEN he paper has been prepared during the visit of the authors at the University Bordeaux supported by Aclimat, PIDEA hans go also to the Czech Science Foundation supporting the project 0/06/086 and to the Ministry of Education, Youth and Sports of the Czech Republic financing the project MSM Received February, 006. R E F E R E N C E S [] J. Antoch and M. Hušová: Estimators of changes. In: Nonparametrics, Asymptotics an ime Series S. Ghosh, ed., M. Deer, New Yor 998, pp [] J. Antoch, M. Hušová, and D. Jarušová: Off-line quality control. In: Multivariate otal Quality Control: Foundations and Recent Advances N. C. Lauro et al. eds., Springer Verlag, Heidelberg 00, pp. 86. [3] R. E. Barlow and F. Proschan: Mathematical heory of Reliability. Wiley, New Yor 964. [4] H. Chernoff and S. Zacs: Estimating the current mean of normal distribution which is subjected to changes in time. Ann. Math. Statist , [5] D. R. Cox and P. A. W. Lewis: he Statistical Analysis of Series of Events. Wiley, New Yor 966. [6] M. Csörgő and L. Horváth: Limit heorems in Change Point Analysis. Wiley, New Yor 997. [7] P. Embrechts, C. Klüppelberg, and. Miosch: Modelling Extremal Events. Springer Verlag, Heildelberg 997. [8] P. Haccou, E. Meelis, and S. van de Geer: he lielihood ratio test for the change point problem for exponentially distributed random variables. Stochastic Process. Appl , 39. [9] J. Háje and Z. Šidá: heory of Ran ests. Academia, Prague 967. [0] Z. Kander and S. Zacs: est procedures for possible changes in parameters of statistical distributions occurring at unnown time points. Ann. Math. Statist , [] J. Kiefer: K-sample analogues of the Kolmogorov Smirnov s and Cramér von Mises tests. Ann. Math. Statist , [] S. Kotz, M. Balarishnan, and N. L. Johnson: Continuous Multivariate Distributions. Volume : Models and Applications. Wiley, New Yor 000. [3] J.. Kvaløy and B. H. Lindqvist: -based tests for trend in repairable systems data. Reliability Engineering and System Safety , 3 8. [4] J.. Kvaløy, B. H. Lindqvist, and H. Malmedal: A statistical test for monotonic and non-monotonic trend in repairable systems. In: Proc. European Conference on Safety and Reliability ESREL 00, orino 00, pp [5] J.. Kvaløy and B. H. Lindqvist: A class of test for renewal process versus monotonic and nonmonotonic trend in repairable systems data. In: Mathematical and Statistical Methods in Reliability B. Lindqvist and K. Dosum, eds., Series on Quality, Reliability and Engineering Statistics, vol. 7, 003, pp , [6] Sigma: Natural Catastrophes and Major Losef in 995. Sigma Publ [7] J. Steinebach. and V. R. Eastwood: On extreme value asymptotics for increments of renewal processes. J. Statist. Plann. Inference

16 430 J. ANOCH AND D. JARUŠKOVÁ [8] J. Steinebach and V. R. Eastwood: Extreme value asymptotics for multivariate renewal processes. J. Multivariate Anal , Jaromír Antoch, Department of Statistics,Charles University of Prague, Faculty of Mathematics and Physics Charles University, Soolovsá 83, Praha 8. Czech Republic. Daniela Jarušová, Department of Mathematics, Faculty of Civil Engineering Czech echnical University, háurova 7, Praha 6. Czech Republic.