CHANGE-POINT DETECTION IN TEMPERATURE SERIES

Transcription

1 CZECH TECHNICAL UNIVERSITY FACULTY OF CIVIL ENGINEERING CHANGE-POINT DETECTION IN TEMPERATURE SERIES PhD. Thesis MONIKA RENCOVÁ Supervisor: Prof. RNDr. Daniela Jarušková, CSc. Department of Mathematics Thákurova 7, Prague 9

2 Abstract This work is devoted to some change-point detection problems in temperature series. Results of this thesis are based on working with real data. The submitted work presents suggestions on how the change-point methods may be applied to detect changes in annual maximal, resp. minimal temperatures and to detect changes in occurrences of unusually hot, resp. cold days. Solving these practical examples we came across some theoretical problems, we tried to work out in this thesis. In the first problem we apply the change-point theory and we will be looking for a change in parameters in a large class of independent random variables with a GEV distribution not satisfying regularity conditions. In the second problem we will focus on dependent variables and show how the change-point theory might be extended from linear processes to strong-mixing sequences. i

3 Acknowledgements It is a real pleasure to express my thanks to Prof. RNDr. Daniela Jarušková, CSc., my thesis supervisor, who introduced me into a field of change-point detection, encouraged me and spent a lot of time in discussions on this topic. I would like to thank the Management and the Staff of Department of Mathematics for providing a good environment for my work on the thesis. However, first of all I thank to my family for supporting me during my work, especially, I would like to appreciate my husband Jirka for his patience. ii

4 Contents Preface Data 7 3 The change-point detection for the GEV distributions 7 3. The change-point detection for the Weibull distributions Main results for the Weibull distributions The change-point detection for the Fréchet distributions Results Conclusion The change-point detection for dependent data The change-point detection for strong-mixing sequences Application Conclusion Block permutation 8 5. Block permutation for strong-mixing sequences Application Comparison Results A Some useful theorems and inequalities A. The test statistic for a change-point detection A. The extreme value distributions A.3 Limit theorems A.4 Strong-mixing sequences A.5 Rank statistics

5 Chapter Preface This work is devoted to some change-point detection problems in temperature series. Results of this thesis are based on working with real data. The broadly accepted hypothesis of global warming stimulated an interest in studying long temperature series. Some scientists assume that changes do not necessarily occur in the mean of the series but rather in some other characteristics, e.g. appearance of some extreme events or increase of difference between summer and winter temperatures etc. This raises an interest in studying statistical properties of extremes of random sequences, see e.g. Embrechts et al. [], Leadbetter et al. []. Our paper presents suggestions on how the change-point methods may be applied to detect changes in annual maximal, resp. minimal temperatures and to detect changes in occurrences of unusually hot, resp. cold days. Solving these practical examples we came across some theoretical problems, we tried to work out in this thesis. The world is filled with changes. We encounter them in economics, medicine, meteorology, climatology etc. A change-point analysis is a statistical method allowing to decide whether an observed stochastic process follows one model or whether the model changes. In the case of a change, we might be interested in following problems: when a change was detected and how many changes have occurred. The change-point detection is formulated in terms of hypotheses testing. The null hypothesis claims that the series is stationary, usually it means that the parameters of the model do not change, while the alternative hypothesis claims that at an unknown time point the model changes. The decision rule for rejecting the null hypothesis is based on test statistics. The earliest change-point studies go back to the 95s, where they arose in the context of quality control. We observe an output of a manufacturing process and assume that a certain characteristic varies around a certain in-control constant a. Sometimes, for example due to a failure of the production device, this constant starts to vary around another out-of-control constant a a and we want to know if and when such a change occurred. Statistical procedures in change-point analysis can be divided into two categories: on-line and off-line procedures. The on-line approach, coming from the manufacturing process, is based on the idea that after each observation we apply a new test and hope to be warned that the change occurred. In this thesis we will work with the off-line analysis when we already have all the observations and we apply a test for

6 . Preface 3 the whole data to decide whether and when the change occurred. This historically first change-point problem can be formulated as follows. For simplicity we assume that the starting value a and variance σ are known and the observations are independent and distributed according to the normal distribution. Moreover we standardize the observations and obtain variables Y i, i =... n with a zero mean and unit variance at the beginning. Then change-point problem formulated by hypotheses testing is: H : Y i = e i, i =,..., n,. A : there exists k {,..., n } such that Y i = e i, i =,..., k, Y i = a + e i, i = k +,..., n, where a. Using likelihood ratio method we obtain a so-called maximum-type statistic { } n max Y k n i.. n k i=k+ However, in practice the distribution of this statistic is very complex, so that it can be computed only for small sample sizes. Therefore, for n large, the asymptotic behavior of the statistic. is of interest. The maximum-type statistic goes to infinity as n a.s., however we can approximate this statistic by a maximum of a standardized Wiener process satisfying = o p t log log n max k n n i= Y i W t sup k /n t The approximate critical values can be calculated from the asymptotic behavior of the probabilities under H: { } n P max Y i > x + b n exp { e x}, x R,.3 k n n k a n where i=k+ a n = log log n, b n = log log n + log log log n log π. This approximation was derived by Darling and Erdös [8] in 956. However, it often happens that the starting value a is unknown. In such case we test the following null hypothesis H against the alternative A: H : Y i = a + e i, i =,..., n,.4 A : there exists k {,..., n } such that Y i = a + e i, i =,..., k, Y i = a + δ + e i, i = k +,..., n.

7 . Preface 4 with δ. Again for i.i.d. random variables distributed according to the normal distribution N, σ with σ known we obtain the maximum-type statistic of a form { } n k max Y k n σ i Y n..5 k n k For n large we may approximate the statistic.5 by the maximum of a standardized Brownian bridges n k max Y k n σ k n k i Y n sup Bt = o p. t t log log n i= i= n t n Yao and Davis [8] proved similar approximation as in.3 for a sequence of independent normal variables. For x R it holds { n k } P max Yi Y n > x + b n exp { e x}. k n σ k n k a n i= For a quite extensive survey on change-point detection we refer to Csörgő Horváth [7]. They used the log-likelihood ratio for the general model working with a sequence of independent random vectors X, X,..., X n with distribution functions F x; θ,..., F x; θ n, respectively, where θ i Θ R d for i =,..., n are parameters of the distribution functions and are assumed to change at unknown time. The general problem tested in Csörgő Horváth [7] has a form: The asymptotics of a testing statistic H : ϕ = ϕ =... = ϕ n A : there exists k {,..., n } such that ϕ =... = ϕ k ϕ k+ =... = ϕ n. max logλ k k n is under null hypothesis again / lim P Alog n max logλ k t + D d log n n k n for all t R, where and where Γt is the gamma function defined Ax = log x D d x = log x + d/ log log x log Γd/, Γt = y t exp ydy. = exp e t

8 . Preface 5 In Chapter 4 of Csörgő Horváth [7] it is also shown that the limit results remain true for a large class of dependent observations. Many articles have been published on change-point detection, see e.g. Antoch et al. [], Antoch J. and Hušková M. [], Gombay E. and Horváth, L. []. For application in climatology we refer to publications of Jarušková [8] and Jandhyala [?]. In this thesis we will apply the change-point methods for two special cases. In the first problem we apply the change-point theory and we will be looking for a change in parameters in a large class of independent random variables with the GEV distribution. In the second problem we will focus on dependent variables and show how the change-point theory might be extended from linear processes to strong-mixing sequences. The material is divided into chapters, sections and paragraphs. The studied data are presented in the second chapter. We summarize the origin of the data sets and provide some statistical characteristics of the observations. The results of the third chapter were obtained while solving change-point detection in annual maximal and minimal temperatures. We are looking for a change in parameters in a large class of random variables with the GEV distribution. The general Theorem.3.. presented by Csörgő Horváth [7], confer Appendix Theorem A.., can not be applied here directly, as extremal distributions do not satisfy conditions on regularity. Since the density function hx; µ, ψ, ξ is defined on the set {x; + ξx µ/ψ > }, the classical regularity conditions for maximum likelihood estimators are not satisfied. The next problem is caused by the conditions C.4 and C.5 of Theorem A.., since they require the continuity of third derivatives. This can be weakened by Smith s theorem, see Appendix Theorem A.3., and we will show that for ξ > there exists a sequence µ n, ψ n, ξ n of solutions of the likelihood equations such that n µ n µ, ψ n ψ, ξ n ξ converge in distribution to a zero mean normal vector with a variance covariance matrix M M is a Fisher information matrix and hence ξ > is still a regular case. From here an idea comes that the assertion of Theorem A.. is still valid. The results on temperature series are presented at the end of the chapter. Many articles have been published on independent observations. Clearly, working with real temperature series, we can not expect that the condition of independency is fulfilled. This problem we encountered solving the second example with occurrences of unusually hot or cold days. While for the annual maximal and minimal temperatures from the first example we might assume that the data form i.i.d. sequence, for occurrences of unusually hot or cold days we have a strong correlation between the temperature values measured at subsequent days, the value of correlation coefficient is for all series very close to.8. In the fourth chapter we define the problem by a model working with data forming strongmixing processes. We generalize the theory for dependent data presented in Csörgő and Horváth [7], and show that the asymptotic distribution of the testing statistic T n t is valid not only for linear processes but for strong-mixing sequences as well. In the context, it is an important question how to estimate σ. We can replace σ with an estimator,

9 where the rate of convergency to σ must be at least o p log log n, which is fulfilled by: ψn σ = R + Ri,.6 where Rj = n n j i= i= Yi Y n Yi+j Y n, Yn = n j n Y j and ψn tends to infinity with a certain speed. The estimator σ is a simplified version of the Bartlett log window estimator σnl = ˆR L + i ˆRi. L For more information about this estimator we refer to Antoch et al. []. i= It is wide known that the rate of convergency to distributional asymptotics under the null hypothesis is very slow, therefore in the fifth chapter we propose a permutation principle for obtaining the corresponding critical values. We generalize the theory presented by Kirch [] from linear processes to strong-mixing processes. We show that the estimator of variance σ LK = KL [ L K Y Kl + k Y n ], l= k= where Y n = n j n Y j, satisfies necessary condition on the rate of convergency. Critical values obtained from our data using the permutation test are listed at the end of the chapter. In Appendix we summarize the known theory for extremal distributions, the theorems concerning change-point analysis for i.i.d. data, Smith s theorem and some results on strong-mixing sequences and rank statistics.

10 Chapter Data The data sets we have studied were taken from a CD-ROM that was a part of the book edited by Camuffo and Jones [6]. The book sums up results of EU research project IM- PROVE. One of the main goals of the project was to produce seven highly reliable daily series Brussels, Cadiz, Milan, Padua, St. Petersburg, Stockholm, Uppsala, extending over more than two centuries, by correcting errors and inhomogeneities caused by changes in measurement style etc. We add one more data set, which attracted our attention the most Prague temperature series obtained from Of course, statisticians enjoy analyzing such long natural series but the length of the series also brings problems. Temperature often started to be measured at famous universities which are now mostly situated in city centers with their heat island effect. The climatologists who analyzed the Milan and Stockholm series tried to remove this effect by comparing the series with the measurements taken in nearby observatories, while the authors of the other series were not able to do it. This is not the only reason why the properties of the studied series are difficult to compare. The other reason is that the way in which the daily averages were calculated differs from place to place. In spite of the effort of the climatologists participating in the project, the series are not complete. Table and show the periods of measurement. period missing data number of obser. Brussels none 4 Cadiz ; Milan none 36 Padua ; 855; 865; 97; 947 St. Petersburg ; ; ; 793; ; 8 85; 846 Stockholm 756 none 45 Uppsala 75 none 66 Prague none 3 Table. Periods of observations together with missing data for annual minima. 7

11 . Data 8 period missing data number of obser. Brussels none 4 Cadiz 87 85; 873; 964; 988; 76 99; Milan none 36 Padua ; 9 9; 947; 954; 4 994; 996 St. Petersburg ; 784; 787; ; ; 8 84 Stockholm 756 none 45 Uppsala 75 none 66 Prague none 3 Table. Periods of observations together with missing data for annual maxima. The following tables summarize basic descriptive statistics of the data. maxima x σ n sk Brussels Cadiz Milan Padua St. Petersburg Stockholm Uppsala Prague Table 3. Descriptive statistics for annual maximal temperatures. minima x σ n sk Brussels Cadiz Milan Padua St. Petersburg Stockholm Uppsala Prague Table 4. Descriptive statistics for annual minimal temperatures. Comparing Tables 3 and 4 we can see that skewness for minimal temperatures is negative and its absolute values are larger, while skewness for maximal temperatures is positive with smaller absolute values. It might suggest that maximal temperatures might be modelled by the normal distribution. For minimal temperatures, the three parameter

12 . Data 9 Weibull distribution fits better, see Rencová [3]. The following figures show the behavior of the series under study Figure. Annual minimal temperatures in C in Brussels Figure. Annual maximal temperatures in C in Brussels Figure 3. Annual minimal temperatures in C in Cadiz. Figure 4. Annual maximal temperatures in C in Cadiz Figure 5. Annual minimal temperatures in C in Milan Figure 6. Annual maximal temperatures in C in Milan.

13 . Data Figure 7. Annual minimal temperatures in C in Padua. Figure 8. Annual maximal temperatures in C in Padua Figure 9. Annual minimal temperatures in C in St. Petersburg Figure. Annual maximal temperatures in C in St. Petersburg Figure. Annual minimal temperatures in C in Stockholm Figure. Annual maximal temperatures in C in Stockholm.

14 . Data Figure 3. Annual minimal temperatures in C in Uppsala. Figure 4. Annual maximal temperatures in C in Uppsala Figure 5. Annual minimal temperatures in C in Prague. Figure 6. Annual maximal temperatures in C in Prague. Figures 7 3 describe the second problem - changes in occurrences of unusually hot, resp. cold days. We provide graphs of sums of exceedances over a chosen level h =.5 and under a chosen level c =.5 for standardized daily series of studied data sets, see Section 4..

15 . Data x 4 Figure 7. Sums of exceedances over a chosen level h =.5 in Brussels x 4 Figure 9. Sums of exceedances over a chosen level h =.5 in Cadiz x 4 Figure 8. Sums of exceedances under a chosen level c =.5 in Brussels x 4 Figure. Sums of exceedances under a chosen level c =.5 in Cadiz x 4 Figure. Sums of exceedances over a chosen level h =.5 in Milan x 4 Figure. Sums of exceedances under a chosen level c =.5 in Milan.

16 . Data x 4 Figure 3. Sums of exceedances over a chosen level h =.5 in Padua x 4 Figure 4. Sums of exceedances under a chosen level c =.5 in Padua x 4 Figure 5. Sums of exceedances over a chosen level h =.5 in St. Petersburg x 4 Figure 6. Sums of exceedances under a chosen level c =.5 in St. Petersburg x 4 Figure 7. Sums of exceedances over a chosen level h =.5 in Stockholm x 4 Figure 8. Sums of exceedances under a chosen level c =.5 in Stockholm.

17 . Data x 4 Figure 9. Sums of exceedances over a chosen level h =.5 in Uppsala x 4 Figure 3. Sums of exceedances under a chosen level c =.5 in Uppsala x 4 Figure 3. Sums of exceedances over a chosen level h =.5 in Prague x 4 Figure 3. Sums of exceedances under a chosen level c =.5 in Prague. Figure 33 shows that there is a strong correlation between the temperature values measured at subsequent days, its value is for all series very close to.8, see the upper stationary graph in Figure 33. The following graphs, from the top to the bottom, depict correlation coefficients between two days with lag equal to, 3 and 4, e.g. the first value in the upper graph is the value of the correlation coefficient between st January and nd January, the first value in the second graph from the top is the value of the correlation coefficient between st January and 3 rd January etc.

18 . Data 5.9 Milan Figure 33. The autocorrelation coefficients between subsequent days in Milan. We can notice that the autocorrelation coefficients for lag equal to oscillate about the value.8 during the whole year, while the autocorrelation coefficients for larger lags are smaller in summer than in winter, see Figure 33.

19 Problem Application of change-point detection for annual maxima and minima

20 Chapter 3 The change-point detection for the GEV distributions In the first part of the thesis we study annual maximal and minimal temperatures. Figures 6 and Tables 3, 4 show behavior of the series under study. Extremes of random sequences are modelled by the GEV distribution, for details confer Appendix, Section A.. Our goal is applying the general Csörgő and Horváth theory for detecting a sudden change Appendix, Section A. of the GEV distribution Hx; µ, ψ, ξ with a density function hx; µ, ψ, ξ = + ξ x µ { ξ exp + ξ x µ } ξ, 3. ψ ψ ψ provided + ξx µ/ψ >. Notice that the support of the density function hx; µ, ψ, ξ depends on the parameters µ, ψ, ξ. Suppose that X,..., X n are independent random variables, we are to test the null hypothesis H against the alternative A : H : X i GEV µ, ψ, ξ, i =,..., n, 3. A : there exists k {,..., n n } such that X i GEV µ, ψ, ξ, i =,..., k, X i GEV µ, ψ, ξ, i = k +,..., n, where the parameters µ, ψ, ξ before the change point are known while µ, ψ, ξ µ, ψ, ξ are unknown or to test the null hypothesis H against the alternative A : A : there exists k { n,..., n n } such that X i GEV µ, ψ, ξ, i =,..., k, 3.3 X i GEV µ, ψ, ξ, i = k +,..., n, where neither the parameters before nor after the change point are known and µ, ψ, ξ µ, ψ, ξ. The constant n may be any fixed integer larger than three. However, to obtain a good estimates of all three parameters we need to have enough observations. For testing the problem 3.3 we may use the twice log likelihood ratio max logλ k = [L k ϕ k + L kϕ k L n ϕ n ], k n 7

21 3. The change-point detection for the GEV distributions 8 while testing the problem 3. yields in a simplified version max k n logλ k = max k n [L k ϕ k L kϕ ]. for more details we refer to Appendix, Remark A... To find critical values we have to find distribution of the test statistics under H. As the exact distribution of max k n logλ k, resp. max k n logλ k, under H are very complex, the approximate critical values can be found using the limit behavior of max k n logλ k, resp. max k n logλ k, see Appendix, Csörgő Horváth theorem. The conditions of Theorem A.. are classical regularity conditions for the existence of the maximum likelihood estimator. Since the density function 3. is defined on the set {x; +ξx µ/ψ > }, the classical regularity conditions for maximum likelihood estimators are not satisfied. The next problem is caused by the conditions C.4 and C.5 of Theorem A.., since they require the continuity of third derivatives. This can be weakened by Smith s Theorem A.3. and we will show that for ξ > there exists a sequence µ n, ψ n, ξ n of solutions of the likelihood equations such that n µ n µ, ψ n ψ, ξ n ξ converge in distribution to a zero mean normal vector with a variance covariance matrix M M is a Fisher information matrix and hence ξ > is still a regular case. We will proceed in two steps. At first we show theory for the three parameter Weibull distributions using the results of Smith s theorem, see Appendix Theorem A.3., and then we will focus on the Fréchet distributions. For those purposes we can use the following reparameterization of the GEV distribution. For ξ <, substituting θ = µ ψ ξ, β = ξ ψ ξ, α = ξ we obtain hx; θ, α, β = αβ x + θ α exp{ β x + θ α } for x θ, = for x < θ. 3.4 It is the three parameter Weibull distribution W eibθ, α, β of a random variable x not the Weibull distribution as a limit distribution for maxima concentrated on, θ from the Fisher Tippet theorem, see Appendix Theorem A... ξ, α =, we obtain a reparameteriza- ξ For ξ >, substituting θ = µ ψ, β = ξ tion ξ ψ hx; θ, α, β = αβx θ α exp{ βx θ α } for x θ, = for x < θ, 3.5 which corresponds to the Fréchet distribution F réch θ, α, β.

22 3. The change-point detection for the GEV distributions 9 Remark 3... Results for the Gumbel distribution from Fisher Tippet theorem A.. corresponding to a case ξ = are obtained by ξ in The change-point detection for the Weibull distributions The general theory presented in Csörgő and Horváth [7] was applied by Jandhyala et al. [?] to develop a test for detecting a sudden change in the two parameter Weibull distribution. However, in the case we use as a model the three parameter Weibull distribution W eib θ, α, β with a density function fx; θ, α, β = x θ α αβ exp{ βx θ α } for x θ, 3.6 = for x < θ, it seems more natural to look for a change in all three parameters. Notice that the support of the density function f is given by the parameter θ. Suppose that X,..., X n are independent random variables, we are to test the null hypothesis H against the alternative A : H : X i W eib θ, α, β, i =,..., n, 3.7 A : there exists k {,..., n n } such that X i W eib θ, α, β, i =,..., k, X i W eib θ, α, β, i = k +,..., n, where the parameters θ, α, β before the change point are known while θ, α, β θ, α, β are unknown or to test the null hypothesis H against the alternative A : A : there exists k { n,..., n n } such that X i W eib θ, α, β, i =,..., k, 3.8 X i W eib θ, α, β, i = k +,..., n, where neither the parameters before nor after the change point are known and θ, α, β θ, α, β. The constant n may be any fixed integer larger than three. For testing the problem 3.8 we may use the twice log likelihood ratio max logλ k = [L k ϕ k + L kϕ k L n ϕ n ], k n while testing the problem 3.7 yields in a simplified version max k n logλ k = max k n [L k ϕ k L kϕ ],

23 3. The change-point detection for the GEV distributions for more details we refer to Appendix. As the exact distribution of max k n logλ k, resp. max k n logλ k, under H are very complex, the approximate critical values can be found using the limit behavior of test statistics max k n logλ k, resp. max k n logλ k, see Appendix, Csörgő Horváth Theorem A... Let θ, α, β be the true values of the parameters under H. We will assume that θ R, α > and β >. The assumptions C.4. and C.5. of Csörgő and Horváth are not satisfied and Theorem A.. cannot be applied directly to get a limit distribution of max k n logλ k, resp. max k n logλ k. On the other hand, Smith showed, confer Appendix Theorem A.3., that for α > there exists a sequence θ n, α n, β n of solutions of the likelihood equations such that n θ n θ, α n α, β n β converge in distribution to a zero mean normal vector with a variance covariance matrix M M is a Fisher information matrix and hence α > is still a regular case. From here an idea comes that the assertion of Csörgő and Horváth theorem A.. is still valid for α >. 3. Main results for the Weibull distributions Our main results concern the asymptotic distribution of the statistic max k n logλ k under H for testing a change in all three parameters, when the parameters before a change point are known while after it they are unknown as well as the statistic max k n logλ k under H for testing a change in all three parameters, when the parameters both before and after a change point are unknown. We start with the characteristics of the log likelihood function L k. The first and second derivatives of L k θ, α, β may be expressed as follows: L k θ = i= L k α = k L k β = L k θ = k [ ] α X i θ + αβx i θ α, 3.9 [ logx i θ + ] α βx i θ α logx i θ, 3. i= k [ ] β X i θ α, 3. k [ ] α X i θ αα βx i θ α, 3. [ ] X i θ + βx i θ α + αβx i θ α logx i θ, 3.3 i= i= L k k θ α = i=

24 3. The change-point detection for the GEV distributions L k k θ β = [ αxi θ α ], 3.4 L k α = i= k i= [ ] α βx i θ α log X i θ, 3.5 L k k α β = [ X i θ α logx i θ], 3.6 L k β = i= k i= For simplicity we denote the true value of parameter [ β ]. 3.7 ϕ = θ, α, β, a maximum likelihood estimator based on X,..., X k when it exists by It holds ϕ k = θ k, α k, β k. E log fxi ; ϕ =, 3.8 θ E log fxi ; ϕ =, α E log fxi ; ϕ =. β Let s denote a Fisher information matrix M on a parameter ϕ = θ, α, β with elements m θθ m θα m θβ M = m αθ m αα m αβ, m βθ m βα m ββ where m θθ = E{ θ logfx i; ϕ θ logfx i; ϕ } = E{ θ logfx i; ϕ }, m αα = E{ α logfx i; ϕ α logfx i; ϕ } = E{ α logfx i; ϕ }, m ββ = E{ β logfx i; ϕ β logfx i; ϕ } = E{ β logfx i; ϕ },

25 3. The change-point detection for the GEV distributions m θα = m αθ = E{ θ logfx i; ϕ α logfx i; ϕ } = E{ θ α logfx i; ϕ }, m θβ = m βθ = E{ θ logfx i; ϕ β logfx i; ϕ } = E{ θ β logfx i; ϕ }, m αβ = m βα = E{ α logfx i; ϕ β logfx i; ϕ } = E{ α β logfx i; ϕ }. 3.9 A maximum likelihood estimator ϕ k = θ k, α k, β k satisfies L k θ ϕ k =, L k α ϕ k =, L k β ϕ k =. 3. The existence of ϕ k = θ k, α k, β k is guaranteed for α > by Theorem A.3.. If we consider positive moments only, then E log fxi ; ϕ r < θ if and only if r < α and for the same r we also have E log fxi ; ϕ r <, α E log fxi ; ϕ r <. β For the second derivatives E log fxi ; ϕ θ s < if and only if s < α and for the same s is also E log fxi ; ϕ s <, θ α E log fxi ; ϕ s <, θ β E log fxi ; ϕ α s <, E log fxi ; ϕ s <, α β E log fxi ; ϕ β s <.

26 3. The change-point detection for the GEV distributions 3 M is a positive definite matrix. According to the Marcinkiewicz-Zygmund law, Appendix - Theorem A.3., for any τ such that < τ < /α lim k kτ k θ L kϕ + m θθ = a.s., lim k kτ k θ α L kϕ + m θα = a.s., lim k kτ k lim k kτ k lim k kτ k lim k kτ k θ β L kϕ + m θβ = a.s., α L kϕ + m αα = a.s., α β L kϕ + m αβ = a.s., β L kϕ + m ββ = a.s. 3. By the law of the iterated logarithm, confer Appendix Theorem A.3.3 lim sup k lim sup k lim sup k L θ kϕ = O k log log k L α kϕ = O k log log k L β kϕ = O k log log k a.s., a.s., a.s. 3. We start with several technical lemmas on the three parameter Weibull distribution W eib θ, α, β. Lemma 3... Let X i W eib θ, α, β, then a for θ R, β > and α > 3 E b for θ R, β > and < α 3 k i= Proof. See Feller []. X i θ <, E X i θ <, E X i θ < 3 E X i θ <, E X i θ <, X i θ 3 = o k 3/α log k 3/α+ a.s. for some >. 3.3

27 3. The change-point detection for the GEV distributions 4 Lemma 3... Let X, X,..., X n are i.i.d. random variables, X i W eib θ, α, β. Under H for any >, the minimum X k and the second minimum X k satisfy Proof. See Jarušková [8]. = o log k + a.s. 3.4 X k θ X k θ Further, we prove the following lemma. We denote for any δ k > } I δk = { θ R, α R, β R; θ θ < δ k, α α < δ k, β β < δ k. Lemma For any sequence {δ k } satisfying δ k k /α+δ for some δ > and for any τ such that < τ < /α lim k kτ sup I δk k θ L k ϕ θ L kϕ = a.s., 3.5 lim k kτ sup I δk k θ α L k ϕ θ α L kϕ = a.s., 3.6 lim k kτ sup I δk k θ β L k ϕ θ β L kϕ = a.s., 3.7 lim k kτ sup I δk k α L k ϕ α L kϕ = a.s., 3.8 lim k kτ sup I δk k α β L k ϕ α β L kϕ = a.s., 3.9 lim k kτ sup k β L k ϕ β L kϕ = a.s. 3.3 I δk Proof. The proof is divided into two parts: the first part, rather lengthy, corresponds to the condition < α 3 and the second part corresponds to the condition α > 3. Let s suppose < α 3. discontinuous at X i = θ. We consider only terms in second derivatives, which are We start with proving 3.5. Substituting 3. into 3.5 and concentrating on terms discontinuous at X i = θ we obtain a following assertion to be examined: k lim k kτ sup I δk k X i θ = a.s., < τ < /α. 3.3 X i θ i=

28 3. The change-point detection for the GEV distributions 5 The proof can be found in Jarušková [8], Lemma 3. Next we look into the assertion 3.6. Substituting 3.3 into 3.6 we get following assertions to be proved k lim k kτ sup I δk k X i θ = a.s. 3.3 X i θ and lim k kτ sup I δk k i= k α βx i θ bα logx i θ i= α β X i θ α logx i θ = a.s We start with 3.3. Similarly as in Smith [6] and Jarušková [8] we write k k X i θ = X i θ i= kx k θ kx k θ k + k i= X ki θ X ki θ First, let X has the three parameter Weibull distribution with the density function 3.6. Then Z = X θ β /α has the Weibull distribution with parameters θ =, β = and α. Random variable Y = has the density Z fy = /y α+ α exp{ /y α } for y, 3.35 = for y <, with a finite moment EY r < for any r < α. Then, according to Theorem A.3.4, we get / k /r X k θ a.s. for any r < α, i.e. the second term on the right side of 3.34 satisfies / k X k θ = ok + r a.s. for any r < α. Second, choose r satisfying /α < /r < /α + δ, then we can write the first term on the right side of 3.34 as follows: Recall that δ k = o k /α+δ then X k θ =. θ θ X k θ b X k θ sup b θ θ <δ k θ θ X k θ = ok/r /k /α+δ = o a.s. 3.36

29 3. The change-point detection for the GEV distributions 6 In the other words, for two first terms in 3.34 sup kx k θ = ok + r a.s kx k θ for any r < α. b θ θ <δ k Further, for the third term in 3.34, again as in Smith [6] and Jarušková [8], using the Taylor expansion, there exists θ that θ θ < θ θ and k k i= X ki θ X ki θ = θ θ k k X ki θ i= We have to distinguish between two cases: i θ ˆθ > and ii θ ˆθ <. We start with i. For θ satisfying ˆθ < θ < θ we have in 3.38 θ θ k k i= X ki θ θ θ k k i= X ki θ. Applying Lemma 3.. for the right side of the above inequality, we obtain sup I δk k for any < υ < α. k X i θ = ok /α+δ O = ok υ a.s. X i θ i= For the proof of ii we use a characteristic of the first minimum that X ki θ X ki X k for every i =,..., n. We obtain following inequalities θ θ k k i= X ki θ θ θ k = θ θ k k i= i= X ki X k k X ki θ X k θ = θ θ k k i= θ θ X k θ k X k θ X ki θ X k θ X ki θ k i= X ki θ.

30 3. The change-point detection for the GEV distributions 7 Applying Lemma 3.. and Lemma 3.. for the factors on the right side of the above inequality, we obtain similarly as in the case i sup I δk k for any < υ < α. k X i θ = ok /α+δ olog k + O X i θ i= = ok υ a.s. Since we assume < α 3, we have α α implying for any < τ < α the rate of convergency of the third term of 3.34 sup I δk k k X i θ = ok τ a.s. X i θ i= and with the result 3.37 we have the assertion of 3.3. Now we prove the assertion The Taylor expansion implies that there exists θ, α, β, such that θ θ < θ θ, α α < α α, β β < β β and k k i= α βx i θ bα logx i θ α β X i θ α logx i θ = k k βx i θ logx α i θ α α i= + α βx i θ log α X i θ α α + αx i θ logx α i θ β β + α α βx i θ logx α i θ θ θ + α βx i θ α θ θ. 3.39

31 3. The change-point detection for the GEV distributions 8 The first term on the right side of 3.39 can be rewritten k k i= = k = k βx i θ logx α i θ α α k i= k i= βx i θ α log X i θ X i θ Xi θ α α α X i θ X i θ βx i θ α + θ θ X i θ α logxi θ + βx i θ α + θ θ α log + θ θ X i θ X i θ From inequality θ θ X i θ < θ θ X k θ for every i =,.... and 3.36 we have θ θ sup θ θ b <δ k X i θ = o a.s. α α. 3.4 Then Similarly sup b θ θ <δ k + θ α θ = O a.s. 3.4 X i θ log + θ θ log + θ θ K θ θ X i θ X k θ X k θ for some K R a.s. and then sup log + θ θ = o a.s. 3.4 θ θ b <δ k X i θ We can suppose that for sufficiently large k, is α > since we have α > and then EX i θ α logx i θ <, EX i θ α <. Therefore the first term in 3.39 satisfies sup I δk k k i= βx i θ logx α i θ α α = ok α +δ.

32 3. The change-point detection for the GEV distributions 9 The second term on the right side of 3.39 can be rewritten k k i= = k = k α βx i θ log α X i θ α α k α βx i θ α Xi θ i= [ k i= α X i θ log X i θ X i θ X i θ α βx i θ α + θ θ X i θ α log X i θ + α βx i θ α + θ θ X i θ α logxi θ log + α βx i θ α + θ θ X i θ α log Using inequalities 3.4, 3.4 and a property we get sup I δk k + θ θ X i θ EX i θ α log X i θ < for α > k i= + θ θ X i θ ] α βx i θ log α X i θ α α = ok α +δ. α α α α Similarly we can rewrite the other terms on the right side of 3.39 for which, supposing again α >, it holds EX i θ α logx i θ <, EX i θ α logx i θ <, EX i θ α <, 3.44 therefore we have for all the terms in 3.39 sup I δk k k i= for any < υ < α. α βx i θ bα logx i θ α β X i θ α logx i θ = ok υ a.s. Since we assume < α 3, we have α α implying for any < τ < α the rate of convergency in 3.33 sup I δk k k i= α βx i θ bα logx i θ α β X i θ α logx i θ = ok τ a.s.

33 3. The change-point detection for the GEV distributions 3 Now we investigate the assertion 3.8. Substituting 3.5 into 3.8 we have to prove k lim k kτ sup I δk k β log X i θx i θ bα + β log X i θ X i θ α i= = a.s From the Taylor expansion we get that there exists θ, α, β, such that θ θ < θ θ, α α < α α, β β < β β and k βx i k θ bα log X i θ β X i θ α log X i θ i= [ = k βx i log θ α 3 X i k θ α α i= + X i log θ α X i θ β β + βx i θ logx α i θ θ θ ] + α βx i θ log α X i θ θ θ Similarly as in the second term of 3.39 we have k sup sup I δk k βx i θ bα log X i θ + β X i θ α log X i θ I δk i= = ok υ a.s. for any < υ <. Since for < α 3 it holds and then α α α k sup sup I δk k βx i θ bα log X i θ + β X i θ α log X i θ I δk i= for any < τ < α. = ok τ a.s. That were all the terms in 3.5, 3.6, 3.7, 3.8, 3.9, 3.3 discontinuous at X i = θ. The proof for the condition α > 3 is trivial, as according to Lemma 3.., all the terms of Taylor expansions have finite expectations. The next theorem gives the convergency of the proposed maximum likelihood estimators.

34 3. The change-point detection for the GEV distributions 3 Theorem There exists a sequence of real number {δ k }, such that δ k k/ log log k and δk k /α+δ for some δ >, and there exists a set A with P A =, such that for any ω A we can find k ω, such that for all k k there exists a local maximum of L k θ, α, β denoted by ϕ k = θ k, α k, β k satisfying θ L k θ k, α k, β k =, α L k θ k, α k, β k =, β L k θ k, α k, β k = and θ k θ δ k, α k α δ k, β k β δ k. Proof. Similarly as in Smith [6], for any sequence {δ k } satisfying assumptions of Theorem 3..4 we define for t R, x R, y R the function f k t, x, y = δ k k L kθ + δ k t, α + δ k x, β o + δ k y. The Taylor expansion for any t R, x R, y R satisfying t + x + y implies that there exist t θ <, x θ <, ỹ θ < such that f k t t, x, y = f k t,, + f k t t θ, x θ, ỹ θ t + f k t x t θ, x θ, ỹ θ x+ + f k t y t θ, x θ, ỹ θ y = = L k δ k k θ ϕ + t L k k θ ϕ θ + x L k k θ α ϕ θ + y L k k θ β ϕ θ = L k δ k k θ ϕ + t L k k θ ϕ θ + x L k k θ α ϕ θ + y L k k θ β ϕ θ + t L k k θ ϕ + x L k k θ α ϕ + y L k k θ β ϕ t L k k θ ϕ x L k k θ α ϕ y L k k θ β ϕ + t m θθ t m θθ + x m θα x m θα + y m θβ y m θβ, 3.47 where m θθ, m θα and m θβ are the elements of the Fisher information matrix M, see the definition 3.9. For t θ <, x θ <, ỹ θ < we denote ϕ θ = θ θ, α θ, β θ such as θ θ = θ + t θ δ k, α θ = α + x θ δ k, βθ = β + ỹ θ δ k satisfying θ θ θ < δ k, α θ α < δ k, β θ β < δ k

35 3. The change-point detection for the GEV distributions 3 and we denote ε k,θ t θ, x θ, ỹ θ = L k δ k k θ ϕ + t L k k θ ϕ θ L k θ ϕ + x L k k θ α ϕ θ L k θ α ϕ + y L k k θ β ϕ θ L k θ β ϕ L k + t k θ ϕ + m θθ L k + x k θ α ϕ + m θα L k + y k θ β ϕ + m θβ. We can then rewrite 3.47 in a form f k t t, x, y = t m θθ x m θα y m θβ + ε k,θ t θ, x θ, ỹ θ, For the term ε k,θ t θ, x θ, ỹ θ it holds ε k,θ t θ, x θ, ỹ θ L k δ k k θ ϕ + t sup L k I δk k θ ϕ θ L k k θ ϕ + x sup L k I δk k θ α ϕ θ L k k θ α ϕ + y sup L k I δk k θ β ϕ θ L k k θ β ϕ + t L k k θ ϕ + m θθ + x L k k θ α ϕ + m θα + y L k k θ β ϕ + m θβ From the law of the iterated logarithm 3. and the characteristics of the sequence δ k that log log k δ k k, we get L k δ k k θ ϕ. Then, combining 3.5, 3.6, 3.7 with 3., we get that also all the next terms in 3.48 tend to and so we obtain ε k,θ t θ, x θ, ỹ θ a.s.

36 3. The change-point detection for the GEV distributions 33 Similarly, f k L k t, x, y = x δ k k α ϕ + t L k k α θ ϕ α + x L k k α ϕ α + y L k k α β ϕ α = t m θα x m αα y m αβ + ε k,α t α, x α, ỹ α, where m θα, m αα and m αβ are the elements of the Fisher information matrix M, see the definition 3.9. For t α <, x α <, ỹ α < we denote ϕ α = θ α, α α, β α, where θ α = θ + t α δ k, α α = α + x α δ k, β α = β + ỹ α δ k satisfying θ α θ < δ k, α α α < δ k, β α β < δ k. For ε k,α t α, x α, ỹ α we obtain following inequalities. ε k,α t α, x α, ỹ α L k δ k k α ϕ + t sup L k I δk k θ α ϕ α L k k θ α ϕ + x sup L k I δk k α ϕ α L k k α ϕ + y sup L k I δk k α β ϕ α L k k α β ϕ + t L k k θ α ϕ + m θα + x L k k α ϕ + m αα + y L k k α β ϕ + m αβ From the law of the iterated logarithm 3. and the characteristics of the sequence δ k that log log k δ k k, we get L k δ k k α ϕ. Then, combining 3.8, 3.6, 3.9 with 3., we get that also all the next terms in 3.49 tend to and so we obtain ε k,α t α, x α, ỹ α a.s. Further, f k L k t, x, y = y δ k k β ϕ + t L k k β θ ϕ β + x L k k α β ϕ β + y L k k β ϕ β = t m θβ x m αβ y m ββ + ε k,β t β, x β, ỹ β,

37 3. The change-point detection for the GEV distributions 34 where m θβ, m αβ and m ββ are the elements of the Fisher information matrix M, see the definition 3.9. For t β <, x β <, ỹ β < we denote ϕ β = θ β, α β, β β, where θ β = θ + t β δ k, α β = α + x β δ k, β β = β + ỹ β δ k satisfying θ β θ < δ k, α β α < δ k, β β β < δ k. For ε k,β t β, x β, ỹ β we have following inequalities ε k,β t β, x β, ỹ β L k δ k k β ϕ + t sup L k I δk k β θ ϕ β L k k β θ ϕ + x sup L k I δk k α β ϕ β L k k α β ϕ + y sup L k I δk k β ϕ β L k k β ϕ + t L k k θ β ϕ + m θβ + x L k k α β ϕ + m αβ + y L k k β ϕ + m ββ. 3.5 From the law of the iterated logarithm 3. and the characteristics of the sequence δ k that log log k δ k k, we get L k δ k k β ϕ. Then, combining 3.7, 3.9, 3.3 with 3., we get that also all the next terms in 3.5 tend to and so we obtain ε k,β t β, x β, ỹ β a.s. Let t + x + y =. Then we have t f k t + x f k x + y f k y = t m θθ x m αα y m ββ x t m θα + y t m θβ + x y m αβ + t ε k,θ t θ, x θ, ỹ θ + x ε k,α t α, x α, ỹ α + y ε k,β t β, x β, ỹ β 3.5 Since we proved that ε k,θ t θ, x θ, ỹ θ a.s., ε k,α t α, x α, ỹ α a.s., ε k,β t β, x β, ỹ β a.s., the expression 3.5 is as k strictly negative by the assumed positivedefiniteness of M. Hence Lemma A.3.5 shows that f k has a local maximum in the range t + x + y < as k.

38 3. The change-point detection for the GEV distributions 35 Clearly, there exists a sequence { θ k, α k, β k } and a set A with P A = such that for any ω A there exists k ω such that for all k k, θ k, α k, β k is a local maximum of L k θ, α, β satisfying θ L k θ k, α k, β k =, α L k θ k, α k, β k =, β L k θ k, α k, β k = and θ k θ δ k, α k α δ k, β k β δ k. The next lemma shows that, in the limit, the proposed maximum likelihood estimators behave as if they were partial sums of random vectors. Lemma For any τ such that < τ < /α and for k it holds L θ kϕ lim k kτ L k log log k α kϕ L β kϕ θk θ k M α k α β k β = a.s. 3.5 Proof. Now, for instance we choose the sequence δ k = log log log k log log k/k} satisfying conditions from Lemma For maximum likelihood estimators ϕ k = θ k, α k, β k we have = θ L k ϕ k = = θ L kϕ k θ k θ m θθ k α k α m θα k β k β m θβ + θ L k ϕ k θ L kϕ θ k θ + θ L k ϕ k θ α L kϕ α k α + θ β L k ϕ k θ β L kϕ β k β + θ L kϕ + k m θθ θ k θ + θ α L kϕ + k m θα α k α + θ β L kϕ + k m θβ β k β 3.53

39 3. The change-point detection for the GEV distributions 36 for ϕ θ = θ θ, α θ, β θ satisfying θ k θ < θ k θ, α k α < α k α, β k β < β k β. Then k log log k θ L kϕ k θ k θ m θθ k α k α m θα k β k β m θβ k = log log k k θ L k ϕ k θ L kϕ θ k θ k log log k k θ L k ϕ k θ α L kϕ α k α k log log k k θ β L k ϕ k θ β L kϕ β k β k log log k k θ L kϕ + k m θθ θ k θ k log log k k θ α L kϕ + k m θα α k α k log log k k θ β L kϕ + k m θβ β k β Using the Marcinkiewicz-Zygmund law 3., characteristics 3.5, 3.6, 3.7 and the characteristics of the sequence { δ k } that log log k θ k θ log log log k, k log log k α k α log log log k, k log log k β k β log log log k k 3.55 for the right side of 3.54 we get lim k kτ k log log k θ L kϕ k m θθ θ k θ k m θα α k α k m θβ β k β = a.s for any τ satisfying < τ < /α, where the coefficients m θθ, m θα, m θβ are the elements of the first line of the matrix M. From the similar expressions for derivatives L α k ϕ k and L β k ϕ k we obtain lim k kτ k log log k α L kϕ km αθ θ k θ km αα α k α km αβ β k β = a.s. 3.57

40 3. The change-point detection for the GEV distributions 37 lim k kτ k log log k β L kϕ km βθ θ k θ km βα α k α km ββ β k β = a.s., 3.58 where coefficients m αθ, m αα, m αβ are the elements of the second line of the matrix M and where coefficients m βθ, m βα, m ββ are the elements of the third line of the matrix M. From 3.56, 3.57, 3.58 we get the assertion 3.5. The next corollary gives the rate of convergency of the proposed maximum likelihood estimators. Corollary The sequence of the proposed maximum likelihood estimators ϕ k = θ k, α k, β k from Theorem 3..4 satisfies k lim sup α k α = O a.s., k log log k k lim sup β k β = O a.s., k log log k lim sup k k log log k θ k θ = O a.s Proof. The proof is an easy consequence of 3.5 and the law of iterated logarithm 3.. Corollary For any τ such that < τ < /α it holds lim k kτ k θ L kϕ, α L kϕ, β L kϕ M L θ kϕ L α kϕ L β kϕ k θk θ, α k α, β θk θ k β M α k α = a.s. 3.6 β k β Proof. For a matrix P, such that P T P = M, we can write equation 3.5 as follows: L lim k kτ P T θ kϕ L log log k α kϕ θk θ kp α k α = a.s. 3.6 k L β kϕ β k β For k we have k θ L kϕ, α L kϕ, β L kϕ P + k θk θ, α k α, β k β P T = O log log k a.s. 3.6 and combining equations 3.6 and 3.6 we obtain the assertion of Corollary 3..7.

41 3. The change-point detection for the GEV distributions 38 Theorem For any τ such that < τ < /α and for k k τ L k ϕ k L k ϕ k θ L kϕ, α L kϕ, β L kϕ M L θ kϕ L α kϕ L β kϕ a.s. Proof. The Taylor expansion L k ϕ k L k ϕ = D cϕk ϕ L k ϕ k + D cϕ k ϕ L k ϕ, 3.63 where D cϕk ϕ L k ϕ k is the first differential at the point ϕ k in the direction ϕ k ϕ, D cϕ k ϕ L k ϕ is the second differential at the point ϕ in the direction ϕ k ϕ and θ θ < θ k θ, α α < α k α, β β < β k β. We can rewrite 3.63 L k ϕ k L k ϕ k θ L kϕ, α L kϕ, β L kϕ M L θ kϕ L α kϕ L β kϕ = D cϕ k ϕ L k ϕ D cϕ k ϕ L k ϕ + D cϕ k ϕ L k ϕ k θk θ, α k α, β θk θ k β M α k α β k β + k θk θ, α k α, β θk θ k β M α k α β k β k θ L kϕ, α L kϕ, For the differences on the right side of 3.64 we obtain: β L kϕ M L θ kϕ L α kϕ L β kϕ. 3.64

42 3. The change-point detection for the GEV distributions 39 - The difference of second differentials is Dcϕ k ϕ L k ϕ Dcϕ k ϕ L k ϕ = θ L k ϕ θ L kϕ θ k θ + α L k ϕ α L kϕ α k α + β L k ϕ β L kϕ β k β + θ α L k ϕ θ α L kϕ θ k θ α k α + θ β L k ϕ θ β L kϕ θ k θ β k β + α β L k ϕ α β L kϕ α k α β k β and its elements are ok τ according to Lemma 3..5 and Corollary Similarly for the difference Dcϕ k ϕ L k ϕ k θk θ, α k α, β θk θ k β M α k α β k β = θ L kϕ + k m θθ θ k θ + α L kϕ + k m αα α k α + β L kϕ + k m ββ β k β + θ α L kϕ + k m θα θ k θ α k α + θ β L kϕ + k m θβ θ k θ β k β + α β L kϕ + k m αβ α k α β k β and its elements are ok τ according to the Marcinkiewicz-Zygmund law 3. and Corollary 3..6.

43 3. The change-point detection for the GEV distributions 4 - The difference k θk θ, α k α, β θk θ k β M α k α β k β k θ L kϕ, α L kϕ, β L kϕ M L θ kϕ L α kϕ L β kϕ is ok τ according to the Corollary Summarizing these three results we obtain that the right side of 3.64 is ok τ a.s. We introduce Ax = log x and D d x = log x+d/ log log x log Γd/, similarly as in Theorem A... Theorem The asymptotic distribution of the maximum likelihood statistic for testing the problem 3.7 under H provided α > is given by / lim P Alog n max n k n logλ k t + D 3 logn = exp e t and for the maximum likelihood statistic for testing the problem 3.8 we have for all t R. / lim P Alog n max logλ k t + D 3 logn n k n = exp e t Proof. Using Theorem 3..8 we can similarly as in Csörgő and Horváth [7] prove that max Lk ϕ k L k ϕ k n max k n k θ L kϕ, α L kϕ, β L kϕ M L θ kϕ L α kϕ L β kϕ = o P log log n and the assertion of Theorem 3..9 is an easy consequence. Now, coming back to the GEV distribution and using parameters µ, ψ, ξ we can write the Theorem 3..9 as follows.

44 3. The change-point detection for the GEV distributions 4 Theorem 3... Provided < ξ <, the asymptotic distribution of the maximum likelihood statistic for testing the problem 3. under H is given by lim P / Alog n max n k n logλ k t + D 3 logn = exp e t and for the maximum likelihood statistic for testing the problem 3.3 we have for all t R. / lim P Alog n max logλ k t + D 3 logn n k n Proof. An easy consequence of the inequality α >. = exp e t 3.3 The change-point detection for the Fréchet distributions Now we concentrate on proving a similar theorem as Theorem 3.. for parameter ξ > corresponding to the Fréchet distribution F réch θ, α, β with the density function 3.5 hx; θ, α, β = αβx θ α exp{ βx θ α } for x θ, = for x < θ. Suppose that X,..., X n are independent random variables, we are to test the null hypothesis H against the alternative A : H : X i F réch θ, α, β, i =,..., n, 3.65 A : there exists k {,..., n n } such that X i F réch θ, α, β, i =,..., k, X i F réch θ, α, β, i = k +,..., n, where the parameters θ, α, β before the change point are known while θ, α, β θ, α, β are unknown or to test the null hypothesis H against the alternative A : A : there exists k { n,..., n n } such that X i F réch θ, α, β, i =,..., k, 3.66 X i F réch θ, α, β, i = k +,..., n, where neither the parameters before nor after the change point are known and θ, α, β θ, α, β. The constant n may be any fixed integer larger than three, α > is an unknown shape parameter, β > is an unknown scale parameter and θ R is an unknown location parameter.

45 3. The change-point detection for the GEV distributions 4 Our goal is to find the limit distribution of max k n logλ k for the problem 3.66, resp. max k n logλ k for the problem At first we show an important characteristic that the right tail of the density function hx; θ, α, β defined in 3.5 decreases faster than any power of x θ. Lemma For every m R Proof. It is an easy consequence of a limit lim x x θm hx; θ, α, β for x θ lim y y p = for every p R. ey The log likelihood of 3.5 is given by k L k θ, α, β = k log α + k log β + α log X i θ i= k β X i θ α i= First and second derivatives of L k θ, α, β are: L k θ = i= L k α = k L k β = L k θ = k [ ] α + X i θ αβx i θ α, [ logx i θ + α βx i θ α logx i θ i= i= i= L k k θ α = k [ ] β X i θ α, k [ ] α + X i θ αα + βx i θ α, [ X i θ βx i θ α αβx i θ α logx i θ i= L k k θ β = L k α = i= k i= [ αxi θ α ], [ ] α βx i θ α log X i θ, ], ],

46 3. The change-point detection for the GEV distributions 43 L k k α β = [ Xi θ α logx i θ ], L k β = i= k i= [ β ] It holds E log hxi ; ϕ =, 3.69 θ E log hxi ; ϕ =, α E log hxi ; ϕ =. β According to Lemma 3.3., for every s R it holds E log hxi ; ϕ θ s <, E log hxi ; ϕ s <, θ α E log hxi ; ϕ s <, θ β E log hxi ; ϕ α s <, E log hxi ; ϕ s <, α β E log hxi ; ϕ β s <. Let s denote a Fisher information matrix M on a parameter ϕ = θ, α, β with elements m θθ m θα m θβ M = m αθ m αα m αβ, m βθ m βα m ββ where m θθ = E{ θ loghx i; ϕ θ loghx i; ϕ } = E{ θ loghx i; ϕ }, m αα = E{ α loghx i; ϕ α loghx i; ϕ } = E{ α loghx i; ϕ },