Statistical Modeling of Non-Metallic Inclusions in Steels and Extreme Value Analysis

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1 Statistical Modelig of No-Metallic Iclusios i Steels ad Extreme Value Aalysis Vo der Fakultät für Mathematik, Iformatik ud Naturwisseschafte der RWTH Aache Uiversity zur Erlagug des akademische Grades eier Doktori der Naturwisseschafte geehmigte Dissertatio vorgelegt vo Diplom-Mathematikeri Aja Bettia Schmiedt aus Oberhause Berichter: Uiversitätsprofessor Dr. Udo Kamps Uiversitätsprofessor Dr. Erhard Cramer Tag der müdliche Prüfug: 9. November 202 Diese Dissertatio ist auf de Iteretseite der Hochschulbibliothek olie verfügbar.

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3 Cotets. Itroductio 7.. Statistics of Extremes No-Metallic Iclusios i Steels Models of Ordered Radom Variables Materials Outlie Extreme Value Theory Extreme Value Theory for Ordiary Order Statistics Extremal Types Theorem Domais of Attractio Extreme Value Theory for Geeralized Order Statistics Preiaries Positive ad Fiite Asymptotic Variace Fiite Asymptotic Expectatio Ifiite Asymptotic Expectatio Ifiite Asymptotic Variace Zero Asymptotic Variace Multivariate Extreme Value Theory for Ordiary Order Statistics The Multivariate Extreme Value Method Simulatio Study Simulated Data Simulatio Results Results for Pareto ad Beta Distributios Results for Log-Normal ad Expoetial Distributios Coclusio Real Data Aalysis Geeralized Model of Ordered Iclusio Sizes Model

4 4 Cotets 4.2. Preiaries Estimatio of Model Parameters Maximum Likelihood Estimatio Maximum Likelihood Estimatio uder Reverse Order Restrictio Real Data Aalysis Multivariate Tests o Model Parameters Model Tests with a Simple Null Hypothesis Model Tests with a Composite Null Hypothesis Real Data Aalysis Log-Liear Lik Fuctio Maximum Likelihood Estimatio of Lik Fuctio Parameters Real Data Aalysis Plug-I Estimatio of Lik Fuctio Parameters Real Data Aalysis Least-Squares Estimatio of Lik Fuctio Parameters Real Data Aalysis Maximum Likelihood Estimatio i Specific Distributios Kow Locatio ad Ukow Scale Parameter Ukow Locatio ad Scale Parameter Extreme Value Aalysis Alterative Model Defiitio Idetificatio of Limit Laws Predictio of Iclusio Sizes Real Data Aalysis Extreme Value Theory for Geeralized Order Statistics Preiaries Positive ad Fiite Asymptotic Variace ad Ifiite Asymptotic Expectatio Ifiite Asymptotic Variace Domai of Geeralized Order Statistics Attractio of Φ,ρ Domai of Geeralized Order Statistics Attractio of Φ 2,ρ Domai of Geeralized Order Statistics Attractio of Φ 3, Vo Mises Coditios Tail Equivalece Compariso of the Domais of oos ad gos Attractio Coclusio Coclusio 73

5 Cotets 5 A. Regular Variatio 75 A.. Regularly Varyig Fuctios A.2. Extesios of Regular Variatio B. Additioal Tables ad Figures 79 C. Abbreviatios ad Notatios 85 C.. Abbreviatios C.2. Notatios C.2.. Basic Notatios C.2.2. Probability C.2.3. Extreme Value Theory Bibliography 89

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7 . Itroductio.. Statistics of Extremes I may practical applicatios extreme values maxima or miima are of particular sigificace. For istace, i ocea egieerig data of highest waves are relevat quatities for the desig of dikes or offshore platforms. I meteorology extreme weather coditios such as very high or low temperatures ad extreme amouts of rai are of iterest due to their ifluece o various aspects of huma life such as agriculture ad the lifetime of some materials. I traffic egieerig a soud estimate of the maximum umber of vehicles passig through certai crossroads at a peak hour provides for a adequate plaig of the traffic flow. I structural egieerig estimates of the maximum earthquake itesity or the maximum wid speed that ca occur at the locatio of a scheduled buildig ifluece its desig characteristics as well as its fial costs. I isurace compaies attetio is focused o the occurrece of largest claims as those could put the solvecy of a portfolio or eve the isurace compay at risk. I fiace the risk maagemet of a bak focuses o securig agaist extreme losses based o fallig prices of certai assets which are issued or held by the bak. Certaily, the precedig listig is by o meas complete, i.e., there are may areas where extreme values play a decisive role, such as corrosio aalysis, hydraulic egieerig, hydrology, logevity of huma life, material stregth, pollutio studies, reliability aalysis, sports, telecommuicatios, ad so o see, e.g., Coles 200; Beirlat et al. 2004; Fikestädt ad Rootzé 2004; Castillo et al. 2005; Reiss ad Thomas We address i detail a further issue from the area of ferrous metallurgy i Sectio.2 that tackles the uavoidable problem of the occurrece of o-metallic iclusios i the course of steel-makig processes. A estimate of the size of the maximum iclusio that ca be foud withi a certai steel compoet is of particular iterest as maximum iclusio sizes are essetial idicators of materials quality. All aforemetioed examples have i commo the eed of extrapolatio. Give a data set, it is required to estimate evets that are more extreme tha those that have already bee observed. SupposethatthegivedataaredescribedasrealizatiosofasampleofradomvariablesX,...,X, N, which are defied o a commo probability space Ω,A,P, ad which are idepedet ad idetically distributed iid with distributio fuctio F. The, the ordered data are deoted by X : X 2: X :, ad X r: is called the r th ordiary order statistic oos. Classical 7

8 8.. Statistics of Extremes extreme value theory deals with the distributio ad its properties of the maximum X : = maxx,...,x, as becomes large, i.e., as. Oe could as well study the miimum rather tha the maximum. As all results obtaied for maxima lead to correspodig results for miima through the relatio X : = mix,...,x = max X,..., X, we itet o maxima i the followig. I theory there is o difficulty i derivig the distributio of X :, deoted subsequetly by F X:, for ay value of. Due to the iid assumptio, it is F X: x = P X : x = P X x,x 2 x,...,x x = F x, x R, N. Nevertheless, the uderlyig distributio fuctio F is ormally ukow i practice. It is therefore reasoable to ivestigate the asymptotic distributioal behavior of X : for, with the objective of approximatig the distributio of X : by a o-degeerate it distributio. Deotig by the right edpoit of F, it is ωf = sup{x R Fx < }, ]. FX: x = P X : x = F x = 0, x < ωf,, x ωf. Hece, the distributio of X : degeerates to a poit mass o ωf as. I order to avoid such a degeerate it distributio, X : has to be appropriately ormalized. This issue is well-kow i the cotextof the cetral itproblem. The latter cosidersthe sum S = X i ad attempts to fid sequeces a N R >0 ad b N R such that a S b coverges i distributio to a odegeerate it. Supposig that both the expected value EX = μ ad the variace VarX = σ 2 exist, the classical cetral it theorem states that the distributio of σ 2 S μ coverges to a stadard ormal oe, i.e., P S μ x = Φx, x R, σ where Φ is the stadard ormal distributio fuctio. To apply this asymptotic approach, it is ot ecessary to be aware of the uderlyig distributio fuctio F i detail. I extreme value theory a similar situatio holds. Cosiderig sample maxima rather tha the average, correspodigly, a two-layered problem arises. O the oe had, oe has to characterize the distributio fuctios F i terms of ecessary ad sufficiet coditios for which there exist sequeces a N R >0 ad b N R ad a odegeerate distributio fuctio G such that F a x+b = P X: b a i= x = Gx, x C G,.2

9 Chapter. Itroductio 9 where C G is the set of all cotiuity poits G. The domai of ordiary order statistics attractio of a o-degeerate it distributio fuctio G will be defied as the set of distributio fuctios F fulfillig.2; see Sectio 2.. O the other had, oe has to idetify all possible odegeerate distributio fuctios G that ca appear as a it i.2. Compared with the cetral it problem, this extreme it problem is ot solved by the stadard ormal distributio, but ay o-degeerate it distributio fuctio i.2 belogs to oe of three possible extreme value distributio families; amely the Gumbel, Fréchet or reversed Weibull family. This fudametal result, i Sectio 2. referred to as the extremal types theorem, was foud first by Fisher ad Tippett 928 ad was completely proved by Gedeko 943. The Gumbel, Fréchet ad reversed Weibull families ca be combied to form the so-called geeralized extreme value GEV family of distributios, havig distributio fuctios G k x = exp k x μ /k, k x μ > 0..3 σ σ Apart from the locatio parameter μ R ad the scale parameter σ R >0, the parametrizatio i.3, which is due to vo Mises 936 ad Jekiso 955, leads to a oe-parameter family with shape parameter k R. The latter is also called the extreme value idex EVI ad determies the behavior of the right tail of the GEV distributio, as illustrated i Figure.. The sub-family for k > 0 correspods to the reversed Weibull case with short-tailed distributio fuctios havig fiite right edpoits. The cases k < 0 ad k = 0 iterpreted as k 0 idicate Fréchet ad Gumbel family with polyomially ad expoetially decreasig desities, respectively, where the right edpoit equals ifiity. I practice values of k i the rage 0.5 < k < 0.5 occur most frequetly cf. Hoskig 985. Sectio 2. of this work gives a short itroductio to extreme value theory for ordiary order statistics. O the oe had, the aforemetioed extremal types theorem is preseted. O the other had, domais of attractio criteria are provided by meas of ecessary ad/or sufficiet coditios for a uderlyig distributio fuctio F to belog to the domai of attractio of a Gumbel, Fréchet or reversed Weibull distributio..2. No-Metallic Iclusios i Steels The metallurgical idustry is faced with the problem of o-metallic iclusios which are kow to affect the reliability ad performace of steel compoets. Of all iclusios, the hard oxides, which are assumed to be early spherical, are most harmful ad most detrimetal to fatigue properties of steels see, e.g., Atkiso ad Shi The crucial geometrical parameter is supposed to be give by the iclusios sizes cf. Murakami 994; Atkiso ad Shi 2003, leadig to particular iterest i the respective distributio from a statistical poit of view.

10 0.2. No-Metallic Iclusios i Steels Gx reversed Weibull k > 0 Gumbel k = 0 Fréchet k < 0 gx reversed Weibull k > 0 Gumbel k = 0 Fréchet k < a x b x Figure.. a GEV distributio ad b GEV desity fuctios with μ = 0, σ = ad k = 0.4,...,0.4 Improvemets i steel-makig techologies have made it possible to progressively reduce the amout ad size of o-metallic iclusios. The resultig steels cotai a few large iclusios ad clouds of small oes cf. Atkiso ad Shi Nevertheless, o-metallic iclusios are still a mai reaso for material defects. Sice fatigue cracks that lead to failure are most likely to be iitiated at the largest iclusio, its size is a essetial idicator of steel quality. Predictig maximum iclusio sizes is thus a key issue of quality egieerig. I order to collect data for a extreme value aalysis, i metallography the method of polished sectios is usually adopted cf. Murakami 994; Murakami ad Beretta 999. That is, o a polished plae surface several areas termed as cotrol or ispectio areas, each of same size, are successively scaed by optical microscopy to detect those iclusios that itersect the surface. Fially, the sizes of their two-dimesioal cross-sectios are measured ad stored, either i terms of the square root of the projected area the so-called area parameter or i terms of a certai diameter; cf. the techical recommedatio ESIS P-02 by Aderso et al The method of polished sectios provides two-dimesioal data. Based o this it is worthwhile to predict the distributio of the largest three-dimesioal iclusio size i a volume. Assumig that iclusios are spherical i ature, some authors atted this classical stereological issue; predictio methods are proposed uder certai model assumptios, such as specific distributios for the size of spheres. For relevat refereces, the reader is referred to Takahashi ad Sibuya 996, 998, 200, 2002, Aderso ad Coles 2002, Aderso et al ad Kaufma et al I avoidace of such model assumptios, other authors cf. Aderso et al. 2002; ASTM Iteratioal 2003; Aderso

11 Chapter. Itroductio et al. 2003; Beretta et al cocetrate o two-dimesioal iclusio sizes, takig the stadpoit that the estimatio of large two-dimesioal iclusio sizes i a certai plae area already gives a useful ad practical idicator for the ratig of steels. As do these authors, the discussio throughout this doctoral thesis is based o two-dimesioal represetatios of iclusios; we do ot treat stereological issues relatig to three-dimesioal sizes. I metallography the applicatio of extreme value theory was pioeered by Y. Murakami. I a wide ragig series of papers, Murakami ad his co-workers established a method, frequetly termed the cotrol area maxima approach, to estimate the size of the largest two-dimesioal iclusio that ca be foud i a certai referece area that is larger tha the ispectio areas used for measurig; see Murakami 994, Murakami et al. 994 ad the refereces therei. I more detail, give the maximum iclusio size of each cotrol area, Murakami s work is based o the Gumbel family of distributios, havig distributio fuctios exp exp x μ, x R,.4 σ with locatio parameter μ R ad scale parameter σ R >0. The fittig of a Gumbel distributio i terms of the maximum likelihood ML priciple to the maximum iclusio sizes of, say, N, N N, cotrol areas has foud etrace i techical recommedatios; see ESIS P-02 by Aderso et al ad E by ASTM Iteratioal Havig oce umerically determied ML estimates ˆμ ad ˆσ of the ukow distributio parameters μ R ad σ R >0, the p quatile estimate ˆx p = ˆμ ˆσl lp is recommeded to be calculated, with p 0,, such as p = E by ASTM Iteratioal As a issue of predictio or rather extrapolatio, such a p quatile is called the characteristic size of the largest iclusio with respect to the retur period T = / p see, e.g., Aderso et al. 2002, beig the size that is expected to be exceeded exactly oce i a referece area that is T times larger tha the cotrol areas, or, i other words, beig the size that is expected to be exceeded by exactly oe maximum iclusio i T cotrol areas. More precisely, modelig the maximum iclusio sizes i T ispectio areas by iid radom variables Z,...,Z T defied o a commo probability space Ω, A, P, ad assumig that these radom variables possess a Gumbel distributio.4 with parameters μ R, σ R >0 ad quatiles x p = μ σl lp, p 0,, it holds T T E {Z i > x p } = P Z i > x p = T p =, i= i= where {Z i > x p } deotes the idicator fuctio of the set {Z i > x p } = {ω Ω Z i ω > x p }. Cocerig the metallographical practice, those quatile estimates are used to predict lower bouds of fatigue it of high stregth steels. The fatigue it is determied as the highest stress amplitude for which the material i questio has a ifiite life cf. Svesso ad de Maré 999. The

12 2.2. No-Metallic Iclusios i Steels correspodig predictive equatio was developed by Murakami ad his co-workers o the basis of the aalysis of may fatigue tests; see Murakami 994, Murakami ad Beretta 999 ad the icluded refereces. As there exist three families of extreme value distributios, amely the Gumbel, Fréchet ad reversed Weibull family, the questio arises whether it is always reasoable to fit a Gumbel distributio to observed cotrol area maxima see, e.g., Svesso ad de Maré 999. The use of a Gumbel distributio has bee frequetly reasoed by the argumet that some measuremets have show that the distributio of iclusio sizes i steels are early described by a log-ormal, expoetial or Weibull distributio cf. Murakami ad Beretta 999; Aderso et al. 2002; Atkiso ad Shi 2003, which belog to the Gumbel domai of attractio. Rather tha adoptig oe predetermied extreme value family, it is owadays more commo to practice o the uificatio of the Gumbel, Fréchet ad reversed Weibull family to the GEV family of distributios, with distributio fuctios give i.3; cf. Coles 200 ad Gomes et al Fittig a GEV distributio to observed maxima, the data itself iteds for the most appropriate extreme value family through iferece o the GEV shape parameter k. I metallographical practice Ekegre ad Bergström 202 recetly made use of this uifyig approach by fittig a GEV distributio to data o failure iducig iclusio sizes from fatigue testig; examples for all three extreme value distributio families were foud ideed i differet steel grades. It is recommeded by the ASTM Iteratioal 2003 stadard practice for extreme value aalysis of o-metallic iclusio sizes to cosider N = 24 cotrol areas, leadig to the ited amout of 24 observed maxima. With the objective of a soud specificatio of the extreme value behavior, it is worthwhile to icorporate more data ito the statistical aalysis tha just cotrol area maxima, if available. I the field of extreme value theory based o iid radom variables, two approaches that model data other tha oly observed maxima are kow, depedig o differet ways to defie extreme observatios, amely the multivariate extreme value MEV ad the peaks over threshold POT approach cf. Gomes et al Referrig to the POT method, a observatio is cosidered to be extreme if it exceeds a certai high threshold u. Provided that this threshold u is take sufficietly high, the appropriate statistical model to approximate the distributio of threshold excesses is give by the family of geeralized Pareto GP distributios, havig distributio fuctios kx u/α /k, kx u/α > 0, x > u, with parameters k R ad α R >0. The case k = 0 is agai iterpreted as the it k 0, leadig to a expoetial distributio. As verified by Pickads 975, a GP distributio appears as a itig distributio for threshold excesses as the threshold icreases if ad oly if iff the uderlyig distributio is i the domai of attractio of oe of the extreme value distributios; worth metioig, the shape parameter k of the GP distributio is equal to that of the associated GEV distributio. A survey of the POT methodology was cotributed by Daviso ad Smith 990. I metallography the applicatio of the POT approach to o-metallic iclusio sizes was developed by Shi et al. 999; a estimatio procedure o the characteristic size of maximum iclusios i.e., the iclusio sizes that

13 Chapter. Itroductio 3 are expected to be exceeded exactly oce i certai proportios of steels was proposed that is based o those two-dimesioal iclusio sizes that exceed a predetermied threshold; see also Aderso et al. 2000, Shi et al. 200 ad Aderso et al. 2003, Naturally, detectig ad measurig all iclusios with sizes larger tha a certai threshold correspod to some measuremet efforts. Applyig the MEV method, the r largest observatios withi each sub-sample are icorporated i a extreme value aalysis, with fixed r N. Theoretically based o the joit itig distributioal behavior of the r largest order statistics, this approach allows for estimatig the GEV distributio parameters o the basis of the r > largest observatios of each sub-sample, istead of just permittig observed maxima. Therefore, more data are used, alog with moderate measuremet efforts. A comprehesive literature search reveals that the MEV method, which is also called the r largest order statistics model, has bee well-recogized i some fields of applicatio such as hydrology, catology ad sports cf. Smith 986; Taw 988; Robiso ad Taw 995; Coles 200; Guedes Soares ad Scotto 2004; A ad Padey 2007; Soukissia ad Kalatzi 2007, whereas we address its use i metallography cf. Schmiedt et al I Chapter 3 of this doctoral thesis a itroductio to the MEV method is provided, ad results from a extesive simulatio study are preseted that gives rise to recommed multivariate extreme value aalysis with the umber r of upper order statistics to be icluded i the estimatio procedure beig sufficietly large i order to avoid miss-specificatios of the extreme value family, which frequetly appear, otherwise. The role of r is illustrated by meas of a real data set of o-metallic iclusio sizes, provided by the Departmet of Ferrous Metallurgy of RWTH Aache Uiversity ad described i Sectio.4. The statistics of extremes applied to the sizes of o-metallic iclusios metioed so far is based o the fudametal assumptio that the icreasig iclusio sizes withi each ispectio area are realizatios of ordiary OS based o iid radom variables. However, withi each cotrol area, usually just a few large ad may smaller iclusios are detected. Hece, it might be possible that the iid assumptio is ot reasoable. Istead, i Chapter 4 of this doctoral thesis we discuss a more flexible model of ordered radom variables that we call geeralized model of ordered iclusio sizes. I the distributio theoretical sese, this model coicides with the model of geeralized order statistics gos, which was itroduced by Kamps 995a, 995b..3. Models of Ordered Radom Variables The model of geeralized order statistics was established by Kamps 995a, 995b as a uifyig approach to various models or appropriately restricted versios of ordered radom variables. Havig itroduced uiform gos see Defiitio. first, quatile trasformatio is used to defie gos with a arbitrary baselie distributio fuctio F see Defiitio.3. The, assumig a absolutely cotiuous distributio fuctio F, the joit desity fuctio of gos see Remark.4 provides a parametric model that embeds several models of ordered radom variables by choosig the

14 4.3. Models of Ordered Radom Variables respective model parameters i a appropriate way Kamps 995a, 995b. Defiitio.. Let N, k > 0, m,...,m R be parameters such that γ j = k+ j+ i=j m i > 0 for all j {,..., }. Let m = m,...,m if 2 ad m R arbitrary if =. If the radom variables U j,, m,k, j, have a joit desity fuctio of the form f U,, m,k,...,u,, m,k u,...,u = k j= γ j u j m j u k.5 o the coe 0 u u < of R, the they are called uiform geeralized order statistics. Geeralized OS based o some arbitrary distributio fuctio F are itroduced i terms of quatile trasformatio, where the pseudo-iverse or quatile fuctio of F is used, which we defie first. Defiitio.2. Give a distributio fuctio F : R [0,], the fuctio F : 0, R with j= F y = if{x R Fx y}, y 0,, is called pseudo-iverse or quatile fuctio of F. Moreover, let F = y F y = ωf ad F 0 = y 0 F y. Defiitio.3. Let the situatio of Defiitio. be give, ad let F be a distributio fuctio. The radom variables Xj,, m,k = F U j,, m,k, j, are called geeralized order statistics based o the distributio fuctio F. By meas of desity trasformatio, the joit desity fuctio of gos based o a absolutely cotiuous distributio fuctio F is the give as follows. Remark.4. Let the assumptios of Defiitio. be give, ad let F be a absolutely cotiuous distributio fuctio with desity fuctio f. The joit desity fuctio of the geeralized order statistics X,, m,k,...,x,, m,k based o the absolutely cotiuous distributio fuctio F has the form f X,, m,k,...,x,, m,k x,...,x = k j= o the coe F 0 < x < < x < F. γ j Fx j m j fx j Fx k fx j=.6

15 Chapter. Itroductio 5 Several structures of ordered radom variables or appropriately restricted versios with differet iterpretatios are icluded i the model of gos i the distributio theoretical sese. That is, by choosig the parameters i.6 appropriately, the correspodig joit desity fuctios are obtaied Kamps 995a, 995b. For istace, the choice m = = m = 0 ad k = i.6 leads to γ j = j +, j, ad provides the joit desity fuctio! j= fx j of ordiary order statistics from iid radom variables with commo distributio fuctio F. Note that it is assumed throughout i this thesis that oos are based o iid radom variables. For both a textbook ad a guide to research literature o oos the reader is referred to the moograph by David ad Nagaraja I reliability theory oos are of particular iterest i modelig ordiary k-out-of- systems, where N ad k. The latter cosist of compoets of the same kid with iid life-legths. All compoets start workig simultaeously ad the system fails if k + or more compoets fail. Hece, k compoets are ecessary for the system to work, ad the k + th oos X k+: i a sample of size represets the life-legth of a k-out-of- systems. If, however, the failure of a compoet may ifluece the life-legth distributio of the remaiig oes, the system is called sequetial k-out-of- system, which is appropriately modeled by sequetial order statistics sos. The latter were itroduced by Kamps 995a, 995b i terms of a triagular scheme of idepedet radom variables. The geeral structure is described as follows. Assume that there are compoets with life-legth distributio F. Observig the first failure at time x, the remaiig compoets are supposed to have a life-legth distributio F 2 that is trucated o the left at x to esure realizatios arraged i ascedig order of magitude, ad so o. If oe restricts oeself to a particular choice of the distributio fuctios F,...,F, amely F j = F α j, j,.7 where F is a absolutely cotiuous baselie distributio fuctio with associated desity fuctio f ad positive model parameters α,...,α R >0, sos are also termed sos with coditioal proportioal hazard rates cf. Bedbur 20; Bedbur et al. 202a. I that case, for ay j, the failure rate of F j is give by α j f/ F ad therefore proportioal to the failure rate of the baselie distributio fuctio F. O the oe had, sos with coditioal proportioal hazard rates ca be regarded as gos i the distributio theoretical sese sice the joit desity fuctio of sos based o F,...,F defied i.7 results from the model of gos by settigi.6 m j = j +α j jα j+, j, ad k = α, leadig to γ j = j +α j, j. O the other had, gos with a absolutely cotiuous baselie distributio fuctio F ca be iterpreted as sos based o F j = F γj/ j+, j, ad F = F k. Thus, the model of gos ad sos with coditioal proportioal hazard rates coicide i the distributio theoretical sese. Whe cosiderig models of ordered radom variables, oe is further led to models of record values, which are subject matter of the moograph by Arold, Balakrisha ad Nagaraja 998.

16 6.3. Models of Ordered Radom Variables By choosig i.6 m = = m = ad k N, leadig to γ j = k, j, oe obtais the joit desity fuctio of the first k th record values based o a sequece X j j N of iid radom variables with absolutely cotiuous distributio fuctio F. The particular case k = correspods to the model of ordiary record values that, motivated by extreme weather coditios, was itroduced by Chadler 952 as a model for successive extremes i a iid sequece of radom variables. Record values are defied by meas of record times at which successively largest values appear, i.e., the radom variables L = ad Lj + = mi { } i > Lj : X i > X Lj, j N, are called record times, ad the ascedigly ordered radom variables X Lj, j N, are called record values. I situatios i which the record values are viewed as outliers, ot the record values themselves but the successively k th largest values are of iterest, which are appropriately described by the precedigly specified model of k th record values. Whereas ordiary record values or k th record values are based o a sequece of iid radom variables, the assumptio of idetical distributios is weakeed i Pfeifer s record model Pfeifer 979, 982. It is based o a double sequece of idepedet but o-idetically distributed radom variables with distributio fuctio F,...,F such that the distributio of the uderlyig radom variables is allowed to chage after each record evet. Deotig by F j, j, a uderlyig absolutely cotiuous distributio fuctio util the j th record occurs, as i the model of sos, oe may restrict oeself to a particular choice of F,...,F, amely F j = F β j, j,.8 with some absolutely cotiuous baselie distributio fuctio F possessig desity fuctio f ad positive real umbers β,...,β R >0. The, by settig i.6 m j = β j β j+, j, ad k = β, i.e., γ j = β j, j, oe gets the joit desity fuctio of the first Pfeifer s record values from o-idetically distributed radom variables based o.8. O the other had, gos with a absolutely cotiuous baselie distributio fuctio F ca always be iterpreted as Pfeifer s record values based o F j = F γ j, j, ad F = F k. Hece, the model of gos, the restricted versio of Pfeifer s record model ad the model of sos with coditioal proportioal hazard rates coicide i the distributio theoretical sese. Note that also i their geeral forms, i the distributio theoretical sese, Pfeifer s record model ad the model of sos coicide. I case of other choices of the parameters i.6, further models of ordered radom variables are icluded i the model of gos i the distributio theoretical sese, such as order statistics with o-itegral sample size ad progressive type-ii cesored order statistics Kamps 995a, 995b. With a view to modelig the sizes of o-metallic iclusios, i Chapter 4 of the preset work we discuss a model of ordered radom variables that we call geeralized model of ordered iclusio sizes. Therei, we permit that the ascedigly ordered iclusio sizes withi each cotrol area possibly

17 Chapter. Itroductio 7 possess differet hazard rates, whereas the uderlyig hazard rate is adjusted accordig to the umber of smaller iclusios. I the distributioal theoretical sese, this model correspods to the model of gos, ad, thus, to both the model of sos with coditioal proportioal hazard rates ad the model of Pfeifer s record values from o-idetically distributed radom variables based o.8, eve though its iterpretatio differs. May authors have ivestigated distributio theory, properties ad statistical applicatios of gos ad sos with coditioal proportioal hazard rates. Geeral accouts of theoretical developmets were give by Kamps 995a, 995b, Kamps ad Cramer 200, Cramer ad Kamps For certai structural results, see, e.g., Cramer 2006, Bieiek 2008 ad Burkschat 2009b. Parametrical iferece with ukow model parameters was addressed by Cramer ad Kamps 996, 998a, whereas Cramer ad Kamps 998a, 998b, 200a dealt with estimatio issues by supposig that the model parameters are kow; a survey article alog with further refereces was provided by Cramer ad Kamps 200b; more recet results ca be foud i Burkschat 2009a ad Schek et al. 20. Balakrisha et al. 2008, Beuter ad Kamps 2009, Burkschat et al. 200 discussed order restricted parametrical iferece. The estimatio of model parameters uder proportioal ad liear lik fuctios was addressed by Balakrisha et al. 20a. Recetly, Bedbur et al. 202a poited out that gos, or equivaletly sos with coditioal proportioal hazard rates, form a expoetial family i the model parameters. This structural fidig has ideed simplified may problems related to parametrical iferece; for more details, the reader is referred to Bedbur 200, 20 ad Bedbur et al. 202a. Although statistical iferece for gos has maily bee cocered with parametric models, o-parametric statistical methods were developed by Beuter 2008, 200. Extreme value theory for models of gos was itroduced by Nasri-Roudsari 996a, 996b ad Cramer Whereas by Nasri-Roudsari 996a, 996b m gos were addressed, Cramer 2003 developed extreme value theory for gos i geeral. As the model of m gos is cotaied i the model ofgosbythechoicem = = m = midefiitio., withm R, leadigtomodelparameters γ j = k+ jm+, j, the results of Cramer 2003 cotai the respective results of Nasri-Roudsari 996a, 996b as a special case. Cramer 2003 showed that with respect to the its t = γj ad t 2 = γj 2 various set-ups have to be distiguished; by aalyzig these j= j= set-ups, possible it distributios for extreme gos were established. I Sectio 2.2 of this thesis the mai results of Cramer 2003 are summarized. Further refereces o recet developmets i extreme value theory for models of gos are give i Chapter 5, ad, based o the fidigs of Cramer 2003, ow cotributios to extreme value theory for gos are preseted that deal with domai of attractio criteria.

18 8.4. Materials.4. Materials Subsequetly, the metallurgical material is described that serves as a basis for statistical aalysis of o-metallic iclusio sizes throughout this doctoral thesis. Note that i Schmiedt et al. 202 it is made use of the same material. The Departmet of Ferrous Metallurgy of RWTH Aache Uiversity has made data of egieerig steel available, which is used, e.g., i forgigs for the automobile idustry. The experimetal material was a sigle rolled roud bar of 6mm diameter. Followig maily the stadards DIN EN ad E by ASTM Iteratioal 2003, a umber of sample areas of 20mm legth by 0mm width were cut from the half radius positio see Figure.2. These plae surfaces of size 200mm 2 were polished a few times. O the respectivepolished plaes, i total 60 cotrol areas, each of Figure.2 Sample preparatio from the steel bar accordig to DIN EN , p. 30 Graphic provided by the Departmet of Ferrous Metallurgy of RWTH Aache Uiversity size mm 2, were determied. Afterwards, each cotrol area was screeed by scaig electro microscopy i order to detect o-metallic iclusios. The sizes of all iclusios with a maximum diameter larger tha 5μm have bee recorded, the threshold of 5μm beig a lower it of detectio due to techical itatios with equipmet. For the experimetal aalysis two major iclusio types have bee differetiated, amely oxides ad sulfides. Scaig electro microscope pictures of these two particle types are show i Figure.3. Typically, the two-dimesioal cross-sectios of oxides are early circular, whereas those of sulfides are rather elogated ad almost ellipsoidal. Subsequetly, we focus o oxides as they are supposed to be more critical with respect to fatigue properties of steel compoets. All iclusios with ratio of maximum ad miimum diameter smaller tha 3 have bee cosidered as oxides. The latter

19 Chapter. Itroductio a 9 b Figure.3. Scaig electro microscope images of o-metallic iclusios: a oxide ad b sulfide Images provided by the Departmet of Ferrous Metallurgy of RWTH Aache Uiversity assumptio is feasible as the egieerig steel had bee highly deformed so that the soft sulfides had bee stretched to a larger extet tha the hard oxides. Based o the described procedure, the umber of recorded oxide iclusios per cotrol area differs from 9 to 8, where the mea umber is about 48, so that the cotrol areas are apparetly differetly structured. We quatify the oxides sizes i terms of the area parameter, i.e., the square root of a iclusio s projected area. The, the largest iclusio measures 𝜇m, ad the 0.9 quatile of all recorded area parameters is give by 𝜇m. I each cotrol area there are may small oxide iclusios ad just a few large oes..5. Outlie The preset work is divided ito five chapters that are orgaized as follows. I Chapter 2 fudametals of extreme value theory are provided that are required for a proper uderstadig of the followig chapters. O the oe had, a brief accout of extreme value theory for ordiary order statistics is give, icludig the extremal types theorem ad domais of attractio criteria. O the other had, the mai results of Cramer 2003 i the cotext of extreme value theory for geeralized order statistics are summarized, ad, i particular, possible it distributios for Deotig by 𝑥 𝑥𝑚 the ascedigly ordered iclusio sizes of all cotrol areas, for ay 𝑝 0,, the 𝑝 quatile is defied as 𝑥𝑘 with 𝑚𝑝 < 𝑘 < 𝑚𝑝 + ad 𝑘 ℕ if 𝑚𝑝 / ℕ, ad as 2 𝑥𝑘 + 𝑥𝑘+ with 𝑘 = 𝑚𝑝 if 𝑚𝑝 ℕ.

20 20.5. Outlie extreme gos are preseted. Chapter 3 deals with multivariate extreme value theory for oos. A brief itroductio to the MEV method is followed by a discussio of the beefit of a multivariate compared to a uivariate extreme value approach. A extesive simulatio study shows that a uivariate set-up may lead to a high proportio of miss-specificatios of the true extreme value distributio, ad that the statistical aalysis is cosiderably improved whe beig based o the respective r > largest observatios of each subsample, with r appropriately chose. Moreover, the simulatio results are illustrated via real data aalysis of o-metallic iclusio sizes, the latter beig obtaied from the metallurgical material described i Sectio.4. The applicatio of the MEV method to o-metallic iclusio sizes is based o the assumptio that the icreasig iclusio sizes withi each cotrol area are realizatios of oos from iid radom variables. However, this fudametal iid assumptio is possibly ot reasoable as large iclusios occur with a sigificatly lower icidece tha smaller oes i real data sets. For this reaso, i Chapter 4 a geeralized model of ordered iclusio sizes is aalyzed which coicides with the model of gos i the distributio theoretical sese. Certai model parameters allow the ascedigly ordered iclusio sizes i each cotrol area to arise from parametrically adjusted hazard rates. Supposig that these model parameters are ukow, methods of statistical iferece are discussed ad applied to real o-metallic iclusio sizes that are obtaied from the metallurgical material described i Sectio.4. Besides, it is made use of extreme value theory for gos i order to predict large iclusio sizes by meas of extrapolatio. Based o the fidigs of Cramer 2003, i Chapter 5 research o extreme value theory for gos is carried o. Beig maily cocered with oe particular set-up with respect to series of powers of the uderlyig model parameters, amely t = γj = t 2 = =, the domais j= j= of attractio of the respective o-degeerate it distributios for extreme gos are extesively aalyzed i terms of ecessary ad/or sufficiet coditios o the baselie distributio fuctio F. Fially, i Chapter 6 the impact ad the cotributios of this thesis are summarized. γ 2 j

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