On the Theory and Application of Model Misspecification Tests in Geodesy

Transcription

1 Istitut für Geodäsie ud Geoiformatio der Uiversität Bo O the Theory ad Applicatio of Model Misspecificatio Tests i Geodesy Iaugural Dissertatio zur Erlagug des akademische Grades Doktor Igeieur Dr. Ig. der Hohe Ladwirtschaftliche Fakultät der Rheiische Friedrich Wilhelms Uiversität zu Bo vorgelegt am 8. Mai 7 vo Dipl. Ig. Boris Kargoll aus Karlsruhe

2 Hauptberichterstatter: Mitberichterstatter: Prof. Dr. tech. W.-D. Schuh Prof. Dr. rer. at. H.-P. Helfrich Tag der müdliche Prüfug:. Jui 7 Gedruckt bei: Diese Dissertatio ist auf dem Hochschulschrifteserver der ULB Bo olie elektroisch publiziert. Erscheiugsjahr: 7

3 O the Theory ad Applicatio of Model Misspecificatio Tests i Geodesy Abstract May geodetic testig problems cocerig parametric hypotheses may be formulated withi the framework of testig liear costraits imposed o a liear Gauss-Markov model. Although geodetic stadard tests for such problems are computatioally coveiet ad ituitively soud, o rigorous attempt has yet bee made to derive them from a uified theoretical foudatio or to establish optimality of such procedures. Aother shortcomig of curret geodetic testig theory is that o stadard approach exists for tacklig aalytically more complex testig problems, cocerig for istace ukow parameters withi the weight matrix. To address these problems, it is prove that, uder the assumptio of ormally distributed observatio, various geodetic stadard tests, such as Baarda s or Pope s test for outliers, multivariate sigificace tests, deformatio tests, or tests cocerig the specificatio of the a priori variace factor, are uiformly most powerful UMP withi the class of ivariat tests. UMP ivariat tests are prove to be equivalet to likelihood ratio tests ad Rao s score tests. It is also show that the computatio of may geodetic stadard tests may be simplified by trasformig them ito Rao s score tests. Fially, testig problems cocerig ukow parameters withi the weight matrix such as autoregressive correlatio parameters or overlappig variace compoets are addressed. It is show that, although strictly optimal tests do ot exist i such cases, correspodig tests based o Rao s Score statistic are reasoable ad computatioally coveiet diagostic tools for decidig whether such parameters are sigificat or ot. The thesis cocludes with the derivatio of a parametric test of ormality as aother applicatio of Rao s Score test. Zur Theorie ud Awedug vo Modell-Misspezifikatiostests i der Geodäsie Zusammefassug Was das Teste vo parametrische Hypothese betrifft, so lasse sich viele geodätische Testprobleme i Form eies Gauss-Markov-Modells mit lieare Restriktioe darstelle. Obwohl geodätische Stadardtests recherisch eifach ud ituitiv verüftig sid, wurde bisher kei streger Versuch uteromme, solche Tests ausgehed vo eier eiheitliche theoretische Basis herzuleite oder die Optimalität solcher Tests zu begrüde. Ei weiteres Defizit im gegewärtige Verstädis geodätischer Testtheorie besteht dari, dass kei Stadardverfahre zum Löse vo aalytisch komplexere Testprobleme exisitiert, welche beispielsweise ubekate Parameter i der Gewichtsmatrix betreffe. Um diese Probleme gerecht zu werde wird bewiese, dass uter der Aahme ormalverteilter Beobachtuge verschiedee geodätische Stadardtests, wie z.b. Baardas oder Popes Ausreissertest, multivariate Sigifikaztests, Deformatiostests, oder Tests bzgl. der Agabe des a priori Variazfaktors, allesamt gleichmäßig beste egl.: uiformly most powerful - UMP ivariate Tests sid. Es wird ferer bewiese dass UMP ivariate Tests äquivalet zu Likelihood-Quotiete-Tests ud Raos Score-Tests sid. Ausserdem wird gezeigt, dass sich die Berechug vieler geodätischer Stadardtests vereifache lässt idem diese als Raos Score-Tests formuliert werde. Abschließed werde Testprobleme behadelt i Bezug auf ubekate Parameter ierhalb der Gewichtsmatrix, beispielsweise i Bezug auf autoregressive Korrelatiosparameter oder überlappede Variazkompoete. I solche Fälle existiere keie im strege Sie beste Tests. Es wird aber gezeigt, dass etsprechede Tests, die auf Raos Score-Statistik beruhe, sivolle ud vom Recheaufwad her güstige Diagose-Tools darstelle um festzustelle, ob Parameter wie die eigags erwähte sigifikat sid oder icht. Am Ede dieser Dissertatio steht mit der Herleitug eies parametrische Tests auf Normalverteilug eie weitere Awedug vo Raos Score-Test.

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5 Cotets Itroductio. Objective Outlie Theory of Hypothesis Testig 3. The observatio model The testig problem The test decisio The size ad power of a test Best critical regios Most powerful MP tests Reductio to sufficiet statistics Uiformly most powerful UMP tests Reductiotoivariatstatistics Uiformly most powerful ivariat UMPI tests Reductio to the Likelihood Ratio ad Rao s Score statistic Theory ad Applicatios of Misspecificatio Tests i the Normal Gauss-Markov Model Itroductio Derivatio of optimal tests cocerig parameters of the fuctioal model Reparameterizatio of the test problem Ceterig of the hypotheses Full decorrelatio/homogeizatio of the observatios Reductio to miimal sufficiet statistics with elimiatio of uisace parameters Reductio to a maximal ivariat statistic Back-substitutio Equivalet forms of the UMPI test cocerig parameters of the fuctioal model Applicatio : Testigforoutliers Baarda s test Pope s test Applicatio : Testigforextesiosofthefuctioalmodel Applicatio 3: Testigforpoitdisplacemets Derivatio of a optimal test cocerig the variace factor Applicatios of Misspecificatio Tests i Geeralized Gauss-Markov models Itroductio Applicatio 5: Testigforautoregressivecorrelatio Applicatio 6: Testigforoverlappigvariacecompoets Applicatio 7: Testigforo-ormalityoftheobservatioerrors Coclusio ad Outlook 86

6 6 Appedix: Datasets Dam Dataset Gravity Dataset Refereces 88

7 Itroductio. Objective Hypothesis testig is the foudatio of all critical model aalyses. Particularly relevat to geodesy is the practice of model misspecificatio testig which has the objective of determiig whether a give observatio model accurately describes the physical reality of the data. Examples of commo testig problems iclude how to detect outliers, how to determie whether estimated parameter values or chages thereof are sigificat, or how to verify the measuremet accuracy of a give istrumet. Geodesists kow how to hadle such problems ituitively usig stadard parameter tests, but it ofte remais uclear i what mathematical sese these tests are optimal. The first goal of this thesis is to develop a theoretical foudatio which allows establishig optimality of such tests. The approach will be based o the theory of Neyma ad Pearso 98, 933, whose celebrated fudametal lemma defies a optimal test as oe which is most powerful amog all tests with some particular sigificace level. As this cocept is applicable oly to very simple problems, tests must be cosidered that are most powerful i a wider sese. A ituitively appealig way to do so is based o the fact that complex testig problems may ofte be reduced to simple problems by exploitig symmetries. Oe mathematical descriptio of symmetry is ivariace, whose applicatio to testig problems the leads to ivariat tests. I this cotext, a uiformly most powerful ivariat test defies a test which is optimal amog all ivariat tests available i the give testig problem. I this thesis, it will be demostrated for the first time that may geodetic stadard tests fit ito this framework ad share the property of beig uiformly most powerful. I order to be useful i practical situatios, a testig procedure should ot oly be optimal, but it must also be computatioally maageable. It is well kow that hypothesis tests have differet mathematical descriptios, which may vary cosiderably i computatioal complexity. Most geodetic stadard tests are usually derived from likelihood ratio tests see, for istace, Koch, 999; Teuisse,. A alterative, oftetimes much simpler represetatio is based o Rao s 948 score test, which has ot bee ackowledged as such by geodesists although it has foud its way ito geodetic practice, for istace, via Baarda s outlier test. To shed light o this importat topic, it is aother major itet of this thesis to describe Rao s score method i a geeral ad systematic way, ad to demostrate what types of geodetic testig problems are ideally hadled by this techique.. Outlie The followig Sectio of this thesis begis with a review of classical testig theory. The focus is o parametric testig problems, that is, hypotheses to be tested are propositios cocerig parameters of the data s probability distributio. We will the follow the classical approach of cosiderig tests with fixed sigificace level ad maximum power. I this cotext, the Neyma-Pearso Lemma ad the resultig idea of a most powerful test will be explaied, ad the cocept of a uiformly most powerful test will be itroduced. The subsequet defiitio of sufficiecy will play a cetral role i reducig the complexity of testig problems. Followig this, we will examie more complex problems that require a simplificatio goig beyod sufficiecy. For this puropse, we will use the priciple of ivariace, which is the mathematical descriptio of symmetry. We will see that ivariat tests are tests with power distributed symmetrically over the space of parameters. This leads us to the otio of a uiformly most powerful ivariat UMPI test, which is a desigated optimal test amog such ivariat tests. Fially, we will explore the relatioships of UMPI tests to likelihood ratio tests ad Rao s score tests. Sectio 3 exteds the ideas developed i Sectio to address the geeral problem of testig liear hypotheses i the Gauss-Markov model with ormally distributed observatios. Here we focus o the case i which the desig matrix is of full rak ad where the weight matrix is kow. The, the testig problem will be reduced by sufficiecy ad ivariace, ad UMPI tests derived for the two cases where the variace of uit weight is either kow or ukow a priori. Emphasis will be placed o demostratig further that these UMPI tests correspod to the tests already used i geodesy. Aother key result of this sectio will be to show how all these tests are formulated as likelihood ratio ad Rao s score tests. The sectio cocludes with a discussio of various geodetic testig problems. It will be show that may stadard tests used so far, such as Baarda s ad Pope s outlier test, multivariate parameter tests, deformatio tests, or tests cocerig the variace of uit weight, are optimal UMPI i a statistical sese, but that computatioal complexity ca ofte be effectively reduced by usig equivalet Rao s score tests istead. Sectio 4 addresses a umber of testig problems i geeralized Gauss-Markov models for which o UMPI

8 INTRODUCTION tests exist, because a reductio by sufficiecy ad ivariace are ot effective. The first problem cosidered will be testig for first-order autoregressive correlatio. Rao s score test will be derived, ad its power agaist several simple alterative hypotheses will be determied by carryig out a Mote Carlo simulatio. The secod applicatio of this sectio will treat the case of testig for a sigle overlappig variace compoet,for which Rao s score test will be oce agai derived. The fial problem cosists of testig whether observatios follow a ormal distributio. It this situatio, Rao s score test will be show to lead to a test which measures the deviatio of the sample s skewess ad kurtosis from the theoretical values of a ormal distributio. Fially, Sectio 5 highlights the mai coclusios of this work ad gives a outlook o promisig extesios to the theory ad applicatios of the approach preseted i this thesis.

9 3 Theory of Hypothesis Testig. The observatio model Let us assume that some data vector y =[y,...,y ] is subject to a statistical aalysis. As this thesis is cocered rather with explorig theoretical aspects of such aalyses, it will be useful to see this data vector as oe of may potetial realizatios of a vector Y of observables Y,...,Y. This is reflected by the fact that measurig the same quatity multiple times does ot result i idetical data values, but rather i some frequecy distributio of values accordig to some radom mechaism. I geodesy, quatities that are subject to observatio or measuremet usually have a geometrical or physical meaig. I this sese, Y, or its realizatio y, will be viewed as beig icorporated i some kid of model ad thereby coected to some other quatities or parameters. Parametric observatio models may be set up for multiple reasos. They are ofte used as a way to reduce great volumes of raw data to low-dimesioal approximatig fuctios. A model might also be used simply because the quatity of primary iterest is ot directly observable, but must be derived from other data. I reality, both aspects ofte go had i had. To give these explaatios a mathematical expressio, let the radom vector Y with values i R be part of a liear model Y = Xβ + E,.- where β R m deotes a vector of ukow o-stochastic parameters ad X R m a kow matrix of ostochastic coefficiets reflectig the fuctioal relatioship. It will be assumed throughout that raka = m ad that >mso that.- costitutes a geuie adjustmet problem. E represets a real-valued radom vector of ukow disturbaces or errors, which are assumed to satisfy EE} = ad ΣE} = σ P ω..- We will occasioally refer to these two coditios as the Markov coditios. The weight matrix P ω may be a fuctio of ukow parameters ω, which allows for certai types of correlatio ad variace-chage or heteroscedasticity models regardig the errors. Wheever such parameters do ot appear, we will use P to deote the weight matrix. To make the followig testig procedures operable, these liear model specificatios must be accompaied by certai assumptios regardig the type of probability distributio cosidered for Y. For this purpose, it will be assumed that ay such distributio P may be defied by a parametric desity fuctio fy; β,σ, ω, c,.-3 which possibly depeds o additioal ukow shape parameters c cotrollig, for istace, the skewess ad kurtosis of the distributio. Now, let the vector θ := [β,σ, ω, c ] comprise the totality of ukow parameters takig values i some u-dimesioal space Θ. The parameter space Θ the correspods to a collectio of desities F = fy; θ :θ Θ},.-4 which i tur defies the cotemplated collectio of distributios W = P θ : θ Θ}..-5 Example.: A agle has bee idepedetly observed times. Each observatio Y,...,Y is assumed to follow a distributio that belogs to the class of ormal distributios W = Nµ, σ :µ R,σ R +}.-6 with mea µ ad variace σ, or i short otatio Y i Nµ, σ. The relatioship betwee Y =[Y,...,Y ] ad the mea parameter µ costitutes the simplest form of a liear model.-, where X is a -vector of oes ad β equals the sigle parameter µ. Furthermore, as the observatios are idepedet with costat mea ad variace, the joit ormal desity fuctio fy; µ, σ may be decomposed i.e. factorized itothe product fy; µ, σ = fy i ; µ, σ.-7

10 4 THEORY OF HYPOTHESIS TESTING of idetical uivariate ormal desity fuctios defied by fy i ; µ, σ = exp } yi µ y i R,µ R,σ R +,,...,..-8 πσ σ Therefore, the class of desities F cosidered for Y may be writte as } } F = πσ / exp σ Y i µ :[µ, σ ] Θ.-9 with two-dimesioal parameter space Θ = R R +.. The testig problem The goal of ay parametric statistical iferece is to extract iformatio from the give data y about the ukow true parameters θ, which refer to the ukow true probability distributio P θ ad the true desity fuctio fy; θ with respect to the observables Y. For this purpose, we will assume that θ, P θ, adfy; θ are uique ad idetifiable elemets of Θ, W, adf respectively. While estimatio aims at determiig the umerical values of θ, that is, selectig oe specific elemet from Θ, the goal of testig is somewhat simpler i that oe oly seeks to determie whether θ is a elemet of a subset Θ of Θ or ot. Despite this seemigly great differece betwee the purpose of estimatio ad testig, which is reflected by a separate treatmet of both topics i most statistical text books, certai cocepts from estimatio will tur out to be idispesable for the theory of testig. As this thesis is focussed o testig, the ecessary estimatio methodology will be itroduced without a detailed aalysis thereof. I order to formulate the test problem, a o-empty ad geuie subset Θ Θ correspodig to some W W ad F F must be specified. The, the ull hypothesis is defied as the propositio H : θ Θ..- Whe the ull hypothesis is such that Θ represets oe poit θ withi the parameter space Θ, the the elemets of θ assig uique umerical values to all the elemets i θ, ad.- simplifies to the propositio H : θ = θ..- I this case, H is called a simple ull hypothesis. O the other had, if at least oe elemet of θ is assiged a whole rage of values, say R +,theh is called a composite ull hypothesis. Isuchacase,aequality relatio as i.- ca clearly ot be established for all the parameters i θ. Ukow parameters whose true values are ot uiquely fixed uder H are also called uisace parameters. Example. Example. cotiued: O the basis of give observed umerical values y =[y,...,y ], we wat to test whether the observed agle is a exact right agle go or ot. Let us ivestigate three differet scearios:. If σ is kow apriorito take the true value σ,theθ = R is oe-dimesioal, ad uder the ull hypothesis H : µ = the subset Θ shriks to the sigle poit Θ = }. Hece, H is a simple ull hypothesis by defiitio.. If µ ad σ are both ukow, the the ull hypothesis, writte as H : µ = σ R +, leaves the uisace parameter σ uspecified. Therefore, the subset Θ =,σ :σ R +} does ot specify a sigle poit, but a iterval of values. Cosequetly, H is composite uder this sceario. 3. If the questio is whether the observed agle is a go ad the stadard deviatio is really 3 mgo e.g. as promised by the producer of the istrumet, the the ull hypothesis H : µ =, σ =.3 refers to the sigle poit Θ =,.3 withiθ. Ithatcase,H is see to be simple.

11 .3 The test decisio 5.3 The test decisio Imagie that the space S of all possible observatios y cosists of two complemetary regios: a regio of acceptace S A, which cosists of all values that support a certai ull hypothesis H, ad a regio of rejectio or critical regio S C, which comprises all the observatios that cotradict H i some sese. A test decisio could the be o based simply observig whether some give data values y are i S A which would imply acceptace of H, or whether y S C which would result i rejectio of H. It will be ecessary to perceive ay test decisio as the realizatio of a radom variable φ which, as a fuctio of Y, takes the value i case of rejectio ad i case of acceptace of H. This mappig, defied as, if y SC, φy =.3-, if y S A, is also called a test or critical fuctio, for it idicates whether a give observatio y falls ito the critical regio or ot..3- ca be viewed as the mathematical implemetatio of a biary decisio rule, whichis typical for test problems. This otio ow allows for the more formal defiitio of the regios S A ad S C as S C = φ = y S φy =},.3-3 S A = φ = y S φy =}..3-4 Example.3 Ex.. cotiued: For simplicity, let Y = be the sigle observatio of a agle, which is assumed to be ormally distributed with ukow mea µ ad kow stadard deviatio σ = σ =3mgo. To test the hypothesis that the observed agle is a right agle H : µ =, a egieer suggests the followig decisio rule: Reject H, whe the observed agle deviates from go by at least five times the stadard deviatio. The critical fuctio reads, if y or y.5 φy =.3-5, if <y<.5. The critical regio is give by S C =, ] [.5, +, ad the regio of acceptace by S A = ,.5. Due to the radom ad biary ature of a test, two differet types of error may occur. The error of the first kid or Type I error arises, whe the data y truly stems from a distributio i W specified by H, but happes to fall ito the regio of rejectio S C.Cosequetly,H is falsely rejected. The error of the secod kid or Type II error occurs, whe the data y does ot stem from a distributio i W, but is a elemet of the regio of acceptace S A. Clearly, H is the accepted by mistake. From Example.3 it is ot clear whether the suggested decisio rule is i fact reasoable. The followig subsectio will demostrate how the two above errors ca be measured ad how they ca be used to fid optimal decisio rules..4 The size ad power of a test As ay test.3- is itself a radom variable derived from the observatios Y, it is straightforward to ask for the probabilities with which these errors occur. Sice tests with small error probabilities appear to be more desirable tha tests with large errors, it is atural to use these probabilities i order to fid optimal test procedures. For this purpose, let α deote the probability of a Type I error, adβ ot to be cofused with the parameter β of the liear model.- the probability of a Type II error. Istead of β, itismore commo to use the complemetary quatity π := β, called the power of a test. Whe H is simple, i.e. whe all the ukow parameter values are specified by H, the the umerical value for α may be computed from.3- by α = P θ [φy =]=P θ Y S C = fy; θ dy..4-6 S C From.4-6 it becomes evidet why α is also called the size of the critical regio, because its value represets the area uder the desity fuctio measured over S C. Notice that for composite H,thevaluefor α will geerally deped o the values of the uisace parameters. I that case, it is appropriate to defie α as a fuctio with αθ =P θ [φy =]=P θ Y S C = fy; θ dy θ Θ..4-7 S C

12 6 THEORY OF HYPOTHESIS TESTING Example.4 Example.3 cotiued: What is the size of the critical regio or the probability of the Type I error for the test defied by.3-5? Recall that µ = is the value assiged to µ by H ad that σ =.3 is the fixed value for σ assumed as kow apriori. The, after trasformig Y ito a N, -distributed radom variable, the values of the stadard ormal distributio fuctio Φ may be obtaied from statistical tables see, for istace, Kreyszig, 998, p to aswer the above questio. α = P θ Y S C =N µ,σ Y or Y.5 = N µ,σ <Y < µ = N, < Y µ <.5 µ σ σ σ = [Φ5 Φ 5]. If σ was ukow, the the umerical value of α would deped o the value of σ. Let us fiish the discussio of the size of a test by observig i Fig.. that differet choices of the critical regio may have the same total probability mass. S A S C S C S A Nµ,σ Nµ,σ α α S C S A S C S A S C S A Nµ,σ α/ α/ Nµ,σ α Fig.. Let Nµ,σ deote the distributio of a sigle observatio Y uder a simple H with kow ad fixed variace σ. This figure presets four out of ifiitely may differet ways to specify a critical regio S C of fixed size α.

13 .4 The size ad power of a test 7 The computatio of the probability of a Type II error is more itricate tha that of α, because the premise of a false H does ot tell us aythig about which distributio we should use to measure the evet that y is i S A. For this very reaso, a alterative class of distributios W W must be specified which cotais the true distributio if H is false. If we let W be represeted by a correspodig o-empty parameter subset Θ Θ, the we may defie the alterative hypothesis as H : θ Θ Θ Θ, Θ Θ =,.4-8 which may be simple or composite i aalogy to H. The coditio Θ Θ = is ecessary to avoid ambiguities due to overlappig hypotheses. Example.5 Example. cotiued: For testig the right agle hypothesis H : µ =, we will assume that σ = σ =.3 is fixed ad kow. Let us cosider the followig three situatios.. Imagie that a map idicates that the observed agle is a right agle, while a secod older map gives a value of say.8 go. I this case, the data y couldbeusedtotesth agaist the alterative H : µ =.8. Θ =.8} represets oe poit i Θ, hece H is simple.. If the right agle hypothesis is doubtful but there is evidece that the agle ca defiitely ot be smaller tha go, the the appropriate alterative reads H : µ >, which is ow composite due to Θ = µ : µ>}, ad it is called oe-sided, because the alterative values for µ are elemets of a sigle iterval. 3. Whe o prior iformatio regardig potetial alterative agle sizes is available, the H : µ is a reasoable choice as we will see later. Sice the alterative values for µ are split up ito two itervals separated by the value uder H, we speak of a two-sided composite H. With the specificatio of a alterative subspace Θ Θ, which the ukow true parameter θ is assumed to be a elemet of if H is false, the probability of a Type II error follows to be either β = P θ [φy =]=P θ Y S A = fy; θ dy.4-9 S A if H is simple i.e. if θ is the uique elemet of Θ, or βθ =P θ [φy =]=P θ Y S A = fy; θ dy θ Θ.4- S A if Θ is composed of multiple elemets. As simple alteratives are rarely ecoutered i practical situatios, the geeral otatio of.4- will be maitaied. As already metioed, it is more commo to use the power of a test, defied as Πθ := P θ Y S A =P θ Y S C =P θ [φy =] θ Θ..4- The umerical values of Π may be iterpreted as the probabilities of avoidig a Type II error. Whe desigig a test, it will be useful to determie the probability of rejectig H as a fuctio defied over the etire parameter space Θ. Such a fuctio may be defied as Pfθ :=P θ [φy =]=P θ Y S C θ Θ.4- ad will be called the power fuctio of a test. Clearly, this fuctio will i particular produce the sizes α for all θ Θ ad the power values Π for all θ Θ. For all the other values of θ, this fuctio will provide the hypothetical power of the test if the true parameter is either assumed to be a elemet of Θ, or of Θ. Example.6 Example.5 cotiued: Recall that the size of this test tured out to be approximately as Ex..4 demostrated. Let us ow ask, what the power of the test would be for testig H : µ = agaist H : µ = µ =.8 with σ = σ =.3 kow apriori. Usig the.4-, we obtai Π= P µ,σ Y S A= N µ,σ <Y < µ = N, < Y µ.3.3 <.5 µ.3 = [Φ Φ ].843.

14 8 THEORY OF HYPOTHESIS TESTING Notice that the larger the differece betwee µ ad µ, the larger the power becomes. For istace, if H had bee specified as µ =., the the power would icrease to Π.977, ad for µ =.4 the power would already be very close to. This is ituitively uderstadable, because very similar hypotheses are expected to be harder to separate o the basis of some observed data tha extremely differet hypotheses. Figure. illustrates this poit. S A S C S A S C Nµ,σ Nµ,σ Nµ,σ Nµ,σ β α β α µ µ µ µ Fig.. The probability of a Type II error β = Π becomes smaller as the distace µ µ with idetical variace σ betwee the ull hypothesis H ad the alterative H icreases. Aother importat observatio to make i this cotext is that, ufortuately, the errors of the first ad secod kid caot be miimized idepedetly. For istace, whe the critical regio S C is exteded towards µ Fig..3 left right, the clearly its size becomes larger. I doig this, S A shriks, ad the error of the secod kid becomes smaller. This effect is explaied by the fact that both errors are measured i complemetary regios ad thereby affect each other s size. Therefore, o critical fuctio ca exist that miimizes both error probabilities simultaeously. The purpose of the followig subsectio is to preset a practical solutio to resolve this coflict. S A S C S A S C Nµ,σ Nµ,σ Nµ,σ Nµ,σ β α β α µ µ Fig..3 Let Nµ,σ adnµ,σ deote the distributios of a sigle observatio Y uder simple H ad H, respectively. Chagig the S C /S A partitioig of the observatio space abscissa ecessarily causes a icrease i probability of oe error type ad a decrease i probability of the other type.

15 .5 Best critical regios 9.5 Best critical regios As poited out i the previous sectio, shiftig the critical regio ad makig oe error type more ulikely always causes the other error to become more probable. Therefore, the probabilities of Type I ad Type II errors caot be miimized simultaeously. Oe way to resolve this coflict is to keep the probability of a Type I error fixed at a relatively small value ad to seek a critical regio that miimizes the probability of a Type II error, or equivaletly that maximizes the power of the test. To make the mathematical cocepts, ecessary for this procedure, ituitively uderstadable, examples will be give maily with respect to the class of observatio models.-6 itroduced i Example.. The remaider of this Sectio.5 is orgaized such that tests with best critical regios will be costructed for testig problems that are progressively complex withi that class of models. The determiatio of optimal critical regios i the cotext of the geeral liear model.- with geeral parametric desities as i.-3 will be subject of detailed ivestigatios i Sectios 3 ad Most powerful MP tests The simplest kid of problem for which a critical regio with optimal power may exist is that of testig a simple H : θ = θ agaist a simple alterative hypothesis H : θ = θ ivolvig a sigle ukow parameter. Usig defiitios.4-6 ad.4-, the problem is to fid a set S C such that the restrictio fy; θ dy = α.5-3 S C is satisfied, where α as a give size is also called the sigificace level, ad fy; θ dy is a maximum..5-4 S C Such a critical regio will be called the best critical regio BCR, ad a test based o the BCR will be deoted as most powerful MP for testig H agaist H at level α. A solutio to this problem may be foud o the basis of the followig lemma of Neyma ad Pearso see, for istace, Rao, 973, p Theorem. Neyma-Pearso Lemma. Suppose that fy ; θ ad fy ; θ are two desities defied o aspaces. LetS C S be ay critical regio with fy; θ dy = α,.5-5 S C where α has a give value. If there exists a costat k α such that for the regio SC S with fy; θ fy; θ >k α if y SC.5-6 fy; θ fy; θ <k α if y / SC, coditio.5-5 is satisfied, the fy; θ dy fy; θ dy..5-7 S C S C Notice if whe fy ; θ adfy ; θ are desities uder simple hypotheses H ad H, ad if the coditios.5-5 ad.5-6 hold for some k α,thesc deotes the BCR for testig H versus H at fixed level α, because.5-7 is equivalet to the desired maximum power coditio.5-4. Also observe that.5-6 the defies the MP test, which may be writte as fy; θ if φy = if fy; θ >k α fy; θ fy; θ <k α..5-8 This coditio.5-8 expresses that i order for a test to be most powerful, the critical regio S C must comprise all the observatios y, for which the so-called desity ratio fy; θ /fy; θ is larger tha some

16 THEORY OF HYPOTHESIS TESTING α-depedet umber k α. This ca be explaied by the followig ituitios of Stuart et al. 999, p. 76. Usig defiitio.4-, the power may be rewritte i terms of the desity ratio as fy; θ Π= fy; θ dy = S C fy; θ fy; θ dy. S C Sice α has a fixed value, maximizig Π is equivalet to maximizig the quatity fy; θ Π α = S C fy; θ fy; θ dy. fy; θ dy S C I order for a test to have maximum power, its critical regio S C must clearly iclude all the observatios y,. for which the itegral value i the deomiator equals α, ad. for which the desity ratio i the omiator produces the largest possible values, whose lower boud may be defied as the umber k α with the values of the additioal factor fy; θ fixed by coditio. These are the very coditios give by the Neyma-Pearso Lemma. A more formal proof may be foud, for istace, i Teuisse, p. 3f.. The followig example demostrates how the BCR may be costructed for a simple test problem by applyig the Neyma-Pearso Lemma. Example.7: Test of the ormal mea with kow variace - Simple alteratives. Let Y,...,Y be idepedetly ad ormally distributed observatios with commo ukow mea µ ad commo kow stadard deviatio σ = σ. What is the BCR for a test of the simple ull hypothesis H : µ = µ agaist the simple alterative hypothesis H : µ = µ at level α? It is assumed that µ, µ, σ ad α have fixed umerical values. I order to costruct the BCR, we will first try to fid a umber k α such that coditio.5-6 about the desity ratio fy; θ /fy; θ holds. As the observatios are idepedetly distributed with commo mea µ ad variace σ, the factorized form of the joit ormal desity fuctio fy accordig to Example. may be applied. This yields the expressio exp } } yi µ exp fy; θ πσ fy; θ = σ πσ σ y i µ exp } = } yi µ exp.5-9 πσ σ πσ σ y i µ for the desity ratio. A applicatio of the ordiary biomial formula allows us to split off a factor that does ot deped o µ, thatis } } exp fy; θ πσ fy; θ = σ yi exp σ y i µ + µ } }. exp πσ σ yi exp.5-3 σ y i µ + µ Now, the first two factors i the omiator ad deomiator cacel out due to their idepedece of µ. Rearragig the remaiig terms leads to } µ exp fy; θ fy; θ = σ y i µ σ }.5-3 µ exp =exp σ µ σ y i µ σ y i µ σ =exp µ µ σ } y i µ σ + µ σ.5-3 } µ µ,.5-33 y i σ

17 .5 Best critical regios which reveals two remarkable facts: the simplified desity ratio depeds o the observatios oly through their sum y i, ad the desity ratio, as a expoetial fuctio, is a positive umber. Therefore, we may choose aother positive umber k α such that } exp µ µ y i µ µ > k α.5-34 σ σ always holds. Takig atural logarithms o both sides of this iequality yields µ µ y i σ σ or, after multiplicatio with σ µ µ µ µ > l k α ad expasio of the left side by, y i > σ l k α + µ µ. Depedig o whether µ > µ or µ < µ, the sample mea ȳ = y i must satisfy or ȳ> σ l k α + µ µ µ µ =: k α if µ >µ ȳ< σ l k α + µ µ µ µ =: k α if µ <µ i order for the secod coditio.5-6 of the Neyma-Pearso Lemma to hold. Note that the quatities σ,,µ,µ are all costats fixed apriori,adk α is a costat whose exact value is still to be determied. Thus, k α is itself a ukow costat. Now, i order for the first coditio.5-5 of the Neyma-Pearso Lemma to hold i additio, S C must have size α uder the ull hypothesis. As metioed above, the critical regio S C may be costructed solely by ispectig the value ȳ, which may be viewed as the outcome of the radom variable Ȳ := Y i. Uder H, Ȳ is ormally distributed with expectatio µ idetical to the expectatio of each of the origial observatios Y,...,Y ad stadard deviatio σ /. Therefore, the size is determied by N µ,σ α = / Ȳ > k α if µ >µ, N µ,σ / Ȳ < k α if µ <µ. It will be more coveiet to stadardize Ȳ because this allows us to evaluate the size i terms of the stadard ormal distributio. The coditio to be satisfied by k α the reads Ȳ µ N, σ α = / > k α µ σ / if µ >µ, Ȳ µ N, σ / < k α µ σ / if µ <µ, or, usig the stadard ormal distributio fuctio Φ, k Φ α µ σ α = / if µ > µ, k Φ α µ σ / if µ < µ. Rewritig this as Φ k α µ σ / = α if µ > µ, α if µ < µ

18 THEORY OF HYPOTHESIS TESTING allows us to determie the argumet of Φ by applyig the iverse stadard ormal distributio fuctio Φ to the previous equatio, which yields k α µ σ / = Φ α ifµ > µ, Φ α ifµ < µ, from which the costat k α is obtaied as µ + σ Φ α ifµ > µ, k α = µ + σ Φ α ifµ < µ, or k α = µ + σ Φ α ifµ > µ, µ σ Φ α ifµ < µ. Cosequetly, depedig o the sig of µ µ, there are two differet values for k α that satisfy the first coditio.5-5 of the Neyma-Pearso Lemma. Whe µ > µ the BCR is see to cosist of all the observatios y S, forwhich ȳ>µ + σ Φ α,.5-35 ad whe µ <µ, the BCR reads ȳ<µ σ Φ α. I the first case µ >µ, the MP test is give by ifȳ>µ + σ Φ α, φ u y = ifȳ<µ + σ Φ α, ad i the secod case µ <µ, the MP test is ifȳ<µ σ Φ α, φ l y = ifȳ>µ σ Φ α Observe that the critical regios deped solely o the value of the oe-dimesioal radom variable Ȳ,which, as a fuctio of the observatios Y, is also called a statistic. As this statistic appears i the specific cotext of hypothesis testig, we will speak of Ȳ as a test statistic. We see from this that it is ot ecessary to actually specify a -dimesioal regio S C used as the BCR, but the BCR may be expressed coveietly i terms of oe-dimesioal itervals. For this purpose, let c u, + ad,c l deote the critical regios with respect to the sample mea ȳ as defied by.5-35 ad The real costats ad c u := µ + σ Φ α.5-39 c l := µ σ Φ α.5-4 are called the upper critical value ad the lower critical value correspodig to the BCR for testig H versus H. I a practical situatio, it will be clear from the umerical specificatio of H which of the tests.5-37 ad.5-38 should be applied. The, the test is carried out by computig the mea ȳ of the give data y ad by checkig how large its value is i compariso to the critical value of.5-37 or.5-38, respectively.

19 .5 Best critical regios 3 Example.8: A most powerful test about the Beta distributio. Let Y,...,Y be idepedetly ad Bα, β-distributed observatios o [, ] with commo ukow parameter ᾱ which i this case is ot to be cofused with the size or level of the test ad commo kow parameter β = ot to be cofused with the probability of a Type II error. What is the BCR for a test of the simple ull hypothesis H :ᾱ = α = agaist the simple alterative hypothesis H :ᾱ = α =atlevelα? The desity fuctio of the uivariate Beta distributio i stadard form is defied by Γα + β fy; α, β = ΓαΓβ yα y β <y<; α, β >,.5-4 see Johso ad Kotz 97b, p. 37 or Koch 999, p. 5. Notice that.5-4 simplifies uder H to fy; α = Γ ΓΓ y y = <y<,.5-4 ad uder H to fy; α = Γ3 ΓΓ y y =y <y<.5-43 where we used the facts that Γ = Γ = ad Γ3 =. The desity.5-4 defies the so-called uiform distributio with parameters a = adb =, see Johso ad Kotz 97b, p. 57 or Koch, p.. We may ow proceed as i Example.7 ad determie the BCR by usig the Neyma-Pearso Lemma Theorem.. For idepedet observatios, the joit desity may be writte as the product of the idividual uivariate desities, which results i the desity ratio fy; α fy; α = y i / = y i,.5-44 whereweassumedthateachobservatiois strictly withi the iterval,. As the desity ratio is a positive umber, we may choose a umber k α such that y i >k α holds. Divisio by ad takig both sides to the power of / yields the equivalet iequality y i / > k α /. Now we have foud a seemigly coveiet coditio about the sample s geometric mea Y := Y i / rather tha about the etire sample Y itself. The the secod coditio.5-6 or equivaletly.5-8 of the Neyma-Pearso Lemma gives if y> k α / =: k α φy = if y< k α / =: k α. To esure that φ has some specified level α, the first coditio.5-5 of the Neyma-Pearso Lemma requires that α equals the probability uder H that the geometric mea exceeds k α. Ufortuately, i cotrast to the arithmetic mea Ȳ of idepedet ormal variables, the geometric mea Y of idepedet stadard uiform variables does ot have a stadard distributio. However, as Stuart ad Ord 3, p. 393 demostrate i their Example.5, the statistic U := l Y = l Y i = l Y i follows a Gamma distributio Gb, p withb =adp =, defied by Equatio.7 i Koch 999, p.. Thus the first Neyma-Pearso coditio reads α = G, U>k α = F G, k α, from which the critical value k α follows to be k α = F G, α, ad which may be obtaied i MATLAB by executig the commad CV = gamiv α,,. I summary, the MP test is give by if uy = l y i >k α = l k α =F G, α, φy = if uy = l y i <k α = l k α =F G, α.

20 4 THEORY OF HYPOTHESIS TESTING.5. Reductio to sufficiet statistics We saw i Example.7 that applyig the coditios of the Neyma-Pearso Lemma to derive the BCR led to a coditio about the sample mea ȳ rather tha about the origial data y. Wemightsaythatitwassufficiet to use the mea value of the data for testig a hypothesis about the parameter µ of the ormal distributio. This raises the importat questio of whether it is always possible to reduce the data i such a way. To geeralize this idea, let F = fy; θ :θ Θ} be a collectio of desities where the parameter θ is ukow. Further, let each fy; θ deped o the value of a radom fuctio or statistic T Y whichis idepedet of θ. If ay iferece about θ, be it estimatio or testig, depeds o the observatios Y oly through the value of T Y, the this statistic will be called sufficiet for θ. This qualitative defiitio of sufficiecy ca be iterpreted such that a sufficiet statistic captures all the relevat iformatio that the data cotais about the ukow parameters. The poit is that the data might have some additioal iformatio that does ot cotribute aythig to solvig the estimatio or test problem. The followig classical example highlights this distictio betwee iformatio that is essetial ad iformatio that is completely egligible for estimatig a ukow parameter. Example.9: Sufficiet statistic i Beroulli s radom experimet. Let Y,...,Y deote idepedet biary observatios withi a idealized settig of Beroulli s radom experimet see, for istace, Lehma, 959a, p The probability p of the elemetary evet success y i = is assumed to be ukow, but valid for all observatios. The probability of the secod possible outcome failure y i =isthe p. Now, it is ituitively clear that i order to estimate the ukow success rate p, it is completely sufficiet to kow how may successes T y := y i occurred i total withi trials. The additioal iformatio regardig which specific observatios were successes or failures does ot cotribute aythig useful for determiig the success rate p. I this sese the use of the statistic T Y reduces the data to a sigle value which carries all the essetial iformatio required to determie p. The cocept of sufficiecy provides a coveiet tool to achieve a data reductio without ay loss of iformatio about the ukow parameters. The defiitio above, however, is ot easily applicable whe oe has to deal with specific estimatio or testig problems. As a remedy, Neyma s Factorizatio Theorem gives a easy-to-check coditio for the existece of a sufficiet statistic i ay give parametric iferece problem. Theorem. Neyma s Factorizatio Theorem. Let F = fy; θ :θ Θ} be a collectio of desities for a sample Y =Y,...,Y. A vector of statistics T Y is sufficiet for θ if ad oly if there exist fuctios gt Y ; θ ad hy such that fy; θ =gt y; θ hy.5-45 holds for all θ Θ ad all y S. Proof. A deeper uderstadig of the sufficiecy cocept ivolves a ivestigatio ito coditioal probabilities which is beyod the scope of this thesis. The reader familiar with coditioal probabilities is referred to Lehma ad Romao 5, p. for a proof of this theorem. It is easy to see that the trivial choice T y :=y, gt y; θ :=fy; θ adhy := is always possible, but achieves o data reductio. Far more useful is the fact that ay reversible fuctio of a sufficiet statistic is also sufficiet for θ cf. Casella ad Berger,, p. 8. I particular, multiplyig a sufficiet statistic with costats yields agai a sufficiet statistic. The followig example will ow establish sufficiet statistics for the ormal desity with both parameters µ ad σ ukow. Example.: Suppose that observatios Y,...,Y are idepedetly ad ormally distributed with commo ukow mea µ ad commo ukow variace σ. Let the sample mea ad variace be defied as Ȳ = Y i/ ad S = Y i Ȳ /, respectively. The joit ormal desity ca the be writte as fy; µ, σ = exp } } πσ σ y i µ =πσ / exp µ σ + µ σ y i σ yi =πσ / exp σ ȳ µ } σ s I R y where T Y :=[Ȳ,S ] is sufficiet for µ, σ adhy :=I R y =withi as the idicator fuctio.

21 .5 Best critical regios 5 The great practical value of Neyma s Factorizatio Theorem i coectio to hypothesis testig lies i the simple fact that ay desity ratio will automatically simplify i the same way as i Example.7 from.5-3 to.5-3. What geerally happes is that the factor hy isthesameforθ ad θ due to its idepedece of ay parameters, ad thereby cacels out i the ratio, that is, fy; θ fy; θ = gt y; θ hy gt y; θ hy = gt y; θ gt y; θ for all y S I additio, this ratio will ow be a fuctio of the observatios Y through a statistic T Y which is usually low-dimesioal, such as [ Ȳ,S ] i Example.. This usually reduces the complexity ad dimesioality of the test problem greatly. Example.7 revisited. Istead of startig the derivatio of the BCR by settig up the desity ratio fy; θ /fy; θ of the raw data as i.5-9, we could save time by first reducig Y to the sufficiet statistic T Y =Ȳ ad by applyig.5-46 i coectio with the distributio Nµ, σ / of the sample mea. The πσ / exp } ȳ µ σ / gȳ; θ gȳ; θ = πσ / exp ȳ µ σ / =exp σ µ µ ȳ } σ µ µ } =exp σ ȳ µ + } σ ȳ µ leads to.5-33 more directly. We have see so far that the sample mea is sufficiet whe µ is the oly ukow parameter, ad that the sample mea ad variace are joitly sufficiet whe µ ad σ are ukow. Now, what is the maximal reductio geerally possible for data that are geerated by a more complex observatio model, such as by.-? Clearly, whe a parametric estimatio or testig problem comprises u ukow parameters that are ot redudat, the a reductio from > uobservatios to u correspodig statistics appears to be maximal. It is difficult to give clear-cut coditios that would ecompass all possible statistical models ad that would also be easily comprehesible without goig ito too may mathematical details. Therefore, the problem will be addressed oly by providig a workig defiitio ad a practical theorem, which will be applicable to most of the test problems i this thesis. Now, to be more specific, we will call a sufficiet statistic T Y miimally sufficiet if, for ay other sufficiet statistic T Y, T Y is a fuctio of T Y. As this defiitio is rather impractial, the followig theorem of Lehma ad Scheffe will be a useful tool. Theorem.3 Lehma-Scheffe. Let fy; θ deote the joit desity fuctio of observatios Y. Suppose there exists a statistic T Y such that, for every two data poits y ad y, the ratio fy ; θ/fy ; θ is costat as a fuctio of θ if ad oly if T y =T y.thet Y is miimally sufficiet for θ. Proof. See Casella ad Berger, p Example.: Suppose that observatios Y,...,Y are idepedetly ad ormally distributed with commo ukow mea µ ad commo ukow variace σ. Let y ad y be two data poits, ad let ȳ,s ad ȳ,s be the correspodig values of the sample mea Ȳ ad variace S. To prove that the sample mea ad variace are miimally sufficiet statistics, the ratio of desities is rewritte as fy ; µ, σ exp fy ; µ, σ = πσ σ y,i µ } πσ / exp [ȳ µ + s exp πσ σ y,i µ } = ]/ σ } πσ / exp [ȳ µ + s ]/ σ } =exp [ ȳ ȳ+µȳ ȳ s s ]/σ }. As this ratio is costat oly if ȳ = ȳ ad s = s,thestatistict Y =Ȳ,S is ideed miimally sufficiet. The observatios Y caot be reduced beyod T Y without losig relevat iformatio.

22 6 THEORY OF HYPOTHESIS TESTING.5.3 Uiformly most powerful UMP tests The cocept of the BCR for testig a simple H agaist a simple H about a sigle parameter, as defied by the Neyma-Pearso Lemma, is dissatifactory isofar that the great majority of test problems ivolves composite alteratives. The questio to be addressed i this subsectio is how a BCR may be defied for such problems. Let us start with the basic premise that we seek a optimal critical fuctio for testig the simple ull H : θ = θ.5-47 versus a composite alterative hypothesis H : θ Θ,.5-48 where the set of parameter values Θ ad θ are disjoit subsets of a oe-dimesioal parameter space Θ. The most straightforward way to establish optimality uder these coditios is to determie the BCR for testig H agaist a fixed simple H : θ = θ for a arbitrary θ Θ ad to check whether the resultig BCR is idepedet of the specific value θ. If this is the case, the all the values θ Θ produce the same BCR, because θ was selected arbitrarily. This critical regio that all the simple alteratives H i H = H : θ = θ with θ Θ }.5-49 have i commo may the be defied as the BCR for testig a simple H agaist a composite H. A test based o such a BCR is called uiformly most powerful UMP for testig H versus H at level α. Now, it seems rather cumbersome to derive the BCR for a composite H by applyig the coditios of the Neyma-Pearso Lemma to each simple H H. The followig theorem replaces this ifeasible procedure by coditios that ca be verified more directly. These coditios say that i order for a UMP test to exist, the test problem may have oly oe ukow parameter, the alterative hypothesis must be oesided, ad 3 each distributio i W must have a so-called mootoe desity ratio. The third coditio meas that, for all θ >θ with θ,θ Θ, the ratio fy; θ /fy; θ or the ratio gt; θ /gt; θ iterms of the sufficiet statistic T Y must be a strictly mootoical fuctio of T Y. The followig example will illumiate this issue. Example.: To show that the ormal distributio Nµ, σ with ukow µ ad kow σ has a mootoe desity ratio, we may directly ispect the simplified desity ratio.5-33 from Example.7. We see immediately that the ratio is a icreasig fuctio of T y := y i whe µ >µ. Theorem.4. Let W be a class of distributios with a oe-dimesioal parameter space ad mootoe desity ratio i some statistic T Y.. Suppose that H : θ = θ is to be tested agaist the upper oe-sided alterative H : θ >θ. The, there exists a UMP test φ u at level α ad a costat C with, if T y >C, φ u T y :=.5-5, if T y <C ad P θ φ u T Y = } = α For testig H agaist the lower oe-sided alterative H : θ <θ, there exists a UMP test φ l at level α ad a costat C with, if T y <C φ l T y :=.5-5, if T y >C ad P θ φ l T Y = } = α..5-53

23 .5 Best critical regios 7 Proof. To prove, cosider first the case of a simple alterative H : θ = θ for some θ >θ. With the values for θ ad θ fixed, the desity ratio ca be writte as fy; θ fy; θ = gt y; θ = ht y, gt y; θ that is, as a fuctio of the observatios aloe. Accordig to the Neyma-Pearso Lemma., the ratio must be large eough, i.e. ht y >kwith k depedig o α. Now, if T y <Ty holds for some y, y S, the certaily also ht y ht y due to the assumptio that the desity ratio is mootoe i T Y. I other words, the observatio y is i both cases at least as suitable as y for makig the ratio h sufficietly large. I this way, the BCR may be equally well costructed by all the data y S for which T y is large eough, for istace T y >C, where the costat C must be determied such that the size of this BCR equals the prescribed value α. As these implicatios are true regardless of the exact value θ, the BCR will be the same for all the simple alteratives with θ >θ. Therefore, the test.5-5 is UMP. The proof of follows the same sequece of argumets with all iequalities reversed. The ext theorem is of great practical value as it esures that most of the stadard distributios used i hypothesis testig have a mootoe desity ratio eve i their o-cetral forms. Theorem.5. The followig P -distributios with possibly additioal kow parameters µ, σ,p ad kow degrees of freedom f, f,, f, have a desity with mootoe desity ratio i some statistic T :. Multivariate idepedet ormal distributios Nµ, σi ad Nµ,σ I,. Gamma distributio Gb, p, 3. Beta distributio Bα, β, 4. No-cetral Studet distributio tf,λ, 5. No-cetral Chi-squared distributio χ f,λ, 6. No-cetral Fisher distributio F f,,f,,λ, Proof. The proofs of ad may be elegatly based o the more geeral result that ay desity that is a member of the oe-parameter expoetial family, defied by fy; θ =hycθexpwθt y}, hy,cθ,.5-54 cf. Olive, 6, for more details has a mootoe desity ratio see Lehma ad Romao, 5, p. 67, ad that the ormal ad Gamma distributios ca be writte i this form The desity fuctio of Nµ, σ I.5-9 ca be rewritte as } fy; µ =πσ / exp } } σ yi exp µ µ σ exp σ y i, where hy :=πσ / exp } σ y i, cθ :=exp µ }, wθ := µ,adt y := σ σ y i satisfy Similarly, the desity fuctio of Nµ,σ I reads, i terms of.5-54, } fy; σ =I R yπσ / exp σ y i µ, where hy correspods to the idicator fuctio I R y with defiite value oe, cθ :=πσ /, wθ := σ,adt y := y i µ.. The Gamma distributio, defied by Equatio.7 i Koch 999, p., with kow parameter p may be directly writte as fy; b = yp Γp bp exp by} b >,p >,y R +, where hy :=y p /Γp, cθ :=b p, wθ :=b, adt y := y satisfy The proofs for these distributios are legthy ad may be obtaied from Lehma ad Romao 5, p. 4 ad 37.