DOCSLIB.ORG

The Copula Information Criterion

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
The Copula Information Criterion Dept. of Math. University of Oslo Statistical Research Report No. 7 ISSN 0806–3842 July 2008 THE COPULA INFORMATION CRITERION STEFFEN GRØNNEBERG AND NILS LID HJORT Abstract. The arguments leading to the classical AIC does not hold for the case of parametric copulae models when using the pseudo maximum likelihood procedure. We derive a proper correction, and name it the Copula Information Criterion (CIC). The CIC is, as the copula, invariant to the underlying marginal distributions, but is fundamentally different from the AIC formula erroneously used by practitioners. We further observe that bias-correction–based information criteria do not exist for a large class of commonly used copulae, such as the Gumbel or Joe copulae. This motivates the investigation of median unbiased information criteria in the fully parametric setting, which then could be ported over to the current, more complex, semi-parametric setting. Contents 1. Introduction and summary 1 2. Approximation Theorems for the Kullback–Leibler divergence 2 2.1. Infinite expectation of qn and rn for certain copula models 8 2.2. AIC-like formula 9 2.3. TIC-like formula 11 3. Illustration 11 Appendix A. Mathematical proofs 14 References 20 1. Introduction and summary Let Xi = (Xi,(1),Xi,(2),...,Xi,(d)) be i.i.d from some joint distribution, for which we intend to employ one or more parametric copulae. Letting Cθ(v) be such a copula, with d density cθ(v) on [0, 1] , in terms of some parameter θ of length p, the pseudo-log-likelihood function is Xn `n(θ) = log cθ(Vˆi) i=1 in terms of the estimated and transformed marginals, say ˆ ˆ ˆ ˆ (1) (2) (d) Vi = (Vi,(1), Vi,(2),..., Vi,(d)) = (Fn (Xi,(1)),Fn (Xi,(2)),...,Fn (Xi,(d)), Date: 25th July 2008. Key words and phrases. AIC, copulae, model selection. 1 2 STEFFEN GRØNNEBERG AND NILS LID HJORT where the traditional choices are 1 Xn F (i)(x) = I{X(i) ≤ x}. n n + 1 i i=1 In particular, the Vˆi lie inside the unit square. The maximizer θˆ of `n(·) is called the maximum pseudo-likelihood estimator. Often in copulae applications one uses ? AIC = 2`n,max − 2length(θ) (1) as a model selection criterion, with `n,max = `n(θˆ) being the maximum pseudo likelihood, # # from the traditional Akaike information criterion AIC = 2`n,max −2length(θ), where `n,max is the usual maximum likelihood for a fully parametric model. One computes this AIC? score for each candidate model and in the end chooses the model with highest score. This cannot be quite correct, however, as the arguments underlying the derivations of the traditional AIC do not apply here – since `n(·) at work here is not a proper log- likelihood function for a model. In other words, the AIC? formula above ignores the noise ? inherent in the transformation step that takes Xi to Vˆi. The AIC -formula would be ◦ ◦ ◦ appropriate only if we could use Ui,(k) = F(k)(Xi,(k)), with known marginals F(k). So this paper is about the appropriate modifications necessary to provide the correct penalisation for complexity. We shall see that the result is the Copula Information Cri- terion ∗ ∗ ∗ CIC = 2`n,max − 2(ˆp +q ˆ +r ˆ ) (2) with a formula for pˆ∗+ˆq∗+ˆr∗ different from (and more complicated than) merely length(θ). These quantities even vary non-trivially with the model parameter – in clear contrast with length(θ) which is invariant to the actual value of θ. Surprisingly, we will see that there does not exist any (first order) bias-correction-based information criterion for the maximum pseudo likelihood estimator for a large class of copulae. This motivates the investigation of a median unbiased information criterion in parametric and semi-parametric estimation. See the further discussion in Section 2.1. We derive the theoretical basis for the Copula Information Criterion in Section 2. As in the fully parametric case, we get different results if we assume the parametric model is in fact correct or not. Subsection 2.2 assumes this and results an AIC-like formula, while Subsection 2.3 does not and results in a TIC-like1 formula. Finally, we give a small numerical illustration of the CIC in Section 3 and conclude with some needed mathematical results in Appendix A. 2. Approximation Theorems for the Kullback–Leibler divergence We start out considering Xn −1 −1 An(θ) = n `n(θ) = n log cθ(Vˆi) i=1 1See Claeskens & Hjort (2008) for the definitions and derivations of the AIC and TIC formulae in the fully parametric case. THE COPULA INFORMATION CRITERION 3 the pseudo-log-likelihood normalized by sample size. We postulate that the true (but unknown) data-generating distribution has a densityf ◦. We then have the unique decom- position Yd ◦ ◦ ◦ ◦ ◦ ◦ f (x1, x2, . , xd) = c (F1 (x1),F2 (x2),...,Fd (xd)) fi (xi). i=1 This motivates naming c◦ the true data generating copula and, more traditionally, name ◦ fi the true data generating marginal distributions. There is convergence almost surely Z a.s. ◦ An(θ) −−−→ A(θ) = c (v) log cθ(v) dv n→∞ [0,1]d for each θ. Let θ0 be the least false value for the family, i.e. the maximizer of A(·), assumed to be unique and an inner point of the parameter space. This is also the minimizer of the Kullback–Leibler distance Z ◦ ◦ c KL(c , cθ) = c log dv [0,1]d cθ from the true to the parametric copula. For the following result, which is proved in Genest et al. (1995), let ∂A (θ ) ∂` (θ ) U = n 0 = n−1 n 0 , n ∂θ ∂θ the normalized pseudo-score function, evaluated at θ0. Also, let ( ) Xd Z ∂φ(v, θ0) ◦ Σ = I + Var (I{ξi ≤ vi} − vi) dC (v) d ∂v i=1 [0,1] i where I is the Information matrix t I = Eφ(ξ, θ0)φ(ξ, θ0) , and where φ(·, θ) = ∂/∂θ log c(·, θ) and ξ ∼ C◦. Lemma 1. Given the regularity assumptions on {cθ : θ ∈ Θ} stated in Genest et al. (1995) and somewhat more explicitly in Tsukahara (2005), we have √ W nUn −−−→ U ∼ Np(0, Σ). n→∞ We shall also need the symmetric matrix Z 00 ∂ log cθ0 (v) J = −A (θ0) = − c(v) t dv, [0,1]d ∂θ∂θ assumed to be of full rank. A useful random process is now √ Hn(s) = n{An(θ0 + s/ n) − An(θ0)}. d √ It is defined for those s ∈ R for which θ0 + s/ n is inside the parameter region; in d particular, for any s ∈ R , Hn(s) is defined for all large n. 4 STEFFEN GRØNNEBERG AND NILS LID HJORT Lemma 2. We have W t 1 t Hn(s) −−−→ H(s) = s U − s Js n→∞ 2 where U is the limit of Un and where the convergence takes place inside each Skorokhod space D([−a, a]d). Proof. Essentially, Taylor expansion demonstrate √ 1 H (s) = st nU − stJ s + o (1), n n 2 n P a.s. where Jn −−−→ J is the empirical version of J. The result then follows from Lemma n→∞ 1. ¤ The first consequence of note is the limiting distribution of the maximum pseudo- likelihood estimator. From continuity of the argmax functional, W Mn = argmax(Hn) −−−→ M = argmax(H). n→∞ But this is the same as √ W −1 −1 −1 n(θˆ − θ0) −−−→ J U ∼ Np(0,J ΣJ ). n→∞ This result duplicates that of Genest et al. (1995) and the slight extension given by Chen & Fan (2005) to allow model mis-specification. Secondly, we investigate the actual Kullback–Leibler distance from the true model to that used from fitting the parametric family. However, we model the marginal distributions fully non-parametrically through the empirical distribution functions, which are strongly consistent under any possible set of true marginal distributions. As long as we confine our selves to consider models estimated through the maximum pseudo likelihood paradigm, we are only interested in the Kullback–Leibler divergence KL(c◦, c(·, θˆ)) between the true data generating copula c◦ and the fitted parametric part of our model c(·, θˆ) given by Z Z KL(c◦, c(·, θˆ)) = c◦ log c◦ dv − c◦(v) log c(v, θˆ) dv. [0,1]d [0,1]d It is rather difficult (but possible) to estimate the first term from data, but we may ignore it, since it is common to all parametric families. For the purposes of model selection it therefore suffices to estimate the second term, which is A(θˆ). We now examine −1 estimator An(θˆ) = n `n,max vis-`a-vis target A(θˆ). Let Zn = n{An(θˆ) − An(θ0)} − n{A(θˆ) − A(θ0)}. Some re-arrangement shows that −1 An(θˆ) − A(θˆ) = n Zn + An(θ0) − A(θ0). (3) Also, 1 Z = H (M ) + n(θˆ − θ )tJ(θˆ − θ ) + o (1), n n n 2 0 0 P THE COPULA INFORMATION CRITERION 5 in which we define the stochastically significant part as pn, giving rise to 1 t W 1 t −1 t −1 pn := Hn(Mn) + n(θˆ − θ0) J(θˆ − θ0) −−−→ H(M) + U J U = U J U =: p. 2 n→∞ 2 We have ¡ ¢ p∗ = Ep = EU tJ −1U = Tr(J −1Σ) = Tr J −1I " Ã !# Xd Z −1 ∂φ(v, θ0) ◦ + Tr J Var (I{ξi ≤ vi} − vi) dC (v) . d ∂v i=1 [0,1] i Remark 3. Note that p∗ ≥ 0 as in the standard parametric case. In the purely parametric case, we have EAn(θ0) − A(θ0) = 0, and so the standard AIC argument simply ends here (modulo stating a reasonable empirical estimate pˆ∗ of p∗), giving ∗ AIC = 2`n,max − 2ˆp as an asymptotic bias-correction for the estimated (relevant part of the) Kullback–Leibler divergence.
Load more

Recommended publications
  • THE COPULA INFORMATION CRITERIA 1. Introduction And
    Dept. of Math. University of Oslo Statistical Research Report No. 7 ISSN 0806{3842 June 2008 THE COPULA INFORMATION CRITERIA STEFFEN GRØNNEBERG AND NILS LID HJORT Abstract. When estimating parametric copula models by the semiparametric pseudo maximum likelihood procedure (MPLE), many practitioners have used the Akaike Information Criterion (AIC) for model selection in spite of the fact that the AIC formula has no theoretical basis in this setting. We adapt the arguments leading to the original AIC formula in the fully parametric case to the MPLE. This gives a significantly different formula than the AIC, which we name the Copula Information Criterion (CIC). However, we also show that such a model-selection procedure cannot exist for a large class of commonly used copula models. We note that this research report is a revision of a research report dated June 2008. The current version encorporates corrections of the proof of Theorem 1. The conclusions of the previous manuscript are still valid, however. 1. Introduction and summary Suppose given independent, identically distributed d-dimensional observations X1;X2;:::;Xn with density f ◦(x) and distribution function ◦ ◦ ◦ F (x) = P (Xi;1 ≤ x1;Xi;2 ≤ x2;:::Xi;d ≤ xd) = C (F?(x)): ◦ ◦ ◦ ◦ Here, C is the copula of F and F? is the vector of marginal distributions of F , that is, ◦ ◦ ◦ F?(x) := (F1 (x1);:::;Fd (xd));Fi(xj) = P (Xi;j ≤ xj): Given a parametric copula model expressed through a set of densities c(u; θ) for Θ ⊆ Rp and d ^ u 2 [0; 1] , the maximum pseudo likelihood estimator θn, also
    [Show full text]
  • Newsletter of the European Mathematical Society
    NEWSLETTER OF THE EUROPEAN MATHEMATICAL SOCIETY S E European M M Mathematical E S Society June 2019 Issue 112 ISSN 1027-488X Feature Interviews Research Center Determinantal Point Processes Peter Scholze Isaac Newton Institute The Littlewood–Paley Theory Artur Avila for Mathematical Sciences Artur Avila (photo courtesy of Daryan Dornelles/Divulgação IMPA) Journals published by the ISSN print 1463-9963 Editors: ISSN online 1463-9971 José Francisco Rodrigues (Universidade de Lisboa, Portugal), Charles M. Elliott (University 2019. Vol. 21. 4 issues of Warwick, Coventry, UK), Harald Garcke (Universität Regensburg,Germany), Juan Luis Approx. 500 pages. 17 x 24 cm Vazquez (Universidad Autónoma de Madrid, Spain) Price of subscription: 390 ¤ online only Aims and Scope 430 ¤ print+online Interfaces and Free Boundaries is dedicated to the mathematical modelling, analysis and computation of interfaces and free boundary problems in all areas where such phenomena are pertinent. The journal aims to be a forum where mathematical analysis, partial differential equations, modelling, scientific computing and the various applications which involve mathematical modelling meet. Submissions should, ideally, emphasize the combination of theory and application. ISSN print 0232-2064 A periodical edited by the University of Leipzig ISSN online 1661-4354 Managing Editors: 2019. Vol. 38. 4 issues J. Appell (Universität Würzburg, Germany), T. Eisner (Universität Leipzig, Germany), B. Approx. 500 pages. 17 x 24 cm Kirchheim (Universität Leipzig, Germany) Price of subscription: 190 ¤ online only Aims and Scope Journal for Analysis and its Applications aims at disseminating theoretical knowledge 230 ¤ print+online The in the field of analysis and, at the same time, cultivating and extending its applications.
    [Show full text]
  • Tilfeldig Gang, Nr. 1, Mai 2020
    ISSN 0803-8953 Tilfeldig GANG Nr. 1, årGANG 37 Mai 2020 UtgitT AV Norsk STATISTISK FORENING Redaksjonelt Fra NFR-lederen....................... 1 Fra redaksjonen....................... 2 TankenøtTER Pusleri nr. 53 ......................... 3 Pusleri nr. 52 (løsning)................... 4 Møter OG KONFERANSER Oppstart NSF Stavanger.................. 7 Medlemsmøte NSF Trondheim.............. 8 Artikler Monitoring the Level and the Slope of the Corona... 9 På leting etter sannheten bak kvantemekanikken.... 22 Tilbakeblikk: Den første nordiske konferanse i matema- tisk statistikk, Aarhus 1965 ............. 28 Kunngjøringer Kontingent.......................... 31 Meldinger Nytt fra NTNU....................... 32 Nytt fra UiB......................... 32 Nytt fra Matematisk institutt ved Universitetet i Oslo. 32 redaksjonelt Fra NFR-lederen Hei alle NFR-medlemmer, Alt vel med dere og nærmeste familie/venner i disse corona-tider håper jeg – alt vel her. Vi lever alle uvanlige liv for tiden. Mennesket er jo et sosialt vesen – og sosiale medier ivaretar bare en flik av det. Jeg har i lengre tid hatt barn som studenter i utlandet – og når vi samles om sommeren på hytta har vi et begrep ‘beyond Skype’ – det betyr en god bamse-klem. Det er behov for gode klemmer – og den tid kommer tilbake. Vi føler vel alle savnet av familie-medlemmer, jobb-kolleger og venner – samt for egen del, savnet av en Kreta-tur i mai. Jeg har, som gammel ‘low-tech’ professor, blitt påtvunget digitalt basert undervisning – og føler meg ikke trygg på at studentene er begeistret. Personlig ser jeg frem til normale undervisnings-tilstander. Statistikken i coronaens tid – lever faktisk i beste velgående. ‘Usikkerhet’ er vel det mest brukte ordet i samfunns-debatten siste måned. Statistiske modeller i epidemiologi er på alles lepper – og diskusjonen går høyt om prøvetaking og test-prosedyrer.
    [Show full text]
  • Tilfeldig Gang Nr. 1/2019 Karl Ove Hufthammer ⋅ Anne Marie Fenstad ⋅ Geir Drage Berentsen
    Tilfeldig gang nr. 1/2019 Karl Ove Hufthammer ⋅ Anne Marie Fenstad ⋅ Geir Drage Berentsen Innhald Redaksjonelt Frå leiaren Frå redaksjonen Medlemskontingent for 2019 Lesarbidrag Statistikk og sannsynligheter i rettspleien Estimering av «mørketall» Manglende uttrykk for manglende data – og samsvar mellom observatører Møte og konferansar Forkurs i visuell formidling av statistikk Års- og medlemsmøte i Statistisk forening i Bergen 2018 Meldingar Nytt frå Noregs teknisk-naturvitskaplege universitet Nytt frå Universitetet i Oslo Nytt frå Nytt fra Oslo senter for biostatistikk og epidemiologi Pusleri Løsning på pusleri nummer 50 Pusleri nummer 51 Statistikkrebuser fra 8. etasje Redaksjonelt Frå leiaren Marie Lilleborge NSF fyller 83 år i april, men NSF regnes også som en fortsettelse av en statistikerklubb som ble stiftet for hundre år siden, 7. januar 1919. En annen fin statistikersak er årets International Prize in Statistics (den andre i rekken), som går til Bradley Efron for «bootstrap»-metoden fra 1977. Og det bringer meg jo lett til våre egne statistikerpriser: Takk for mange gode nominasjoner til Sverdrup prisene. Komitéen fikk en utfordrende oppgave, og det er akkurat sånn det skal være! Takk for at dere er gode statistikere, og takk for at dere ser hverandre. Jeg ser frem til Sverdruppris-foredrag på statistikermøtet til sommeren. Årets statistikermøte er det tjuende i rekken: Det blir stas å treffes på det 20. norske statistikermøtet på Sola Strand Hotel tirsdag 18. til torsdag 20. juni. Merk at påmeldingsfristen er 26. april. Vi kan se frem til inviterte foredrag fra Sigrunn Holbek Sørbye, Hans Julius Skaug og Manuela Zucknick. Det Marie Lilleborge. er også et spennende tilbud om forkurs i visuell formidling av statistikk med Kathrine Frey Frøslie 17.–18.
    [Show full text]
  • Parametric Or Nonparametric: the Fic Approach
    Statistica Sinica 27 (2017), 951-981 doi:https://doi.org/10.5705/ss.202015.0364 PARAMETRIC OR NONPARAMETRIC: THE FIC APPROACH Martin Jullum and Nils Lid Hjort University of Oslo Abstract: Should one rely on a parametric or nonparametric model when analysing a given data set? This classic question cannot be answered by traditional model selection criteria like AIC and BIC, since a nonparametric model has no likelihood. The purpose of the present paper is to develop a focused information criterion (FIC) for comparing general non-nested parametric models with a nonparamet- ric alternative. It relies in part on the notion of a focus parameter, a population quantity of particular interest in the statistical analysis. The FIC compares and ranks candidate models based on estimated precision of the different model-based estimators for the focus parameter. It has earlier been developed for several classes of problems, but mainly involving parametric models. The new FIC, including also nonparametrics, is novel also in the mathematical context, being derived without the local neighbourhood asymptotics underlying previous versions of FIC. Cer- tain average-weighted versions, called AFIC, allowing several focus parameters to be considered simultaneously, are also developed. We concentrate on the standard i.i.d. setting and certain direct extensions thereof, but also sketch further generalisa- tions to other types of data. Theoretical and simulation-based results demonstrate desirable properties and satisfactory performance. Key words and phrases: Asymptotic theory, focused information criterion, model selection. 1. Introduction Statistical model selection is the task of choosing a good model for one's data. There is of course a vast literature on this broad topic, see e.g.
    [Show full text]
  • Focustat: Focus Driven Statistical Inference with Complex Data
    FocuStat: Focus Driven Statistical Inference with Complex Data CONTENT: FocuStat: The Task Force 2 The Tasks and Themes 3 The Task Focused Statistical Methods 4 Force Statistical Stories 12 Workshops and the VVV Conference 20 The FocuStat Research Kitchens 26 VIPs, Guests, Collaborators, Networking 28 FocuStat Excursions 30 The FocuStat Blog 32 Awards & Prizes 35 FocuStat in the Media 36 The core group, at a book discussion evening: G.H. Hermansen, E.Aa. Stoltenberg, V. Ko, Talks & Presentations 40 S.-E. Walker, B. Cuda, C. Cunen, K.H. Hellton, N.L. Hjort. Publications 44 FocuStat after FocuStat 48 FOCUSTAT HAS BEEN a five- 2016, 2018; Ko, Stoltenberg year project within the Statistics and Walker will defend theirs Division of the Department of in 2019. After their PostDoc Mathematics, University of Oslo terms, Hellton is a senior (UiO), funded by the Research researcher with the Norwegian Council of Norway, operating Computing Centre, Herman- from January 2014 to December sen a First Amanuensis with 2018. The directly funded core the BI Norwegian Business group has consisted of Professor School; also, Cunen is from 2019 Nils Lid Hjort (project leader), onwards a PostDoc with the Kristoffer Herland Hellton and Department of Mathematics, Gudmund Horn Hermansen UiO. Former Master students (PostDocs), Céline Cunen and Josephina Argyrou, Jonas Moss, Sam-Erik Walker (PhDs). Riccardo Parviero, Leiv Tore Salte Rønneberg (again, super- SEVERAL ASSOCIATED PhD vised or co-supervised by Hjort, students and Master students, and each of them having taken along with further colleagues up PhD positions afterwards) and collaborators, have also have worked with FocuStat been close to the core group.
    [Show full text]
  • Centre for Ecological and Evolutionary Synthesis
    COE - CENTRE FOR ECOLOGICAL AND EVOLUTIONARY SYNTHESIS Centre for Ecological and Evolutionary Synthesis Chair: Nils Chr. Stenseth, Department of Biology, University of Oslo LINKING ECOLOGY AND EVOLUTION Ecological and evolutionary processes are inescapably intertwined. Environmental changes affect the ecology of species causing novel selection pressures to which the species respond evolutionarily. Ever since the industrial revolution the influence of human activity on earth has accelerated, and today anthropogenic impacts on the biota are of great concern to politicians, academics, and laypeople. In order to comprehend how such distortion of the environment may affect tomorrow's nature, we need more and better knowledge of how ecology determines the course of evolution, which again determines future ecological dynamics. Understanding how living organisms respond and adapt to environmental changes remains a major and most urgent scientific challenge. Individual organisms constantly face challenges data from the field and the lab. Through this to which they respond through behavioural Centre of Excellence (CoE) we will for the next mechanisms, physiological plasticity and habitat 10 years counter the trend towards selection. Populations may contract or expand fragmentation of sciences by having several their geographic ranges, change in densities, interacting research foci. These will be divide or merge with other populations, or adapt organized around three mutually dependent evolutionarily to new conditions. To achieve a Themes: better understanding of these subjects, some 1) The role of population structuring in adaptive major biological questions have to be answered: evolution. How do ecological structures and processes, as 2) The potential for adaptation. well as intrinsic processes, act as drivers of, or 3) The evolution of reproductive isolation.
    [Show full text]
  • Information Bulletin Nº 2
    INFORMATION BULLETIN Nº 2 60th World Statistics Congress of the International Statistical Institute Riocentro | Rio de Janeiro | Brazil 26-31 July 2015 www.isi2015.org CONTENTS Welcome Messages IBGE President .................................................................................................................... 01 ISI President ......................................................................................................................... 02 Committees Scientific Programme Committee (SPC) ................................................................... 03 Local Programme Committee (LPC) ........................................................................... 04 Short Course Committee (SCC) .................................................................................... 05 Lunch Roundtable Discussions Committee (LRTDC) ........................................... 05 National Honorary Committee (NHC) ......................................................................... 06 Local Organising Committee (LOC) ............................................................................ 07 ISI Governance ..................................................................................................................... 08 Conference Schedule ........................................................................................................ 09 Venue Floor Plans ............................................................................................................... 10 Scientific Programme ......................................................................................................
    [Show full text]
  • Bayesian, Fiducial and Frequentist Statistics Bo Henry Lindqvist and Gunnar Taraldsen
    Mathematical Foundations of Distribution Estimators - Bayesian, Fiducial and Frequentist Statistics Bo Henry Lindqvist and Gunnar Taraldsen 1 Relevance relative to the call for proposals This project is a Researcher Project based on national and international collaboration of es- tablished researchers, and having an additional recruitment component in the form of two PhD fellowships and one PostDoc position. According to the proposal, the FRIPRO scheme shall promote scientific quality and boldness in scientific thinking and innovation. The fundamental research problems in the proposed project are given by foundational issues in statistics, and in particular the long standing problem of usage of improper priors without a theoretical justification. Further merits of the project are given by the proposed international collaboration with four well established and world leading researchers. 2 Aspects relating to the research project 2.1 Background and status of knowledge 2.1.1 Bayesian, fiducial and frequentist statistics Recent developments of powerful data acquisition technologies has led to a data explosion which demands a need for efficient methodology and new computing tools for analyzing data and drawing inference. This has led to an increased importance of statistics in recent decades. Driven by the increasing demand for statistics, and by an effort of statisticians of multiple generations, there have been several major ground-breaking advancements in the respective fields of Bayesian, fiducial and frequentist inferences in recent years. Bayesian computing using MCMC is now the major computational tool in complex statistical analyses. Furthermore, there are developed so-called objective Bayesian methods, which are oriented towards the development of prior distributions that can be used automatically, with solid links to fiducial and frequentist methods.
    [Show full text]
  • Utgitt Av Norsk Statistisk Forening INNHOLD
    ISSN 0803-8953 TILFELDIG GANG Nr. 1, årgang 24 Mars 2007 Utgitt av Norsk Statistisk Forening INNHOLD Side Fra redaktøren, Inge Helland .................................................................................................... 1 Brev Scandinavian Journal of Statistics, Ørnulf Borgan og Bo Lindqvist ....................................... 2 Fra presidenten i European Mathematical Society, Ari Laptev ............................................... 3 Artikler Grete U. Fenstad – 50 år på Blindern. Ingrid Glad og Bent Natvig ........................................ 5 Om kultur, om statistikk og om bakgrunn for egne meninger. Inge Helland .......................... 9 Gambling og sannsynlighetsteori. Kjell Stordahl ................................................................... 11 Bruno de Finetti – a great probabilist and a great man. Giuseppe Arichini ............................ 16 Puslerispalten Løsning til pusleri nr. 20. Jostein Lillestøl .............................................................................. 21 Pusleri nr. 21. Jostein Lillestøl ................................................................................................ 22 Stipendiatpresentasjoner Bayesian p-values. Gunnhildur H. Steinbakk ......................................................................... 23 Tools for improved total value chain analysis... Nils F. Haavardsson .................................. 24 Notiser Statistikkpakker med kontor i Norge. Inge Helland ............................................................... 26 Statistisk
    [Show full text]
  • Confidence, Likelihood, Probability
    i i “Schweder-Book” — 2015/10/21 — 17:46 — page i — #1 i i Confidence, Likelihood, Probability This lively book lays out a methodology of confidence distributions and puts them through their paces. Among other merits, they lead to optimal combinations of confidence from different sources of information, and they can make complex models amenable to objective and indeed prior-free analysis for less subjectively inclined statisticians. The generous mixture of theory, illustrations, applications and exercises is suitable for statisticians at all levels of experience, as well as for data-oriented scientists. Some confidence distributions are less dispersed than their competitors. This concept leads to a theory of risk functions and comparisons for distributions of confidence. Neyman–Pearson type theorems leading to optimal confidence are developed and richly illustrated. Exact and optimal confidence distributions are the gold standard for inferred epistemic distributions in empirical sciences. Confidence distributions and likelihood functions are intertwined, allowing prior distributions to be made part of the likelihood. Meta-analysis in likelihood terms is developed and taken beyond traditional methods, suiting it in particular to combining information across diverse data sources. TORE SCHWEDER is a professor of statistics in the Department of Economics and at the Centre for Ecology and Evolutionary Synthesis at the University of Oslo. NILS LID HJORT is a professor of mathematical statistics in the Department of Mathematics at the University of Oslo. i i i i i i “Schweder-Book” — 2015/10/21 — 17:46 — page ii — #2 i i CAMBRIDGE SERIES IN STATISTICAL AND PROBABILISTIC MATHEMATICS Editorial Board Z. Ghahramani (Department of Engineering, University of Cambridge) R.
    [Show full text]
  • Essays in Policy Evaluation
    UC Irvine UC Irvine Electronic Theses and Dissertations Title Essays in Policy Evaluation Permalink https://escholarship.org/uc/item/6218s8mm Author Button, Patrick Publication Date 2015 Peer reviewed|Thesis/dissertation eScholarship.org Powered by the California Digital Library University of California UNIVERSITY OF CALIFORNIA, IRVINE Essays in Policy Evaluation DISSERTATION submitted in partial satisfaction of the requirements for the degree of DOCTOR OF PHILOSOPHY in Economics by Patrick James Harold Button Dissertation Committee: Professor David Neumark, Chair Professor Marianne Bitler Professor Jan Brueckner 2015 Chapter 1 c 2015 De Gruyter All other materials c 2015 Patrick James Harold Button DEDICATION I dedicate this dissertation to my parents, Gene and David Button. I could not have gotten this far without both of you. I have asked much of you to survive this ordeal and you have never hesitated to go above and beyond what was required to make sure I was taken care of. I hope I can repay your kindness and patience. I would like to thank my dissertation committee for all their support. I am particularly indebted to David Neumark for being my role model for so many years. I never felt like a real economist until I was able to work with you. I can't imagine what my life would be like had I not had the privilege of working with you. I thank Marianne for being an inspiration to me as someone who strives to achieve the most rigorous standards in research. I hope to continue to achieve rigorous empirical standards in my research by asking myself: \What would Marianne do?" I thank Jan Brueckner for sharing his wisdom with me, especially when I was writing my paper on tax incentives for the film industry.
    [Show full text]