Multivariate negative binomial models for insurance claim counts

Size: px
Start display at page:

Download "Multivariate negative binomial models for insurance claim counts"

Transcription

1 Multivariate negative binomial models for insurance claim counts Peng Shi (Northern Illinois University) and Emiliano A. Valdez (University of Connecticut) 9 November 0, Montréal, Quebec Université de Montréal Seminar

2 Université de Montréal Seminar 0 Background For short term insurance contracts, there are generally two major components to decompose: cost of claims = frequency severity Many believe that the frequency component: may be the more important of these two components reveals, to an extent, more of the riskiness of the policyholder Model improvements can be made on the prediction on frequency: usual accounting for heterogeneity - risk classification variables using additional information available in data recorded e.g. types of claim additional insight on the association between the types of claim Shi, P. and Valdez, E.A. /

3 Université de Montréal Seminar 0 The multivariate claim count data Assume we have a portfolio of m policyholders observed over a fixed time period (cross-sectional). For each policyholder, we observe a vector of claim counts expressed as (N i, N i,..., N ik ), where N ij is the claim count associated with type j, with j =,,...,k, and i =,,...,m. Each policyholder also reveals a set of observable covariates x i useful to sub-divide the portfolio into classes of risks with homogeneous characteristics. We present and compare various alternatives for modeling multivariate claim counts using classical methods of using common shocks and the modern methods of using copulas. Traditional count regression models are used in the modeling of the marginal component. Shi, P. and Valdez, E.A. /

4 Université de Montréal Seminar 0 Alternative multivariate models Some literature on incorporating dependence structure in a multivariate count outcomes: use of a common additive error: Kocherlakota and Kocherlakota (99), Johnson et al. (997), Winkelmann (000), Karlis and Meligkotsidou (00), and Bermúdez and Karlis (0) Poisson, zero-inflated Poisson, and negative binomial to capture overdispersion mixture model with a multiplicative error: Hausmann et al. (98), Day and Chung (99) may be restricted to non-negative covariance structure - alternatives here are Aitchison and Ho (989) and van Ophem (999) Shi, P. and Valdez, E.A. /

5 Université de Montréal Seminar 0 Multivariate models using copulas Construct general discrete multivariate distribution to support more complex correlation structures through the use of copulas: although believed still in its infancy, increasingly been popular applications extend to several disciplines: Economics: Prieger (00), Cameron et al. (00), Zimmer and Trivedi (00) Biostatistics: Song et al. (008), Madsen and Fang (00) Actuarial science: Purcaru and Denuit (00), Shi and Valdez (0), Shi and Valdez (0) Boucher, Denuit and Guillén (008) provides a survey of models for insurance claim counts with time dependence. Genest and Nešlehová (007) says be pre-cautious when using copulas: non-uniqueness of the copula, vague interpretation of the nature of dependence. Shi, P. and Valdez, E.A. /

6 Negative binomial distribution model Université de Montréal Seminar 0 N is said to follow a negative binomial distribution if its pmf may be expressed as ( ) Γ(η + n) η ( ) ψ n Pr(N = n) =, n = 0,,, Γ(η)Γ(n + ) + ψ + ψ and is denoted as N NB(ψ, η) for ψ, η > 0. Mean E(N) = ηψ and variance Var(N) = ηψ( + ψ) Unlike the Poisson distribution, the NB accommodates overdispersion via parameter ψ. As ψ 0, overdispersion vanishes and the NB converges to the Poisson distribution. Shi, P. and Valdez, E.A. /

7 Université de Montréal Seminar 0 Negative binomial regression models For regression, consider the mean specified in terms of covariates λ = ηψ = exp(x β), which may come in two different parameterizations: NB-I model: η = σ exp(x β) NB-II model: η = σ Both assume same mean structure but different dispersion φ such that Var(N x) = φe(n x). The NB-I model implies a constant dispersion φ = + σ. The NB-II model allows for subject heterogeneity in the dispersion φ = + σ exp(x β). See Winkelmann (008). Shi, P. and Valdez, E.A. 7 /

8 Université de Montréal Seminar 0 Empirical data For our empirical analysis, claims data are obtained from one randomly selected automobile insurance company in Singapore: claims experience for a single year focus on non-fleet policy considered only policyholders with comprehensive coverage Summary of response and explanatory variables Variable Description Mean StdDev Responses N number of claims of third-party bodiliy injury N number of claims of own damage N number of claims of third-party property damage Covariates young equals if age is less than lowncd equals if NCD is less than vage vehicle age.98.0 private equals for private car vlux equals for luxuary car smallcap equals if vehicle capacity is small Shi, P. and Valdez, E.A. 8 /

9 Goodness of fit of the marginal models Université de Montréal Seminar 0 N Observed Fitted Poisson ZIP NegBin-I NegBin-II χ N Observed Fitted Poisson ZIP NegBin-I NegBin-II χ N Observed Fitted Poisson ZIP NegBin-I NegBin-II χ Shi, P. and Valdez, E.A. 9 /

10 Multivariate models considered Université de Montréal Seminar 0 Multivariate models constructed based on common shocks as follows: N i = U i + U i + U i N i = U i + U i + U i N i = U i + U i + U i Multivariate models based on copulas: with F i (n, n, n x i ) = C(F (n x i ), F (n x i ), F (n x i ) x i ;Θ) f i (n, n, n x i ) = l =0 l =0 l =0 ( ) l +l +l C(u i,l, u i,l, u i,l x i ;Θ) Shi, P. and Valdez, E.A. 0 /

11 Families of copulas examined Université de Montréal Seminar 0 The class of mixtures of max-id copulas: extends construction of the Archimedean copula by mixing bivariate max-id copulas a multivariate distribution, say H, is max-id if H γ is a CDF for all γ > 0, Joe (99) allows to capture a global together with pair-wise dependence the copula appears complex but in manageable explicit form: C(u, u, u ) = ϕ X j<k log R jk e ν jϕ (u j ), e ν kϕ (u k ) X j= «ω j ν j ϕ (u j ) Joe and Hu (997) Shi, P. and Valdez, E.A. /

12 Université de Montréal Seminar 0 Families of copulas examined - continued The class of elliptical copulas: has been popular because of straightforward calculation of the copula while at the same time, allowing for flexible correlation structures considered both Gaussian and t copulas, but ended up choosing Gaussian as a better model of the two (based on our empirical data) Gaussian copula: ) C(u, u, u ) = Φ Σ (Φ (u ), Φ (u ), Φ (u ) to circumvent the problem often encountered with using copulas for multivariate discrete data, we continuitize the discrete form using jitters Shi, P. and Valdez, E.A. /

13 Continuous extension with jitters Université de Montréal Seminar 0 Define Ñij = N ij U ij for j {,, } where U ij Uniform(0, ). The joint pdf of jittered counts for the ith policyholder Ñ i = (Ñi, Ñi, Ñi) may be expressed as: ef i (en i, en i, en i x i ) = c( e F (en i x i ), e F (en i x i ), e F (en i x i ) x i ;Θ) Y j= ef ij (en ij x ij ) Retrieve the joint pmf of (N i, N i, N i ) by averaging over the jitters: f i (n i, n i, n i x i ) =E Ui c( e F (n i U i x i ), e F (n i U i x i ), e F (n i U i x i ) x i ;Θ) Based on relations: Y j= «ef j (n ij U ij x ij ) F j (ñ ij x ij ) = F j ([ñ ij ] x ij ) + (ñ ij [ñ ij ])f j ([ñ ij + ] x ij ) f j (ñ ij x ij ) = f j ([ñ ij + ] x ij ) Shi, P. and Valdez, E.A. /

14 Multivariate NB-I with common shocks Université de Montréal Seminar 0 Bodily Injury Own Damage Property Damage Estimate t-ratio Estimate t-ratio Estimate t-ratio intercept young lowncd vage vage vage private vlux smallcap Estimate 9% CI lnλ -.79 (-., -.) lnλ (-.899, -.8) lnλ -.7 (-.7, -.08) lnσ -.97 (-7.87, -.007) Goodness-of-Fit Loglikelihood -9.9 AIC BIC 9.7 Shi, P. and Valdez, E.A. /

15 Multivariate NB-I using mixtures of max-id copulas Université de Montréal Seminar 0 Bodily Injury Own Damage Property Damage Estimate t-ratio Estimate t-ratio Estimate t-ratio NB-I intercept young lowncd vage vage vage private vlux smallcap Estimate 9% CI lnσ -.9 (-.0, -.) lnσ -. (-., -.) lnσ -.9 (-.7, -.08) θ.9 (.77,.8) θ.7 (.89,.) θ.7 (.,.) Goodness-of-Fit Loglikelihood -87. AIC 8. BIC 88. Shi, P. and Valdez, E.A. /

16 Multivariate NB-II using mixtures of max-id copulas Université de Montréal Seminar 0 Bodily Injury Own Damage Property Damage Estimate t-ratio Estimate t-ratio Estimate t-ratio NB-II intercept young lowncd vage vage vage private vlux smallcap Estimate 9% CI lnσ 0.9 (-0.07, 0.8) lnσ -0.9 (-0.79, 0.) lnσ 0. (-0.09,.9) θ. (.79,.8) θ.80 (.99,.) θ. (.,.07) Goodness-of-Fit Loglikelihood -8. AIC 8.7 BIC 88.8 Shi, P. and Valdez, E.A. /

17 Multivariate NB-I using Gaussian copula Université de Montréal Seminar 0 Bodily Injury Own Damage Property Damage Estimate t-ratio Estimate t-ratio Estimate t-ratio NB-I intercept young lowncd vage vage vage private vlux smallcap Estimate 9% CI lnσ -.90 (-.77, -.) lnσ (-.80, -.7) lnσ -.99 (-.9, -.00) ρ ( 0.770, 0.8) ρ 0.80 ( 0.79, 0.8) ρ 0.8 ( 0., 0.) Goodness-of-Fit Loglikelihood -9.8 AIC 87. BIC 88.0 Shi, P. and Valdez, E.A. 7 /

18 Multivariate NB-II using Gaussian copula Université de Montréal Seminar 0 Bodily Injury Own Damage Property Damage Estimate t-ratio Estimate t-ratio Estimate t-ratio NB-II intercept young lowncd vage vage vage private vlux smallcap Estimate 9% CI lnσ -0.7 (-0.9, 0.9) lnσ -0.7 (-., 0.0) lnσ -0.0 (-.,.00) ρ (0.77, 0.87) ρ 0.80 (0.770, 0.88) ρ 0.8 (0., 0.7) Goodness-of-Fit Loglikelihood -9.8 AIC 8. BIC 88.8 Shi, P. and Valdez, E.A. 8 /

19 Predictive applications Université de Montréal Seminar 0 This application examines the financial losses of third party bodily injury, own damage, and third party property damage, in a comprehensive coverage. For demonstration purposes, we look into the variable L i = N i + N i + N i that is the basis of the premium for risk class i. Five hypothetical risk classes: Ranking from high to low risk levels, they are Excellent, Very Good, Good, Fair, and Poor. For example, the policyholders in class Excellent are older than, have a NCD score above 0, and drive a 0-year old private small luxury car. Hypothetical risk classification profile young lowncd vage private vlux smallcap Excellent Very Good 0 0 Good Fair Poor Shi, P. and Valdez, E.A. 9 /

20 Prediction estimates of claim frequency Université de Montréal Seminar 0 Excellent Very Good MVNB MaxID Gaussian Product MVNB MaxID Gaussian Product ooo oo o+o oo o o o Good Fair MVNB MaxID Gaussian Product MVNB MaxID Gaussian Product ooo oo o+o oo o o o Poor MVNB MaxID Elliptical Product ooo oo o+o oo o o o Shi, P. and Valdez, E.A. 0 /

21 Mean and variance of total claims by risk class Université de Montréal Seminar 0 Excellent Very Good Good Fair Poor E(L i ) Var(L i ) E(L i ) Var(L i ) E(L i ) Var(L i ) E(L i ) Var(L i ) E(L i ) Var(L i ) MVNB MaxID-I MaxID-II Gauss-I Gauss-II Indep-I Indep-II Shi, P. and Valdez, E.A. /

22 Predictive distributions for various models Université de Montréal Seminar 0 Percent of Total Percent of Total MaxID Copula NB I Excellent Very Good Good Fair Poor Expected Count Elliptical Copula NB I Excellent Very Good Good Fair Percent of Total Percent of Total MaxID Copula NB II Excellent Very Good Good Fair Poor Expected Count Elliptical Copula NB II Excellent Very Good Good Fair Poor Poor Expected Count Expected Count Shi, P. and Valdez, E.A. /

23 Concluding remarks Université de Montréal Seminar 0 We considered alternative approaches to construct multivariate count regression models based on the negative binomial distributions. The trivariate negative binomial models was proposed in order to accommodate the dependency among three types of claims: the third party bodily injury, own damage, and third-party property damage. Copulas provide flexibility to allow modeling various dependence structures, allowing to separate the effects of peculiar characteristics of the margins such as thickness of tails. In contrast, the class of multivariate negative binomial models based on common shocks rely on the additivity of the NB-I distribution and require a common dispersion for all marginals. We found that the superiority of the copula approach was supported by the better fit in the empirical analysis. Shi, P. and Valdez, E.A. /

24 Université de Montréal Seminar 0 - Merci bien - Shi, P. and Valdez, E.A. /

Counts using Jitters joint work with Peng Shi, Northern Illinois University

Counts using Jitters joint work with Peng Shi, Northern Illinois University of Claim Longitudinal of Claim joint work with Peng Shi, Northern Illinois University UConn Actuarial Science Seminar 2 December 2011 Department of Mathematics University of Connecticut Storrs, Connecticut,

More information

Generalized linear mixed models (GLMMs) for dependent compound risk models

Generalized linear mixed models (GLMMs) for dependent compound risk models Generalized linear mixed models (GLMMs) for dependent compound risk models Emiliano A. Valdez joint work with H. Jeong, J. Ahn and S. Park University of Connecticut 52nd Actuarial Research Conference Georgia

More information

Generalized linear mixed models for dependent compound risk models

Generalized linear mixed models for dependent compound risk models Generalized linear mixed models for dependent compound risk models Emiliano A. Valdez joint work with H. Jeong, J. Ahn and S. Park University of Connecticut ASTIN/AFIR Colloquium 2017 Panama City, Panama

More information

Peng Shi. Actuarial Research Conference August 5-8, 2015

Peng Shi. Actuarial Research Conference August 5-8, 2015 Multivariate Longitudinal of Using Copulas joint work with Xiaoping Feng and Jean-Philippe Boucher University of Wisconsin - Madison Université du Québec à Montréal Actuarial Research Conference August

More information

Ratemaking with a Copula-Based Multivariate Tweedie Model

Ratemaking with a Copula-Based Multivariate Tweedie Model with a Copula-Based Model joint work with Xiaoping Feng and Jean-Philippe Boucher University of Wisconsin - Madison CAS RPM Seminar March 10, 2015 1 / 18 Outline 1 2 3 4 5 6 2 / 18 Insurance Claims Two

More information

Generalized linear mixed models (GLMMs) for dependent compound risk models

Generalized linear mixed models (GLMMs) for dependent compound risk models Generalized linear mixed models (GLMMs) for dependent compound risk models Emiliano A. Valdez, PhD, FSA joint work with H. Jeong, J. Ahn and S. Park University of Connecticut Seminar Talk at Yonsei University

More information

Multivariate count data generalized linear models: Three approaches based on the Sarmanov distribution. Catalina Bolancé & Raluca Vernic

Multivariate count data generalized linear models: Three approaches based on the Sarmanov distribution. Catalina Bolancé & Raluca Vernic Institut de Recerca en Economia Aplicada Regional i Pública Research Institute of Applied Economics Document de Treball 217/18 1/25 pág. Working Paper 217/18 1/25 pág. Multivariate count data generalized

More information

GB2 Regression with Insurance Claim Severities

GB2 Regression with Insurance Claim Severities GB2 Regression with Insurance Claim Severities Mitchell Wills, University of New South Wales Emiliano A. Valdez, University of New South Wales Edward W. (Jed) Frees, University of Wisconsin - Madison UNSW

More information

Marginal Specifications and a Gaussian Copula Estimation

Marginal Specifications and a Gaussian Copula Estimation Marginal Specifications and a Gaussian Copula Estimation Kazim Azam Abstract Multivariate analysis involving random variables of different type like count, continuous or mixture of both is frequently required

More information

Ratemaking application of Bayesian LASSO with conjugate hyperprior

Ratemaking application of Bayesian LASSO with conjugate hyperprior Ratemaking application of Bayesian LASSO with conjugate hyperprior Himchan Jeong and Emiliano A. Valdez University of Connecticut Actuarial Science Seminar Department of Mathematics University of Illinois

More information

Parametric Modelling of Over-dispersed Count Data. Part III / MMath (Applied Statistics) 1

Parametric Modelling of Over-dispersed Count Data. Part III / MMath (Applied Statistics) 1 Parametric Modelling of Over-dispersed Count Data Part III / MMath (Applied Statistics) 1 Introduction Poisson regression is the de facto approach for handling count data What happens then when Poisson

More information

A finite mixture of bivariate Poisson regression models with an application to insurance ratemaking

A finite mixture of bivariate Poisson regression models with an application to insurance ratemaking A finite mixture of bivariate Poisson regression models with an application to insurance ratemaking Lluís Bermúdez a,, Dimitris Karlis b a Risc en Finances i Assegurances-IREA, Universitat de Barcelona,

More information

On a simple construction of bivariate probability functions with fixed marginals 1

On a simple construction of bivariate probability functions with fixed marginals 1 On a simple construction of bivariate probability functions with fixed marginals 1 Djilali AIT AOUDIA a, Éric MARCHANDb,2 a Université du Québec à Montréal, Département de mathématiques, 201, Ave Président-Kennedy

More information

Mohammed. Research in Pharmacoepidemiology National School of Pharmacy, University of Otago

Mohammed. Research in Pharmacoepidemiology National School of Pharmacy, University of Otago Mohammed Research in Pharmacoepidemiology (RIPE) @ National School of Pharmacy, University of Otago What is zero inflation? Suppose you want to study hippos and the effect of habitat variables on their

More information

Modelling multivariate count data using copulas

Modelling multivariate count data using copulas Modelling multivariate count data using copulas Dimitris Karlis, Aristidis K. Nikoloulopoulos To cite this version: Dimitris Karlis, Aristidis K. Nikoloulopoulos. Modelling multivariate count data using

More information

arxiv: v1 [stat.ap] 16 Nov 2015

arxiv: v1 [stat.ap] 16 Nov 2015 Simulating Posterior Distributions for Zero Inflated Automobile Insurance Data arxiv:1606.00361v1 [stat.ap] 16 Nov 2015 J.M. Pérez Sánchez a and E. Gómez Déniz b a Department of Quantitative Methods, University

More information

Modelling Dependent Credit Risks

Modelling Dependent Credit Risks Modelling Dependent Credit Risks Filip Lindskog, RiskLab, ETH Zürich 30 November 2000 Home page:http://www.math.ethz.ch/ lindskog E-mail:lindskog@math.ethz.ch RiskLab:http://www.risklab.ch Modelling Dependent

More information

DOCUMENT DE TREBALL XREAP Mixture of bivariate Poisson regression models with an application to insurance

DOCUMENT DE TREBALL XREAP Mixture of bivariate Poisson regression models with an application to insurance DOCUMENT DE TREBALL XREAP2011-10 Mixture of bivariate Poisson regression models with an application to insurance Lluís Bermudez (RFA-IREA) Dimitris Karlis Mixture of bivariate Poisson regression models

More information

A measure of radial asymmetry for bivariate copulas based on Sobolev norm

A measure of radial asymmetry for bivariate copulas based on Sobolev norm A measure of radial asymmetry for bivariate copulas based on Sobolev norm Ahmad Alikhani-Vafa Ali Dolati Abstract The modified Sobolev norm is used to construct an index for measuring the degree of radial

More information

Tail negative dependence and its applications for aggregate loss modeling

Tail negative dependence and its applications for aggregate loss modeling Tail negative dependence and its applications for aggregate loss modeling Lei Hua Division of Statistics Oct 20, 2014, ISU L. Hua (NIU) 1/35 1 Motivation 2 Tail order Elliptical copula Extreme value copula

More information

DEEP, University of Lausanne Lectures on Econometric Analysis of Count Data Pravin K. Trivedi May 2005

DEEP, University of Lausanne Lectures on Econometric Analysis of Count Data Pravin K. Trivedi May 2005 DEEP, University of Lausanne Lectures on Econometric Analysis of Count Data Pravin K. Trivedi May 2005 The lectures will survey the topic of count regression with emphasis on the role on unobserved heterogeneity.

More information

A Goodness-of-fit Test for Copulas

A Goodness-of-fit Test for Copulas A Goodness-of-fit Test for Copulas Artem Prokhorov August 2008 Abstract A new goodness-of-fit test for copulas is proposed. It is based on restrictions on certain elements of the information matrix and

More information

Modelling Dependence with Copulas and Applications to Risk Management. Filip Lindskog, RiskLab, ETH Zürich

Modelling Dependence with Copulas and Applications to Risk Management. Filip Lindskog, RiskLab, ETH Zürich Modelling Dependence with Copulas and Applications to Risk Management Filip Lindskog, RiskLab, ETH Zürich 02-07-2000 Home page: http://www.math.ethz.ch/ lindskog E-mail: lindskog@math.ethz.ch RiskLab:

More information

Notes for Math 324, Part 20

Notes for Math 324, Part 20 7 Notes for Math 34, Part Chapter Conditional epectations, variances, etc.. Conditional probability Given two events, the conditional probability of A given B is defined by P[A B] = P[A B]. P[B] P[A B]

More information

Simulating Realistic Ecological Count Data

Simulating Realistic Ecological Count Data 1 / 76 Simulating Realistic Ecological Count Data Lisa Madsen Dave Birkes Oregon State University Statistics Department Seminar May 2, 2011 2 / 76 Outline 1 Motivation Example: Weed Counts 2 Pearson Correlation

More information

The Credibility Estimators with Dependence Over Risks

The Credibility Estimators with Dependence Over Risks Applied Mathematical Sciences, Vol. 8, 2014, no. 161, 8045-8050 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2014.410803 The Credibility Estimators with Dependence Over Risks Qiang Zhang

More information

Gauge Plots. Gauge Plots JAPANESE BEETLE DATA MAXIMUM LIKELIHOOD FOR SPATIALLY CORRELATED DISCRETE DATA JAPANESE BEETLE DATA

Gauge Plots. Gauge Plots JAPANESE BEETLE DATA MAXIMUM LIKELIHOOD FOR SPATIALLY CORRELATED DISCRETE DATA JAPANESE BEETLE DATA JAPANESE BEETLE DATA 6 MAXIMUM LIKELIHOOD FOR SPATIALLY CORRELATED DISCRETE DATA Gauge Plots TuscaroraLisa Central Madsen Fairways, 996 January 9, 7 Grubs Adult Activity Grub Counts 6 8 Organic Matter

More information

Risk Aggregation with Dependence Uncertainty

Risk Aggregation with Dependence Uncertainty Introduction Extreme Scenarios Asymptotic Behavior Challenges Risk Aggregation with Dependence Uncertainty Department of Statistics and Actuarial Science University of Waterloo, Canada Seminar at ETH Zurich

More information

Dependence. Practitioner Course: Portfolio Optimization. John Dodson. September 10, Dependence. John Dodson. Outline.

Dependence. Practitioner Course: Portfolio Optimization. John Dodson. September 10, Dependence. John Dodson. Outline. Practitioner Course: Portfolio Optimization September 10, 2008 Before we define dependence, it is useful to define Random variables X and Y are independent iff For all x, y. In particular, F (X,Y ) (x,

More information

Polynomial approximation of mutivariate aggregate claim amounts distribution

Polynomial approximation of mutivariate aggregate claim amounts distribution Polynomial approximation of mutivariate aggregate claim amounts distribution Applications to reinsurance P.O. Goffard Axa France - Mathematics Institute of Marseille I2M Aix-Marseille University 19 th

More information

Bonus Malus Systems in Car Insurance

Bonus Malus Systems in Car Insurance Bonus Malus Systems in Car Insurance Corina Constantinescu Institute for Financial and Actuarial Mathematics Joint work with Weihong Ni Croatian Actuarial Conference, Zagreb, June 5, 2017 Car Insurance

More information

Simulating Exchangeable Multivariate Archimedean Copulas and its Applications. Authors: Florence Wu Emiliano A. Valdez Michael Sherris

Simulating Exchangeable Multivariate Archimedean Copulas and its Applications. Authors: Florence Wu Emiliano A. Valdez Michael Sherris Simulating Exchangeable Multivariate Archimedean Copulas and its Applications Authors: Florence Wu Emiliano A. Valdez Michael Sherris Literatures Frees and Valdez (1999) Understanding Relationships Using

More information

On the Importance of Dispersion Modeling for Claims Reserving: Application of the Double GLM Theory

On the Importance of Dispersion Modeling for Claims Reserving: Application of the Double GLM Theory On the Importance of Dispersion Modeling for Claims Reserving: Application of the Double GLM Theory Danaïl Davidov under the supervision of Jean-Philippe Boucher Département de mathématiques Université

More information

EXAMINATIONS OF THE HONG KONG STATISTICAL SOCIETY GRADUATE DIPLOMA, Statistical Theory and Methods I. Time Allowed: Three Hours

EXAMINATIONS OF THE HONG KONG STATISTICAL SOCIETY GRADUATE DIPLOMA, Statistical Theory and Methods I. Time Allowed: Three Hours EXAMINATIONS OF THE HONG KONG STATISTICAL SOCIETY GRADUATE DIPLOMA, 008 Statistical Theory and Methods I Time Allowed: Three Hours Candidates should answer FIVE questions. All questions carry equal marks.

More information

Non parametric estimation of Archimedean copulas and tail dependence. Paris, february 19, 2015.

Non parametric estimation of Archimedean copulas and tail dependence. Paris, february 19, 2015. Non parametric estimation of Archimedean copulas and tail dependence Elena Di Bernardino a and Didier Rullière b Paris, february 19, 2015. a CNAM, Paris, Département IMATH, b ISFA, Université Lyon 1, Laboratoire

More information

Estimation of Copula Models with Discrete Margins (via Bayesian Data Augmentation) Michael S. Smith

Estimation of Copula Models with Discrete Margins (via Bayesian Data Augmentation) Michael S. Smith Estimation of Copula Models with Discrete Margins (via Bayesian Data Augmentation) Michael S. Smith Melbourne Business School, University of Melbourne (Joint with Mohamad Khaled, University of Queensland)

More information

A simple bivariate count data regression model. Abstract

A simple bivariate count data regression model. Abstract A simple bivariate count data regression model Shiferaw Gurmu Georgia State University John Elder North Dakota State University Abstract This paper develops a simple bivariate count data regression model

More information

Discrete Choice Modeling

Discrete Choice Modeling [Part 6] 1/55 0 Introduction 1 Summary 2 Binary Choice 3 Panel Data 4 Bivariate Probit 5 Ordered Choice 6 7 Multinomial Choice 8 Nested Logit 9 Heterogeneity 10 Latent Class 11 Mixed Logit 12 Stated Preference

More information

Multivariate Distributions

Multivariate Distributions IEOR E4602: Quantitative Risk Management Spring 2016 c 2016 by Martin Haugh Multivariate Distributions We will study multivariate distributions in these notes, focusing 1 in particular on multivariate

More information

A strategy for modelling count data which may have extra zeros

A strategy for modelling count data which may have extra zeros A strategy for modelling count data which may have extra zeros Alan Welsh Centre for Mathematics and its Applications Australian National University The Data Response is the number of Leadbeater s possum

More information

Statistical Models for Defective Count Data

Statistical Models for Defective Count Data Statistical Models for Defective Count Data Gerhard Neubauer a, Gordana -Duraš a, and Herwig Friedl b a Statistical Applications, Joanneum Research, Graz, Austria b Institute of Statistics, University

More information

Financial Econometrics and Volatility Models Copulas

Financial Econometrics and Volatility Models Copulas Financial Econometrics and Volatility Models Copulas Eric Zivot Updated: May 10, 2010 Reading MFTS, chapter 19 FMUND, chapters 6 and 7 Introduction Capturing co-movement between financial asset returns

More information

Modelling Dropouts by Conditional Distribution, a Copula-Based Approach

Modelling Dropouts by Conditional Distribution, a Copula-Based Approach The 8th Tartu Conference on MULTIVARIATE STATISTICS, The 6th Conference on MULTIVARIATE DISTRIBUTIONS with Fixed Marginals Modelling Dropouts by Conditional Distribution, a Copula-Based Approach Ene Käärik

More information

Parameters Estimation Methods for the Negative Binomial-Crack Distribution and Its Application

Parameters Estimation Methods for the Negative Binomial-Crack Distribution and Its Application Original Parameters Estimation Methods for the Negative Binomial-Crack Distribution and Its Application Pornpop Saengthong 1*, Winai Bodhisuwan 2 Received: 29 March 2013 Accepted: 15 May 2013 Abstract

More information

Hybrid Copula Bayesian Networks

Hybrid Copula Bayesian Networks Kiran Karra kiran.karra@vt.edu Hume Center Electrical and Computer Engineering Virginia Polytechnic Institute and State University September 7, 2016 Outline Introduction Prior Work Introduction to Copulas

More information

Modelling the risk process

Modelling the risk process Modelling the risk process Krzysztof Burnecki Hugo Steinhaus Center Wroc law University of Technology www.im.pwr.wroc.pl/ hugo Modelling the risk process 1 Risk process If (Ω, F, P) is a probability space

More information

The Instability of Correlations: Measurement and the Implications for Market Risk

The Instability of Correlations: Measurement and the Implications for Market Risk The Instability of Correlations: Measurement and the Implications for Market Risk Prof. Massimo Guidolin 20254 Advanced Quantitative Methods for Asset Pricing and Structuring Winter/Spring 2018 Threshold

More information

Multivariate Distribution Models

Multivariate Distribution Models Multivariate Distribution Models Model Description While the probability distribution for an individual random variable is called marginal, the probability distribution for multiple random variables is

More information

Correlation: Copulas and Conditioning

Correlation: Copulas and Conditioning Correlation: Copulas and Conditioning This note reviews two methods of simulating correlated variates: copula methods and conditional distributions, and the relationships between them. Particular emphasis

More information

Copula Regression RAHUL A. PARSA DRAKE UNIVERSITY & STUART A. KLUGMAN SOCIETY OF ACTUARIES CASUALTY ACTUARIAL SOCIETY MAY 18,2011

Copula Regression RAHUL A. PARSA DRAKE UNIVERSITY & STUART A. KLUGMAN SOCIETY OF ACTUARIES CASUALTY ACTUARIAL SOCIETY MAY 18,2011 Copula Regression RAHUL A. PARSA DRAKE UNIVERSITY & STUART A. KLUGMAN SOCIETY OF ACTUARIES CASUALTY ACTUARIAL SOCIETY MAY 18,2011 Outline Ordinary Least Squares (OLS) Regression Generalized Linear Models

More information

Notes for Math 324, Part 19

Notes for Math 324, Part 19 48 Notes for Math 324, Part 9 Chapter 9 Multivariate distributions, covariance Often, we need to consider several random variables at the same time. We have a sample space S and r.v. s X, Y,..., which

More information

Introduction to Dependence Modelling

Introduction to Dependence Modelling Introduction to Dependence Modelling Carole Bernard Berlin, May 2015. 1 Outline Modeling Dependence Part 1: Introduction 1 General concepts on dependence. 2 in 2 or N 3 dimensions. 3 Minimizing the expectation

More information

Unconditional Distributions Obtained from Conditional Specification Models with Applications in Risk Theory

Unconditional Distributions Obtained from Conditional Specification Models with Applications in Risk Theory Unconditional Distributions Obtained from Conditional Specification Models with Applications in Risk Theory E. Gómez-Déniz a and E. Calderín Ojeda b Abstract Bivariate distributions, specified in terms

More information

Modelling dependent yearly claim totals including zero-claims in private health insurance

Modelling dependent yearly claim totals including zero-claims in private health insurance Modelling dependent yearly claim totals including zero-claims in private health insurance Vinzenz Erhardt, Claudia Czado Technische Universität München, Zentrum Mathematik, Boltzmannstr. 3, 85747 Garching,

More information

Multivariate Non-Normally Distributed Random Variables

Multivariate Non-Normally Distributed Random Variables Multivariate Non-Normally Distributed Random Variables An Introduction to the Copula Approach Workgroup seminar on climate dynamics Meteorological Institute at the University of Bonn 18 January 2008, Bonn

More information

Varieties of Count Data

Varieties of Count Data CHAPTER 1 Varieties of Count Data SOME POINTS OF DISCUSSION What are counts? What are count data? What is a linear statistical model? What is the relationship between a probability distribution function

More information

Contents 1. Contents

Contents 1. Contents Contents 1 Contents 6 Distributions of Functions of Random Variables 2 6.1 Transformation of Discrete r.v.s............. 3 6.2 Method of Distribution Functions............. 6 6.3 Method of Transformations................

More information

Statistical Model Of Road Traffic Crashes Data In Anambra State, Nigeria: A Poisson Regression Approach

Statistical Model Of Road Traffic Crashes Data In Anambra State, Nigeria: A Poisson Regression Approach Statistical Model Of Road Traffic Crashes Data In Anambra State, Nigeria: A Poisson Regression Approach Nwankwo Chike H., Nwaigwe Godwin I Abstract: Road traffic crashes are count (discrete) in nature.

More information

Using copulas to model time dependence in stochastic frontier models

Using copulas to model time dependence in stochastic frontier models Using copulas to model time dependence in stochastic frontier models Christine Amsler Michigan State University Artem Prokhorov Concordia University November 2008 Peter Schmidt Michigan State University

More information

Analysis of Count Data A Business Perspective. George J. Hurley Sr. Research Manager The Hershey Company Milwaukee June 2013

Analysis of Count Data A Business Perspective. George J. Hurley Sr. Research Manager The Hershey Company Milwaukee June 2013 Analysis of Count Data A Business Perspective George J. Hurley Sr. Research Manager The Hershey Company Milwaukee June 2013 Overview Count data Methods Conclusions 2 Count data Count data Anything with

More information

CHAPTER 1 INTRODUCTION

CHAPTER 1 INTRODUCTION CHAPTER 1 INTRODUCTION 1.0 Discrete distributions in statistical analysis Discrete models play an extremely important role in probability theory and statistics for modeling count data. The use of discrete

More information

Reducing Model Risk With Goodness-of-fit Victory Idowu London School of Economics

Reducing Model Risk With Goodness-of-fit Victory Idowu London School of Economics Reducing Model Risk With Goodness-of-fit Victory Idowu London School of Economics Agenda I. An overview of Copula Theory II. Copulas and Model Risk III. Goodness-of-fit methods for copulas IV. Presentation

More information

Default Priors and Effcient Posterior Computation in Bayesian

Default Priors and Effcient Posterior Computation in Bayesian Default Priors and Effcient Posterior Computation in Bayesian Factor Analysis January 16, 2010 Presented by Eric Wang, Duke University Background and Motivation A Brief Review of Parameter Expansion Literature

More information

Construction and estimation of high dimensional copulas

Construction and estimation of high dimensional copulas Construction and estimation of high dimensional copulas Gildas Mazo PhD work supervised by S. Girard and F. Forbes Mistis, Inria and laboratoire Jean Kuntzmann, Grenoble, France Séminaire Statistiques,

More information

Multivariate Normal-Laplace Distribution and Processes

Multivariate Normal-Laplace Distribution and Processes CHAPTER 4 Multivariate Normal-Laplace Distribution and Processes The normal-laplace distribution, which results from the convolution of independent normal and Laplace random variables is introduced by

More information

Trivariate copulas for characterisation of droughts

Trivariate copulas for characterisation of droughts ANZIAM J. 49 (EMAC2007) pp.c306 C323, 2008 C306 Trivariate copulas for characterisation of droughts G. Wong 1 M. F. Lambert 2 A. V. Metcalfe 3 (Received 3 August 2007; revised 4 January 2008) Abstract

More information

High-Throughput Sequencing Course

High-Throughput Sequencing Course High-Throughput Sequencing Course DESeq Model for RNA-Seq Biostatistics and Bioinformatics Summer 2017 Outline Review: Standard linear regression model (e.g., to model gene expression as function of an

More information

On the existence of maximum likelihood estimators in a Poissongamma HGLM and a negative binomial regression model

On the existence of maximum likelihood estimators in a Poissongamma HGLM and a negative binomial regression model Int. Statistical Inst.: Proc. 58th World Statistical Congress, 2011, Dublin Session CPS027) p.6433 On the existence of maximum likelihood estimators in a Poissongamma HGLM and a negative binomial regression

More information

CTDL-Positive Stable Frailty Model

CTDL-Positive Stable Frailty Model CTDL-Positive Stable Frailty Model M. Blagojevic 1, G. MacKenzie 2 1 Department of Mathematics, Keele University, Staffordshire ST5 5BG,UK and 2 Centre of Biostatistics, University of Limerick, Ireland

More information

MULTIDIMENSIONAL POVERTY MEASUREMENT: DEPENDENCE BETWEEN WELL-BEING DIMENSIONS USING COPULA FUNCTION

MULTIDIMENSIONAL POVERTY MEASUREMENT: DEPENDENCE BETWEEN WELL-BEING DIMENSIONS USING COPULA FUNCTION Rivista Italiana di Economia Demografia e Statistica Volume LXXII n. 3 Luglio-Settembre 2018 MULTIDIMENSIONAL POVERTY MEASUREMENT: DEPENDENCE BETWEEN WELL-BEING DIMENSIONS USING COPULA FUNCTION Kateryna

More information

Lecture 2: Repetition of probability theory and statistics

Lecture 2: Repetition of probability theory and statistics Algorithms for Uncertainty Quantification SS8, IN2345 Tobias Neckel Scientific Computing in Computer Science TUM Lecture 2: Repetition of probability theory and statistics Concept of Building Block: Prerequisites:

More information

Report and Opinion 2016;8(6) Analysis of bivariate correlated data under the Poisson-gamma model

Report and Opinion 2016;8(6)   Analysis of bivariate correlated data under the Poisson-gamma model Analysis of bivariate correlated data under the Poisson-gamma model Narges Ramooz, Farzad Eskandari 2. MSc of Statistics, Allameh Tabatabai University, Tehran, Iran 2. Associate professor of Statistics,

More information

Graduate Econometrics I: What is econometrics?

Graduate Econometrics I: What is econometrics? Graduate Econometrics I: What is econometrics? Yves Dominicy Université libre de Bruxelles Solvay Brussels School of Economics and Management ECARES Yves Dominicy Graduate Econometrics I: What is econometrics?

More information

Vine copulas with asymmetric tail dependence and applications to financial return data 1. Abstract

Vine copulas with asymmetric tail dependence and applications to financial return data 1. Abstract *Manuscript Vine copulas with asymmetric tail dependence and applications to financial return data 1 Aristidis K. Nikoloulopoulos 2, Harry Joe 3 and Haijun Li 4 Abstract In Aas et al. (2009) and Aas and

More information

Correlation & Dependency Structures

Correlation & Dependency Structures Correlation & Dependency Structures GIRO - October 1999 Andrzej Czernuszewicz Dimitris Papachristou Why are we interested in correlation/dependency? Risk management Portfolio management Reinsurance purchase

More information

Hidden Markov models for time series of counts with excess zeros

Hidden Markov models for time series of counts with excess zeros Hidden Markov models for time series of counts with excess zeros Madalina Olteanu and James Ridgway University Paris 1 Pantheon-Sorbonne - SAMM, EA4543 90 Rue de Tolbiac, 75013 Paris - France Abstract.

More information

Package CopulaRegression

Package CopulaRegression Type Package Package CopulaRegression Title Bivariate Copula Based Regression Models Version 0.1-5 Depends R (>= 2.11.0), MASS, VineCopula Date 2014-09-04 Author, Daniel Silvestrini February 19, 2015 Maintainer

More information

The LmB Conferences on Multivariate Count Analysis

The LmB Conferences on Multivariate Count Analysis The LmB Conferences on Multivariate Count Analysis Title: On Poisson-exponential-Tweedie regression models for ultra-overdispersed count data Rahma ABID, C.C. Kokonendji & A. Masmoudi Email Address: rahma.abid.ch@gmail.com

More information

MODELING COUNT DATA Joseph M. Hilbe

MODELING COUNT DATA Joseph M. Hilbe MODELING COUNT DATA Joseph M. Hilbe Arizona State University Count models are a subset of discrete response regression models. Count data are distributed as non-negative integers, are intrinsically heteroskedastic,

More information

2. Inference method for margins and jackknife.

2. Inference method for margins and jackknife. The Estimation Method of Inference Functions for Margins for Multivariate Models Harry Joe and James J. Xu Department of Statistics, University of British Columbia ABSTRACT An estimation approach is proposed

More information

Describing Contingency tables

Describing Contingency tables Today s topics: Describing Contingency tables 1. Probability structure for contingency tables (distributions, sensitivity/specificity, sampling schemes). 2. Comparing two proportions (relative risk, odds

More information

STAT5044: Regression and Anova

STAT5044: Regression and Anova STAT5044: Regression and Anova Inyoung Kim 1 / 18 Outline 1 Logistic regression for Binary data 2 Poisson regression for Count data 2 / 18 GLM Let Y denote a binary response variable. Each observation

More information

VaR bounds in models with partial dependence information on subgroups

VaR bounds in models with partial dependence information on subgroups VaR bounds in models with partial dependence information on subgroups L. Rüschendorf J. Witting February 23, 2017 Abstract We derive improved estimates for the model risk of risk portfolios when additional

More information

Lifetime Dependence Modelling using a Generalized Multivariate Pareto Distribution

Lifetime Dependence Modelling using a Generalized Multivariate Pareto Distribution Lifetime Dependence Modelling using a Generalized Multivariate Pareto Distribution Daniel Alai Zinoviy Landsman Centre of Excellence in Population Ageing Research (CEPAR) School of Mathematics, Statistics

More information

Probability and Stochastic Processes

Probability and Stochastic Processes Probability and Stochastic Processes A Friendly Introduction Electrical and Computer Engineers Third Edition Roy D. Yates Rutgers, The State University of New Jersey David J. Goodman New York University

More information

Unobserved Heterogeneity and the Statistical Analysis of Highway Accident Data. Fred Mannering University of South Florida

Unobserved Heterogeneity and the Statistical Analysis of Highway Accident Data. Fred Mannering University of South Florida Unobserved Heterogeneity and the Statistical Analysis of Highway Accident Data Fred Mannering University of South Florida Highway Accidents Cost the lives of 1.25 million people per year Leading cause

More information

Calculation of Bayes Premium for Conditional Elliptical Risks

Calculation of Bayes Premium for Conditional Elliptical Risks 1 Calculation of Bayes Premium for Conditional Elliptical Risks Alfred Kume 1 and Enkelejd Hashorva University of Kent & University of Lausanne February 1, 13 Abstract: In this paper we discuss the calculation

More information

Fall 2017 STAT 532 Homework Peter Hoff. 1. Let P be a probability measure on a collection of sets A.

Fall 2017 STAT 532 Homework Peter Hoff. 1. Let P be a probability measure on a collection of sets A. 1. Let P be a probability measure on a collection of sets A. (a) For each n N, let H n be a set in A such that H n H n+1. Show that P (H n ) monotonically converges to P ( k=1 H k) as n. (b) For each n

More information

Mixture distributions in Exams MLC/3L and C/4

Mixture distributions in Exams MLC/3L and C/4 Making sense of... Mixture distributions in Exams MLC/3L and C/4 James W. Daniel Jim Daniel s Actuarial Seminars www.actuarialseminars.com February 1, 2012 c Copyright 2012 by James W. Daniel; reproduction

More information

Creating New Distributions

Creating New Distributions Creating New Distributions Section 5.2 Stat 477 - Loss Models Section 5.2 (Stat 477) Creating New Distributions Brian Hartman - BYU 1 / 18 Generating new distributions Some methods to generate new distributions

More information

Point process with spatio-temporal heterogeneity

Point process with spatio-temporal heterogeneity Point process with spatio-temporal heterogeneity Jony Arrais Pinto Jr Universidade Federal Fluminense Universidade Federal do Rio de Janeiro PASI June 24, 2014 * - Joint work with Dani Gamerman and Marina

More information

Copula modeling for discrete data

Copula modeling for discrete data Copula modeling for discrete data Christian Genest & Johanna G. Nešlehová in collaboration with Bruno Rémillard McGill University and HEC Montréal ROBUST, September 11, 2016 Main question Suppose (X 1,

More information

Nonparametric Estimation of the Dependence Function for a Multivariate Extreme Value Distribution

Nonparametric Estimation of the Dependence Function for a Multivariate Extreme Value Distribution Nonparametric Estimation of the Dependence Function for a Multivariate Extreme Value Distribution p. /2 Nonparametric Estimation of the Dependence Function for a Multivariate Extreme Value Distribution

More information

Stat 704 Data Analysis I Probability Review

Stat 704 Data Analysis I Probability Review 1 / 39 Stat 704 Data Analysis I Probability Review Dr. Yen-Yi Ho Department of Statistics, University of South Carolina A.3 Random Variables 2 / 39 def n: A random variable is defined as a function that

More information

Tail Dependence of Multivariate Pareto Distributions

Tail Dependence of Multivariate Pareto Distributions !#"%$ & ' ") * +!-,#. /10 243537698:6 ;=@?A BCDBFEHGIBJEHKLB MONQP RS?UTV=XW>YZ=eda gihjlknmcoqprj stmfovuxw yy z {} ~ ƒ }ˆŠ ~Œ~Ž f ˆ ` š œžÿ~ ~Ÿ œ } ƒ œ ˆŠ~ œ

More information

Modeling Longitudinal Count Data with Excess Zeros and Time-Dependent Covariates: Application to Drug Use

Modeling Longitudinal Count Data with Excess Zeros and Time-Dependent Covariates: Application to Drug Use Modeling Longitudinal Count Data with Excess Zeros and : Application to Drug Use University of Northern Colorado November 17, 2014 Presentation Outline I and Data Issues II Correlated Count Regression

More information

Some results on Denault s capital allocation rule

Some results on Denault s capital allocation rule Faculty of Economics and Applied Economics Some results on Denault s capital allocation rule Steven Vanduffel and Jan Dhaene DEPARTMENT OF ACCOUNTANCY, FINANCE AND INSURANCE (AFI) AFI 0601 Some Results

More information

On the Systemic Nature of Weather Risk

On the Systemic Nature of Weather Risk Martin Odening 1 Ostap Okhrin 2 Wei Xu 1 Department of Agricultural Economics 1 Ladislaus von Bortkiewicz Chair of Statistics C.A.S.E. Center for Applied Statistics and Economics 2 Humboldt Universität

More information

Individual loss reserving with the Multivariate Skew Normal framework

Individual loss reserving with the Multivariate Skew Normal framework May 22 2013, K. Antonio, KU Leuven and University of Amsterdam 1 / 31 Individual loss reserving with the Multivariate Skew Normal framework Mathieu Pigeon, Katrien Antonio, Michel Denuit ASTIN Colloquium

More information

The combined model: A tool for simulating correlated counts with overdispersion. George KALEMA Samuel IDDI Geert MOLENBERGHS

The combined model: A tool for simulating correlated counts with overdispersion. George KALEMA Samuel IDDI Geert MOLENBERGHS The combined model: A tool for simulating correlated counts with overdispersion George KALEMA Samuel IDDI Geert MOLENBERGHS Interuniversity Institute for Biostatistics and statistical Bioinformatics George

More information