Contents

Cover

Half Title page

Title page

Copyright page

Preface

Chapter 1: Introduction

Chapter 2: Coxian and Related Distributions

2.1 Introduction

2.2 Combinations of Exponentials

2.3 Coxian-2 Distributions

Chapter 3: Mixed Erlang Distributions

3.1 Introduction

3.2 Members of the Mixed Erlang Class

3.3 Distributional Properties

3.4 Mixed Erlang Claim Severity Models

Chapter 4: Extreme Value Distributions

4.1 Introduction

4.2 Distribution of the Maximum

4.3 Stability of the Maximum of the Extreme Value Distribution

4.4 The Fisher–Tippett theorem

4.5 Maximum Domain of Attraction

4.6 Generalized Pareto Distributions

4.7 Stability of Excesses of the Generalized Pareto

4.8 Limiting Distributions of Excesses

4.9 Parameter Estimation

Chapter 5: Analytic and Related Methods for Aggregate Claim Models

5.1 Introduction

5.2 Elementary Approaches

5.3 Discrete Analogues

5.4 Right-Tail Asymptotics for Aggregate Losses

Chapter 6: Computational Methods for Aggregate Models

6.1 Recursive Techniques for Compound Distributions

6.2 Inversion Methods

6.3 Calculations with Approximate Distributions

6.4 Comparison of Methods

6.5 The Individual Risk Model

Chapter 7: Counting Processes

7.1 Nonhomogeneous Birth Processes

7.2 Mixed Poisson Processes

Chapter 8: Discrete Claim Count Models

8.1 Unification of the (a, b, 1) and Mixed Poisson Classes

8.2 A Class of Discrete Generalized Tail-Based Distributions

8.3 Higher Order Generalized Tail-Based Distributions

8.4 Mixed Poisson Properties of Generalized Tail-Based Distributions

8.5 Compound Geometric Properties of Generalized Tail-Based Distributions

Chapter 9: Compound Distributions with Time Dependent Claim Amounts

9.1 Introduction

9.2 A Model for Inflation

9.3 A Model for Claim Payment Delays

Chapter 10: Copula Models

10.1 Introduction

10.2 Sklar’s Theorem and Copulas

10.3 Measures of Dependency

10.4 Tail Dependence

10.5 Archimedean Copulas

10.6 Elliptical Copulas

10.7 Extreme Value Copulas

10.8 Archimax Copulas

10.9 Estimation of Parameters

10.10 Simulation from Copula Models

Chapter 11: Continuous-Time Ruin Models

11.1 Introduction

11.2 The Adjustment Coefficient and Lundberg’s Inequality

11.3 An Integrodifferential Equation

11.4 The Maximum Aggregate Loss

11.5 Cramer’s Asymptotic Ruin Formula and Tijms’ Approximation

11.6 The Brownian Motion Risk Process

11.7 Brownian Motion and the Probability of Ruin

Chapter 12: Interpolation and Smoothing

12.1 Introduction

12.2 Interpolation with Splines

12.3 Extrapolating with Splines

12.4 Smoothing with Splines

Appendix A: An Inventory of Continuous Distributions

A.1 Introduction

A.2 Transformed Beta Family

A.3 Transformed Gamma Family

A.4 Distributions for Large Losses

A.5 Other Distributions

A.6 Distributions with Finite Support

Appendix B: An Inventory of Discrete Distributions

B.1 Introduction

B.2 The (a, b, 0) Class

B.3 The (a, b, 1) Class

B.4 The Compound Class

B.5 A Hierarchy of Discrete Distributions

Appendix C: Discretization of the Severity Distribution

C.1 The Method of Rounding

C.2 Mean Preserving

C.3 Undiscretization of a Discretized Distribution

Appendix D: Solutions to Exercises

D.1 Chapter 4

D.2 Chapter 5

D.3 Chapter 6

D.4 Chapter 7

D.5 Chapter 8

D.6 Chapter 10

D.7 Chapter 11

D.8 Chapter 12

References

Index

LOSS MODELS

WILEY SERIES IN PROBABILITY AND STATISTICS

ESTABLISHED BY WALTER A. SHEWHART AND SAMUEL S. WILKS

Editors: David J. Balding, Noel A. C. Cressie, Garrett M. Fitzmaurice, Harvey Goldstein, Iain M. Johnstone, Geert Molenberghs, David W. Scott, Adrian F. M. Smith, Ruey S. Tsay, Sanford Weisberg

Editors Emeriti: Vic Barnett, J. Stuart Hunter, Joseph B. Kadane, Jozef L. Teugels

The Wiley Series in Probability and Statistics is well established and authoritative. It covers many topics of current research interest in both pure and applied statistics and probability theory. Written by leading statisticians and institutions, the titles span both state-of-the-art developments in the field and classical methods.

Reflecting the wide range of current research in statistics, the series encompasses applied, methodological and theoretical statistics, ranging from applications and new techniques made possible by advances in computerized practice to rigorous treatment of theoretical approaches.

This series provides essential and invaluable reading for all statisticians, whether in academia, industry, government, or research.

^† ABRAHAM and LEDOLTER · Statistical Methods for Forecasting

AGRESTI · Analysis of Ordinal Categorical Data, Second Edition

AGRESTI · An Introduction to Categorical Data Analysis, Second Edition

AGRESTI · Categorical Data Analysis, Second Edition

ALTMAN, GILL, and McDONALD · Numerical Issues in Statistical Computing for the Social Scientist

AMARATUNGA and CABRERA · Exploration and Analysis of DNA Microarray and Protein Array Data

ANDĚL · Mathematics of Chance

ANDERSON · An Introduction to Multivariate Statistical Analysis, Third Edition

* ANDERSON · The Statistical Analysis of Time Series

ANDERSON, AUQUIER, HAUCK, OAKES, VANDAELE, and WEISBERG · Statistical Methods for Comparative Studies

ANDERSON and LOYNES · The Teaching of Practical Statistics

ARMITAGE and DAVID (editors) · Advances in Biometry

ARNOLD, BALAKRISHNAN, and NAGARAJA · Records

* ARTHANARI and DODGE · Mathematical Programming in Statistics

* BAILEY · The Elements of Stochastic Processes with Applications to the Natural Sciences

BAJORSKI · Statistics for Imaging, Optics, and Photonics

BALAKRISHNAN and KOUTRAS · Runs and Scans with Applications

BALAKRISHNAN and NG · Precedence-Type Tests and Applications

BARNETT · Comparative Statistical Inference, Third Edition

BARNETT · Environmental Statistics

BARNETT and LEWIS · Outliers in Statistical Data, Third Edition

BARTHOLOMEW, KNOTT, and MOUSTAKI · Latent Variable Models and Factor Analysis: A Unified Approach, Third Edition

BARTOSZYNSKI and NIEWIADOMSKA-BUGAJ · Probability and Statistical Inference, Second Edition

BASILEVSKY · Statistical Factor Analysis and Related Methods: Theory and Applications

BATES and WATTS · Nonlinear Regression Analysis and Its Applications

BECHHOFER, SANTNER, and GOLDSMAN · Design and Analysis of Experiments for Statistical Selection, Screening, and Multiple Comparisons

BEIRLANT, GOEGEBEUR, SEGERS, TEUGELS, and DE WAAL · Statistics of Extremes: Theory and Applications

BELSLEY · Conditioning Diagnostics: Collinearity and Weak Data in Regression

^† BELSLEY, KUH, and WELSCH · Regression Diagnostics: Identifying Influential Data and Sources of Collinearity

BENDAT and PIERSOL · Random Data: Analysis and Measurement Procedures, Fourth Edition

BERNARDO and SMITH · Bayesian Theory

BHAT and MILLER · Elements of Applied Stochastic Processes, Third Edition

BHATTACHARYA and WAYMIRE · Stochastic Processes with Applications

BIEMER, GROVES, LYBERG, MATHIOWETZ, and SUDMAN · Measurement Errors in Surveys

BILLINGSLEY · Convergence of Probability Measures, Second Edition

BILLINGSLEY · Probability and Measure, Anniversary Edition

BIRKES and DODGE · Alternative Methods of Regression

BISGAARD and KULAHCI · Time Series Analysis and Forecasting by Example

BISWAS, DATTA, FINE, and SEGAL · Statistical Advances in the Biomedical Sciences: Clinical Trials, Epidemiology, Survival Analysis, and Bioinformatics

BLISCHKE and MURTHY (editors) · Case Studies in Reliability and Maintenance

BLISCHKE and MURTHY · Reliability: Modeling, Prediction, and Optimization

BLOOMFIELD · Fourier Analysis of Time Series: An Introduction, Second Edition

BOLLEN · Structural Equations with Latent Variables

BOLLEN and CURRAN · Latent Curve Models: A Structural Equation Perspective

BOROVKOV · Ergodicity and Stability of Stochastic Processes

BOSQ and BLANKE · Inference and Prediction in Large Dimensions

BOULEAU · Numerical Methods for Stochastic Processes

* BOX and TIAO · Bayesian Inference in Statistical Analysis

BOX · Improving Almost Anything, Revised Edition

* BOX and DRAPER · Evolutionary Operation: A Statistical Method for Process Improvement

BOX and DRAPER · Response Surfaces, Mixtures, and Ridge Analyses, Second Edition

BOX, HUNTER, and HUNTER · Statistics for Experimenters: Design, Innovation, and Discovery, Second Editon

BOX, JENKINS, and REINSEL · Time Series Analysis: Forcasting and Control, Fourth Edition

BOX, LUCEÑO, and PANIAGUA-QUIÑONES · Statistical Control by Monitoring and Adjustment, Second Edition

* BROWN and HOLLANDER · Statistics: A Biomedical Introduction

CAIROLI and DALANG · Sequential Stochastic Optimization

CASTILLO, HADI, BALAKRISHNAN, and SARABIA · Extreme Value and Related Models with Applications in Engineering and Science

CHAN · Time Series: Applications to Finance with R and S-Plus^®, Second Edition

CHARALAMBIDES · Combinatorial Methods in Discrete Distributions

CHATTERJEE and HADI · Regression Analysis by Example, Fourth Edition

CHATTERJEE and HADI · Sensitivity Analysis in Linear Regression

CHERNICK · Bootstrap Methods: A Guide for Practitioners and Researchers, Second Edition

CHERNICK and FRIIS · Introductory Biostatistics for the Health Sciences

CHILÈS and DELFINER · Geostatistics: Modeling Spatial Uncertainty, Second Edition

CHOW and LIU · Design and Analysis of Clinical Trials: Concepts and Methodologies, Second Edition

CLARKE · Linear Models: The Theory and Application of Analysis of Variance

CLARKE and DISNEY · Probability and Random Processes: A First Course with Applications, Second Edition

* COCHRAN and COX · Experimental Designs, Second Edition

COLLINS and LANZA · Latent Class and Latent Transition Analysis: With Applications in the Social, Behavioral, and Health Sciences

CONGDON · Applied Bayesian Modelling

CONGDON · Bayesian Models for Categorical Data

CONGDON · Bayesian Statistical Modelling, Second Edition

CONOVER · Practical Nonparametric Statistics, Third Edition

COOK · Regression Graphics

COOK and WEISBERG · An Introduction to Regression Graphics

COOK and WEISBERG · Applied Regression Including Computing and Graphics

CORNELL · A Primer on Experiments with Mixtures

CORNELL · Experiments with Mixtures, Designs, Models, and the Analysis of Mixture Data, Third Edition

COX · A Handbook of Introductory Statistical Methods

CRESSIE · Statistics for Spatial Data, Revised Edition

CRESSIE and WIKLE · Statistics for Spatio-Temporal Data

CSÖRGŐ and HORVÁTH · Limit Theorems in Change Point Analysis

DAGPUNAR · Simulation and Monte Carlo: With Applications in Finance and MCMC

DANIEL · Applications of Statistics to Industrial Experimentation

DANIEL · Biostatistics: A Foundation for Analysis in the Health Sciences, Eighth Edition

* DANIEL · Fitting Equations to Data: Computer Analysis of Multifactor Data, Second Edition

DASU and JOHNSON · Exploratory Data Mining and Data Cleaning

DAVID and NAGARAJA · Order Statistics, Third Edition

* DEGROOT, FIENBERG, and KADANE · Statistics and the Law

DEL CASTILLO · Statistical Process Adjustment for Quality Control

DEMARIS · Regression with Social Data: Modeling Continuous and Limited Response Variables

DEMIDENKO · Mixed Models: Theory and Applications

DENISON, HOLMES, MALLICK and SMITH · Bayesian Methods for Nonlinear Classification and Regression

DETTE and STUDDEN · The Theory of Canonical Moments with Applications in Statistics, Probability, and Analysis

DEY and MUKERJEE · Fractional Factorial Plans

DILLON and GOLDSTEIN · Multivariate Analysis: Methods and Applications

* DODGE and ROMIG · Sampling Inspection Tables, Second Edition

* DOOB · Stochastic Processes

DOWDY, WEARDEN, and CHILKO · Statistics for Research, Third Edition

DRAPER and SMITH · Applied Regression Analysis, Third Edition

DRYDEN and MARDIA · Statistical Shape Analysis

DUDEWICZ and MISHRA · Modern Mathematical Statistics

DUNN and CLARK · Basic Statistics: A Primer for the Biomedical Sciences, Fourth Edition

DUPUIS and ELLIS · A Weak Convergence Approach to the Theory of Large Deviations

EDLER and KITSOS · Recent Advances in Quantitative Methods in Cancer and Human Health Risk Assessment

* ELANDT-JOHNSON and JOHNSON · Survival Models and Data Analysis

ENDERS · Applied Econometric Time Series, Third Edition

^† ETHIER and KURTZ · Markov Processes: Characterization and Convergence

EVANS, HASTINGS, and PEACOCK · Statistical Distributions, Third Edition

EVERITT, LANDAU, LEESE, and STAHL · Cluster Analysis, Fifth Edition

FEDERER and KING · Variations on Split Plot and Split Block Experiment Designs

FELLER · An Introduction to Probability Theory and Its Applications, Volume I, Third Edition, Revised; Volume II, Second Edition

FITZMAURICE, LAIRD, and WARE · Applied Longitudinal Analysis, Second Edition

* FLEISS · The Design and Analysis of Clinical Experiments

FLEISS · Statistical Methods for Rates and Proportions, Third Edition

^† FLEMING and HARRINGTON · Counting Processes and Survival Analysis

FUJIKOSHI, ULYANOV, and SHIMIZU · Multivariate Statistics: High-Dimensional and Large-Sample Approximations

FULLER · Introduction to Statistical Time Series, Second Edition

^† FULLER · Measurement Error Models

GALLANT · Nonlinear Statistical Models

GEISSER · Modes of Parametric Statistical Inference

GELMAN and MENG · Applied Bayesian Modeling and Causal Inference from ncomplete-Data Perspectives

GEWEKE · Contemporary Bayesian Econometrics and Statistics

GHOSH, MUKHOPADHYAY, and SEN · Sequential Estimation

GIESBRECHT and GUMPERTZ · Planning, Construction, and Statistical Analysis of Comparative Experiments

GIFI · Nonlinear Multivariate Analysis

GIVENS and HOETING · Computational Statistics

GLASSERMAN and YAO · Monotone Structure in Discrete-Event Systems

GNANADESIKAN · Methods for Statistical Data Analysis of Multivariate Observations, Second Edition

GOLDSTEIN · Multilevel Statistical Models, Fourth Edition

GOLDSTEIN and LEWIS · Assessment: Problems, Development, and Statistical Issues

GOLDSTEIN and WOOFF · Bayes Linear Statistics

GREENWOOD and NIKULIN · A Guide to Chi-Squared Testing

GROSS, SHORTLE, THOMPSON, and HARRIS · Fundamentals of Queueing Theory, Fourth Edition

GROSS, SHORTLE, THOMPSON, and HARRIS · Solutions Manual to Accompany Fundamentals of Queueing Theory, Fourth Edition

* HAHN and SHAPIRO · Statistical Models in Engineering

HAHN and MEEKER · Statistical Intervals: A Guide for Practitioners

HALD · A History of Probability and Statistics and their Applications Before 1750

^† HAMPEL · Robust Statistics: The Approach Based on Influence Functions

HARTUNG, KNAPP, and SINHA · Statistical Meta-Analysis with Applications

HEIBERGER · Computation for the Analysis of Designed Experiments

HEDAYAT and SINHA · Design and Inference in Finite Population Sampling

HEDEKER and GIBBONS · Longitudinal Data Analysis

HELLER · MACSYMA for Statisticians

HERITIER, CANTONI, COPT, and VICTORIA-FESER · Robust Methods in Biostatistics

HINKELMANN and KEMPTHORNE · Design and Analysis of Experiments, Volume 1: Introduction to Experimental Design, Second Edition

HINKELMANN and KEMPTHORNE · Design and Analysis of Experiments, Volume 2: Advanced Experimental Design

HINKELMANN (editor) · Design and Analysis of Experiments, Volume 3: Special Designs and Applications

HOAGLIN, MOSTELLER, and TUKEY · Fundamentals of Exploratory Analysis of Variance

* HOAGLIN, MOSTELLER, and TUKEY · Exploring Data Tables, Trends and Shapes

* HOAGLIN, MOSTELLER, and TUKEY · Understanding Robust and Exploratory Data Analysis

HOCHBERG and TAMHANE · Multiple Comparison Procedures

HOCKING · Methods and Applications of Linear Models: Regression and the Analysis of Variance, Second Edition

HOEL · Introduction to Mathematical Statistics, Fifth Edition

HOGG and KLUGMAN · Loss Distributions

HOLLANDER, WOLFE, and CHICKEN · Nonparametric Statistical Methods, Third Edition

HOSMER and LEMESHOW · Applied Logistic Regression, Second Edition

HOSMER, LEMESHOW, and MAY · Applied Survival Analysis: Regression Modeling of Time-to-Event Data, Second Edition

HUBER · Data Analysis: What Can Be Learned From the Past 50 Years

HUBER · Robust Statistics

^† HUBER and RONCHETTI · Robust Statistics, Second Edition

HUBERTY · Applied Discriminant Analysis, Second Edition

HUBERTY and OLEJNIK · Applied MANOVA and Discriminant Analysis, Second Edition

HUITEMA · The Analysis of Covariance and Alternatives: Statistical Methods for Experiments, Quasi-Experiments, and Single-Case Studies, Second Edition

HUNT and KENNEDY · Financial Derivatives in Theory and Practice, Revised Edition

HURD and MIAMEE · Periodically Correlated Random Sequences: Spectral Theory and Practice

HUSKOVA, BERAN, and DUPAC · Collected Works of Jaroslav Hajek—with Commentary

HUZURBAZAR · Flowgraph Models for Multistate Time-to-Event Data

JACKMAN · Bayesian Analysis for the Social Sciences

^† JACKSON · A User’s Guide to Principle Components

JOHN · Statistical Methods in Engineering and Quality Assurance

JOHNSON · Multivariate Statistical Simulation

JOHNSON and BALAKRISHNAN · Advances in the Theory and Practice of Statistics: A Volume in Honor of Samuel Kotz

JOHNSON, KEMP, and KOTZ · Univariate Discrete Distributions, Third Edition

JOHNSON and KOTZ (editors) · Leading Personalities in Statistical Sciences: From the Seventeenth Century to the Present

JOHNSON, KOTZ, and BALAKRISHNAN · Continuous Univariate Distributions, Volume 1, Second Edition

JOHNSON, KOTZ, and BALAKRISHNAN · Continuous Univariate Distributions, Volume 2, Second Edition

JOHNSON, KOTZ, and BALAKRISHNAN · Discrete Multivariate Distributions

JUDGE, GRIFFITHS, HILL, LÜTKEPOHL, and LEE · The Theory and Practice of Econometrics, Second Edition

JUREK and MASON · Operator-Limit Distributions in Probability Theory

KADANE · Bayesian Methods and Ethics in a Clinical Trial Design

KADANE AND SCHUM · A Probabilistic Analysis of the Sacco and Vanzetti Evidence

KALBFLEISCH and PRENTICE · The Statistical Analysis of Failure Time Data, Second Edition

KARIYA and KURATA · Generalized Least Squares

KASS and VOS · Geometrical Foundations of Asymptotic Inference

^† KAUFMAN and ROUSSEEUW · Finding Groups in Data: An Introduction to Cluster Analysis

KEDEM and FOKIANOS · Regression Models for Time Series Analysis

KENDALL, BARDEN, CARNE, and LE · Shape and Shape Theory

KHURI · Advanced Calculus with Applications in Statistics, Second Edition

KHURI, MATHEW, and SINHA · Statistical Tests for Mixed Linear Models

* KISH · Statistical Design for Research

KLEIBER and KOTZ · Statistical Size Distributions in Economics and Actuarial Sciences

KLEMELÄ · Smoothing of Multivariate Data: Density Estimation and Visualization

KLUGMAN, PANJER, and WILLMOT · Loss Models: From Data to Decisions, Third Edition

KLUGMAN, PANJER, and WILLMOT · Loss Models: Further Topics

KLUGMAN, PANJER, and WILLMOT · Solutions Manual to Accompany Loss Models: From Data to Decisions, Third Edition

KOSKI and NOBLE · Bayesian Networks: An Introduction

KOTZ, BALAKRISHNAN, and JOHNSON · Continuous Multivariate Distributions, Volume 1, Second Edition

KOTZ and JOHNSON (editors) · Encyclopedia of Statistical Sciences: Volumes 1 to 9 with Index

KOTZ and JOHNSON (editors) · Encyclopedia of Statistical Sciences: Supplement Volume

KOTZ, READ, and BANKS (editors) · Encyclopedia of Statistical Sciences: Update Volume 1

KOTZ, READ, and BANKS (editors) · Encyclopedia of Statistical Sciences: Update Volume 2

KOWALSKI and TU · Modern Applied U-Statistics

KRISHNAMOORTHY and MATHEW · Statistical Tolerance Regions: Theory, Applications, and Computation

KROESE, TAIMRE, and BOTEV · Handbook of Monte Carlo Methods

KROONENBERG · Applied Multiway Data Analysis

KULINSKAYA, MORGENTHALER, and STAUDTE · Meta Analysis: A Guide to Calibrating and Combining Statistical Evidence

KULKARNI and HARMAN · An Elementary Introduction to Statistical Learning Theory

KUROWICKA and COOKE · Uncertainty Analysis with High Dimensional Dependence Modelling

KVAM and VIDAKOVIC · Nonparametric Statistics with Applications to Science and Engineering

LACHIN · Biostatistical Methods: The Assessment of Relative Risks, Second Edition

LAD · Operational Subjective Statistical Methods: A Mathematical, Philosophical, and Historical Introduction

LAMPERTI · Probability: A Survey of the Mathematical Theory, Second Edition

LAWLESS · Statistical Models and Methods for Lifetime Data, Second Edition

LAWSON · Statistical Methods in Spatial Epidemiology, Second Edition

LE · Applied Categorical Data Analysis, Second Edition

LE · Applied Survival Analysis

LEE · Structural Equation Modeling: A Bayesian Approach

LEE and WANG · Statistical Methods for Survival Data Analysis, Fourth Edition

LEPAGE and BILLARD · Exploring the Limits of Bootstrap

LESSLER and KALSBEEK · Nonsampling Errors in Surveys

LEYLAND and GOLDSTEIN (editors) · Multilevel Modelling of Health Statistics

LIAO · Statistical Group Comparison

LIN · Introductory Stochastic Analysis for Finance and Insurance

LITTLE and RUBIN · Statistical Analysis with Missing Data, Second Edition

LLOYD · The Statistical Analysis of Categorical Data

LOWEN and TEICH · Fractal-Based Point Processes

MAGNUS and NEUDECKER · Matrix Differential Calculus with Applications in Statistics and Econometrics, Revised Edition

MALLER and ZHOU · Survival Analysis with Long Term Survivors

MARCHETTE · Random Graphs for Statistical Pattern Recognition

MARDIA and JUPP · Directional Statistics

MARKOVICH · Nonparametric Analysis of Univariate Heavy-Tailed Data: Research and Practice

MARONNA, MARTIN and YOHAI · Robust Statistics: Theory and Methods

MASON, GUNST, and HESS · Statistical Design and Analysis of Experiments with Applications to Engineering and Science, Second Edition

McCULLOCH, SEARLE, and NEUHAUS · Generalized, Linear, and Mixed Models, Second Edition

McFADDEN · Management of Data in Clinical Trials, Second Edition

* McLACHLAN · Discriminant Analysis and Statistical Pattern Recognition

McLACHLAN, DO, and AMBROISE · Analyzing Microarray Gene Expression Data

McLACHLAN and KRISHNAN · The EM Algorithm and Extensions, Second Edition

McLACHLAN and PEEL · Finite Mixture Models

McNEIL · Epidemiological Research Methods

MEEKER and ESCOBAR · Statistical Methods for Reliability Data

MEERSCHAERT and SCHEFFLER · Limit Distributions for Sums of Independent Random Vectors: Heavy Tails in Theory and Practice

MENGERSEN, ROBERT, and TITTERINGTON · Mixtures: Estimation and Applications

MICKEY, DUNN, and CLARK · Applied Statistics: Analysis of Variance and Regression, Third Edition

* MILLER · Survival Analysis, Second Edition

MONTGOMERY, JENNINGS, and KULAHCI · Introduction to Time Series Analysis and Forecasting

MONTGOMERY, PECK, and VINING · Introduction to Linear Regression Analysis, Fifth Edition

MORGENTHALER and TUKEY · Configural Polysampling: A Route to Practical Robustness

MUIRHEAD · Aspects of Multivariate Statistical Theory

MULLER and STOYAN · Comparison Methods for Stochastic Models and Risks

MURTHY, XIE, and JIANG · Weibull Models

MYERS, MONTGOMERY, and ANDERSON-COOK · Response Surface Methodology: Process and Product Optimization Using Designed Experiments, Third Edition

MYERS, MONTGOMERY, VINING, and ROBINSON · Generalized Linear Models. With Applications in Engineering and the Sciences, Second Edition

NATVIG · Multistate Systems Reliability Theory With Applications

^†NELSON · Accelerated Testing, Statistical Models, Test Plans, and Data Analyses

^†NELSON · Applied Life Data Analysis

NEWMAN · Biostatistical Methods in Epidemiology

NG, TAIN, and TANG · Dirichlet Theory: Theory, Methods and Applications

OKABE, BOOTS, SUGIHARA, and CHIU · Spatial Tesselations: Concepts and Applications of Voronoi Diagrams, Second Edition

OLIVER and SMITH · Influence Diagrams, Belief Nets and Decision Analysis

PALTA · Quantitative Methods in Population Health: Extensions of Ordinary Regressions

PANJER · Operational Risk: Modeling and Analytics

PANKRATZ · Forecasting with Dynamic Regression Models

PANKRATZ · Forecasting with Univariate Box-Jenkins Models: Concepts and Cases

PARDOUX · Markov Processes and Applications: Algorithms, Networks, Genome and Finance

PARMIGIANI and INOUE · Decision Theory: Principles and Approaches

* PARZEN · Modern Probability Theory and Its Applications

PEÑA, TIAO, and TSAY · A Course in Time Series Analysis

PESARIN and SALMASO · Permutation Tests for Complex Data: Applications and Software

PIANTADOSI · Clinical Trials: A Methodologic Perspective, Second Edition

POURAHMADI · Foundations of Time Series Analysis and Prediction Theory

POWELL · Approximate Dynamic Programming: Solving the Curses of Dimensionality, Second Edition

POWELL and RYZHOV · Optimal Learning

PRESS · Subjective and Objective Bayesian Statistics, Second Edition

PRESS and TANUR · The Subjectivity of Scientists and the Bayesian Approach

PURI, VILAPLANA, and WERTZ · New Perspectives in Theoretical and Applied Statistics

^† PUTERMAN · Markov Decision Processes: Discrete Stochastic Dynamic Programming

QIU · Image Processing and Jump Regression Analysis

* RAO · Linear Statistical Inference and Its Applications, Second Edition

RAO · Statistical Inference for Fractional Diffusion Processes

RAUSAND and HØYLAND · System Reliability Theory: Models, Statistical Methods, and Applications, Second Edition

RAYNER, THAS, and BEST · Smooth Tests of Goodnes of Fit: Using R, Second Edition

RENCHER and SCHAALJE · Linear Models in Statistics, Second Edition

RENCHER and CHRISTENSEN · Methods of Multivariate Analysis, Third Edition

RENCHER · Multivariate Statistical Inference with Applications

RIGDON and BASU · Statistical Methods for the Reliability of Repairable Systems

* RIPLEY · Spatial Statistics

* RIPLEY · Stochastic Simulation

ROHATGI and SALEH · An Introduction to Probability and Statistics, Second Edition

ROLSKI, SCHMIDLI, SCHMIDT, and TEUGELS · Stochastic Processes for Insurance and Finance

ROSENBERGER and LACHIN · Randomization in Clinical Trials: Theory and Practice

ROSSI, ALLENBY, and McCULLOCH · Bayesian Statistics and Marketing

^† ROUSSEEUW and LEROY · Robust Regression and Outlier Detection

ROYSTON and SAUERBREI · Multivariate Model Building: A Pragmatic Approach to Regression Analysis Based on Fractional Polynomials for Modeling Continuous Variables

* RUBIN · Multiple Imputation for Nonresponse in Surveys

RUBINSTEIN and KROESE · Simulation and the Monte Carlo Method, Second Edition

RUBINSTEIN and MELAMED · Modern Simulation and Modeling

RYAN · Modern Engineering Statistics

RYAN · Modern Experimental Design

RYAN · Modern Regression Methods, Second Edition

RYAN · Statistical Methods for Quality Improvement, Third Edition

SALEH · Theory of Preliminary Test and Stein-Type Estimation with Applications

SALTELLI, CHAN, and SCOTT (editors) · Sensitivity Analysis

SCHERER · Batch Effects and Noise in Microarray Experiments: Sources and Solutions

* SCHEFFE · The Analysis of Variance

SCHIMEK · Smoothing and Regression: Approaches, Computation, and Application

SCHOTT · Matrix Analysis for Statistics, Second Edition

SCHOUTENS · Levy Processes in Finance: Pricing Financial Derivatives

SCOTT · Multivariate Density Estimation: Theory, Practice, and Visualization

* SEARLE · Linear Models

^† SEARLE · Linear Models for Unbalanced Data

^† SEARLE · Matrix Algebra Useful for Statistics

^† SEARLE, CASELLA, and McCULLOCH · Variance Components

SEARLE and WILLETT · Matrix Algebra for Applied Economics

SEBER · A Matrix Handbook For Statisticians

^† SEBER · Multivariate Observations

SEBER and LEE · Linear Regression Analysis, Second Edition

^† SEBER and WILD · Nonlinear Regression

SENNOTT · Stochastic Dynamic Programming and the Control of Queueing Systems

* SERFLING · Approximation Theorems of Mathematical Statistics

SHAFER and VOVK · Probability and Finance: It’s Only a Game!

SHERMAN · Spatial Statistics and Spatio-Temporal Data: Covariance Functions and Directional Properties

SILVAPULLE and SEN · Constrained Statistical Inference: Inequality, Order, and Shape Restrictions

SINGPURWALLA · Reliability and Risk: A Bayesian Perspective

SMALL and McLEISH · Hilbert Space Methods in Probability and Statistical Inference

SRIVASTAVA · Methods of Multivariate Statistics

STAPLETON · Linear Statistical Models, Second Edition

STAPLETON · Models for Probability and Statistical Inference: Theory and Applications

STAUDTE and SHEATHER · Robust Estimation and Testing

STOYAN · Counterexamples in Probability, Second Edition

STOYAN, KENDALL, and MECKE · Stochastic Geometry and Its Applications, Second Edition

STOYAN and STOYAN · Fractals, Random Shapes and Point Fields: Methods of Geometrical Statistics

STREET and BURGESS · The Construction of Optimal Stated Choice Experiments: Theory and Methods

STYAN · The Collected Papers of T. W. Anderson: 1943–1985

SUTTON, ABRAMS, JONES, SHELDON, and SONG · Methods for Meta-Analysis in Medical Research

TAKEZAWA · Introduction to Nonparametric Regression

TAMHANE · Statistical Analysis of Designed Experiments: Theory and Applications

TANAKA · Time Series Analysis: Nonstationary and Noninvertible Distribution Theory

THOMPSON · Empirical Model Building: Data, Models, and Reality, Second Edition

THOMPSON · Sampling, Third Edition

THOMPSON · Simulation: A Modeler’s Approach

THOMPSON and SEBER · Adaptive Sampling

THOMPSON, WILLIAMS, and FINDLAY · Models for Investors in Real World Markets

TIERNEY · LISP-STAT: An Object-Oriented Environment for Statistical Computing and Dynamic Graphics

TSAY · Analysis of Financial Time Series, Third Edition

TSAY · An Introduction to Analysis of Financial Data with R

UPTON and FINGLETON · Spatial Data Analysis by Example, Volume II: Categorical and Directional Data

^† VAN BELLE · Statistical Rules of Thumb, Second Edition

VAN BELLE, FISHER, HEAGERTY, and LUMLEY · Biostatistics: A Methodology for the Health Sciences, Second Edition

VESTRUP · The Theory of Measures and Integration

VIDAKOVIC · Statistical Modeling by Wavelets

VIERTL · Statistical Methods for Fuzzy Data

VINOD and REAGLE · Preparing for the Worst: Incorporating Downside Risk in Stock Market Investments

WALLER and GOTWAY · Applied Spatial Statistics for Public Health Data

WEISBERG · Applied Linear Regression, Third Edition

WEISBERG · Bias and Causation: Models and Judgment for Valid Comparisons

WELSH · Aspects of Statistical Inference

WESTFALL and YOUNG · Resampling-Based Multiple Testing: Examples and Methods for p-Value Adjustment

* WHITTAKER · Graphical Models in Applied Multivariate Statistics

WINKER · Optimization Heuristics in Economics: Applications of Threshold Accepting

WOODWORTH · Biostatistics: A Bayesian Introduction

WOOLSON and CLARKE · Statistical Methods for the Analysis of Biomedical Data, Second Edition

WU and HAMADA · Experiments: Planning, Analysis, and Parameter Design Optimization, Second Edition

WU and ZHANG · Nonparametric Regression Methods for Longitudinal Data Analysis

YIN · Clinical Trial Design: Bayesian and Frequentist Adaptive Methods

YOUNG, VALERO-MORA, and FRIENDLY · Visual Statistics: Seeing Data with Dynamic Interactive Graphics

ZACKS · Stage-Wise Adaptive Designs

* ZELLNER · An Introduction to Bayesian Inference in Econometrics

ZELTERMAN · Discrete Distributions—Applications in the Health Sciences

ZHOU, OBUCHOWSKI, and McCLISH · Statistical Methods in Diagnostic Medicine, Second Edition

*Now available in a lower priced paperback edition in the Wiley Classics Library.

^†Now available in a lower priced paperback edition in the Wiley–Interscience Paperback Series.

Copyright © 2013 by John Wiley & Sons, Inc. All rights reserved.

Published by John Wiley & Sons, Inc., Hoboken, New Jersey.
Published simultaneously in Canada.

No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 750-4470, or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008, or online at http://www.wiley.com/go/permission.

Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages.

For general information on our other products and services or for technical support, please contact our Customer Care Department within the United States at (800) 762-2974, outside the United States at (317) 572-3993 or fax (317) 572-4002.

Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic formats. For more information about Wiley products, visit our web site at www.wiley.com.

Library of Congress Cataloging-in-Publication Data:

Klugman, Stuart A., 1949–
  Loss models : further topics / Stuart A. Klugman, Society of Actuaries, Schaumburg, IL, Harry H. Panjer,
Department of Statistics and Actuarial Science, University of Waterloo, Ontario, Canada, Gordon E. Willmot,
Department of Statistics and Actuarial Science, University of Waterloo, Ontario, Canada.
     pages cm
  Includes bibliographical references and index.
  ISBN 978-1-118-34356-2 (cloth)
  1. Insurance—Statistical methods. 2. Insurance—Mathematical models. I. Panjer, Harry H. II. Willmot, Gordon E.,– III. Title.
  HG8781.K584 2013
  368’.01—dc23
  2013009457

PREFACE

Readers who have been with us since the first edition may have noticed that each edition added several new topics while few were dropped. By the third edition, a limit had been reached and it was time to rethink how the material was presented. With the encouragement of our publisher, we decided to produce two books. The first, published in 2012, is the fourth edition, continuing to be called Loss Models: From Data to Decisions [58]. In that book we included all the topics currently covered on the examinations of the Casualty Actuarial Society and the Society of Actuaries (with some updates to specific topics). We also included a few topics we think may be worth adding in the future (and that we like to teach). When designing this companion book, we wanted to do two things. The first was to cover the topics from the third edition that had been excluded from the fourth edition. These are:

Extreme value distributions (with expanded material on tail calculations)

Computational methods for aggregate models [including an (a, b, m) recursion]

Counting processes

Copula models

Continuous-time ruin models

Interpolation and smoothing

The second was to add new material, particularly with regard to expanding the number of models presented and demonstrating how they apply to actuarial problems. The new topics are (though some include material that was in the third edition):

Coxian and related distributions

Mixed Erlang distributions

Analytic methods for aggregate claim models

More discrete claim count models

Compound distributions with time dependent claim amounts

We have viewed this companion book as more of a practitioner’s and researcher’s resource than a textbook and thus have only created exercises where additional concepts are introduced. However, for material brought over from the third edition, those exercises have been retained. Solutions to all exercises are in an Appendix. Together with the fourth edition, we believe the two books present a comprehensive look at the current state of this aspect of actuarial work. We are thankful for the continued support and encouragement from John Wiley & Sons and the Society of Actuaries. We also thank Joan Hatton for her expert typing and Mirabelle Huynh who did a thorough job of proofreading our writing.

S. A. KLUGMAN, H. H. PANJER, G. E. WILLMOT

Schaumburg, IL and Waterloo, Ontario

CHAPTER 1

INTRODUCTION

As noted in the preface, the purpose of this book is to provide information on topics not covered in the fourth edition of Loss Models: From Data to Decisions [59]. In general, the emphasis here is less on data and decisions and more on what is in between, namely the vast array of models available for actuarial work. In this introduction we give a brief overview of the models covered. The material can be broken up into six sets of topics.

Univariate models for loss amounts

Three chapters are devoted to classes of univariate models. The first is the class of Coxian distributions (Chapter 2). These distributions have the desirable property that their Laplace transform (or, equivalently, their moment generating function) is a ratio of polynomials. Thus, when used as a claim size distribution, convenient explicit expressions for the associated aggregate or compound distribution may sometimes be derived. The second is the class of mixed Erlang distributions (Chapter 3). These distributions are notable because they can approximate any positive continuous distribution to an arbitrary degree of accuracy. Moreover, the mixed Erlang class contains a large number of distributions, including some whose mixed Erlang structure is not obvious. Also, calculations of most quantities of interest in an insurance loss context are computationally straightforward. The third chapter (Chapter 4) covers the two classes of extreme value distributions. This material is largely reproduced from the third edition [58] with some additional material on tail calculations.

As the name implies, these models are especially useful for management of risks that may produce large losses.

Calculation of aggregate losses

The basic methods for these calculations are covered in the fourth edition. This book contains two enhancements. Some of the univariate models introduced in the early chapters allow for exact calculation of aggregate loss probabilities. The formulas are developed in Chapter 5 along with asymptotic formulas for the right tail. Computational methods left out of the fourth edition are provided in Chapter 6. These include inversion methods, calculating with approximate distributions, and calculating from the individual risk model (which was in the second edition, but not the third). A new item is a presentation of the recursive formula when the frequency distribution is a member of the (a, b, m) class of distributions.

Loss model applications of discrete counting models

The next three chapters focus on various issues that are of interest in the loss modeling context. The first chapter (Chapter 7) introduces counting processes and, as in the third edition, deals with nonhomogeneous birth processes and mixed Poisson processes, which are useful for modeling the development of claim counts over time. Chapter 8 is new and considers properties of discrete counting distributions that are of interest in connection with loss model concepts such as deductibles and limits, recursions for compound distributions, evaluation of stop-loss moments, and computation of the risk measures VaR and TVaR in a discrete setting. The third chapter (Chapter 9) deals with models where the claim amounts depend on the time of incurral of the claim. Examples include inflation and claim payment delays.

Multivariate models

Chapter 10 covers the analysis of multivariate models based on copula functions. The material is taken from the third edition. Methods for simulation that were in a later chapter of the third edition were moved to this chapter.

Continuous-time ruin models

The material in Chapter 11 is taken directly from the third edition. It contains the classic analysis of the infinite-time ruin problem.

Interpolation and smoothing

While this material was covered in the third edition, two changes have been made for Chapter 12. First, some of the earlier material has been eliminated or streamlined. The goal is to efficiently arrive at the smoothing spline, the method most suitable for actuarial problems. More emphasis is placed on the most common application, the smoothing of data from experience studies. A traditional actuarial method, Whittaker–Henderson, has been added along with discussion of its similarity to smoothing splines.

CHAPTER 2

COXIAN AND RELATED DISTRIBUTIONS

2.1 Introduction

For the analysis of aggregate claims, the typical models involve compound distributions, which result in analytical complexities. A useful feature of compound distributions is the simplicity of the probability generating function (for discrete cases) and the Laplace transform (for continuous cases). This characteristic can be exploited to obtain useful results from either a mathematical or computational viewpoint. Because the class of Coxian distributions is defined through its Laplace transform, members of the class are well suited for use as claim amount distributions in aggregate claims models. In this chapter we briefly discuss two fairly broad classes of models that have been used in applications involving loss models. Both are subclasses of the class of Coxian distributions, which we now define.

Definition 2.1 A distribution is from the Coxian-n class if its Laplace transform (s) = dx may be expressed as

(2.1)

where λ_i > 0 for i = 1, 2,…, m and (without loss of generality) we assume that λ_i ≠ λ_j for i ≠ j. We further assume that n_i is a nonnegative integer for i = 1, 2, …, m and that n = ∑^m_i=1 n_i > 0. Also, a(s) is a polynomial of degree n − 1 or less.

As (0) = 1, it follows that . Furthermore, a partial fraction expansion of (2.1) yields

(2.2)

where

(2.3)

Note that

(2.4)

where

(2.5)

is the probability density function (pdf) of an Erlang-j random variable with scale parameter λ_i. Then, from (2.2) and also (2.4), the Coxian-n class has pdf of the form

with e_λi,j (x) given by (2.5), which is a finite combination of Erlang pdfs. We now discuss the special case when n_i = 1 for i = 1, 2,…, m.

2.2 Combinations of exponentials

Suppose that X has pdf of the form

(2.6)

where λ_i ≥ 0 for i = 1, 2, …, n. The condition that = 1 implies that ∑ⁿ_i=1 q_i = 1, and if 0 ≤ q_i ≤ 1 for i = 1, 2, …, n then (2.6) may be interpreted as a mixture of exponential pdfs.

But (2.6) is a pdf in many cases even if some of the q_i are negative, in which case (2.6) may be referred to as a combination (or a generalized mixture) of exponentials. Two examples where this occurs are now given.

EXAMPLE 2.1 Generalized Erlang distribution

The use of the partial fraction expansion in the previous example is essentially equivalent to Lagrange’s polynomial representation. That is, if x₁, x₂, …, x_n are distinct numbers and g(x) is a polynomial of degree n − 1 or less, then g(x) may be expressed in terms of the functional values g(x_i) for i = 1, 2, …, n as

(2.10)

An important special case of (2.10) is when g(x) = 1, yielding the identity

(2.11)

If x = 0 in (2.11) and x_j = λ_j, it follows immediately that ∑ⁿ_i=1 q_i = 1, where q_i is given by (2.8), a condition necessary for (2.9) to be a pdf. Also, (2.11) may be viewed as a polynomial identity in x. The left-hand side is a polynomial of degree n − 1 in x, with coefficient of xⁿ⁻¹ satisfying

(2.12)

because the coefficient of xⁿ⁻¹ on the right-hand side of (2.11) is 0. This proves the outer equality in (2.12), and the left-hand equality in (2.12) follows by multiplication by (−1)ⁿ⁻¹. It is clear from (2.8) and (2.9) that

and thus (2.12) with x_j = λ_j implies that f_Sn (0) = 0 if n = 2, 3,….

We now consider a second example of a combination of exponentials.

EXAMPLE 2.2 A logbeta distribution

The class of combinations of exponentials is an important class of distributions as it is dense in the set of probability distributions on [0, ∞), implying that any such probability distribution may be approximated by a combination of exponentials. Dufresne [19] considers this approximation problem and uses logbeta pdfs of the type considered in Example 2.2 in this context. Interestingly, the terminology “logbeta” is also due to Dufresne [19] and is more appropriate than the use of the term “lognormal” in that the log (not the exponential) of a lognormal random variable is normally distributed.

Assuming without loss of generality that λ₁ < λ₂ < · · · < λ_n, necessary conditions for (2.6) to be a valid pdf are that q₁ > 0 and ∑ⁿ_i=1 q_iλ_i ≥ 0, and these conditions are also sufficient if there is not more than one sign change in the sequence {q₁,q₂, …, q_n}, obviously the case if n = 2. See Steutel and van Harn [89, pp. 338–339] for further details. Again assuming that λ₁ < λ₂ < · · · < λ_n, Bartholomew [7] shows that alternative sufficient conditions for (2.6) to be a valid pdf are that ∑^k_i=1 q_iλ_i ≥ 0 for k = 1, 2, …, n.

2.3 Coxian-2 distributions

In the Coxian-n case with n = 2, a(s) is a linear function of s, and thus (2.1) may be expressed as

(2.16)

where λ₁ > 0, λ₂ > 0, and λ₁ = λ₂ is possible. We wish to consider values of p for which (2.16) is the Laplace transform of a pdf. First, note that if p = 0 then (s) = λ₁/(λ₁ + s), which is the Laplace transform of an exponential (with mean 1/λ₁) pdf. Similarly, if p = 1 − λ₂/λ₁, that is, λ₂ = λ₁ (1 − p), then (s) = λ₂/(λ₂ + s), again of exponential form. Thus we exclude the cases with p = 0 and p = 1 − λ₂/λ₁ in what follows.

It is clear from (2.16) that

(2.17)

which implies that

(2.18)

where

(2.19)

Clearly, h(x) is easy to evaluate, but its form depends on whether λ₁ = λ₂ or not. In any event, h(0) = 0 from (2.19), implying from (2.18) that f(0) = λ₁(1 − p), and so the condition p ≤ 1 is required for f(x) to be a valid pdf. The Laplace transform of the tail is, from (2.17),

from which it follows that

Thus, again using (2.19),

(2.20)

If λ₁ ≥ λ₂ then from (2.19) lim_x→∞ h(x) = ∞, and thus from (2.20) it is clear that p ≥ 0 because if p < 0 then would become negative for large x. But it was assumed that p ≠ 0, and thus if λ₁ ≥ λ₂ it follows that 0 < p ≤ 1. Thus if λ₁ = λ₂ = λ, (2.18) and (2.19) yield

(2.21)

which is the pdf of the mixture of two Erlang pdfs, both with the same scale parameter λ. We remark that pdfs of the form (2.21) will be discussed in much detail later.

If λ₁ < λ₂ then from (2.19)

and thus from (2.20)

This limit obviously cannot be negative, and it follows that λ₂ − λ₁ + pλ₁ ≥ 0, i.e., p ≥ 1 − λ₂/λ₁, which is equivalent to λ₂ ≥ λ₁ (1 − p). But again it is assumed that p ≠ 1 − λ₂/λ₁, and therefore if λ₁ < λ₂ then 1 − λ₂/λ₁ < p ≤ 1 but p ≠ 0.

If λ₁ ≠ λ₂ then

(2.22)

which follows directly or from (2.7) and (2.8). Substitution of (2.22) into (2.17) yields

That is, if λ₁ ≠ λ₂,

(2.23)

where

(2.24)

If 0 < p ≤ 1 then (2.23) is either a mixture or a combination of two exponential pdfs. However, if p < 0 then one must have λ₁ < λ₂, and α > 0 from (2.24). But if p is negative, one must have 1 − λ₂/λ₁ < p, or equivalently λ₁ − λ₂ < λ₁p, and because λ₁ − λ₂ must also be negative, 1 > λ₁p/(λ₁ − λ₂), i.e., α < 1. Thus, if p < 0 then 0 < α < 1 and (2.23) is a mixture.

To summarize, when λ₁ ≠ λ₂, the pdf f(x) is given by (2.23) with α given by (2.24). If p > 0 then (2.23) is either a mixture or a combination of two exponential pdfs, whereas if p < 0 then (2.23) is a mixture.

Again from (2.17)

that is,

(2.25)

With s = 0, (2.25) gives the mean, namely,

(2.26)

The equilibrium pdf thus has Laplace transform, from (2.25) and (2.26), given by

That is,

(2.27)

where

(2.28)