Generalized Diagonal Band Copulae with Two-Sided Generating - PowerPoint PPT Presentation

Generalized Diagonal Band Copulae with Two-Sided Generating Densities "Presentation Short Course: Beyond Beta and Applications" November 20th, 2018 , La Sapienza Samuel Kotz and J. René van Dorp 1 Faculty Web-Page: www.seas.gwu.edu/~dorpjr 1 Corresponding Author, Department of Engineering Management and Systems Engineering, The George Washington University, Washington D.C., USA BEYOND BETA SHORT COURSE: La Sapienza J.R. van Dorp; dorpjr@gwu.edu - Page 1

OUTLINE 1. INTRODUCTION 2. COPULA CONSTRUCTION 3. GENERALIZED DIAGONAL BAND EXAMPLES 4. SAMPLING PROCEDURE 5. ORDINAL MEASURES OF ASSOCIATION 6. COPULA PARAMETER ELICITATION 7. A VALUE OF INFORMATION EXAMPLE 8. SELECTED REFERENCES BEYOND BETA SHORT COURSE: La Sapienza J.R. van Dorp; dorpjr@gwu.edu - Page 1

1. INTRODUCTION... Copula • w , w : Continuous random variables such that w w \ ] \ µ KÐ † Ñß ] µ LÐ † Ñ • , H( ): Cumulative distribution functions - cdf's. KÐ † Ñ † w w • The mapping Uniform is called the \ Ä \ œ KÐ\ Ñ Ê \ µ Ò!ß " Ó probability integral transformation e.g. Nelsen (1999). • Any bivariate joint distribution of w w can be transformed to a bivariate Ð\ ß ] Ñ w w copula { } - Sklar (1959). Ð\ß ] Ñ œ KÐ\ Ñß LÐ] Ñ • Thus, a bivariate copula is a bivariate distribution with uniform marginals. • As such, many authors studied copulae indirectly. • Gaussian and Student-t Copulae (of this construct) were studied explicitly Þ BEYOND BETA SHORT COURSE: La Sapienza J.R. van Dorp; dorpjr@gwu.edu - Page 2

1. INTRODUCTION... Archimedean Copula • Genest and Mackay (1986) used an algabraic method for copula construction. • , a convex decreasing function with - The generator : À Ð!ß "Ó Ä Ò!ß ∞Ñ : Ð"Ñ œ ! function. • They possess joint cdf and probability density function (pdf) :  " : Ö ÐBÑ  : : ÐCÑ× : ÐBÑ  : ÐCÑ Ÿ ! GÖBß Cl Ð † Ñ× œ : Ð"Ñ elsewhere ! '' ' ' : ÖGÐBß CÑ× : ÐBÑ : ÐCÑ -ÖBß Cl Ð † Ñ× œ  : Ð#Ñ ' $ Ò : ÖGÐBß CÑ×Ó BEYOND BETA SHORT COURSE: La Sapienza J.R. van Dorp; dorpjr@gwu.edu - Page 3

1. INTRODUCTION... Diagonal Band Copula • Cooke and Waij (1986) used a geometric method for copula construction A A B B (1, 2−θ) (1, 2−θ) y y (1, 1) (1, 1) (1, 0) (1, 0) 4.00 4.00 DB 3 DB 3 3.00 3.00 (1, θ) (1, θ) 2.00 2.00 DB 2 DB 2 1 1 1.00 1.00 (0, 1−θ) (0, 1−θ) DB 1 DB 1 0.00 0.00 (0, 0) (0, 0) 1 1 0 0 (0, 1) (0, 1) y y 0 0 (0, θ−1) (0, θ−1) x x Figure 1: A: Gray area support of a copula comprised of sub-areas HFÐ Ñ ) B: Example of a copula. HF ß 3 œ "ß #ß $à HFÐ!Þ&Ñ 3 BEYOND BETA SHORT COURSE: La Sapienza J.R. van Dorp; dorpjr@gwu.edu - Page 4

1. INTRODUCTION... Diagonal Band Copula • Diagonal Band (DB) copula possess pdf:   "ÎÐ"  Ñ ) ÐBß CÑ − HF ∪ HF " $  GÖBß Cl × œ ) "ÎÖ#Ð"  Ñ× ) ÐBß CÑ − HF Ð$Ñ # elsewhere ! • Analagous to Archimedean copula Bojarski (2001) generalized , copula via a HFÐ Ñ ) generator function | . 0Ð † ) Ñ • Generator function | is a symmetric pdf with support . 0Ð † ) Ñ Ò )  "ß "  ) Ó • Lewandowski (2005) showed that Bojarski's (2001) GDB Copulae are equivalent to Fergusons (1995) family of copulae with joint pdf: " , 1Ð † Ñ pdf on -ÐBß CÑ œ Ö1ÐlB  Cl  1Ð"  l"  B  ClÑ× Ò!ß "Ó Ð%Ñ # BEYOND BETA SHORT COURSE: La Sapienza J.R. van Dorp; dorpjr@gwu.edu - Page 5

1. INTRODUCTION... Generalized DB Copula • For sampling efficiency inverse cdf of generator would be desirable. 0Ð † l Ñ ) • Consider Van Dorp and Kotz's (2003) symmetric Two-Sided (TS) pdf's À  , for " :ÐD  "l Ñ G  "  D Ÿ !ß { 0 Dl:Ð † l Ñ× œ G ‚ Ð&Ñ , for :Ð"  Dl Ñ G !  D  "ß # that too uses the generating pdf concept. Pdf has support :ÐDÑ :ÐDÑ Ò!ß "ÓÞ • The inverse cdf (or quantile function) associated with Ð%Ñ  " " , for T Ð#?l Ñ  " G !  ? Ÿ ß " { # J ?l:Ð † l Ñ× œ G Ð'Ñ " " , for "  T Ð#  #?l Ñ G  ?  "ß # " where is the quantile function of T Ð † l Ñ < :Ð † l ÑÞ G BEYOND BETA SHORT COURSE: La Sapienza J.R. van Dorp; dorpjr@gwu.edu - Page 6

OUTLINE 1. INTRODUCTION 2. COPULA CONSTRUCTION 3. GENERALIZED DIAGONAL BAND EXAMPLES 4. SAMPLING PROCEDURE 5. ORDINAL MEASURES OF ASSOCIATION 6. COPULA PARAMETER ELICITATION 7. A VALUE OF INFORMATION EXAMPLE 8. SELECTED REFERENCES BEYOND BETA SHORT COURSE: La Sapienza J.R. van Dorp; dorpjr@gwu.edu - Page 7

2. CONSTRUCTION... GDB Copula with TS Gen. PDF • Bivariate pdf is constructed where and the conditional 1ÐBß CÑ ß \ µ YÒ!ß "Ó pdf 1ÐClBÑ has the following form À , , 1ÖClBß :Ð † l Ñ G × œ 0ÖB  C l:Ð † l Ñ× G B  " Ÿ C Ÿ B  " Ð(Ñ • From , and TS framework pdf Ð%Ñ it follows that: \ µ YÒ!ß "Ó Ð(Ñ  " :Ð"  B  C l Ñ G ß  "  B  C Ÿ !ß 1ÖBß Cl :Ð † l Ñ× G œ ‚ Ð)Ñ :Ð"  B  C l Ñ G ß !  B  C  "ß # • From , a bivariate pdf | is constucted on the unit square Ð)Ñ -ÐBß C :Ð † l Ñ G Ñ # by folding back the probability masses of outside Ò!ß "Ó 1ÖBß Cl :Ð † l Ñ× G Ò!ß "Ó # the unit square onto it, using "folding" lines and . C œ " C œ ! BEYOND BETA SHORT COURSE: La Sapienza J.R. van Dorp; dorpjr@gwu.edu - Page 8

2. CONSTRUCTION... GDB Copula with TS Gen. PDF (0,1) B (1,1) x + y = 2 A y 3 A A 1 4 A 1 2 x – y = – 1 (0,0) (1,0) C x 2.00 1 x – y = 0 x + y = 1 1.00 1 A 3 A x + y = 0 A 4 y 1 A 2 0.00 0 1 x – y = 1 x 0 Figure 1. A: pdf B: Areas 1ÐBß CÑ Ð)Ñà E ß 3 œ "ß á ß %à 3 C: pdf with on . -ÖBß Cl :Ð † l G Ñ × Ð"!Ñ :ÐDÑ œ #D Ò!ß "Ó BEYOND BETA SHORT COURSE: La Sapienza J.R. van Dorp; dorpjr@gwu.edu - Page 9

2. CONSTRUCTION... GDB Copula with TS Gen. PDF • Relationship between and in 8 -ÖBß Cl :Ð † l G Ñ × 1ÖBß Cl :Ð † l G Ñ× Ð Ñ À -ÖBß Cl :Ð † l Ñ G × œ Ð*Ñ  , 1ÖBß Cl :Ð † l Ñ× G  1ÖBß  Cl :Ð † l Ñ× G ß !  B  C Ÿ " . 1ÖBß Cl :Ð † l Ñ× G  1ÖBß #  Cl :Ð † l Ñ× G ß "  B  C Ÿ # • Combining with Ð)Ñ now yields À Ð*Ñ  -ÖBß Cl :Ð † l Ñ G × œ Ð"!Ñ   , :Ð "  B  C l Ñ  :Ð"  G B  C l Ñ G ß ÐBß CÑ − E "  " ,  :Ð "  B  C l Ñ  :Ð"  G B  C l Ñ G ß ÐBß CÑ − E  # # ‚  , :Ð B  C  " l Ñ  :Ð"  G B  C l Ñ G ß ÐBß CÑ − E $ :Ð B  C  " l Ñ  G :Ð"  B  C l Ñ G ß ÐBß CÑ − E Þ % • Note in Ð"!Ñ -ÐCß B œ -ÐBß CÑ . Hence, Ñ \ µ Y Ò!ß "Ó Ê ] µ Y Ò!ß "Ó BEYOND BETA SHORT COURSE: La Sapienza J.R. van Dorp; dorpjr@gwu.edu - Page 10

2. CONSTRUCTION... Joint CDF • Pdf of GDB copula with TS pdf with generating pdf :ÐDl G Ñ À  -ÖBß Cl :Ð † l Ñ G × œ   , :Ð "  B  C l Ñ  :Ð"  G B  C l Ñ G ß ÐBß CÑ − E "  , "  :Ð "  B  C l Ñ  :Ð"  G B  C l Ñ G ß ÐBß CÑ − E  # # ‚  , :Ð B  C  " l Ñ  :Ð"  G B  C l Ñ G ß ÐBß CÑ − E $ :Ð B  C  " l Ñ  G :Ð"  B  C l Ñ G ß ÐBß CÑ − E Þ % • Cdf of GDB copula with TS gen. pdf :ÐDl G Ñ and cdf T ÐDl G follows as: Ñ       " B C  " , B  TÐDl Ñ.D G ß ÐBß CÑ − E  " # " BC  " BC " , C  TÐDl Ñ×.D G ß ÐBß CÑ − E   #  # " BC GÖBß Cl :Ð † l Ñ G × œ  Ð""Ñ  " BC  " , B  TÐDl Ñ.D G ß ÐBß CÑ − E #  BC" $  # B C" " BC " C  TÐDl Ñ.D G ß ÐBß CÑ − E Þ % BEYOND BETA SHORT COURSE: La Sapienza J.R. van Dorp; dorpjr@gwu.edu - Page 11

OUTLINE 1. INTRODUCTION 2. COPULA CONSTRUCTION 3. GENERALIZED DIAGONAL BAND EXAMPLES 4. SAMPLING PROCEDURE 5. ORDINAL MEASURES OF ASSOCIATION 6. COPULA PARAMETER ELICITATION 7. A VALUE OF INFORMATION EXAMPLE 8. SELECTED REFERENCES BEYOND BETA SHORT COURSE: La Sapienza J.R. van Dorp; dorpjr@gwu.edu - Page 12