Robust and Efficient Fitting of Claim Severity Distributions Vytaras Brazauskas a,b University of Wisconsin-Milwaukee 44th Actuarial Research Conference Madison, WI, July 30–August 1, 2009 a In collaboration with Jones and Zitikis (University of Western Ontario) b Supported by a grant from the Actuarial Foundation, SOA, and CAS
Outline 1. Introduction – Preliminaries – Motivation 2. Method of Trimmed Moments – Definition – Asymptotic Properties – Examples – Simulations 3. Illustrations and Conclusions – Real-Data Examples – Concluding Remarks
1. I NTRODUCTION Preliminaries 1. Introduction Preliminaries • Claim Severity Distributions ⊲ S TATISTICAL O BJECTIVE : + Accurate model fit ⊲ A CTUARIAL O BJECTIVES : + Risk evaluations + Ratemaking + Reserve calculations 1
1. I NTRODUCTION Preliminaries • Standard Estimation & Fitting Techniques ⊲ E MPIRICAL N ONPARAMETRIC + Simple approach + Weak assumptions 2
1. I NTRODUCTION Preliminaries • Standard Estimation & Fitting Techniques ⊲ E MPIRICAL N ONPARAMETRIC + Simple approach + Weak assumptions – Lack of smoothness – Limited to the range of observed data 2
1. I NTRODUCTION Preliminaries • Standard Estimation & Fitting Techniques ⊲ E MPIRICAL N ONPARAMETRIC + Simple approach + Weak assumptions – Lack of smoothness – Limited to the range of observed data ⊲ P ARAMETRIC + Efficiency + Smoothness + Stretchability beyond the range of observed data + Special distributional features (e.g., mode at 0) 2
1. I NTRODUCTION Preliminaries • Standard Estimation & Fitting Techniques ⊲ E MPIRICAL N ONPARAMETRIC + Simple approach + Weak assumptions – Lack of smoothness – Limited to the range of observed data ⊲ P ARAMETRIC + Efficiency + Smoothness + Stretchability beyond the range of observed data + Special distributional features (e.g., mode at 0) – Strong assumptions – Outliers (e.g., loss that receives an extensive media attention) 2
1. I NTRODUCTION Motivation Motivation ⊲ N OT R OBUST : maximum likelihood, method-of-moments ⊲ R OBUST : M -, L -, R -statistics 3
1. I NTRODUCTION Motivation Motivation ⊲ N OT R OBUST : maximum likelihood, method-of-moments ⊲ R OBUST : M -, L -, R -statistics ⊲ M ( maximum likelihood type) + Most popular + Easy to generalize 3
1. I NTRODUCTION Motivation Motivation ⊲ N OT R OBUST : maximum likelihood, method-of-moments ⊲ R OBUST : M -, L -, R -statistics ⊲ M ( maximum likelihood type) + Most popular + Easy to generalize – Computationally complex – Lack of transparency 3
1. I NTRODUCTION Motivation Motivation ⊲ N OT R OBUST : maximum likelihood, method-of-moments ⊲ R OBUST : M -, L -, R -statistics ⊲ M ( maximum likelihood type) + Most popular + Easy to generalize – Computationally complex – Lack of transparency ⊲ L ( linear combinations of order statistics) – Not easy to generalize 3
1. I NTRODUCTION Motivation Motivation ⊲ N OT R OBUST : maximum likelihood, method-of-moments ⊲ R OBUST : M -, L -, R -statistics ⊲ M ( maximum likelihood type) + Most popular + Easy to generalize – Computationally complex – Lack of transparency ⊲ L ( linear combinations of order statistics) – Not easy to generalize + Computer friendly + Transparent 3
2. M ETHOD OF T RIMMED M OMENTS Definition 2. Method of Trimmed Moments Definition • Assumptions & Notation ⊲ D ATA : X 1 , . . . , X n i.i.d. with cdf F ⊲ CDF : + F is continuous + F depends on θ 1 , . . . , θ k ( unknown parameters) ⊲ O RDERED D ATA : X 1: n ≤ · · · ≤ X n : n 4
2. M ETHOD OF T RIMMED M OMENTS Definition • Three-Step Procedure 1. S AMPLE T RIMMED M OMENTS : n − m ∗ n ( j ) X 1 µ j = b h j ( X i : n ) n − m n ( j ) − m ∗ n ( j ) i = m n ( j )+1 j = 1 , . . . , k , with m n ( j ) /n ≈ a j , m ∗ n ( j ) /n ≈ b j chosen trimming proportions, h j chosen function. 5
2. M ETHOD OF T RIMMED M OMENTS Definition • Three-Step Procedure 1. S AMPLE T RIMMED M OMENTS : n − m ∗ n ( j ) X 1 µ j = b h j ( X i : n ) n − m n ( j ) − m ∗ n ( j ) i = m n ( j )+1 j = 1 , . . . , k , with m n ( j ) /n ≈ a j , m ∗ n ( j ) /n ≈ b j chosen trimming proportions, h j chosen function. 2. P OPULATION T RIMMED M OMENTS : Z 1 − b j 1 h j ( F − 1 ( u )) d u µ j := µ j ( θ 1 , . . . , θ k ) = 1 − a j − b j a j j = 1 , . . . , k . 5
2. M ETHOD OF T RIMMED M OMENTS Definition 3. M ATCH & S OLVE : 8 > µ 1 ( θ 1 , . . . , θ k ) = µ 1 , b > > < . . . > > > : µ k ( θ 1 , . . . , θ k ) = µ k . b 6
2. M ETHOD OF T RIMMED M OMENTS Definition 3. M ATCH & S OLVE : 8 > µ 1 ( θ 1 , . . . , θ k ) = µ 1 , b > > < . . . > > > : µ k ( θ 1 , . . . , θ k ) = µ k . b • MTM estimators of θ 1 , . . . , θ k � . . . . . . , � θ 1 = g 1 ( � µ k ) , θ k = g k ( � µ k ) . µ 1 , . . . , � µ 1 , . . . , � 6
2. M ETHOD OF T RIMMED M OMENTS Asymptotic Properties Asymptotic Properties � � � ( θ 1 , . . . , θ k ) , n − 1 DΣD ′ � θ 1 , . . . , � � is AN θ k 7
2. M ETHOD OF T RIMMED M OMENTS Asymptotic Properties Asymptotic Properties � � � ( θ 1 , . . . , θ k ) , n − 1 DΣD ′ � θ 1 , . . . , � � is AN θ k ˛ ˛ where D k × k with d ij = ∂g i ( µ 1 ,...,µ k ) and Σ k × k with ˛ ∂ b µ j 1 σ 2 = ij (1 − a i − b i )(1 − a j − b j ) Z 1 − b i Z 1 − b j ` ´ ` ´ ` ´ F − 1 ( v ) F − 1 ( u ) × min { u, v } − uv d h j d h i a i a j 7
2. M ETHOD OF T RIMMED M OMENTS Examples Examples • Location-Scale Families ⊲ CDF , QF : � x − θ � F ( x ) = F 0 − ∞ < x < ∞ , , σ F − 1 ( u ) = θ + σF − 1 ( u ) , 0 < u < 1 . 0 where θ ∈ R , σ > 0 , and F 0 is the standard version of F . ⊲ F UNCTIONS h : h 2 ( t ) = t 2 . h 1 ( t ) = t, 8
2. M ETHOD OF T RIMMED M OMENTS Examples ⊲ S AMPLE TM S : n − m ∗ X n 1 X j µ j = b i : n , j = 1 , 2 n − m n − m ∗ n i = m n +1 ⊲ P OPULATION TM S : Z 1 − b 1 F − 1 ( u ) d u µ 1 = 1 − a − b a θ + σ × c 1 = Z 1 − b h i 2 1 F − 1 ( u ) µ 2 = d u 1 − a − b a θ 2 + 2 θσ × c 1 + σ 2 × c 2 = 9
2. M ETHOD OF T RIMMED M OMENTS Examples ⊲ MTM of ( θ, σ ) : 8 > b > θ MTM = µ 1 − c 1 b σ MTM b < q > ‹ > : µ 2 ( c 2 − c 2 µ 2 − b σ MTM b = ( b 1 ) 1 ) ⊲ A SYMPTOTICS : „ « ( θ, σ ) , σ 2 `b ´ is AN θ MTM , b σ MTM n S ⊲ E XAMPLES of F 0 and log F 0 : Cauchy, Gumbel, Laplace, Logistic, Normal, Student’s t ; and log-Cauchy, Weibull, log-Laplace, loglogistic, lognormal, log- t . 10
2. M ETHOD OF T RIMMED M OMENTS Examples • Lognormal Model ⊲ CDF , QF : „ log( x − x 0 ) − θ « F ( x ) = Φ σ log( F − 1 ( u ) − x 0 ) = θ + σ Φ − 1 ( u ) θ ∈ R , σ > 0 , x > x 0 ( known deductible), 0 < u < 1 , and Φ , Φ − 1 are CDF , QF of N (0 , 1) . ⊲ F UNCTIONS h : h 2 ( t ) = log 2 ( t − x 0 ) h 1 ( t ) = log( t − x 0 ) , 11
2. M ETHOD OF T RIMMED M OMENTS Examples „ « `b ´ ( θ, σ ) , σ 2 θ MLE , b σ MLE is AN n S 0 p `b T ABLE 1: ARE (( b | S 0 | / | S | . θ MTM , b σ MTM ) , θ MLE , b σ MLE )) = b a 0 0.05 0.15 0.49 0.70 0 1 .932 .821 .502 .312 0.05 .872 .771 .470 .286 0.15 .676 .390 .208 0.49 .074 – 0.70 – 12
2. M ETHOD OF T RIMMED M OMENTS Examples „ « `b ´ ( θ, σ ) , σ 2 θ MLE , b σ MLE is AN n S 0 p `b T ABLE 1: ARE (( b | S 0 | / | S | . θ MTM , b σ MTM ) , θ MLE , b σ MLE )) = b a 0 0.05 0.15 0.49 0.70 0 1 0.05 .932 .872 0.15 .821 .771 .676 0.49 .502 .470 .390 .074 0.70 .312 .286 .208 – – 12
2. M ETHOD OF T RIMMED M OMENTS Examples „ « `b ´ ( θ, σ ) , σ 2 θ MLE , b σ MLE is AN n S 0 p `b T ABLE 1: ARE (( b | S 0 | / | S | . θ MTM , b σ MTM ) , θ MLE , b σ MLE )) = b a 0 0.05 0.15 0.49 0.70 0 1 .932 .821 .502 .312 0.05 .932 .872 .771 .470 .286 0.15 .821 .771 .676 .390 .208 0.49 .502 .470 .390 .074 – 0.70 .312 .286 .208 – – 12
2. M ETHOD OF T RIMMED M OMENTS Examples • Pareto Model ⊲ CDF , QF : � x � − α F ( x ) = 1 − , x 0 F − 1 ( u ) = x 0 (1 − u ) − 1 /α α > 0 , x > x 0 ( known deductible), 0 < u < 1 . ⊲ F UNCTION h 1 : h 1 ( t ) = log t 13
2. M ETHOD OF T RIMMED M OMENTS Examples ⊲ MTM of α : α MTM = const 1 b µ 1 b ⊲ A SYMPTOTICS : „ « α, α 2 α MTM is AN b n Const 1 ⊲ C OMPARISON with MLE: „ « α, α 2 n P n AN α MLE = b is i =1 log( X i /x 0 ) n 14
2. M ETHOD OF T RIMMED M OMENTS Examples T ABLE 2: ARE ( b α MTM , b α MLE ) = 1 /Const 1 . b a 0 0.05 0.10 0.15 0.25 0.49 0.70 0 1 .918 .847 .783 .666 .423 .238 0.05 .918 .848 .783 .667 .425 .242 0.10 .848 .785 .669 .430 .250 0.15 .787 .672 .437 .261 0.25 .679 .452 .285 0.49 .487 – 0.70 – 15
2. M ETHOD OF T RIMMED M OMENTS Examples T ABLE 2: ARE ( b α MTM , b α MLE ) = 1 /Const 1 . b a 0 0.05 0.10 0.15 0.25 0.49 0.70 0 1 0.05 1.00 .918 0.10 1.00 .918 .848 0.15 .999 .919 .850 .787 0.25 .995 .918 .851 .790 .679 0.49 .958 .897 .839 .786 .688 .487 0.70 .857 .824 .781 .738 .659 – – 15
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