Generalized method of moments for an extended Gamma process Zeina - PowerPoint PPT Presentation

Generalized method of moments for an extended Gamma process Zeina Al Masry, Sophie Mercier, Ghislain Verdier Laboratoire de Mathématiques et de leurs Applications Pau UMR CNRS 5142 Université de Pau et des Pays de l’Adour, Pau, France Fourth AMMSI Project Workshop 20 January 2016, Grenoble

Outline 1 Motivation 2 Generalized method of moments (GMM) 3 GMM for an extended Gamma process 4 Numerical comparisons 5 Summary

Motivation 1 Motivation 2 Generalized method of moments (GMM) 3 GMM for an extended Gamma process 4 Numerical comparisons 5 Summary Zeina Al Masry, Sophie Mercier, Ghislain Verdier AMMSI Fourth Project Workshop 1 / 22

Motivation Standard Gamma process(SGP) Definition Let Y = ( Y t ) t ≥ 0 be a stochastic process, A ( t ) an increasing continuous function and let b 0 > 0. Y is said to be a SGP ( Y ∼ Γ 0 ( A ( t ) , b 0 ) ) if - Y 0 = 0, - for 0 ≤ t 1 < ... < t n , Y t 1 , Y t 2 − Y t 1 , ..., Y t n − Y t n − 1 are independents, - for all s < t , Y t − Y s ∼ Γ 0 ( A ( t ) − A ( s ) , b 0 ) . The pdf at time t is given by b A ( t ) Γ( A ( t )) x A ( t ) − 1 exp ( − b 0 x ) , ∀ x ≥ 0 . 0 f t ( x ) = ▲ A SGP is not always a proper choice to model the evolution of the cumulative deterioration of a system over time. Zeina Al Masry, Sophie Mercier, Ghislain Verdier AMMSI Fourth Project Workshop 1 / 22

Motivation Extended Gamma process(EGP) Definition [Cinlar 1980] Let X = ( X t ) t ≥ 0 be a stochastic process, A ( t ) an increasing continuous function and b ( t ) a measurable positive function such � t 1 that b ( s ) a ( s ) ds < ∞ , for all t > 0, with a ( t ) the derivative of 0 A ( t ) . X is an EGP ( X ∼ Γ( A ( t ) , b ( t ))) if : � t dY s X t = b ( s ) with Y ∼ Γ 0 ( A ( t ) , 1 ) . 0 ➤ The increments are independent, ➤ for all t , λ ≥ 0 , h > 0, � t + h � � � � λ L X t + h − X t ( λ ) = exp − ln 1 + a ( s ) ds , t b ( s ) � t � t a ( s ) ds a ( s ) ds ➤ E ( X t ) = b ( s ) and V ( X t ) = b ( s ) 2 . 0 0 Zeina Al Masry, Sophie Mercier, Ghislain Verdier AMMSI Fourth Project Workshop 2 / 22

Motivation Technical tools for the use of an EGP ▲ Technical difficulties of an EGP : ✦ no exact stochastic simulation ✦ no explicit formula for the probability density function(pdf) and cumulative distribution function(cdf) ✔ An approximate EGP with a piecewise constant scale function : ✦ simulate approximate paths ✦ compute the cdf of a general EGP at a known precision Z. Al Masry, S. Mercier, G. Verdier,“Approximate simulation techniques and distribution of an extended Gamma process," Methodology and Computing in Applied Probability (2015). Zeina Al Masry, Sophie Mercier, Ghislain Verdier AMMSI Fourth Project Workshop 3 / 22

Motivation Parameter estimation of an EGP Let θ ∈ Θ ⊆ R p a p parameter vector. X ∼ Γ( A ( t , θ ) , b ( t , θ )) . Estimate the parameters of an EGP : ✖ Standard maximum likelihood estimation is not possible ✔ The moments and an explicit form of the Laplace transform are known ☞ Generalized method of moments L. P. Hansen,“Large sample properties of generalized method of moments estimators," Econometrica 50(4), 1029-1054(1982) Zeina Al Masry, Sophie Mercier, Ghislain Verdier AMMSI Fourth Project Workshop 4 / 22

Generalized method of moments (GMM) 1 Motivation 2 Generalized method of moments (GMM) 3 GMM for an extended Gamma process 4 Numerical comparisons 5 Summary Zeina Al Masry, Sophie Mercier, Ghislain Verdier AMMSI Fourth Project Workshop 5 / 22

Generalized method of moments (GMM) General approach Let W be a random vector of dimension d and { W n , n = 1 , . . . , N } a set of i.i.d random vectors having the same distribution as W . Let f : R d × Θ → R r ( r ≥ p ) be a function such that  f ( 1 ) ( w ( 1 ) , θ )  .  .  f ( w , θ ) =  , .    f ( d ) ( w ( d ) , θ ) � w ( 1 ) , . . . , w ( d ) � and f ( i ) ( w ( i ) , θ ) , i = 1 , . . . , d a where w = vector of dimension k . Zeina Al Masry, Sophie Mercier, Ghislain Verdier AMMSI Fourth Project Workshop 5 / 22

Generalized method of moments (GMM) General approach Definition (Population moment condition) Let θ 0 be the true unknown vector to be estimated. The population moment condition is defined E [ f ( W , θ 0 )] = 0 . Definition (Sample moment condition) The sample moment condition is derived from the average population moment condition, N g N ( θ ) = 1 � ˆ f ( W n , θ ) . N n = 1 Zeina Al Masry, Sophie Mercier, Ghislain Verdier AMMSI Fourth Project Workshop 6 / 22

Generalized method of moments (GMM) General approach GMM estimator is defined as follows : Definition Let ( P N ) be a sequence of positive semi–definite weighting matrices that converges in probability to a constant positive definite matrix P . Then, the GMM estimator based on these population moments conditions is the value of θ that minimizes ˆ g N ( θ ) T P N ˆ θ N = arg min θ ∈ Θ ˆ g N ( θ ) . Zeina Al Masry, Sophie Mercier, Ghislain Verdier AMMSI Fourth Project Workshop 7 / 22

Generalized method of moments (GMM) Asymptotic properties W. K. Newey and D. L. McFadden,“Handbook of Econometrics, Large sample estimation and hypothesis testing," Elsevier Science Publishers, Amsterdam, The Netherlands 4, 2113-247(1994). Consistency p Under technical assumptions, ˆ − → θ 0 . θ N Asymptotic normality Under technical assumptions, √ � p � ˆ → N ( 0 , HSH T ) θ N − θ 0 − N � ∂ f ( W , θ 0 ) � where H = ( D T 0 PD 0 ) − 1 D T 0 P , D 0 = E and ∂ θ T � f ( W , θ 0 ) f ( W , θ 0 ) T � S = E . Zeina Al Masry, Sophie Mercier, Ghislain Verdier AMMSI Fourth Project Workshop 8 / 22

Generalized method of moments (GMM) Optimal choice of weighting matrix A. R. Hall,“Generalized method of moments," Oxford University Press, Oxford, UK (2005). Theorem If assumptions of the asymptotic normality hold and S is non-singular, then the minimum asymptotic variance of ˆ θ N is � − 1 � D T 0 S − 1 D 0 V = and this can be obtained by setting P = S − 1 . Zeina Al Masry, Sophie Mercier, Ghislain Verdier AMMSI Fourth Project Workshop 9 / 22

Generalized method of moments (GMM) Two-step estimator Two-step estimator : 1. Set P N = I and solve ( 1 ) g N ( θ ) T ˆ ˆ N = arg min θ ∈ Θ ˆ g N ( θ ); θ ( 1 ) 2. Construct a consistent estimator of S based on ˆ θ N N S N = 1 ( 1 ) ( 1 ) ˆ f ( W n , ˆ N ) f ( W n , ˆ � N ) T . θ θ N n = 1 The estimator of θ 0 is given by g N ( θ ) T ˆ − 1 ˆ θ N = arg min θ ∈ Θ ˆ S N ˆ g N ( θ ) . Zeina Al Masry, Sophie Mercier, Ghislain Verdier AMMSI Fourth Project Workshop 10 / 22

GMM for an extended Gamma process 1 Motivation 2 Generalized method of moments (GMM) 3 GMM for an extended Gamma process 4 Numerical comparisons 5 Summary Zeina Al Masry, Sophie Mercier, Ghislain Verdier AMMSI Fourth Project Workshop 11 / 22

GMM for an extended Gamma process Two kinds of GMM : ① GMM based on the moments ② GMM based on the Laplace transform Zeina Al Masry, Sophie Mercier, Ghislain Verdier AMMSI Fourth Project Workshop 11 / 22

GMM for an extended Gamma process We define the increments of X by W ( i ) = X t i − X t i − 1 , i = 1 , 2 , .., d where t 0 = 0 < t 1 < · · · < t d = T . W ( 1 )   n .  .  W n =  . .    W ( d ) n Zeina Al Masry, Sophie Mercier, Ghislain Verdier AMMSI Fourth Project Workshop 12 / 22

GMM for an extended Gamma process GMM based on the moments The sample moment condition is given by m ( 1 ) ( θ ) − m ( 1 ) ( θ )  ˆ  v ( 1 ) ( θ ) − v ( 1 ) ( θ ) ˆ    .  . g N ( θ ) = ˆ   .     m ( d ) ( θ ) − m ( d ) ( θ ) ˆ   v ( d ) ( θ ) − v ( d ) ( θ ) ˆ � t i � t i a ( s , θ ) ds a ( s , θ ) ds where m ( i ) ( θ ) = b ( s , θ ) , v ( i ) ( θ ) = b ( s , θ ) 2 , t i − 1 t i − 1 N N � 2 � � � m ( i ) ( θ ) = 1 W ( i ) v ( i ) ( θ ) = 1 W ( i ) − m ( i ) ( θ ) ˆ n , ˆ . n N N n = 1 n = 1 GMM estimator is ˆ g N ( θ ) T P N ˆ θ N = arg min θ ∈ Θ ˆ g N ( θ ) . Zeina Al Masry, Sophie Mercier, Ghislain Verdier AMMSI Fourth Project Workshop 13 / 22

GMM for an extended Gamma process GMM based on the Laplace transform The sample moment condition is given by ˆ L ( 1 ) ( λ 1 , θ ) − L ( 1 ) ( λ 1 , θ )   ˆ L ( 1 ) ( λ 2 , θ ) − L ( 1 ) ( λ 2 , θ )     ˆ L ( 1 ) ( λ 3 , θ ) − L ( 1 ) ( λ 3 , θ )     .   . ˆ g N ( θ ) = ,   .     L ( d ) ( λ 1 , θ ) − L ( d ) ( λ 1 , θ ) ˆ     ˆ L ( d ) ( λ 2 , θ ) − L ( d ) ( λ 2 , θ )     ˆ L ( d ) ( λ 3 , θ ) − L ( d ) ( λ 3 , θ ) N L ( i ) ( λ k , θ ) = 1 where ˆ � exp ( − λ k W ( i ) n ) , k = 1 , 2 , 3. N n = 1 ➤ λ 2 and λ 3 are multiples of λ 1 . Zeina Al Masry, Sophie Mercier, Ghislain Verdier AMMSI Fourth Project Workshop 14 / 22