Blockwise empirical likelihood and efficiency for semi-Markov - PowerPoint PPT Presentation

Blockwise empirical likelihood and efficiency for semi-Markov processes Wolfgang Wefelmeyer Mathematical Institute University of Cologne jointly with Cindy Greenwood (Arizona State University) and Uschi M¨ uller (Texas A&M University)

Empirical likelihood and empirical estimators in the i.i.d. case. Let X 1 , . . . , X n be i.i.d. with a distribution fulfilling a linear constraint Ph = E [ h ( X )] = 0. The empirical likelihood of Owen (1988, 2001) uses a weighted empirical distribution that fulfills this constraint: n P w h = 1 � w j h ( X j ) = 0 . n j =1 A linear functional Pf = E [ f ( X )] is then estimated by the weighted empirical estimator n P w f = 1 � w j f ( X j ) . n i =1 Take f and h one-dimensional. The weights are of the form w j = 1 / (1 + µh ( X j )), and one can show that µ = P h/ P h 2 + o P ( n − 1 / 2 ), where P h = 1 � n j =1 h ( X j ) denotes the usual empirical estimator. n

The weighted empirical estimator has the stochastic expansion P w f = P f + Pfh Ph 2 P h + o P ( n − 1 / 2 ) . (1) The asymptotic variances of P f and P w f are Var f ( X ) and Var f ( X ) − ( Pfh ) 2 /Ph 2 . The reduction can be considerable. From (1) we derive an alternative to P w f , the additively corrected empirical estimator P add f = P f − P fh P h 2 P h. Both estimators are asymptotically efficient. For dependent data, an efficient estimator must fulfill (1) with Ph 2 and Pfh replaced by variance and covariance of P h and P f . But (1) continues to hold for the weighted empirical estimator, which is therefore not efficient any more. We will now see that blockwise weighting gives the right result.

Empirical likelihood for Markov renewal processes. Let ( X 0 , T 0 ) , . . . , ( X n , T n ) be observations of a Markov renewal pro- cess. (The results carry over to semi-Markov processes.) Write V j = T j − T j − 1 for the inter-arrival times. Then ( X 1 , V 1 ) , . . . , ( X n , V n ) follow a Markov chain with transition distribution not depending on the previous inter-arrival time, S ( x ; dy, dv ) = Q ( x ; dy ) R ( x, y ; dv ). The empirical estimator for Pf = E [ f ( X, Y, V )] is n P f = 1 � f ( X j − 1 , X j , V j ) . n j =1 If the embedded chain is exponentially ergodic, P f has the martingale approximation P f − Pf = P Af + o P ( n − 1 / 2 ) with Af ( x, y, v ) = f ( x, y, v ) − Sf ( x ) + Sf ( y ) − QSf ( x ) ∞ ( Q t Sf ( y ) − Q t +1 Sf ( x )) . � + t =1

Assume the linear constraint Ph = E [ h ( X, Y , V )] = 0. By MSW (2001), an efficient estimator ˆ ϑ for Pf is characterized by ϑ = P f − PAfAh P ( Ah ) 2 P h + o P ( n − 1 / 2 ) . ˆ Such an estimator is the additively corrected empirical estimator P add f = P f − ˆ γ σ 2 P h ˆ with m n − k 1 � � � ˆ γ = P fh + h ( X j − 1 , X j , V j ) f ( X j + k − 1 , X j + k , V j + k ) n − k j =1 k =1 � + h ( X j + k − 1 , X j + k , V j + k ) f ( X j − 1 , X j , V j ) , m n − k 1 σ 2 = P h 2 + 2 � � ˆ h ( X j − 1 , X j , V j ) h ( X j + k − 1 , X j + k , V j + k ) . n − k k =1 j =1

An efficient estimator of Pf = E [ f ( X, Y, V )] is also obtained with the blockwise empirical likelihood, introduced by Kitagawa (1997) for different purposes. Let n = νm with m → ∞ slowly. Take averages over blocks, m F i = 1 � f ( X ( i − 1) m + k − 1 , X ( i − 1) m + k , V ( i − 1) m + k ) , m k =1 m H i = 1 � h ( X ( i − 1) m + k − 1 , X ( i − 1) m + k , V ( i − 1) m + k ) . m k =1 The empirical estimator of Pf can be written P f = 1 � ν i =1 F i . ν Define blockwise weights w i as solutions of ν P w h = 1 � w i H i = 0 . ν i =1 The blockwise weighted empirical estimator is ν P w f = 1 � w i F i . ν i =1

We show that this blockwise weighted empirical estimator ν P w f = 1 � with weights P w h = 0 w i F i ν i =1 is asymptotically equivalent to the blockwise additively corrected empirical estimator � ν i =1 F i H i P block f = P f − P h. � ν i =1 H 2 i This, in turn, is asymptotically equivalent to the above additively corrected empirical estimator P add f = P f − ˆ γ σ 2 P h, ˆ which we know to be efficient. Blocks are also used to bootstrap dependent data. For empirical likelihood, we need not separate blocks by gaps. The blocks may even overlap.