Consistent Variance Estimation in Semiparametric Models with Application to Interval-Censored Data - Interval-Censored-Time-to-Event Data: Methods and Applications - page 261

By (9.27) and assumption (A2), using the Cauchy{Schwarz inequality, the

last four terms on the right side in the above equation converge to zero in
probability. So we have

P` 2 (; b n ; b n ) = P` 2 (; 0 ; 0 ) + o p (1) ! p I( 0 ):

This combined with assumptions (A1) and (A2) imply that

P n ` 2 (; b n ; b n ) = (P n P)` 2 (; b n ; b n ) + P` 2 (; b n ; b n ) ! p P` 2 (; 0 ; 0 ) = I( 0 ):

This completes the proof for the case of d = 1.
When d > 1, let ` j (x; ;h) be the j-th element of `(x; ;h). We need to
show that

P n ` j (; b n ; b n )` k (; b n ; b n ) ! p P` j (; 0 ; 0 )` k (; 0 ; 0 ); 1 j;k d: (9.28)

When j = k, this follows from the proof for d = 1. When j 6= k, then by

assumption (A1), we have

(P n P)` j (; b n ; b n )` k (; b n ; b n ) ! p 0:

(9.29)

Write

P[` j (; b n ; b n )` k (; b n ; b n ) ` j (; 0 ; 0 )` k (; 0 ; 0 )]

Pf` j (; b n ; b n )[` k (; b n ; b n ) ` k (; 0 ; 0 )]g
+Pf` k (; 0 ; 0 )[` j (; b n ; b n ) ` j (; 0 ; 0 )]g

=

T 1n + T 2n :

By Equation (9.24), P` j (; b n ; b n ) = O p (1). Thus, by the Cauchy-Schwarz
inequality,

T 1n P` j (; b n ; b n )P[` k (; b n ; b n ) ` k (; 0 ; 0 )] 2 ! p 0

(9.30)

and

T 2n P` k (; 0 ; 0 )P[` j (; b n ; b n ) ` j (; 0 ; 0 )] 2 ! p 0:

(9.31)

Now (9.28) follows from (9.29) to (9.31). This completes the proof.