Page 107 - Textos de Matemática Vol. 47

ESTIMATION OF THE CONDITIONAL QUANTILE FUNCTION 97
By inverting this estimator, we then derive a natural estimate of the conditional quantile function and its consistency is established.
2. Assumptions and notation
Let {(Xt, Yt) , t 2 [0, +1[ } be a Rd ⇥ R-valued continuous time measurable stochastic process on the probability space (⌦, A, P ).
It is assumed that {(Xt, Yt) , t 2 [0, +1[ } is a strictly stationary and ergodic process with absolutely continuous marginal distributions.
The density function of Xt will be denoted by g and, for each x 2 Rd such that g(x) > 0, fYt/Xt(·/x) ⌘ f(·/x) and FYt/Xt(·/x) ⌘ F(·/x) denote the density and the distribution function of Yt given Xt = x, respectively. We also consider
q↵(x)=inf{z2R:F(z/x) ↵}, ↵2]0,1[,
the conditional quantile function of Yt given Xt = x. Our estimator of FZ(·/x) is defined by
+1 d FT(y/x)= 1] 1,y](z)fT(z/x)dz, y2R, x2R ,
1
where fT (·/x) is the kernel estimator of the conditional density fT (·/x) pro- posed by Didi [13]:
1 Z T ✓ z Yt ◆ ✓ x Xt ◆ d+1 K0 K dt
fT(z/x)=
ThT0 hT hT
, z2R.
1 ZT ✓x Xt◆
d K dt
ThT0 hT
As usual, K0 and K are density functions on R and Rd, respectively, and hT ⌘ h(T ), T > 0, is a real positive function.
More precisely, FT takes the form
FT(y/x)= NT (x,y),
with
and
1 Z T ✓ y Yt ◆ ✓ x Xt ◆
NT (x,y)=
DT(x)=dK dt,
DT (x)
K dt ThT0 hT hT
d G
1 ZT ✓x Xt◆
ThT0 hT
where G is the distribution function corresponding to K0.