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Contents of PMS, Vol. 4, Fasc. 2,
pages 221 - 236
 

TWO APPROACHES TO CONSTRUCTING SIMULTANEOUS CONFIDENCE BOUNDS FOR QUANTILES

M. Csörgő
P. Révész

Abstract: Given some regularity conditions on the distribution function F of a random sample X ,X ,...,X , 1 2 n the sequence of quantile processes (n1/2f(Q(y))(Q (y) -Q(y));0 < y < 1) n behaves like a sequence of Brownian bridges (B (y);0 < y < 1), n where Q(y) := F -1(y), the inverse of F (.) , and Q (y) = X n k:n if (k- l)/n < y < k/n (k = 1,2,..,n) with the order statistics X < X < ...< X 1:n 2:n n:n of the above sample. First, a sequence of consistent direct estimators is proposed for the quantile-density function Q'(y) = 1/f(Q(y)). The latter then also enables us to construct simultaneous confidence bounds for an unknown quantile function Q(y). The second approach makes frequently misused heuristic steps like

1- a = P(F (x) - n-1/2c(a) < Fn(x) < F(x)+ n1/2c(a);- oo < x < oo ) -1/2 -1 -1/2 = P(y - n c(a) < Fn(F (y)) < y+ n c(a); F -1(0) < F- 1(y) < F-1(1)) = P(F -1(y - n-1/2c(a)) < F -1(y) < F- 1(y + n-1/2c(a));0 < y < 1) n n
precise for large n , where Fn is the empirical distribution function of the above random sample, and for a (- (0,1), c(a) is defined by

P (0<suyp<1 |B(y)|< c(a))= 1 - a
for a Brownian bridge B(.).

2000 AMS Mathematics Subject Classification: Primary: -; Secondary: -;

Key words and phrases: -

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