Journal of Statistical Planning and Inference 137 (2007) 133 – 147
www.elsevier.com/locate/jspi
Conditional saddlepoint approximations for non–continuous and
non–lattice distributions
John E. Kolassa
a , ∗
, John Robinson
b
a
Department of Statistics, Rutgers University, Piscataway, NJ 08855, USA
b
School of Mathematics and Statistics, University of Sydney, NSW 2006, Australia
Received 29 January 2005; received in revised form 21 September 2005; accepted 7 November 2005
Available online 19 December 2005
Abstract
This manuscript presents an approximation to the distribution function of a smooth transformation of a random vector, conditional
on the event that values of other smooth transformations of the same random vector lie in a small rectangle. This approximation
is used to extend the application of standard saddlepoint conditional tail area approximations in circumstances beyond continuous
and lattice cases currently justified in the literature. We consider application to two examples, finite sampling and score testing in
logistic regression, where conditioning on a rectangle is essential.
© 2005 Elsevier B.V.All rights reserved.
MSC: Primary 60F10; secondary 60E99
Keywords: Saddlepoint approximation; Conditional inference
1. Introduction
We calculate a saddlepoint approximation to conditional tail probabilities associated with a smooth function of means
¯
Z of independent, but not necessarily identically distributed, random vectors Z
1
,..., Z
n
,... in R
d
. Specifically, for a
smooth function G, let V = G(
¯
Z), and choose a vector v, and
j
> 0 possibly depending on n, but bounded. We construct
an expression of the form
P [V
1
v
1
|V
j
∈ v
j
±
j
/n, j = 2,...,m + 1]
=
¯
(
√
n ˆ w
1
) +
(
√
n ˆ w
1
)
√
n
1
ˆ
1
-
1
ˆ w
1
(1 + R
n
), (1)
for m + 1 d , and
R
n
=
O(1/n) if
¯
Z has a density, if G
1
is linear, or if d = m + 1,
O
log(n)
(d+1-m)/2
√
n
otherwise.
(2)
∗
Corresponding author.
E-mail address: kolassa@stat.rutgers.edu (J.E. Kolassa).
0378-3758/$ - see front matter © 2005 Elsevier B.V. All rights reserved.
doi:10.1016/j.jspi.2005.11.003