Saddlepoint Approximations for the Difference of Order Statistics and Studentized Sample Quantiles

By CHUNSHENG MA and JOHN ROBINSON School of Mathematics and Statistics The University of Sydney, N.S.W. 2006, Australia

SUMMARY

For a sample from a given distribution the difference of two order statistics and the studentized quantile are statistics whose distribution is needed to obtain tests and confidence intervals for quantiles and quantile differences. This paper gives saddlepoint approximations for densities and saddlepoint approximations of the Lugannani-Rice form for tail probabilities of these statistics. The relative errors of the approximations are n^{−1} uniformly in a neighbourhood of the parameters and this uniformity is global if the densities are log-concave.

Keywords: SADDLEPOINT APPROXIMATION; LAPLACE METHOD; DIFFERENCE OF ORDER STATISTICS; STUDENTIZED SAMPLE QUANTILE.

1 INTRODUCTION

Suppose that F (x) is an absolutely continuous distribution function with density f, X_{1}, ⋯, X_{n} are a random sample from F (x), and X_{1:n} ≤ ⋯ ≤ X_{n:n} are the corresponding order statistics. The purpose of this paper is to derive saddlepoint approximations for the difference of order statistics and studentized sample quantiles by using the Laplace approximation. Similar approximations for other statistics have been considered in the literature; see for instance, Tierney, Kass and Kadane (1989), DiCiccio, Field and Fraser (1990), Daniels and Young (1991), Diciccio and Martin (1991), Jing and Robinson (1994), and Jensen (1995).

The difference of two order statistics

Z_{r_{1},r_{2}} = X_{r_{2}:n} − X_{r_{1}:n},

is often used to construct a confidence interval for the corresponding population quantile difference, say ξ_{p_{02}} − ξ_{p_{01}}, where r_{i} = [np_{0i}] + 1, i = 1, 2 and [x] stands for the largest integer not exceeding x. Such quantities may be of interest in their own right; X_{n:n} − X_{1:n}, the range of a distribution, X_{\frac{3}{4} n:n} − X_{\frac{1}{4} n:n}, the interquartile range and X_{\frac{9}{10} n:n} − X_{\frac{1}{10} n:n}, the interdecile of a distribution. These measures evidently give some approximate idea of “spread” of a distribution. However, for most densities f, the density of Z_{r_{1},r_{2}} is impossible to determine analytically, and its asymptotic normality is often used; see for example David (1981). Harvill and Newton (1995) obtain the saddlepoint approximation for the density of the difference of order statistics by using the approach of Easton and Ronchetti (1986), where the major difficulty arises in suitably approximating the related cumulant generating function so that the solution of the approximating saddlepoint equation exists and is unique. By using the Laplace method in Section 2 we derive saddlepoint approximations for both the density and the tail probability of the difference of order statistics. These approximations have relative errors of order n^{−1} uniformly in a neighbourhood of the difference of the population quantiles. Further, if the density f is log-concave, the relative errors are uniformly of order n^{−1}; results of this type for means are discussed in Jensen (1991 and 1995, Chapter 6). Interestingly enough, our saddlepoint approximation for the density of Z_{r_{1},r_{2}} is “exact” in the case of the exponential distribution and the uniform distribution. This is related to the results of Daniels (1954, 1980) and Blaesild and Jensen (1985) who obtained the only three “exact” saddlepoint approximations for the density of an i.i.d. sum.

Another quantity of interest is of the form

\frac{X_{r:n} − ξ_{p_{0}}}{Z_{r_{1},r_{2}}} ,

where r_{1} < r < r_{2} are integers, ξ_{p_{0}} is the p_{0}-th population quantile (0 < p_{0} < 1) and

r = \{ \begin{matrix}np_{0}, & if np_{0} is an integer, \\ [np_{0}] + 1, & otherwise. \end{matrix}

It is called the studentized sample quantile, since its structure is somewhat similar to that of Student’s t in the sense that

(i) its numerator is the difference between the sample quantile X_{r:n} and the population quantile ξ_{p_{0}} and as an estimate of a scale parameter, its denominator is the difference of order statistics; and

(ii) it is invariant under linear transformations, so that its distribution is independent of location and scale parameters. The studentized sample quantile has been investigated by Siddiqui (1960), Bloch and Gastwirth (1968), Birnbaum and Vincze (1973), Hall and Sheather (1988), amongst others. In particular, Hall and Sheather (1988) derived the Edgeworth expansion for the studentized sample quantile with specified r_{1} = r −r_{0}, r_{2} = r +r_{0} and applied it to construct confidence intervals. In Section 3 we will derive saddlepoint approximations for both the density and the tail probability of the studentized sample quantile, with relative errors uniformly of order n^{−1} in a neighbourhood of the quantile. Again,if the density f is log-concave, the relative errors are uniformly of order n^{−1}. A numerical example shows the great accuracy of the saddlepoint approximation.

Let ϕ(·) and Φ(·) represent the standard normal density and distribution functions respectively. For a multivariate function h(x_{1}, ⋯, x_{k}), denote its derivatives as follows

ḣ_{x_{i}}(x_{1}, ⋯, x_{k}) = \frac{∂h(x_{1}, ⋯, x_{k})}{∂x_{i}},
ḧ_{x_{i}x_{j}}(x_{1}, ⋯, x_{k}) = \frac{∂^{2}h(x_{1}, x_{j})}{∂x_{i}∂x_{j}}, i, j = 1, ⋯, k,

and let ∇h(x_{1}, ⋯, x_{k}) stands for its the Hessian matrix.

2 SADDLEPOINT APPROXIMATION FOR THE DIFFERENCE OF ORDER STATISTICS

Given 0 < p_{01} < p_{02} < 1, consider the difference of two order statistics

Z_{r_{1},r_{2}} = X_{r_{2}:n} − X_{r_{1}:n},

where r_{i} = [np_{0i}] + 1, i = 1, 2. Denote the density and tail probability of Z_{r_{1},r_{2}} by f_{Z_{r_{1},r_{2}}}(z) and F̄_{Z_{r_{1},r_{2}}}(z), respectively.

The joint density of (X_{r_{1}:n}, X_{r_{2}:n}) is

f(x_{1}, x_{2}) = \frac{n!f(x_{1})f(x_{2})}{(r_{1} − 1)!(r_{2} − r_{1} − 1)!(n − r_{2})!}F^{r_{1}−1}(x_{1})\{F (x_{2}) − F (x_{1})\}^{r_{2}−r_{1}−1}\{1 − F (x_{2})\}^{n−r_{2}},

x_{1} < x_{2}.

Based on f(x_{1}, x_{2}), the joint density of (X_{r_{1}:n}, Z_{r_{1},r_{2}}) is f(t, t + z), z ≥ 0, which gives the exact density of Z_{r_{1},r_{2}} as

f_{Z_{r_{1},r_{2}}}(z) = ∫_{−∞}^{∞} f(t, t + z)dt, z ≥ 0.

Write p_{i} = r_{i}/n, i = 1, 2, where we have suppressed the dependence on n to simplify the notation.

The density f(x_{1}, x_{2}) may be re-expressed as

f(x_{1}, x_{2}) = c_{n} exp{−n · h(x_{1}, x_{2})}g(x_{1}, x_{2}), x_{1} < x_{2},

where

h(x_{1}, x_{2}) = p_{1} ln p_{1} + (p_{2} − p_{1}) ln(p_{2} − p_{1}) + (1 − p_{2}) ln(1 − p_{2})
−{p_{1} ln F (x_{1}) + (p_{2} − p_{1}) ln(F (x_{2}) − F (x_{1})) + (1 − p_{2}) ln(1 − F (x_{2}))},
g(x_{1}, x_{2}) = {F (x_{1})(F (x_{2}) − F (x_{1}))}^{−1}f(x_{1})f(x_{2}),

and

c_{n} = \frac{n!}{(r_{1} − 1)!(r_{2} − r_{1} − 1)!(n − r_{2})!}
· exp{n[p_{1} ln p_{1} + (p_{2} − p_{1}) ln(p_{2} − p_{1}) + (1 − p_{2}) ln(1 − p_{2})]},

and again we have suppressed the dependence of h(x_{1}, x_{2}) and g(x_{1}, x_{2}) on n. So

f_{Z_{r_{1},r_{2}}}(z) = c_{n} ∫_{−∞}^{∞} exp{−n · h(t, t + z)}g(t, t + z)dt, z ≥ 0.

Applying Stirling’s formula to each factorial, an estimate of c_{n} is then given by

c_{n} = \frac{n}{2π} c\{1 + O(n^{−1})\},

where

c = \{\frac{p_{1}(p_{2} − p_{1})}{1 − p_{2}}\}^{\frac{1}{2}} .

As a result, f_{Z_{r_{1},r_{2}}}(z) is approximated by

f_{Z_{r_{1},r_{2}}}(z) = \frac{nc}{2π} ∫_{−∞}^{∞} exp\{−n · h(t, t + z)\}g(t, t + z)dt\{1 + O(n^{−1})\},  z ≥ 0. (2.1)

Next it will be further approximated by using a Laplace approximation.

To state the results, some notation is needed. Let z^{0} = x_{2}^{0} − x_{1}^{0}, where (x_{1}^{0}, x_{2}^{0}) is given by

F (x_{1}^{0}) = p_{1}, F (x_{2}^{0}) = p_{2}.

For each z let t̃= t(z) be the solution of the equation ∂h(t, t + z)/∂t = 0, namely,

(\frac{p_{1}}{F (t̃)} − \frac{p_{2} − p_{1}}{F (t̃+ z) − F (t̃)})f(t̃) = (\frac{1 − p_{2}}{1 − F (t̃+ z)} − \frac{p_{2} − p_{1}}{F (t̃+ z) − F (t̃)})f(t̃+ z). (2.2)

Our saddlepoint approximations for the density and tail probability of Z_{r_{1},r_{2}} are presented in the following theorem.

Theorem 1 Suppose that f′′(x) exists and is bounded, and f(x_{i}^{0}) > 0 (i = 1, 2). Then uniformly in a neighbourhood of z^{0},

(i) f_{Z_{r_{1},r_{2}}}(z) = \sqrt{\frac{n}{2π}} exp\{−n · H(z)\}G(z)\{1 + O(n^{−1})\},  (2.3)

where

H(z) = h(t(z), t(z) + z),
G(z) = cg(t(z), t(z) + z)ḧ_{t}^{− \frac{1}{2}} (t(z); z),

and

ḧ_{t}(t; z) = \frac{∂^{2}h(t, t + z)}{∂t^{2}}.

(ii) F̄_{Z_{r_{1},r_{2}}}(z) = 1 − Φ(\sqrt{n}w(z)) − \frac{ϕ(\sqrt{n}w(z))}{\sqrt{n}}\{\frac{1 − ψ(z)}{w(z)} + O(n^{−1})\},  (2.4)

or alternatively,

F̄_{Z_{r_{1},r_{2}}}(z) = [1 − Φ(\sqrt{n}w^{∗}(z))]\{1 + O(n^{−1})\},  (2.5)

where

w(z) = {2H(z)}^{1/2}sgn(z − z^{0}),
w^{∗}(z) = w(z) − \frac{1}{nw(z)} log\{ψ(z)\},

and

ψ(z) = G(z) | \frac{w(z)}{H′(z)} | .

To prove Theorem 1, we require some lemmas, whose proofs are given in an Appendix. Lemma 1 (i) below ensures that H(z) is always nonnegative so that w(z) in (2.4) and (2.5) is well-defined. To apply the Laplace method, it has to be shown that for large n, the behaviour of the integral in (2.1) is determined by behaviour near the minimum of h(t, t + z), where z > 0 is fixed. This can be seen from Lemma 1 (iii).

Lemma 1

(i) h(x_{1}, x_{2}) is nonnegative and attains its minimum 0 at (x_{1}^{0}, x_{2}^{0}).

(ii) For any constant 0 ≤ λ_{0} ≤ \frac{1}{2} min\{p_{1}, 1 − p_{2}\},

h(x_{1}, x_{2}) ≥ λ_{0}{[F (x_{1}) − p_{1}]^{2} + [F (x_{2}) − p_{2}]^{2}}, x_{1} < x_{2}.

(iii) For fixed z > 0 and any d_{1} < z^{0} < d_{2},

(∫_{−∞}^{d_{1}} + ∫_{d_{2}}^{∞}) exp{−n · h(t, t + z)}g(t, t + z)dt ≤ c_{1}e^{−nc_{2}},

where c_{1}, c_{2} are positive constants independent of n.

In particular, for z^{0} = x_{2}^{0} − x_{1}^{0}, t(z^{0}) = x_{1}^{0} and from Lemma 1 (i), h(t, t + z^{0}) is minimized at x_{1}^{0}.

Lemma 2

(i) | ∇h(t(z), t(z) + z) |= ḧ_{t}(t(z); z)H′′(z).

(ii) ∇h(t(z), t(z) + z) is positive definite if and only if one of the following statements is true (1) ḧ_{t}(t(z); z) > 0 and H′′(z) > 0;

(2) ḧ_{t}(t(z); z) > 0 and | ∇h(t(z), t(z) + z) |> 0;

(3) H′′(z) > 0 and | ∇h(t(z), t(z) + z) |> 0.

It is easy to see that under the conditions described in Lemma 2 (ii), for each given z, h(t, t + z) as a function of t has its unique minimum at t̃= t(z), and H′′(z) > 0.

Proof of Theorem 1 (i) Note that

ḧ_{t}(t(z^{0}); z^{0}) = \frac{f^{2}(x_{1}^{0}))}{p_{1}} + \frac{[f(x_{2}^{0}) − f(x_{1}^{0})]^{2}}{p_{2} − p_{1}} + \frac{f(x_{2}^{0})}{1 − p_{2}} ,
| ∇h(x_{1}^{0}, x_{2}^{0}) | = \frac{n^{3}}{r_{1}(r_{2} − r_{1})(n − r_{2})}f^{2}(x_{1}^{0})f^{2}(x_{2}^{0}),

which implies that ∇h(t(z^{0}), t(z^{0}) + z^{0}) is positive definite by Lemma 2 (ii). So there exists a neighbourhood of z^{0} such that for every fixed z > 0 in that region, h(t, t+z) as a function of t is locally convex for | t − x_{1}^{0} |< δ with δ > 0 small, and takes its unique minimum at t̃= t(z). Outside that region Lemma 1(iii) says that the integrand of (2.1) is exponentially small and the integral is likewise exponentially small. Consequently, as n becomes large, the major contribution to the integral in (2.1) comes from that neighbourhood of z^{0}. The Laplace method is then applicable (see e.g. Jensen (1995)),

f_{Z_{r_{1},r_{2}}}(z) = \frac{n}{2π} c ∫_{x_{1}^{0}−δ}^{x_{1}^{0}+δ} exp\{−n · h(t, t + z)\}g(t, t + z)dt\{1 + O(n^{−1})\}

+\frac{n}{2π} c(∫_{−∞}^{x_{1}^{0}−δ} + ∫_{x_{1}^{0}+δ}^{∞}) exp\{−n · h(t, t + z)\}g(t, t + z)dt\{1 + O(n^{−1})\}

= \sqrt{\frac{n}{2π}} exp\{−n · H(z)\}G(z)\{1 + O(n^{−1})\} + \frac{n}{2π} cc_{1}e^{−nc_{2}},

and so (2.3) is obtained.

(ii) We derive (2.4) by Temme’s method; see Temme (1981) or Barndorff-Nielsen and Cox (1989, p82). Obviously,

\frac{dw}{dz} = \frac{H′(z)}{w(z)},

and from (2.3),

F_{Z_{r_{1},r_{2}}}(z) = \sqrt{\frac{n}{2π}} ∫_{w(z)}^{∞} ψ(w)e^{− \frac{n}{2} w^{2}}dw\{1 + O(n^{−1})\},  (2.6)

where

ψ(w) = G(z)| \frac{dw}{dz} |^{−1} = c\frac{ḧ_{t}^{− \frac{1}{2}} (t(z); z)f(t(z) + z) | w(z) |}{| p_{1}F (t(z) + z) − p_{2}F (t(z)) |}.

Specifically, ψ(0) = 1. To see this, note that w → 0 is equivalent to z → z^{0}, and

lim_{z→z^{0}} | \frac{dw}{dz} |= \{H′′(z^{0})\}^{\frac{1}{2}} .

From Lemma 2 (i), one has

lim_{w→0} ψ(w) = G(z^{0}) lim_{z→z^{0}} | \frac{dw}{dz} |^{−1} = | \frac{H′′(z^{0})ḧ_{t}(x_{1}^{0}; z^{0})}{| ∇h(x_{1}^{0}, x_{2}^{0}) |} |^{− \frac{1}{2}} = 1.

Replacing ψ(w) by 1 − {1 − ψ(w)}, and then integrating by parts, from (2.6) we have

F̄_{Z_{r_{1},r_{2}}}(z) = 1 − Φ(\sqrt{n}w(z)) − \frac{1}{\sqrt{2π}} ∫_{w(z)}^{∞} we^{− \frac{n}{2} w^{2}}\{\frac{1}{w} − \frac{ψ(w)}{w}\}dw\{1 + O(n^{−1})\}.

Finally, (2.4) is obtained by integrating by parts.

The approximation in (2.6) is equivalent to (2.5) by an approach of Jensen (1992). The proof of Theorem 1 is established.

An important special case is that f(x) is log-concave. Such a density is also called a P F_{2} density. Typical examples include: normal, truncated normal, exponential, double exponential (or Laplace), logistic, uniform, triangular, some of the Gamma, Beta and Weibull, and so on. For properties we refer to Ibragimov (1956), Karlin (1968) and Barlow and Proschan (1981). The following one is proved by Pratt (1981, p105).

Lemma 3 If f(x) is log-concave, then

(i) both F (x) and 1 − F (x) are log-concave;

(ii) F (x) − F (y) is log-concave as a function of (x, y) ∈ {(x_{1}, x_{2}) : x_{1} ≥ x_{2}}; (iii) for constants a_{i} and b_{i} (i = 1, 2), F (a_{1}x + b_{1}y) − F (a_{2}x + b_{2}y) is log-concave

as a function of (x, y) on such a set that (a_{1}x + b_{1}y, a_{2}x + b_{2}y) ∈ {(x_{1}, x_{2}) : x_{1} ≥ x_{2}}.

Corollary Under the assumption of Lemma 3, f(x_{1}, x_{2}) is log-concave; and for every fixed z > 0, h(t, t + z) is a convex function of t and takes its unique minimum at t̃ = t(z), and H′′(z) > 0. Furthermore, (2.3)-(2.5) hold uniformly for every z if f(·) is log-concave.

Example 1 Consider an exponential distribution with the hazard rate λ > 0 and

f(x) = λ exp(−λx), x ≥ 0.

It is well-known that the normalized spacings

T_{k} = λ(n − k + 1)(X_{k:n} − X_{k−1:n}),  k = 1, ⋯, n,  (2.7)

are independent and exponentially distributed with the hazard rate 1, where X_{0:n} ≡ 0. Consequently,

Z_{r_{1},r_{2}} = \frac{1}{λ} ∑_{i=r_{1}+1}^{r_{2}} \frac{T_{i}}{n − i + 1} =^{d} \frac{1}{λ} ∑_{i=1}^{r_{2}−r_{1}} \frac{T_{i}}{(n − r_{1}) − i + 1} =^{d} X_{r_{2}−r_{1}:n−r_{1}},

where =^{d} signifies that the two sides have the same distribution. So the exact density of Z_{r_{1},r_{2}} is

f_{Z_{r_{1},r_{2}}}(z) = \frac{(n − r_{1})!}{(r_{2} − r_{1} − 1)!(n − r_{2})!}[F (z)]^{r_{2}−r_{1}−1}[1 − F (z)]^{n−r_{2}}f(z),  z ≥ 0. (2.8)

For a given z, solving (2.2) gives

t(z) = −\frac{1}{λ} log(1 − \frac{r_{1}}{n}) ≡ x_{1}^{0}.

Also

ḧ_{t}(t(z), t(z) + z) = \frac{1 − p_{1}}{p_{1}} λ^{2}

is independent of z.

H(z) = −{(p_{2} − p_{1}) log F (z) + (1 − p_{2}) log[1 − F (z)]} − c,

where

c = (p_{2} − p_{1}) log \frac{1 − p_{1}}{p_{2} − p_{1}} + (1 − p_{2}) log \frac{1 − p_{1}}{1 − p_{2}} ,

and

G(z) = \{\frac{(1 − p_{1})(p_{2} − p_{1})}{1 − p_{2}}\}^{\frac{1}{2}} \frac{f(z)}{F (z)}.

Hence, from (2.3),

f̂_{Z_{r_{1},r_{2}}}(z) = \frac{1}{\sqrt{2π}} \begin{matrix}(n − r_{1}) & n−r_{1}+ \frac{1}{2} \\ (r_{2} − r_{1}) & r_{2}−r_{1}− \frac{1}{2} & (n − r_{2}) & n−r_{2}+ \frac{1}{2} \end{matrix} [F (z)]^{r_{2}−r_{1}−1}[1 − F (z)]^{n−r_{2}}f(z),

z ≥ 0,

which is the same as (2.8) except for a scale constant, obtained on replacing the factorials by Stirling’s approximation.

The exact tail probabilities are compared in Table 1 with the saddlepoint approximation provided by the formula (2.5). Saddlepoint approximations are seen to have remarkably small relative errors.

[Insert Table 1 about here.]

Example 2 For a uniform distribution

f(x) = 1, 0 ≤ x ≤ 1,

it is easy to verify that Z_{r_{1},r_{2}} =^{d}X_{r_{2}−r_{1}:n}, and so

f_{Z_{r_{1},r_{2}}}(z) = \frac{n!}{(r_{2} − r_{1} − 1)!(n − r_{2} + r_{1})!}z^{r_{2}−r_{1}−1}(1 − z)^{n−r_{2}+r_{1}},  0 ≤ z ≤ 1. (2.9)

From (2.3), the saddlepoint approximation for the density of Z_{r_{1},r_{2}} is

f̂_{Z_{r_{1},r_{2}}}(z) = \frac{1}{\sqrt{2π}} \frac{n^{n+ \frac{1}{2}}}{(r_{2} − r_{1})^{r_{2}−r_{1}− \frac{1}{2}} (n − r_{2} + r_{1})^{n−r_{2}+r_{1}+ \frac{1}{2}}} · z^{r_{2}−r_{1}−1}(1 − z)^{n−r_{2}+r_{1}},

0 ≤ z ≤ 1,

which is the same as (2.9) except for a scale constant, obtained on replacing the factorials by Stirling’s approximation.

Table 2 compares the exact tail probability F̄_{Z_{r_{1},r_{2}}}(z) with the saddlepoint approximation (2.5), which has remarkably small relative error.

[Insert Table 2 about here.]

3 SADDLEPOINT APPROXIMATION FOR THE STUDENTIZED SAMPLE QUANTILE

For a given p_{0} ∈ (0, 1), denote the p_{0}-th population quantile by ξ_{p_{0}}; namely, F (ξ_{p_{0}}) = p_{0}. We are interested in the distribution of the studentized sample quantile

Y = \sqrt{n}\frac{X_{r:n} − ξ_{p_{0}}}{σ_{p}S_{n,r_{0}}} ,

where r and r_{0} are integers, r = [np_{0}] + 1, r_{0} = [c_{1}n^{1−δ_{0}}] + 1 for some c_{1} > 0 and 0 < δ_{0} < 1 so that \frac{r_{0}}{n} → 0 (n → ∞), σ_{p}^{2} = p(1 − p) and

S_{n,r_{0}} = \frac{n}{2r_{0}} (X_{r+r_{0}:n} − X_{r−r_{0}:n}).

For simplicity of notation, denote p = r/n and p_{0}^{∗} = r_{0}/n. Y is alternatively expressed as

Y = s(X_{r−r_{0}:n}, X_{r:n}, X_{r+r_{0}:n}),

where

s(x_{1}, x_{2}, x_{3}) = σ_{n,r_{0}}^{−1} \frac{x_{2} − ξ_{p}}{x_{3} − x_{1}} , x_{1} < x_{2} < x_{3},

and

σ_{n,r_{0}} = \frac{\sqrt{n}σ_{p}}{2r_{0}} .

The joint density of (X_{r−r_{0}:n}, X_{r:n}, X_{r+r_{0}:n}) is, for x_{1} < x_{2} < x_{3},

f(x_{1}, x_{2}, x_{3}) = \frac{n!}{(r − r_{0} − 1)!(r_{0} − 1)!(r_{0} − 1)!(n − r − r_{0})!}f(x_{1})f(x_{2})f(x_{3})

F^{r−r_{0}−1}(x_{1}){F (x_{2}) − F (x_{1})}^{r_{0}−1}{F (x_{3}) − F (x_{2})}^{r_{0}−1}{1 − F (x_{3})}^{n−r−r_{0}},

or alternatively,

f(x_{1}, x_{2}, x_{3}) = c_{n} exp{−n · h(x_{1}, x_{2}, x_{3})}g(x_{1}, x_{2}, x_{3}),

where

c_{n} = \frac{n!}{(r − r_{0} − 1)!(r_{0} − 1)!(r_{0} − 1)!(n − r − r_{0})!}

· exp{−[(r − r_{0}) ln(p − p_{0}^{∗}) + 2r_{0} ln p_{0}^{∗} + (n − r − r_{0}) ln(1 − p − p_{0}^{∗})]},

g(x_{1}, x_{2}, x_{3}) = [F (x_{1})(F (x_{2}) − F (x_{1}))(F (x_{3}) − F (x_{2}))]^{−1}f(x_{1})f(x_{2})f(x_{3}),

and

h(x_{1}, x_{2}, x_{3}) = (p − p_{0}^{∗}) ln \frac{p − p_{0}^{∗}}{F (x_{1})} + p_{0}^{∗} ln \frac{p_{0}^{∗}}{F (x_{2}) − F (x_{1})} + p_{0}^{∗} ln \frac{p_{0}^{∗}}{F (x_{3}) − F (x_{2})}

+(1 − p − p_{0}^{∗}) ln \frac{1 − p − p_{0}^{∗}}{1 − F (x_{3})} .

The joint density of (X_{r−r_{0}:n}, X_{r+r_{0}:n}, Y) is derived from f(x_{1}, x_{2}, x_{3}) by making a transformation

\{ \begin{matrix}t_{1} = x_{1}, \\ t_{2} = x_{3}, \\ y = s(x_{1}, x_{2}, x_{3}), \end{matrix}

whose Jacobian is

| J |= | \frac{∂s(x_{1}, x_{2}, x_{3})}{∂x_{2}} |^{−1} = σ_{n,r_{0}}(x_{3} − x_{1}) = σ_{n,r_{0}}(t_{2} − t_{1}),

under which f(x_{1}, x_{2}, x_{3}) transforms to

c_{n} exp{−n · h̃(t_{1}, t_{2}, y)}g̃(t_{1}, t_{2}, y) | J |,

where

h̃(t_{1}, t_{2}, y) = h(t_{1}, ξ_{p} + (t_{2} − t_{1})σ_{n,r_{0}}y, t_{2}),

and

g̃(t_{1}, t_{2}, y) = g(t_{1}, ξ_{p} + (t_{2} − t_{1})σ_{n,r_{0}}y, t_{2}).

Hence, the exact density of Y = s(X_{r−r_{0}:n}, X_{r:n}, X_{r+r_{0}:n}) is

f_{Y} (y) = c_{n} ∫ ∫_{D(t_{1},t_{2};y)} exp{−n · h̃(t_{1}, t_{2}, y)}g̃(t_{1}, t_{2}, y) | J | dt_{1}dt_{2},

where D(t_{1}, t_{2}; y) is a convex set defined by

D(t_{1}, t_{2}; y) = {(t_{1}, t_{2}) : t_{1} ≤ ξ_{p} + (t_{2} − t_{1})σ_{n,r_{0}}y ≤ t_{2}}.

Applying Stirling’s approximation to each factorial of c_{n} gives

c_{n} = (\frac{n}{2π})^{\frac{3}{2}} c^{∗}\{1 + O(r_{0}^{−1})\},

where

c^{∗} = (\frac{p − p_{0}^{∗}}{1 − p − p_{0}^{∗}})^{\frac{1}{2}} p_{0}^{∗}.

Hence,

f_{Y} (y) = (\frac{n}{2π})^{\frac{3}{2}} c^{∗} ∫ ∫_{D(t_{1},t_{2};y)} exp\{−n · h̃(t_{1}, t_{2}, y)\}g̃(t_{1}, t_{2}, y) | J | dt_{1}dt_{2}\{1 + O(r_{0}^{−1})\},

(3.1)

which will be approximated by using a Laplace approximation. In the sequel, for each y, let (t̃_{1}, t̃_{2}) = (t_{1}(y), t_{2}(y)) be the solution of the equations

\{ \begin{matrix}ḣ_{x_{1}}(t̃_{1}, ξ_{p} + (t̃_{2} − t̃_{1})σ_{n,r_{0}}y, t̃_{2}) − σ_{n,r_{0}}yḣ_{x_{2}}(t̃_{1}, ξ_{p} + (t̃_{2} − t̃_{1})σ_{n,r_{0}}y, t̃_{2}) = 0, \\ ḣ_{x_{3}}(t̃_{1}, ξ_{p} + (t̃_{2} − t̃_{1})σ_{n,r_{0}}y, t̃_{2}) + σ_{n,r_{0}}yḣ_{x_{2}}(t̃_{1}, ξ_{p} + (t̃_{2} − t̃_{1})σ_{n,r_{0}}y, t̃_{2}) = 0. \end{matrix}
(3.2)

Let y^{0} = s(x_{1}^{0}, x_{2}^{0}, x_{3}^{0}), where F (x_{1}^{0}) = p − p_{0}^{∗}, F (x_{2}^{0}) = p, F (x_{3}^{0}) = p + p_{0}^{∗}.

Clearly, x_{i}^{0} → ξ_{p} (i = 1, 2, 3) as n → ∞. It is easy to verify that for a fixed y, the Hessian matrix of h̃(t_{1}, t_{2}, y) is

∇h̃(t_{1}, t_{2}; y) = ( \begin{matrix}1 & 0 \\ −σ_{n,r_{0}}y & σ_{n,r_{0}}y \\ 0 & 1 \end{matrix} )^{T} ∇h(x_{1}, x_{2}, x_{3}) ( \begin{matrix}1 & 0 \\ −σ_{n,r_{0}}y & σ_{n,r_{0}}y \\ 0 & 1 \end{matrix} ) ,

where x_{1} = t_{1}, x_{2} = ξ_{p} + (t_{2} − t_{1})σ_{n,r_{0}}y, x_{3} = t_{2}.

Our saddlepoint approximations are demonstrated in the following theorem.

Theorem 2 Suppose that f′′(x) exists and is bounded, and f(ξ_{p}) > 0. Then uniformly in a neighbourhood of ξ_{p},

(i) f_{Y} (y) = \sqrt{\frac{n}{2π}} exp\{−n · H(y)\}G(y)\{1 + O(r_{0}^{−1})\},  (3.3)

where

H(y) = h̃(t_{1}(y), t_{2}(y), y),
G(y) = c^{∗}g̃(t_{1}(y), t_{2}(y), y) | J | ·| ∇h̃(t_{1}(y), t_{2}(y); y) |^{− \frac{1}{2}} ,

and

| J |= σ_{n,r_{0}}(t_{2}(y) − t_{1}(y)).
(ii) F̄_{Y} (y) = 1 − Φ(\sqrt{n}w(y)) − \frac{ϕ(\sqrt{n}w(y))}{\sqrt{n}}\{\frac{1 − ψ(y)}{w(y)} + O(r_{0}^{−1})\},  (3.4)

or alternatively,

F̄_{Y} (y) = [1 − Φ(\sqrt{n}w^{∗}(y))]\{1 + O(r_{0}^{−1})\},  (3.5)

where

w(y) = {2H(y)}^{1/2}sgn(y − y^{0}),
w^{∗}(y) = w(y) − \frac{1}{nw(y)} log\{ψ(y)\},

and

ψ(y) = G(y) | \frac{w(y)}{H′(y)} | .

Corollary (3.3)-(3.5) hold uniformly for every y if f(·) is log-concave.

The following Lemmas 4 and 5 are used to prove Theorem 2. The proofs are similar to those of Lemma 1 and 2 and are thus omitted.

Lemma 4

(i) h(x_{1}, x_{2}, x_{3}) is nonnegative and attains its minimum 0 at (x_{1}^{0}, x_{2}^{0}, x_{3}^{0}).

(ii) For any constant 0 ≤ λ_{0} ≤ \frac{p−p_{0}^{∗}}{2} ,

h(x_{1}, x_{2}, x_{3}) ≥ λ_{0}{[F (x_{1}) − (p − p_{0}^{∗})]^{2} + [F (x_{2}) − p]^{2} + [F (x_{3}) − (p + p_{0}^{∗})]^{2}},

x_{1} < x_{2} < x_{3}.

Let

∇h[y] = ∇h(x_{1}, x_{2}, x_{3}),

at x_{1} = t_{1}(y), x_{2} = ξ_{p} + (t_{2}(y) − t_{1}(y))σ_{n,r_{0}}y, x_{3} = t_{2}(y).

Lemma 5 H′′(y), t′_{1}(y) and t′_{2}(y) exist, and

| ∇h[y] |=| ∇h̃(t_{1}(y), t_{2}(y); y) | ·\frac{H′′(y)}{| J |}.  (3.6)

∇h[y] is positive definite if and only if H′′(y) > 0 and ∇h̃(t_{1}(y), t_{2}(y); y) is positive definite.

Lemma 6

(i) Assume that f(x) is continuously differentiable near ξ_{p} and f(ξ_{p}) > 0. Then there exists a neighbourhood of ξ_{p} such that for every fixed y > 0 in that region, h̃(t_{1}, t_{2}, y) as a function of (t_{1}, t_{2}) is locally convex with | t_{1} − x_{1}^{0} | and | t_{2} − x_{3}^{0} | small and attains its unique minimum at (t_{1}(y), t_{2}(y)), where t̃_{1} = t_{1}(y), t̃_{2} = t_{2}(y) are the solution of the equations (3.2). In addition, H′′(y) > 0 for y in that region and H′(y) ≠ 0 unless y = y^{0}.

(ii) If f(x) is continuously differentiable and log-concave, then for a fixed y, h̃(t_{1}, t_{2}, y) is convex on D(t_{1}, t_{2}, y) and attains its unique minimum at (t_{1}(y), t_{2}(y)). Furthermore, H′′(y) > 0 and H′(y) = 0 if and only if y = y^{0}.

Proof of Theorem 2 (i) Obviously, (t_{1}(y^{0}), t_{2}(y^{0})) = (x_{1}^{0}, x_{3}^{0}), at which h̃(t_{1}, t_{2}, y^{0}) takes its minimum. Lemma 4 and Lemma 6 (i) imply that there exists a neighbourhood of ξ_{p_{0}} such that as n becomes large, the major contribution to the integral in (3.1) comes from such a region. By using the Laplace approximation (see e.g. Barndorff-Nielsen and Cox (1989), Ch.6) , we approximate f_{Y} (y) by (3.3).

(ii) From the definition of w(y), one has

w^{2}(y) = H(y), w(y)\frac{dw}{dy} = H′(y),

and so

\frac{dw}{dy} = \frac{1}{w(y)} | J | ḣ_{x_{2}}(t_{1}(y), ξ_{p} + (t_{2}(y) − t_{1}(y))σ_{n,r_{0}}y, t_{2}(y)).

Therefore, from (3.3),

F̄_{Y} (y) = \sqrt{\frac{n}{2π}} ∫_{w(y)}^{∞} ψ(w)e^{− \frac{n}{2} w^{2}}dw\{1 + O(n^{−1})\},  (3.7)

where

ψ(y) = G(y) · | \frac{dw}{dy} |^{−1}
= \frac{c^{∗}}{p_{0}^{∗}} \frac{f(t_{1}(y))f(t_{2}(y))w(y)| ∇h̃(t_{1}(y), t_{2}(y); y) |^{− \frac{1}{2}}}{F (t_{1}(y)) | 2F (ξ_{p} + (t_{2}(y) − t_{1}(y))σ_{n,r_{0}}y) − F (t_{1}(y)) − F (t_{2}(y)) |}.

Observe that w → 0 is equivalent to y → y^{0}, and

lim_{y→y^{0}} | \frac{dw}{dy} |^{2} = H′′(y^{0}).

We obtain, using (3.6),

ψ(0) = lim_{w→0} ψ(w) = G(y^{0}) lim_{y→y^{0}} | \frac{dw}{dy} |^{−1} =| J_{0} | \{\frac{| ∇h̃(x_{1}^{0}, x_{3}^{0}; y^{0}) | H′′(y^{0}) | J_{0} |}{| ∇h(x_{1}^{0}, x_{2}^{0}, x_{3}^{0}) |}\}^{− \frac{1}{2}} = 1.

Proceeding as in the proof of Theorem 1, one obtains (3.4) and (3.5) from (3.7).

Example 3 For an exponential distribution with the hazard rate λ > 0 and

f(x) = λ exp(−λx), x ≥ 0,

the evaluation of the distribution of the studentized sample quantile Y reduces to that of a weighted sum of independent exponentials; more precisely,

P (Y ≥ y) = P ( ∑_{i=1}^{r+r_{0}} γ_{j}(y)T_{j} ≥ λξ_{p}),  (3.8)

where T_{1}, ⋯, T_{n} are defined in (2.7), and

γ_{j}(y) = \{ \begin{matrix}\frac{1}{n−j+1}, & j = 1, ⋯, r − r_{0}, \\ \frac{1−yσ_{n,r_{0}}}{n−j+1} , & j = r − r_{0} + 1, ⋯, r, \\ −\frac{yσ_{n,r_{0}}}{n−j+1}, & j = r + 1, ⋯, r + r_{0}. \end{matrix}

The probability of the right side of (3.8) can be calculated exactly by using a formula of Knight and Satchell (1996); see also Johnson and Kotz (1970, Ch.18). Except for the extreme case where y satisfies

(j−j′) = yσ_{n,r_{0}}(n−j′+1), (n−j′+1) = −yσ_{n,r_{0}}(j−j′), or n−j+1 = −yσ_{n,r_{0}}(n−j′+1),

for some positive integers j > j′, the γ’s are distinct, and thus

P (Y ≥ y) = \{ \begin{matrix}∑_{j=1}^{r+r_{0}} [ ∏_{k≠j}^{r+r0}(1 − \frac{θ_{j}(y)}{θ_{k}(y)})]^{−1}(1 − p)^{λθ_{j}(y)}, & y < 0, \\ ∑_{j=1}^{r} [ ∏_{k≠j}^{r+r0}(1 − \frac{θ_{j}(y)}{θ_{k}(y)})]^{−1}(1 − p)^{λθ_{j}(y)}, & 0 ≤ y < σ_{n,r_{0}}^{−1}, \\ ∑_{j=1}^{r−r_{0}} [ ∏_{k≠j}^{r+r0}(1 − \frac{θ_{j}(y)}{θ_{k}(y)})]^{−1}(1 − p)^{λθ_{j}(y)}, & y ≥ σ_{n,r_{0}}^{−1}, \end{matrix}

where θ_{i}(y) = γ_{i}^{−1}(y), i = 1, …, r + r_{0}.

Table 3 compares the exact tail probability F̄_{Y} (y) with the saddlepoint approximation (3.5) and the Edgeworth expansion given by Hall and Sheather (1988). As usual, the Edgeworth expansion works only in the centre of the distribution. The saddlepoint approximation has remarkably small relative error.

[Insert Table 3 about here.]

AP P ENDIX : P ROOF S OF LEMMAS

Proof of Lemma 1

Part (i) is a special case of Part (ii) with λ_{0} = 0. To prove (ii), write

h_{u}(u_{1}, u_{2}) = p_{1} ln p_{1} + (p_{2} − p_{1}) ln(p_{2} − p_{1}) + (1 − p_{2}) ln(1 − p_{2})
−{p_{1} ln u_{1} + (p_{2} − p_{1}) ln(u_{2} − u_{1}) + (1 − p_{2}) ln(1 − u_{2}))},

where 0 < u_{1} < u_{2} < 1. Then

h(x_{1}, x_{2}) = h_{u}(F (x_{1}), F (x_{2})).

So it suffices to consider the uniform distribution with F (x) = x, 0 < x < 1, and show

H_{u}(u_{1}, u_{2}) ≥ 0, 0 < u_{1} < u_{2} < 1,

with equality if and only if u_{i} = p_{i}, i = 1, 2, where

H_{u}(u_{1}, u_{2}) = h_{u}(u_{1}, u_{2}) − λ_{0}{(u_{1} − p_{1})^{2} + (u_{2} − p_{2})^{2}}, 0 < u_{1} < u_{2} < 1.

The Hessian matrix of H_{u}(u_{1}, u_{2}) is

∇H_{u}(u_{1}, u_{2}) = ( \begin{matrix}\frac{p_{2}−p_{1}}{(u_{2}−u_{1})^{2}} + \frac{p_{1}}{u_{1}^{2}} − 2λ_{0} & \frac{p_{2}−p_{1}}{(u_{2}−u_{1})^{2}} \\ \frac{p_{2}−p_{1}}{(u_{2}−u_{1})^{2}} & \frac{p_{2}−p_{1}}{(u_{2}−u_{1})^{2}} + \frac{1−p_{2}}{(1−u_{2})^{2}} − 2λ_{0} \end{matrix} ),

which is positive definite since for 0 < u_{1} < u_{2} < 1,

\frac{p_{2} − p_{1}}{(u_{2} − u_{1})^{2}} + \frac{p_{1}}{u_{1}^{2}} − 2λ_{0} > \frac{p_{2} − p_{1}}{(u_{2} − u_{1})^{2}} + p_{1} − 2λ_{0} ≥ \frac{p_{2} − p_{1}}{(u_{2} − u_{1})^{2}} ,

\frac{p_{2} − p_{1}}{(u_{2} − u_{1})^{2}} + \frac{1 − p_{2}}{(1 − u_{2})^{2}} − 2λ_{0} > \frac{p_{2} − p_{1}}{(u_{2} − u_{1})^{2}} + (1 − p_{2}) − 2λ_{0} ≥ \frac{p_{2} − p_{1}}{(u_{2} − u_{1})^{2}} ,

and | ∇H_{u}(u_{1}, u_{2}) |> 0. Thus H_{u}(u_{1}, u_{2}) is convex and attains its unique minimum 0 at (p_{1}, p_{2}), where [∂H_{u}(u_{1}, u_{2})/∂u_{i}]_{u_{i}=p_{i}} = 0, i = 1, 2.

The proof of (iii) is obtained by using (ii). In fact, for t < d_{1} one has p_{1} − F (t) ≥ p_{1} − F (d_{1}), and

h(t, t + z) ≥ λ_{0}{[F (t) − p_{1}]^{2} + [F (t + z) − p_{2}]^{2}} ≥ λ_{0}{[p_{1} − F (d_{1})]^{2}},

where λ_{0} > 0. So

∫_{−∞}^{d_{1}} exp{−n · h(t, t + z)}g(t, t + z)dt
= ∫_{−∞}^{d_{1}} exp{−(n − 2) · h(t, t + z)} · exp{−2 · h(t, t + z)}g(t, t + z)dt
≤ exp{−(n − 2)λ_{0}[p_{1} − F (d_{1})]^{2}} · ∫_{−∞}^{d_{1}} exp{−2h(t, t + z)}g(t, t + z)dt,

and similarly,

∫_{d_{2}}^{∞} exp{−n · h(t, t + z)}g(t, t + z)dt
≤ exp{−(n − 2)λ_{0}[F (d_{2}) − p_{2}]^{2}} · ∫_{d_{2}}^{∞} exp{−2h(t, t + z)}g(t, t + z)dt.

Thus, on writing c_{2} = λ_{0} min{[p_{1} − F (d_{1})]^{2}, [F (d_{2}) − p_{2}]^{2}}, we have

(∫_{−∞}^{d_{1}} + ∫_{d_{2}}^{∞}) exp{−n · h(t, t + z)}g(t, t + z)dt ≤ c_{1}e^{−nc_{2}},

where

c_{1} = e^{2c_{2}} ∫_{−∞}^{∞} exp{−2h(t, t + z)}g(t, t + z)dt < ∞.

The proof of Lemma 1 is complete.

Proof of Lemma 2

We will prove the following identity, from which Lemma 2 follows immediately.

( \begin{matrix}1 & t′(z) \\ 0 & 1 \end{matrix} )^{T} ( \begin{matrix}1 & 0 \\ 1 & 1 \end{matrix} )^{T} ∇h(t(z), t(z)+z) ( \begin{matrix}1 & 0 \\ 1 & 1 \end{matrix} ) ( \begin{matrix}1 & t′(z) \\ 0 & 1 \end{matrix} ) = ( \begin{matrix}ḧ_{t}(t(z); z) & 0 \\ 0 & H′′(z) \end{matrix} ) .

(A.1)

In fact, one has

H′(z) = {ḣ_{x_{1}}(t(z), t(z) + z) + ḣ_{x_{2}}(t(z), t(z) + z)}t′(z) + ḣ_{x_{2}}(t(z), t(z) + z)
= ḣ_{x_{2}}(t(z), t(z) + z)
= \frac{f(t(z)) [p_{1}F (t(z) + z) − p_{2}F (t(z))]}{F (t(z)) [F (t(z) + z) − F (t(z))]},

where the second equality follows from (2.2), and

ḧ_{t}(t(z); z) = ḧ_{x_{1}x_{1}}(t(z), t(z) + z) + 2ḧ_{x_{1}x_{2}}(t(z), t(z) + z) + ḧ_{x_{2}x_{2}}(t(z), t(z) + z)
= p_{1}[\frac{f(t(z))}{F (t(z))}]^{2} + (p_{2} − p_{1})[\frac{f(t(z) + z) − f(t(z))}{F (t(z) + z) − F (t(z))}]^{2} + (1 − p_{2})[\frac{f(t(z) + z)}{1 − F (t(z) + z)}]^{2}

−(\frac{p_{1}}{F (t(z))} − \frac{p_{2} − p_{1}}{F (t(z) + z) − F (t(z))})f′(t(z))

+(\frac{1 − p_{2}}{1 − F (t(z) + z)} − \frac{p_{2} − p_{1}}{F (t(z) + z) − F (t(z))})f′(t(z) + z)

= p_{1}[\frac{f(t(z))}{F (t(z))}]^{2} + (p_{2} − p_{1})[\frac{f(t(z) + z) − f(t(z))}{F (t(z) + z) − F (t(z))}]^{2}

+(1 − p_{2})[\frac{f(t(z) + z)}{1 − F (t(z) + z)}]^{2} − H′(z)[\frac{f′(t(z))}{f(t(z))} − \frac{f′(t(z) + z)}{f(t(z) + z)}],

where the last equality follows from (2.2).

Obviously, both H′′(z) and t′(z) exist, and

H′′(z) = \frac{d}{dz} H′(z) = \frac{d}{dz} ḣ_{x_{2}}(t(z), t(z) + z)
= ḧ_{x_{2}x_{2}}(t(z), t(z) + z) + [ḧ_{x_{1}x_{2}}(t(z), t(z) + z) + ḧ_{x_{2}x_{2}}(t(z), t(z) + z)]t′(z).

Observe that (2.2) is equivalent to

ḣ_{x_{1}}(t(z), t(z) + z) + ḣ_{x_{2}}(t(z), t(z) + z) = 0.

Differentiating both sides of the above equality with respect to z produces

{ḧ_{x_{1}x_{1}}(t(z), t(z) + z) + 2ḧ_{x_{1}x_{2}}(t(z), t(z) + z) + ḧ_{x_{2}x_{2}}(t(z), t(z) + z)}t′(z)

+ ḧ_{x_{1}x_{2}}(t(z), t(z) + z) + ḧ_{x_{2}x_{2}}(t(z), t(z) + z) = 0,

that is,

ḧ_{t}(t(z); z)t′(z) + ḧ_{x_{1}x_{2}}(t(z), t(z) + z) + ḧ_{x_{2}x_{2}}(t(z), t(z) + z) = 0.

(A.1) is then derived by the matrix multiplication.

Proof of Lemma 6

(i) It is easy to verify that

∇h(x_{1}^{0}, x_{2}^{0}, x_{3}^{0}) = \frac{1}{p_{0}^{∗}} ( \begin{matrix}\frac{p}{p−p_{0}^{∗}} f^{2}(x_{1}^{0}) & −f(x_{1}^{0})f(x_{2}^{0}) & 0 \\ −f(x_{1}^{0})f(x_{2}^{0}) & 2f^{2}(x_{2}^{0}) & −f(x_{2}^{0})f(x_{3}^{0}) \\ 0 & −f(x_{2}^{0})f(x_{3}^{0}) & \frac{1−p}{1−p−p_{0}^{∗}} f^{2}(x_{3}^{0}) \end{matrix} ) ,

and

| ∇h(x_{1}^{0}, x_{2}^{0}, x_{3}^{0}) |= \frac{1}{p_{0}^{∗}^{2}(p − p_{0}^{∗})(1 − p − p_{0}^{∗})}f^{2}(x_{1}^{0})f^{2}(x_{2}^{0})f^{2}(x_{3}^{0}).

Since f(ξ_{p}) > 0, there exists a neighbourhood of ξ_{p}, say | x − ξ_{p} |< δ with small δ > 0, such that for each x in this region f(x) > 0. Thus one may assume that f(x_{i}^{0}) > 0 (i = 1, 2, 3), which implies that ∇h(x_{1}^{0}, x_{2}^{0}, x_{3}^{0}) is positive definite, and furthermore, ∇h(x_{1}, x_{2}, x_{3}) is also positive definite for | x_{i} −ξ_{p} |< δ (i = 1, 2, 3). Lemma 5 in turns implies that for each y in a certain neighbourhood of ξ_{p}, ∇h̃(t_{1}, t_{2}; y) is positive definite with | t_{1} − x_{1}^{0} | and | t_{2} − x_{3}^{0} | small, and H′′(y) > 0.

(ii) From Lemma 3, F (t_{1}), F (ξ_{p} + (t_{2} − t_{1})σ_{n,r_{0}}y) − F (t_{1}), F (t_{2}) − F (ξ_{p} + (t_{2} − t_{1})σ_{n,r_{0}}y) and 1 − F (t_{2}) are convex on D(y) for each fixed y. So is h̃(t_{1}, t_{2}, y).

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TABLE 1 Comparison of the exact value of F̄_{Z_{r_{1},r_{2}}}(z) with the saddlepoint approximation (2.5). (λ = 1, n = 12, r_{1} = 4, r_{2} = 8)

z exact tail prob. saddlepoint approx relative error (%)

0.5  .6092  .6079  .2153
1.0  .1280  .1270  .7490
1.5  .0168  .0165  1.463
2.0  1.778 × 10^{−3}  1.736 × 10^{−3}  2.349
2.5  1.688 × 10^{−4}  1.630 × 10^{−4}  3.411
3.0  1.509 × 10^{−5}  1.439 × 10^{−5}  4.644
3.5  1.303 × 10^{−6}  1.224 × 10^{−6}  6.039
4.0  1.102 × 10^{−7}  1.019 × 10^{−7}  7.577
4.5  9.214 × 10^{−9}  8.363 × 10^{−9}  9.238

TABLE 2 Comparison of the exact value of F̄_{Z_{r_{1},r_{2}}}(z) with the saddlepoint approximation (2.5). (n = 20, r_{1} = 7, r_{2} = 16)

z exact tail prob. saddlepoint approx relative error (%)

.46  .3793  .3790  .0772
.50  .2517  .2515  .1025
.55  .1308  .1306  .1405
.60  .0565  .0564  .1869
.65  .0196  .0195  .2441
.70  5.138 × 10^{−3}  5.122 × 10^{−3}  .3160
.75  9.354 × 10^{−4}  9.316 × 10^{−4}  .4089
.80  1.017 × 10^{−4}  1.012 × 10^{−4}  .5344
.85  4.983 × 10^{−6}  4.947 × 10^{−6}  .7156
.90  5.815 × 10^{−8}  5.756 × 10^{−8}  1.009
.95 2.108 × 10^{−11}  2.074 × 10^{−11}  1.613

TABLE 3 Comparison of the exact value of F̄_{Y} (y) with the saddlepoint approximation (3.5). (λ = 1, n = 25, r = 13, r_{0} = 5, p = 0.5)

y exact tail prob. saddlepoint app. r.e. (%)

Edgeworth

r.e. (%)

0.2  .4165  .4094  1.706  .3964  4.836
0.5  .2989  .2990  .0396  .2793  6.535
2.2  .0141  .0143  1.224  .0183  29.73
2.5 7.253 × 10^{−3}  7.355 × 10^{−3}  1.411  .0112  54.59
3.2 1.579 × 10^{−3}  1.607 × 10^{−3}  1.762 2.846 × 10^{−3} 80.23
3.5 8.493 × 10^{−4}  8.651 × 10^{−4}  1.863 1.362 × 10^{−3} 60.41
4.2 2.210 × 10^{−4}  2.254 × 10^{−4}  1.999 1.636 × 10^{−4} 26.00
4.5 1.296 × 10^{−4}  1.323 × 10^{−4}  2.038 5.540 × 10^{−5} 57.26
5.2 4.091 × 10^{−5}  4.178 × 10^{−5}  2.114 2.968 × 10^{−6} 92.74
5.5 2.586 × 10^{−5}  2.642 × 10^{−5}  2.138 7.150 × 10^{−7} 97.24
6.2 9.540 × 10^{−6}  9.752 × 10^{−6}  2.221 1.750 × 10^{−8} 99.82