Georgian Mathematical Journal Volume 8 (2001), Number 2, 401–414

ON ACCURACY OF IMPROVED χ^{2}-APPROXIMATIONS

V. V. ULYANOV AND Y. FUJIKOSHI

Abstract. For a statistic S whose distribution can be approximated by χ^{2}- distributions, there is a considerable interest in constructing improved χ^{2}- approximations. A typical approach is to consider a transformation T = T (S) based on the Bartlett correction or a Bartlett type correction. In this paper we consider two cases in which S is expressed as a scale mixture of a χ^{2}- variate or the distribution of S allows an asymptotic expansion in terms of χ^{2}-distributions. For these statistics, we give sufficient conditions for T to have an improved χ^{2}-approximation. Furthermore, we present a method for obtaining its error bound.

2000 Mathematics Subject Classification: 62E20.

Key words and phrases: Asymptotic expansion, error bound, improved χ^{2}-approximations, scale mixture of χ^{2}-variate, transformation.

1. Introduction

Suppose that a statistic S has an asymptotic χ^{2}-approximation as some parameter n tends to infinity. In this case, it is of considerable interest to construct improved χ^{2}-approximations for the statistic S. A typical approach is to consider a transformation T = T (S) based on the Bartlett correction or a Bartlett type correction. For a Bartlett type correction, see, e.g., the works by Cordeiro and Ferrari [2] and by Fujikoshi [4]. In addition to that T has a limiting χ^{2}-distribution with q degrees of freedom, we can expect in some cases that

P (T ≤ x) = G(x) + O(n^{−2})

while

P (S ≤ x) = G(x) + O(n^{−1}),

where G is a distribution function of a χ^{2}-variate χ_{q}^{2} with q degrees of freedom. We say that T has an improved χ^{2}-approximation.

Our aim is to construct an improved χ^{2}-approximation and to obtain its error bound. First we consider the case in which S is expressed as a scale mixture of a χ^{2}-variate, i.e.,

S = Y^{−1}χ_{q}^{2},  (1.1)

ISSN 1072-947X / $8.00 / \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag www.heldermann.de

where Y is a positive random variable independent of χ_{q}^{2}. In this case, the order of the remainder term depends on the closeness of Y to 1. Next, we consider the case in which S allows an asymptotic expansion such that

P (S ≤ x) = G_{q}(x) + \frac{1}{n} ∑_{j=0}^{k} a_{j}G_{q+2j}(x) + R_{k},  (1.2)

where R_{k} satisfies the inequality

|R_{k}| ≤ c_{k}/n^{2}  (1.3)

with some positive constant c_{k}. For these statistics, we give sufficient conditions for T (S) to have an improved χ^{2}-approximation in terms of the inverse function to T (x). Furthermore, we present a method for obtaining an error bound of the improved approximation.

In Section 5 we consider a special case in which S = χ_{q}^{2}/Y with Y = n^{−1}χ_{n}^{2} and Y, χ_{n}^{2} are independent. We show what kind of results with computed values of absolute constants can be derived in this case from the general theorems of Section 2. We also consider examples of transformations of S = nχ_{q}^{2}/χ_{n}^{2} with independent χ_{q}^{2} and χ_{n}^{2} which provide better approximations. We also compare the transformations.

Note that a more general approach to constructing transformed statistics T (S) with improved Pearson type approximations but without error bounds can be found in [1].

2. Scale Mixtures of χ^{2}-Variates

In this section, let S = χ_{q}^{2}/Y be a mixture of a χ^{2}-variate defined by (1.1). Set G(x) = P{χ_{q}^{2} ≤ x}, and

α_{i} = E(Y − 1)^{i} for i = 1, …, 4, and β = max{|α_{3}|, α_{4}}.

Under the condition α_{1} = 0 it is easy to show (see, e.g. [7]) that

|P{S ≤ x} − G(x)| ≤ cα_{2}

with a constant c depending only on q.

In order to improve the appoximation we consider a transformation T that is an increasing non-negative function defined on [0, +∞). We denote by b the function which is inverse to T (x), i.e.,

b(T (x)) = T (b(x)) = x for all x ≥ 0.

Theorem 2.1. Let S = χ_{q}^{2}/Y, where Y is a positive random variable independent of χ_{q}^{2}. Suppose that α_{1} = 0 and that there exist positive constants B_{i} = B_{i}(q), i = 1, 2, 3, depending only on q such that B_{1} ≤ 1 and, for all x > 0, one has

b(x) ≥ B_{1}x,  (2.1)
|b(x) − x| ≤ A(x)\sqrt{β},  (2.2)

ON ACCURACY OF IMPROVED χ^{2}-APPROXIMATIONS 403

and

|G′(x)(b(x) − x) + \frac{α_{2}}{2} G′′(x)x^{2}| ≤ B_{3}β,  (2.3)

where

A(x) = B_{2}x exp (B_{1}x/16) .  (2.4)

Then we have for q ≥ 2

|P{T (S) ≤ x} − G(x)| ≤ cβ,  (2.5)

where c is a constant depending only on q and B_{i} with i = 1, 2, 3 (see (3.24)).

Remark 2.2. It is easy to see that the class of positive increasing functions b on [0, +∞) which satisfy (2.1)–(2.3) is not empty. It is enough to take

b(x) = x(1 − \frac{α_{2}}{4} (q − 2)) + \frac{α_{2}}{4} x^{2}.  (2.6)

If

\frac{\sqrt{β}}{4} (q − 2) > 1,

then (2.5) easily follows with c = ((q − 2)/4)^{2}. Thus, we assume

\frac{\sqrt{β}}{4} (q − 2) ≤ 1.

Therefore, since α_{2} ≤ \sqrt{β}, we obtain

1 − \frac{α_{2}}{4} (q − 2) ≥ 0

and b(x) is increasing. Moreover, for b defined by (2.6), we have that (2.1) and (2.3) hold with

B_{1} = 1 − \frac{α_{2}}{4} (q − 2) and B_{3} = 0,

respectively. Condition (2.2) is also trivially satisfied (cf. Remark 4.2 after the proof of Theorem 4.1).

Remark 2.3. In fact, it is possible to obtain an inequality similar to (2.5) omitting the condition q ≥ 2 and replacing two conditions (2.2) and (2.3) by only one condition. However, in this case we have also to replace the constant c in (2.5) by a larger one. Namely, the following theorem holds.

Theorem 2.4. Let S = χ_{q}^{2}/Y, where Y is a positive random variable independent of χ_{q}^{2}. Suppose that α_{1} = 0 and that there exist positive constants B_{1} and B_{4} depending only on q such that B_{1} ≤ 1 and, for all x > 0, the condition (2.1) is satisfied and

|b(x) − x + \frac{α_{2}}{4} x^{2}(\frac{q − 2}{x} − 1)| ≤ B_{4}x exp(B_{1}x/16)β.  (2.7)

Then we have for all q ≥ 1 that

|P{T (S) ≤ x} − G(x)| ≤ c_{1}β,  (2.8)

where c_{1} is a constant depending only on q, B_{1}, and B_{4}.

Remark 2.5. In Theorems 2.1 and 2.4, we give uniform error bounds. This means that the right-hand sides of (2.5) and (2.8) do not depend on x. However, by using an approach developed in [9], it is possible to construct the so-called non-uniform bounds when the right-hand sides tend to 0 as x → +∞. For example, the following theorem holds (see, e.g., Corollary 6 in [9]).

Theorem 2.6. Let S = χ_{q}^{2}/Y, where Y is a positive random variable independent of χ_{q}^{2} and EY^{−4} < ∞. Suppose that α_{1} = 0 and we put

T_{0}(x) = \frac{q − 2}{2} − \frac{2}{α_{2}} + (\frac{1}{4}(q − 2 − \frac{4}{α_{2}} )^{2} + \frac{4x}{α_{2}})^{1/2} .

Then we have for all q ≥ 2 and x > 0 that

|P\{T_{0}(S) ≤ x\} − G(x)| ≤ \frac{c_{2}}{1 + x^{4}} (β + EY^{−4}1_{\{Y <1/2\}}) ,

where c_{2} is a constant depending only on q and 1_{A} is the indicator function of an event A.

Remark 2.7. The proof of Theorem 2.6 and more general results with nonuniform bounds will be published separately.

3. Proofs of Theorems 2.1 and 2.4

Proof of Theorem 2.1. We have

P{T (χ_{q}^{2}/Y) ≤ x} = P{χ_{q}^{2} ≤ b(x) · Y } = EG(Y · b).  (3.1)

We consider the function G(y·b) as a function of two variables F (y, b) = G(y·b). Since G(x) is smooth for x > 0, we can expand F (y, b) at the point (1, x) so that the remainder term is of order O(β). Note that F (1, x) = G(x) and

F (y, b) = F (y, x) + G′(yx)(b − x)y + \frac{1}{2}G′′(yx′)(b − x)^{2}y^{2},  (3.2)
F (y, x) = F (1, x) + G′(x)x(y − 1) + \frac{1}{2}G′′(x)x^{2}(y − 1)^{2}
+\frac{1}{6}G′′′(x)x^{3}(y − 1)^{3} + \frac{1}{24}G^{(4)}(y′x)x^{4}(y − 1)^{4},  (3.3)
G′(yx) = G′(x) + G′′(x)x(y − 1) + \frac{1}{2}G′′′(x)(y′′x)x^{2}(y − 1)^{2}, (3.4)

where x′ ∈ (b ∧ x, b ∨ x), y′ ∈ (y ∧ 1, y ∨ 1), y′′ ∈ (y ∧ 1, y ∨ 1) and as usual b ∧ x = min(b, x), b ∨ x = max(b, x).

Combining (3.2)–(3.4), we arrive at the following representation:

F (y, b) = G(x) + G′(x)((b − x)y + x(y − 1))

+\frac{1}{2}G′′(x)(x^{2}(y − 1)^{2} + 2x(b − x)y(y − 1))

+\frac{1}{6}G′′′(x)x^{3}(y − 1)^{3} + \frac{1}{2}G′′(yx′)(b − x)^{2}y^{2}

+\frac{1}{2}G′′′(y′′x)x^{2}(y − 1)^{2}(b − x)y + \frac{1}{24}G^{(4)}(y′x)x^{4}(y − 1)^{4}. (3.5)

Our aim is to approximate EG(Y · b) (see (3.1)) by G(x) and to prove (2.5). Therefore, considering y in (3.5) as a random variable Y, we subtract from each term on the right-hand side of (3.5) its expectation excluding the terms of order O(β). Then we arrive at an expression that we denote by △(y, b, x), namely

△(y, b, x) ≡ F (y, b) − G(x) − G′(x)(b − x)(y − 1) − G′(x)x(y − 1)

− \frac{1}{2}G′′(x)x^{2}(y − 1)^{2} + \frac{α_{2}}{2} G′′(x)x^{2} − G′′(x)x(b − x)y(y − 1)

+ α_{2}G′′(x)x(b − x) − \frac{1}{6}G′′′(x)x^{3}(y − 1)^{3} + \frac{α_{3}}{6} G′′′(x)x^{3}.  (3.6)

It follows from (3.5) and (3.6) that △(y, b, x) can also be written in the form

△(y, b, x) = G′(x)(b − x) + \frac{α_{2}}{2} G′′(x)x^{2} + α_{2}G′′(x)x(b − x)

+\frac{α_{3}}{6} G′′′(x)x^{3} + \frac{1}{2}G′′(yx′)(b − x)^{2}y^{2}

+\frac{1}{2}G′′′(y′′x)x^{2}(b − x)(y − 1)^{2}y + \frac{1}{24}G^{(4)}(y′x)x^{4}(y − 1)^{4}. (3.7)

Since α_{1} = 0 we have α_{2} = EY (Y − 1). Then it follows from (3.1) and (3.6) that

E△(Y, b, x) = EF (Y, b) − G(x) = P{T (χ_{q}^{2}/Y) ≤ x} − G(x).

Therefore, in order to prove the theorem, it is enough to show that

E|△(Y, b, x)| ≤ c · β.  (3.8)

Let us fix an arbitrary d such that 0 < d < 1, and write

E|△(Y, b, x)| = I_{1} + I_{2},

where I_{1} = E|△(Y, b, x)| · 1_{{|Y −1|>d}}, I_{2} = E|△(Y, b, x)| · 1_{{|Y −1|≤d}}. We prove the upper bounds for I_{1} and I_{2} with the help of (3.6) and (3.7), respectively.

First we consider I_{1}. By (3.6), we have, for all positive x and y with |y−1| > d, that

|△(y, b, x)| ≤ 1 + |y − 1| sup G′(x)x + sup |G′(x)(b − x) + \frac{α_{2}}{2} G′′(x)x^{2}|

+\frac{1}{2}(y − 1)^{2} sup |G′′(x)|x^{2} + y sup G′(x)|b − x|

+ (y|y − 1| + α_{2}) sup |G′′(x)(b − x)|x

+\frac{1}{6} (|y − 1|^{3} + |α_{3}|) sup |G′′′(x)|x^{3}

≤ (y − 1)^{4}(d^{−4} + d^{−3} sup G′(x)x + \frac{1}{2}d^{−2} sup |G′′(x)|x^{2})

+(1 + d)d^{−2}(y − 1)^{2} sup G′(x)|b − x|

+ sup |G′(x)(b − x) + \frac{α_{2}}{2} G′′(x)x^{2}|

+(\frac{1 + d}{d}(y − 1)^{2} + α_{2}) sup |G′′(x)(b − x)|x

+\frac{1}{6}(d^{−1}(y − 1)^{4} + |α_{3}|) sup |G′′′(x)|x^{3},  (3.9)

where all supremums in (3.9) are taken over all positive x.

Next, we consider I_{2}. By (3.7), we obtain for all positive x and y with |y − 1| ≤ d that

|△(y, b, x)| ≤ sup |G′(x)(b − x) + \frac{α_{2}}{2} G′′(x)x^{2}|

+α_{2} sup |G′′(x)(b − x)|x + \frac{|α_{3}|}{6} sup |G′′′(x)|x^{3}

+ \frac{(1 + d)^{2}}{2} sup |G′′(yx′)|(b − x)^{2}

+ \frac{1 + d}{2}(y − 1)^{2} sup |G′′′(y′′x)(b − x)|x^{2}

+ \frac{(y − 1)^{4}}{24} sup |G^{(4)}(y′x)|x^{4},  (3.10)

where all supremums in (3.10) are taken over all positive x and y with |y−1| ≤ d, x′ ∈ (b ∧ x, b ∨ x), y′ ∈ (y ∧ 1, y ∨ 1), y′′ ∈ (y ∧ 1, y ∨ 1).

Let us recall that

G′(x) = [2^{q/2}Γ(q/2)]^{−1}x^{q/2−1}e^{−x/2} for x ≥ 0.

Let γ_{i} = sup_{x>0} |G^{(i)}(x)x^{i}| for i = 1, 2, 3. For example,

γ_{1} = Γ^{−1}(q/2)(\frac{q}{2e})^{q/2}.

It is clear that γ_{2} and γ_{3} are also functions of q. Now we show how other supremums in (3.9) and (3.10) can be calculated by using the properties of b described in (2.1)–(2.3). We consider the most complicated expression

sup |G′′(yx′)|(b − x)^{2}.

Note that

G′′(x)2^{q/2}Γ(q/2) = e^{−x/2}x^{q/2−2} (−\frac{x}{2} + \frac{q}{2} − 1) .  (3.11)

We shall employ below the following inequality: for any positive numbers a and b, we have |a − b| ≤ a ∨ b. We consider two cases.

Case 1: b(x) ≤ x. Then x′ ∈ (b, x) and, by (2.1), we have for q ≥ 2

exp ( − \frac{yx′}{2} ) ≤ exp ( − \frac{(1 − d)B_{1}}{2} x),  (3.12)
(yx′)^{q/2−2}| − \frac{yx′}{2} + \frac{q}{2} − 1| ≤ \frac{1}{2}(1 + d)^{q/2−2}x^{q/2−2}
×[((1 + d)x) ∨ (q − 2)].  (3.13)

Case 2: b(x) > x. Then x′ ∈ (x, b) and we have for q ≥ 2

exp ( − \frac{yx′}{2} ) ≤ exp ( − \frac{(1 − d)}{2}x′),  (3.14)
(yx′)^{q/2−2}| − \frac{yx′}{2} + \frac{q}{2} − 1| ≤ \frac{1}{2}(1 + d)^{q/2−2}(x′)^{q/2−2}
×[((1 + d)x′) ∨ (q − 2)].  (3.15)

Since the function A(x) in condition (2.2) of the theorem is increasing, we have in Case 2 that

sup |G′′(yx′)|(b − x)^{2} ≤ \frac{β}{2} (1 + d)^{q/2−2}

× sup_{x} exp ( − \frac{(1 − d)}{2}x)A^{2}(x)x^{q/2−2}[[((1 + d)x) ∨ (q − 2)]. (3.16)

Let us take d = 1/2. Combining (2.4), (3.11)–(3.16) we obtain

2^{q/2}Γ(q/2) sup |G′′(yx′)|(b − x)^{2} ≤ \frac{βB_{2}^{2}}{2} (\frac{3}{2})^{q/2−2}

× [(\frac{3}{2}c(B_{1}/8, q/2 + 1)) ∨ ((q − 2)c(B_{1}/8, q/2))]

≡ c_{1}(q, B_{1})B_{2}^{2}β,  (3.17)

where

c(α, γ) = sup_{x>0}(x^{γ} exp\{−αx\}) = (\frac{γ}{αe})^{γ} for positive α and γ.

408 V. V. ULYANOV AND Y. FUJIKOSHI

Since

2^{q/2}Γ(q/2)G′′′(x)
= e^{−x/2}x^{q/2−3}(\frac{x^{2}}{4} − (\frac{q}{2} − 1) x + (\frac{q}{2} − 1) (\frac{q}{2} − 2))  (3.18)

and

2^{q/2}Γ(q/2)G^{(4)}(x) = e^{−x/2}x^{q/2−4}(−\frac{x^{3}}{8} + \frac{3}{4} (\frac{q}{2} − 1) x^{2}

− \frac{3}{2} (\frac{q}{2} − 1) (\frac{q}{2} − 2) x + (\frac{q}{2} − 1) (\frac{q}{2} − 2) (\frac{q}{2} − 3)),  (3.19)

similarly to (3.17) we obtain

sup G′(x)|b − x| ≤ (2^{q/2}Γ(q/2))^{−1}B_{2}\sqrt{β} c(7/16, q/2),  (3.20)

2^{q/2}Γ(q/2) sup |G′′(x)(b − x)|x

≤ \frac{1}{2}[c(7/16, q/2 + 1) ∨ ((q − 2)c(7/16, q/2))]B_{2}\sqrt{β}

≡ c_{2}(q)B_{2}\sqrt{β},  (3.21)

2^{q/2}Γ(q/2) sup |G′′′(y′′x)(b − x)|x^{2}

≤ (\frac{1}{4}c(3/16, q/2 + 2) + |\frac{q}{2} − 1| c(3/16, q/2 + 1)

+ |(\frac{q}{2} − 1) (\frac{q}{2} − 2)| c(3/16, q/2))B_{2}\sqrt{β} ≡ c_{3}(q)B_{2}\sqrt{β}, (3.22)

2^{q/2}Γ(q/2) sup |G^{(4)}(y′x)|x^{4}

≤ \frac{1}{8}c(1/4, q/2 + 3) + \frac{3}{4} |\frac{q}{2} − 1| c(1/4, q/2 + 2)

+ \frac{3}{2} |(\frac{q}{2} − 1) (\frac{q}{2} − 2)| c(1/4, q/2 + 1)

+ |(\frac{q}{2} − 1) (\frac{q}{2} − 2) (\frac{q}{2} − 3)| c(1/4, q/2) ≡ c_{4}(q).  (3.23)

Since E(y − 1)^{2} = α_{2} ≤ \sqrt{β}, we obtain from (3.9), (3.10), and (3.17)–(3.23) that (3.8) holds with

c = 16 + 8γ_{1} + 2γ_{2} + \frac{1}{2}γ_{3} + B_{3}

+B_{2} (2^{q/2}Γ(q/2))^{−1} [6c(7/16, q/2) + 4c_{2}(q)

+\frac{9}{8}B_{2}c_{1}(q, B_{1}) + \frac{3}{4}c_{3}(q) + \frac{1}{24}c_{4}(q)],  (3.24)

where c_{1}(q, B_{1}), c_{2}(q), c_{3}(q) and c_{4}(q) are defined in (3.17), (3.21), (3.22), and (3.23), respectively. Thus, Theorem 2.1 is proved.

Proof of Theorem 2.4. It is enough to verify that (2.7) implies (2.2) and (2.3). Note that

x exp(B_{1}x/16) exp(−x/2)x^{q/2−1} ≤ c(1/2 − B_{1}/16, q/2).

Therefore, (2.7) implies (2.3) with

B_{3} = B_{4}[2^{q/2}Γ(q/2)]^{−1}c(1/2 − B_{1}/16, q/2).

Furthermore, without loss of generality, we may assume that β ≤ 1. Otherwise (2.8) holds with c_{1} = 1. It is easy to see that (2.2) follows from (2.7) if we take in (2.4)

B_{2} = B_{4} + \frac{1}{4}(|q − 2| + c(B_{1}/16, 1)).

Moreover, now q can be equal to 1. In this case, the expression q − 2 and the symbol ∨ in (3.13), (3.15)–(3.17), and (3.21) must be replaced by 1 and the symbol +, respectively. Thus, Theorem 2.4 is proved.

4. Statistic with an Error Estimate

In this section we consider a statistic satisfying (1.2). Certain transformations which improve approximation for S have been proposed by Cordeiro and Ferrari [2], Kakizawa [6], Fujisawa [5], Fujikoshi [4], and others. We give sufficient conditions under which there exists a transformation T that improves approximation for S. We shall find a transformation T in the class of positive increasing functions defined on [0, +∞). Fujisawa [5] has shown that a monotone increasing transformation improving approximation exists. Sufficient conditions will be formulated in terms of the function b(x) inverse to T. Our aim is also to give a method of finding an error bound for this improved approximation.

Theorem 4.1. Suppose that there exists a positive, increasing function b defined on [0, +∞) such that, for some positive constants D_{i} = D_{i}(q, k), i = 1, 2, 3, 4: D_{1} ≤ 1, D_{3} < D_{1}/4, the following conditions are satisfied for all x > 0:

b(x) ≥ D_{1}x  (4.1)
|b(x) − x| ≤ \frac{D_{2}x}{n} exp (D_{3}x)  (4.2)
G′_{q}(x)|b(x) − x + \frac{x}{n} ∑_{j=1}^{k} a_{j} ∑_{m=0}^{j−1} \frac{(x/2)^{m}}{∏_{l=0}^{m}(q/2 + l)}| ≤ D_{4}/n^{2},  (4.3)

where a_{j}’s are the same as in (1.2).

If P(S ≤ x) can be written in form (1.2), then

|P(T (S) ≤ x) − G_{q}(x)| ≤ \frac{c}{n^{2}} ,  (4.4)

where T is the inverse function to b and c is a positive constant depending on q, D_{j}, j = 1, …, 4, and c_{k} in (1.3) (see (4.12)).

Proof. Since G_{q}(x) is smooth for all x > 0, we can write

G_{q}(b(x)) = G_{q}(x) + G′_{q}(x)(b − x) + \frac{1}{2}G′′(x′)(b − x)^{2},  (4.5)

where x′ ∈ (b ∧ x, b ∨ x). It is known that

G_{q+2}(x) = G_{q}(x) + \frac{(x/2)^{q/2}e^{−x/2}}{Γ(q/2 + 1)} .  (4.6)

Note that in (1.2), it is necessary that ∑_{j=0}^{k} a_{j} = 0. By using (1.2), (1.3), (4.5), and (4.6), we obtain

P(T (S) ≤ x) = P(S ≤ b(x)) = G_{q}(x) + G′_{q}(x)(b − x)

+ \frac{1}{n} ∑_{j=1}^{k} a_{j}e^{−x/2}(x/2)^{q/2} ∑_{m=1}^{j} \frac{(x/2)^{m−1}}{Γ(q/2 + m)}

+ \frac{1}{2}G′′_{q}(x′)(b − x)^{2} + \frac{1}{n} ∑_{j=0}^{k} a_{j}G′_{q+2j}(x′′)(b − x) + R_{k},  (4.7)

where x′′ ∈ (b ∧ x, b ∨ x).

Now we construct a uniform bound for G′′_{q}(x′)(b − x)^{2}. We consider two cases. Case 1: b ≤ x. Then x′ ∈ (b, x). Since

G′′_{q}(x) = (2^{q/2}Γ(q/2))^{−1}e^{−x/2}x^{q/2−2} (−\frac{x}{2} + \frac{q}{2} − 1) ,

we obtain from (4.1) and (4.2) that

sup_{1}|G′′_{q}(x′)|(b − x)^{2} ≤ \frac{(2^{q/2}Γ(q/2))^{−1}}{2n^{2}} D_{2}^{2}
×sup_{2} exp(−x(D_{1}/2 − 2D_{3}))x^{q/2} (x + |q − 2|)
≡ c_{1}(k, q, D_{1}, D_{3})/n^{2},  (4.8)

where the first supremum on the left-hand side is taken over x such that b(x) ≤ x and the second supremum is taken over all x > 0. By the hypotheses of the theorem, we have c_{1}(k, q, D_{1}, D_{3}) < ∞.

Case 2: b > x. Then x′ ∈ (x, b) and, by (4.2), we obtain

sup_{1}|G′′_{q}(x′)|(b − x)^{2} ≤ \frac{1}{n^{2}} sup_{2}|G′′_{q}(x′)|D_{2}^{2}(x′)^{2} exp(2D_{3}x′)

≤ \frac{(2^{q/2}Γ(q/2))^{−1}}{2n^{2}} D_{2}^{2}sup_{3} exp(−x(1/2 − 2D_{3}))|x|^{q/2}(|x| + |q − 2|)

≡ c_{2}(k, q, D_{1}, D_{3})/n^{2},  (4.9)

where the first two supremums are taken over x such that b(x) > x and the third one is taken over all x > 0. By the hypotheses of the theorem, we have

c_{2}(k, q, D_{1}, D_{3}) < ∞.

Since c_{2}(k, q, D_{1}, D_{3}) ≤ c_{1}(k, q, D_{1}, D_{3}), we obtain

sup G′′_{q}(x′)(b − x)^{2} ≤ c_{1}(k, q, D_{1}, D_{3})/n^{2},  (4.10)

where the supremum is taken over all x > 0. By using (4.6), we can obtain similarly to (4.10) that, for all x > 0, one has

| ∑_{j=0}^{k} a_{j}G′_{q+2j}(x′′)(b − x)| = |b − x|| ∑_{j=1}^{k} a_{j} ∑_{m=1}^{j} \frac{((x′′/2)^{\frac{q+2m}{2} −1}e^{−x′′/2})′}{Γ(\frac{q+2m}{2})} |

≤ \frac{D_{2}x}{2^{q/2}Γ(q/2)n}(x′′)^{q/2−1} exp(D_{3}x − \frac{x′′}{2} )

× ∑_{j=1}^{k} |a_{j}| ∑_{m=1}^{j} \frac{x′′ + q + 2m − 2}{∏_{l=0}^{m−1}(q/2 + l)} (\frac{x′′}{2} )^{m−1}

≤ \frac{D_{2}}{n} (2^{q/2}Γ(q/2))^{−1} sup_{x>0} [ exp(−x(D_{1}/2 − D_{3})x^{q/2}

× ∑_{j=1}^{k} |a_{j}| ∑_{m=0}^{j−1} \frac{x + q + 2m}{∏_{l=0}^{m}(q/2 + l)} (\frac{x}{2})^{m}]

≡ c_{3}(k, q, D_{1}, D_{3})/n.  (4.11)

Combining (4.7), (4.10), (4.3), (4.12), and (1.3), we obtain (4.4) with

c = D_{4} + \frac{1}{2}c_{1}(k, q, D_{1}, D_{3}) + c_{3}(k, q, D_{1}, D_{3}) + c_{k}.  (4.12)

This brings our proof to the end.

Remark 4.2. It is clear that a positive function b that satisfies (4.1)–(4.3) may not be increasing. Therefore, we have to require the existence of an increasing function b in Theorem 4.1. We have shown in the previous sections that the required function b exists.

Theorem 4.3. Suppose that there exist a positive, increasing function b(x) defined on [0, +∞) and positive constants D_{i} = D_{i}(q, k), j = 1, 3, 5, such that (4.1) holds and, for all x > 0, one has

|b(x) − x + \frac{x}{n} ∑_{j=1}^{k} a_{j} ∑_{m=0}^{j−1} \frac{(x/2)^{m}}{∏_{l=0}^{m}(q/2 + l)}| ≤ \frac{D_{5}}{n^{2}} x exp(D_{3}x)  (4.13)

with D_{3} < D_{1}/4. If a statistic S admits the representation (1.2), then we have (4.4), where T is the inverse function to b and c is a positive constant depending on q, k, and D_{1}, D_{3}, D_{5}.

Proof. The claim is proved similarly to Theorem 2.4.

5. Examples

At first we consider the case in which S = χ_{q}^{2}/Y with Y = n^{−1}χ_{n}^{2} and Y, χ_{n}^{2} are independent. In this case S can be represented in form (1.2) with

k = 2, a_{0} = −\frac{1}{4}q(q − 2), a_{1} = \frac{1}{2}q^{2}, a_{2} = −\frac{1}{4}q(q + 2),

and a uniform bound for the remainder term of type (1.3) can be obtained (see, e.g., [7]). Therefore we can apply Theorem 4.1. However, it would be preferable to apply Theorem 2.1, since it yields a better constant c in a bound of type (4.4). This is due to the fact that in Theorem 2.1 we have used the properties of a χ^{2}-variate mixture, but not representation (1.2). At the same time, it is easily seen that some parts of the proofs of Theorem 2.1 and Theorem 4.1 are similar. By using Theorem 2.1, we can obtain the following result.

Corollary 5.1. Let Y = n^{−1}χ_{n}^{2}. Suppose that conditions (2.1) and (2.2) of Theorem 2.1 are satisfied and that (2.3) is replaced by the following:

|b(x) − x + \frac{x^{2}}{2n}((q − 2)/x − 1)| ≤ \frac{1}{n^{2}} 14.824B_{3}2^{q/2}Γ(q/2)x^{1−q/2}e^{x/2}, (5.1)

where B_{3} is the same as in (2.3). Then we have

|P{T (χ_{q}^{2}/Y) ≤ x} − G(x)| ≤ 14.824c/n^{2},  (5.2)

provided that Y and χ_{q}^{2} are independent and c is defined by (3.24).

Proof. It is enough to note that under the hypotheses of this corollary, we have

α_{2} = E(Y − 1)^{2} = 2/n, α_{3} = E(Y − 1)^{3} = 8/n^{2},
α_{4} = E(Y − 1)^{4} = 12/n^{2} + 48/n^{3}.  (5.3)

Since c in (3.24) is not less than 18.94, we may assume that n ≥ 17. Then, by (5.3), we have

β = max {|α_{3}|, α_{4}} = α_{4} ≤ 14.824n^{2}.

Therefore, this corollary follows from Theorem 2.1.

Now we consider examples of transformations of S = nχ_{q}^{2}/χ_{n}^{2} with independent χ_{q}^{2} and χ_{n}^{2} which provide better approximations:

T_{1}(x) = (n + (q − 2)/2) log(1 + x/n),
T_{2}(x) = n exp (\frac{q − 2}{2n})(1 − exp ( − \frac{x}{n})),
T_{3}(x) = \frac{q − 2}{2} − n + ((n − \frac{q − 2}{2})^{2} + 2nx)^{1/2}.

The transformations T_{1}(x) and T_{2}(x) have been introduced, respectively, in [4] and [3]. The transformation T_{3}(x) is a new one (cf. T_{0}(x) in Section 2. We show what bounds of type (5.2) can be obtained for these transformations.

In what follows we assume that q = 4. Then we obtain

b_{1}(x) = n( exp (\frac{x}{n + 1}) − 1),
b_{2}(x) = −n log (1 − \frac{x}{n} exp ( − \frac{1}{n})),
b_{3}(x) = x(1 − \frac{1}{n}) + \frac{x^{2}}{2n}

as the inverse functions for T_{1}, T_{2}, and T_{3}, respectively. Note that b_{2}(x) is defined only for

x: x < n exp(1/n),

whereas b_{1}(x) and b_{3}(x) are defined for all x.

Moreover, if we take x_{ε} = n exp(1/n)(1 − ε), then b_{2}(x_{ε}) ↑ ∞ as ε ↓ 0. This means that condition (2.2) is not satisfied for b_{2}(x), and the approach proposed in Theorem 2.1 does not lead to a uniform estimate of type (5.2) in this case. Therefore, we exclude T_{2} from our considerations here.

In the case of T_{1}, it is easy to verify that (2.1), (2.2), and (5.2) are satisfied with

B_{1} = 79/80, B_{2} = 0.97065, B_{3} = 0.67556,

provided that n > 78. Therefore (5.2) holds for T = T_{1} with

c = c(T_{1}) = 424.16.

However, in the case of the transformation T_{3}, we obtain

B_{1} = 64/65, B_{2} = 0.77633, B_{3} = 0.

Therefore (5.2) holds for T = T_{3} with

c = c(T_{3}) = 321.46.

Thus T_{3} yields better bounds for the remainder term than T_{1}.

Moreover, by using the above simple expression for b_{3} and arguing similarly to the proof of Theorem 2.1, we can prove (5.2) with

c(T_{3}) = 132.34.

In particular, one has to take d = 0.34 in the proof.

Acknowledgement

Research was supported by RFBR (grant No. 00–01–00406) and JSPS.

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(Received 15.12.2000)

Authors’ addresses:

V. V. Ulyanov Faculty of Computational Mathematics and Cybernetics Moscow State University Moscow 119899, Russia E-mail: vladim195@mtu-net.ru

Y. Fujikoshi Faculty of Science, Hiroshima University Higashi–Hiroshima 739–8526 Japan