## Log 0in Y

We have now all the tools in hand to prove the following intermediary result of independent interest.

Theorem 2.2 For each g Ç.G, we have, as n n-1/2Vnn(g) ^ N (û, ((2) - 1+a2g) . (48)

Proof. Step 1. Recall the notation (32)-(33)-(34). To illustrate the arguments of our proof, we start by considering the simple case where g(u) = a+bu is an affine function. Under this assumption, we infer from (30)-(31) that ¡g = a + b and ag = b2. Thus, by (28) and (34), we obtain that n ni 1

This, in turn, shows that

is the partial sum of order n of an i.i.d. sequence of random variables. Setting w = W1,n, an easy calculus (see the Appendix in the sequel) shows that

Tests of Fit based on Products of Spacings 129

e({ w - 1} ^ =1. (54) We readily infer from the above equalities that

Va^f ( - log w - y +(w - 1)) + b(w - 1)1 = Z(2) - 1 + b2 = Z(2) - 1 + a2g. (55)

Given (50) and (55), the proof of (48) for the particular choice of g(u) = a+bu is a simple consequence of the central limit theorem.

Step 2. In this step, we consider the setup where g is proportional to the indicator function of an interval, namely of the form

'0 for 0 < u < c, g(u) = A]I(c,d](u) = ^ A for c < u < d, (56)

0 for u > d, where A G R, c and d are specified constants fulfilling 0 < c < d < &. We will set, for convenience, C = 1 - e-c and D = 1 - e-d, and observe that the constants C,D are such that 0 < C < D < 1. An easy calculus based upon (30)-(31) shows that, under (56), r d

= Ae-udu = A(D - C) and ag = Ag(D - C)(1 - D + C). (57)

Letting g gG be as in (56), we infer from (34), (42) and (43), in combination with Fact 1, that, almost surely as n 4 &,

= A^ - log(1 - Ui) G (c, d] : 1 < i < n} - nA(D - C)

Denote by \u~\ >u> \u~\ - 1 the upper integer part of u. Then, by (42), n

!3n(t)= n1/2(Vn(t) - t)= u1/2(Ulnt-],n - t) for 0 <t < 1. (59) We readily infer from (58) and (59) that, almost surely as n ^to,

Now, making use of (46), we may write nU¡nD] ,n - nU¡nc] ,n - ( \nD\ - \nC\ )

An application of the central limit theorem shows, in turn, that, as n ^ to, a 1 1 n+i

1 + - + -Y,\0i,n - l} = 1 + Op(n-1/2). nn n n n i=i

Therefore, we obtain that, as n ^ to,

By all this, it follows that nU\nD],u — nU\no^n — (\nD] — \nC})

= E {9,n — ^ — (D — C) Y, {0in — 0P(1), i=\nC1+i i=\nC1 + i

This, when combined with (60), entails that

\nD1

+X(D - C ) É [0i, n - 1} + Or(n1/4(log n)3/4). (61)

Cn,n(g) = -X E - a(D - C)E {°i,n - ^, i=\nG]+1 i=1

A direct application of (53) and (57) shows that, as n — to,

Var (c , n + zn , n(g)) = Var(c, n) + Varfe , Jff))

= (1 + o(1))nX2(D - C)(1 - D + C) = (1 + o(l))a2. (62)

Now, in view of (61), given that Zn n + Zn n(g) is a linear combination of three partial sums of i.i.d. centered random variables, an application of the central limit theorem shows that, as n — ^

n-1/2{ Zn , n + Zn ,n(d)} = n-1'2 { Z*n , n + C , n(ff)}+ Op(n-1'4(log n)3'4) - N(0,a2g).

In view of (32), (33) and (34), we thus obtain (48) in this particular case.

Step 3. We are now ready to establish the most general version of our theorem. In the first place, we consider the case where ql € G is a step-function, of the form

where a,1, ...,o,l and 0 < C1 < ¿1 < ... < cl < ¿l < 1 are specified constants. By repeating the arguments of Step 2 in this case, we readily obtain that the weak convergence n-1'2Yn,n(9L) - N(0, Z(2) - 1 + alL), (64)

holds. Now, if g € G = L2(R+,e-udu) is arbitrary, for each e > 0, we may select L > 1, together with A1, ...,XL and 0 < c1 < d1 < ... < cl < dL < 1, such that jg = jgL and

132 Paul Deheuvels and Gérard Derzko By combining (35) with (64)-(65), we see that

/ 2\ E( n-1/2Yn,n(g) - n-1/2Ynin(gL) J < e. (66)

Since £ > 0 in (65)-(66) may be chosen arbitrarily small, we conclude to the validity of (48) by routine arguments.□

Proof of Theorem 1. Our assumptions imply that F(Zin) = Uin for i = l,...,n. Therefore, we infer readily (25) from (48), by setting in this last relation g(y) = f (Q(1 - e-y)) for y> 0, and making use of Lemma 2.1.□

Proof of Theorem 1.1. Denote by Q(t) = inf {z : F (z) > t}, for 0 <t < 1, the quantile function pertaining to F. Our assumptions imply that Q has a continuous derivative on (0,1) given by

By Taylor's formula, we obtain readily that

- log {n(Zi,n - Zi-!,n)J = - log{ n(Q(F (Zi,n)) - Q(F (Zi-1,n)))J

where Z*n lies within the interval (Zi-in,Zin). We now make use of the following well-known fact (see, e.g. Csorgo, Haeusler and Mason [CsHM88], and the references therein). Let Vi,...,Vn be an i.i.d. sequence of replic^of a random variable V with finite expectation p = E(V) and variance 0 < a2 = \&r(V) < to. Denote by V1,n < ... < Vn,n the order statistics of V1,..., Vn. Then, independently of the fixed integers p > 1 and q > 1, we have n-q+1

Let us now assume, for the sake of simplicity, that f is nonincreasing on (a, b). In this case, we see that n n sn := -E log f (Zi-!,n) < Rn := "E bg f Z,n)

Set f = E(log f (Z)) and a2 = Var(log f (Z)). In view of (68)-(69), we have, for each t G R, lim w(n-1/2(S'n - nf) < ^ < v(^n-1/2(Rn - nf) < t)

< P^n-1/2(S'n - nf) < ^ = \$(t), where @(t) is the N(0,1) distribution function. Thus, we may apply Theorem 1 in combination with Remark 2.1 to obtain (8). The proof of this result when f is monotone only in the neighborhood of the end-points is very similar, and obtained by splitting the range of Z into three component intervals, in combination with an application of (69). We omit details.□

### 2.2 Appendix.

Let w denote a unit exponential random variable. The present sub-section is devoted to the computation of some moments of interest of slected functions of w. We first observe that, for each s G R such that s < 1,

Recalling the expansion (see, e.g. Abramowitz and Stegun [AS70])

we obtain readily that the k-th cumulant Kk of - log w is given by ki = y and Kk = (k - 1)!Z(k) for k > 2, whence

Jo rn2

Var(logw)= (log s)2e-sds - y2 = K2 = Z(2) = —. (73)

Let now w1 and w2 denote two independent unit exponential random variables. The following result, of independent interest, has potential applications in two-sample tests. This problem will be considered elsewhere.

Lemma 2.2 The random variable R = \og(u\/u2) follows a logistic law, with distribution and moments given by

the moment-generating function of R is given by

E(exp(sR)) = r(1 — s)r(1 + s) for |s| < 1, (75)

and the k-th cumulant of R is given by

In particular, we get n2

An easy calculus yields

= [—(x log x — x)e-x]x=œ + (log s)e-sds + 1 = —7 + 1. (78)

x Jo

Likewise, we get

= [—x(log x)2e-x]l=œ + (log s)2e-sds + 2l (log s)e-sds x Jo Jo

[—x2(log x)2e~^x=o' + 2 I s (log s)2e-sds + 2 / s(log s)e-s ds

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