Strong Limit Theorems for Sums of Logarithms of rath Order Spacings
Magnus Ekström
Department of Mathematical Stat istics Umeå University
S-901 87 Umeå, Sweden
Abstract
Several strong limit theorems axe proved for sums of logarithms of mth order spacings from general distributions. In all given results, the order of the spac
ings is allowed to increase to infinity with the sample size. These results provide a nonparametric strongly consistent estimator of entropy as well a s a charac
terization of the uniform distribution on [0,1]. Furthermore, it is shown that Cressie's (1976) goodness of fit test is strongly consistent against all continuous alternatives. 1
Key words and phrases: Spacings, Strong limit theorems, Entropy estimation, Uniform distribution, Goodness of fit
1991 AMS subject classification: 60F15, 62G05, 94A17
1 Research was supported by The Bank of Sweden Tercentenary Foundation.
1 Introduction
Let X i , X2, .. . , Xn be an i.i.d. sample of random variables on [0,1] with distri
bution F(x). Denote the order statistics by X(i) < X(2) < ... < X(„) a nd define X(o) = 0 and X(n+i) = 1. For an integer m > 1 the mth order spacings, sometimes called mth order gaps, are defined by
Dì* = X(j+m) - X(j), j = 0,1,..., n - m + 1.
When m = 1, the mth order spacings reduce to simple spacings (or one-step spacings). A v olumous literature exists on simple spacings, see e.g. the reviews by Pyke (1965, 1972), or D'Agostino and Stephens (1986).
Consider the statistic
The statistic Lhas been popular for testing uniformity and was first investi
gated by Darling (1953). Attention has mostly been focused on the case m = 1, see the reviews mentioned above. For literature on testing uniformity using statistics like Lbased on high order spacings, see Cressie (1976, 1978, 1979), Dudewicz and Van der Meulen (1981), Kuo and Rao (1981) and Hall (1986).
Furthermore, a general method of estimating parameters in continuous univari
ate distributions, called the maximum spacing method, is based on LW. This method was introduced by Cheng and Amin (1983) and independently by Ran- neby (1984). That functions other than the log-function can be used to obtain consistent estimators is shown in Ranneby and Ekström (1997).
The asymptotic properties of statistics like Lhave been investigated by many authors, mainly for the particular case where the underlying distribution is uniform on [0,1], or for a sequence of restricted alternatives approaching the uniform distribution as n increases.
For general distributions F, convergence in probability of Lis shown in Hall (1984) (m fixed) and in Khashimov (1989) (m -» oo as n —> oo). General almost sure convergence of Lis given in Van Es (1992). For m even, is related to Vasicek's (1976) entropy estimator, for which Beirlant and Van Zuijlen (1985) give a strong limit theorem.
In the present paper several one- and two-sided strong limit theorems for Lwill be established, where m is allowed to increase with the sample size. In section 2, the main result is a strong limit theorem for m = o(n), based on uniform spacings. Further, a characterization of t he uniform distribution on [0,1] is given. In section 3 several one-sided and two-sided strong limit results for statistics related to Lm = o(ra/logrc), will be shown, including some results for the statistic (1) multiplied on the right by the indicator function Ib{Xqj), where B Ç [0,1] is a measurable set. For the results for general continuous
distributions related to Van Es (1992), we will use less restrictive assumptions on the densities, e.g. we do not assume that the densities are bounded away from zero on their support. In Van Es (1992) the approach of Vasicek (1976), using Stieltjes sums, was used. Here we adopt a different approach, related to that of Shao and Hahn (1995) who gave results for
We add one remark to guard against misapprehension. To say m = o(s(n)) or m = 0(s(n)), where s(n) -» oo as n —• oo, does not exclude that m = m(n) can be a bounded sequence.
2 Strong limit theorems for based on uni
form spacings
Throughout this section, 0 = U(o) < E/(i) < ... < £/(n) < t/(n+1) = 1 denotes the order statistics from an i.i.d. n-sample £/i, #2? •••> Un of uniformly distributed random variables on [0,1], and
Theorem 1 Let {mn} be a sequence of positive integers such that mn = o(n).
Then,
Remark. If mn is fixed or if mn/logn -» 00, Theorem 1 is a special case of Theorem 2 in Van Es (1992). For comments on Van Es' (1992) conditions, see the remark after Corollary 4.
For the proof of Theorem 1, we will use a well known relationship between uniform spacings and standard exponential random variables (Pyke (1965)), and a useful inequality for standard exponential random variables obtained by Van Es (1992), i.e. the two lemmas below.
Lemma 1 Let Z \, Z2,Zn+i be independent standard exponential random vari
ables. Set W j = Zj/ ( Z i + ... + Zn+1), j = l,...,n+ 1. Then (Wu W2,Wn+i) is distributed as the set ofn + 1 simple spacings determined by n independent uniform random variables on [0,1].
L<?\U) =
n — m + 2 1
log r(x) is the digamma function and T the gamma function.
Proof. See Pyke (1965).
Lemma 2 Let Zi,...,Zn+i be independent standard exponential random vari
ables. Then, if mn/ log n —¥ oo as n —• oo, for any e > 0 and for n large enough,
p( sup logf^- >4<^-
y><j<n-mn+1 \ n t=j+1 / /
Proof. For any J > 0 and for rnn such that mn/ log n —^ oo, with n large enough, Van Es (1992) obtained the inequality,
P I sup
\0<i<n-mn+l
i i+mn
- E s - i >* < 4,
from which the lemma follows. •
Proof of Theorem 1. By the Borel-Cantelli lemma it sufficies to show, for every positive Sy
E ^ (|£imB)(f0 - ^(™») + logmn| > <j) < c». (2) For a sequence Z\y Zn+i of i.i.d. standard exponentially distributed random variables, define
I i+m„ \
Otj,n = log I — J2 Zi , ßn = log m*n n i=;+l
Then, by Lemma 1,
P (|4m«>([/)-V(mn)+logmn| > S)
= P
tirl'4
< P
1 n — m„ + 2
1 n — m„ + 2
n-m„+l
E ajin-ßn-ip(mn)+log mn i=o
>S
n-mn+1
E a j<n-il>(mn)+logmn
j=o >2 +P ( l W > | )
An application of Lemma 2 with mn = n + 1 yields Yin (\ßn\ > S /2) < oo. To check that £„ P(|(n-mn+2)-1 J?jZon+1 <Xj,n-*l>(mn) + logmn| > S/2) <oo, two separate cases will be considered.
First the case mn < n1'4. Here the high order moments of t he otjj n's will be used. As Zj+i + Zj+2 + ... + Zj+mn is r(mn, l)-distributed it follows, for all positive integers mn and r, that
and especially that
E [ûtj.n] = ip(mn) - log mn. Clearly, there exist constants Cr such that, for mn = 1,
E [lai,n|r] < J |1°6 x\r e~xdx < Cr xe~xdx + J |log x\r dx < Cr + r!,
and, for mn > 2,
|log^|rx--'e-^
< r?r \ / ( — + — ) xm n~1e ~zd x i ( mn) Jo \ r an X J
V / T { mn + 1 ) + r ( _ < 3 a
*») V m» /
r(m,
Thus, ctj> n has finite moments of all orders with an upper bound independent of n. Moreover, for all positive integers rj,r2,r, and r = rj + ... + r„
•••<»] <>-E[Ku|r] <r(3C.+r!). (3) Note that {ûj,n}"r™"+1 is a (mn — Independent sequence, that is a;,n and aJ>n
axe independent for all i and j such that |t — j\ > m„ — 1. This together with the inequality (3) implies that there exist positive constants K and N such that, for all n > iV,
E \n - mn + 2
n-m„+l ^
J 2 ai." - <Mmn) + log mn
j=0 I
= 0 ("") + (n-ml + 2)' I24E n ta» - + '"S™") (o»i,n - V>(mn) + log mn) JJ (aij,n ~ ^(mn) + log mn) j
i=l J/
«1 ^ «2»«1 i1 «3
*2 < «3
t 2
Thus, by Chebyshev's inequality, for some constant K i , mn < n1'4 and for n large enough, P(|(n-m„+2)_1 £"=™n+1 a,>-V>(mn)+logmn|>£/2) < Ki/n3'2. Next, consider the second case, mn > n1/4. Since — logm -> 0 as m ->
oo, an application of Lemma 2 yields, if mn > n1/4 and if n is large enough, P(|(n -mn + 2)"1 E"=o,n+1 <*j,n - V'C"»».) + logmn| > 8/2) < 2/n2. Thus, the inequality (2) holds and the theorem is proved. •
Theorem 2 Let XiìX2y ...,Xn be i.i.d. random variables on [0,1] with distri
bution function F(*), and let {mn} be a sequence of positive integers such that mn = o(n/log n). Then F is the uniform distribution if and only if
Jim (L(„mn)(X) - r/>(mn) + log mn) = 0 (a.s.)- Proof See next section.
3 Strong limit theorems for statistics related to l4m) based on non-uniform spacings
The context of this section will generalize several results obtained for LM by Shao and Hahn (1995) to high order spacings.
Theorem 3 Let Ui,U2,...,Un be an i. i.d. sequence of uniformly distributed ran
dom variables on [0,1] and let {mn} be a sequence of positive integers such that mn = o(n/ log n). Then, for any measurable set B Ç [0,1] and for any nonde- creasing bounded functi on G( - ) o n [0,1],
b m i o zJ log —St ~ I J r - M g ( x ) d x (a.s.),
n—foo n mn + 2 j£ j g y U(j) J J &
where g(x) = G'(x) a.e.} and Jß = {j : U(j) € B, 0< j <n — mn+l}. If the index set is redefined as Jß = {j : [t/y ), ^(j+mn)] Q mn+1}; the inequality above holds if B is a finite union of in tervals.
To show t hat the majority of the quotients (G(U(j+mn))—G(U(j)))/(U(j+mn) — U(j)) are arbitrarily close to g(U(j)) when n -> oo, the following two lemmas will be used.
Lemma 3 Let {/i, C/2,Un be an i.i.d. sequence of uniform ly distributed random variables on [0,1]. Then there i s a co nstant C such that
lim —f—
n —• 00 log logn l<i<n<•?<%, ("» - "«-•>) - ^ +l V w w '/ n Proof. See Slud (1978). •
< C (a.s.).
Lemma 4 Let G(-) be any nondecreasing bounde d function on [0,1] and let Mg be the set of all points where the derivative g of G exists. Further} let {an} be a
sequence of positive real numbers such that lim^oo an = 0. Then, given e, 6 > 0, there exists an integer N such that the set
An = {x G [0,1] \ Mg • sup G ^X +^ -^ ( a r ) | < e l ,
t fce»( o,ow) n I J
where 6(0, a) denotes a ball with center 0 and radius a, has Lebesgue measure greater than 1 — S.
Proof. The proof follows by Egorov's theorem and is essentially the same as the proof of Lemma 3.1. in Shao and Hahn (1995). •
Proof of Theorem 3. We follow t he same approach as in Shao and Hahn (1995), where this one-sided strong limit result is proved for the special case mn = 1 and B = [0,1].
Only the case when Jg = { j : U{j) € #,0 < j < n — mn + 1} will b e proved here. The derivation of the second part, where Jß is redefined, is similar because if B is a finite union of intervals then for some positive constant I
caxd{; : [tfyj, f/(j+m„)] Q B , 0 < j < n-m„+l}
< card{jF : t/y) 6 ß, 0 < j < n — mn+ 1}
< card{j : t/(;+mn)] C S, 0 < j < n-mn+1} + lmn.
Let kn = ( mn/ n ) log n. Then, by Lemma 3, there exists a constant C and a sufficiently large (random) N% such that for all n >
Ä, ("»-I - "») < Ck- ("•)•
and by Lemma 4, for any positive numbers e, 5 < 1, there exists an integer N such that
An = \ x € [0,1] \ Mg : sup ^ + ^- 0(x)| < e}
t hçb(o,ckN) ri I J
has Lebesgue measure greater than 1 — S. Let Bn = B fl An, — B \ Bn, Jn = {j : U(j) € Bn, 0 < j < n-m„ + l}, = {0,l,...,n-m„ + l}\.//v and let .7^ = card Jff. Then for n large enough,
3GJb V (i+m") (i) /
i€Jv V UU+ mn) - UU) J jçjc \ U( j + m n )- U{j ) )
< E iQg (d (uu ) ) +£) +E ios
Ì€^JV i€J^ \3NW0+mn) U(}))J
< È'»S fa(f,)+£) '«„(Cj) - m.log£+Ä lo8 (^ft/fo+l^)
-""E"'«® (fw-.) - «Ol)) Ib -JV.m). (4) By the strong law of larg e numbers,
Ä t> 1 4.9 è n—m„+2 j=0 loe (9 { U j ) + e ) M U j ) = f log ( g ( x ) + e) <fx (a.s.) (5) and since jfr — 1 < card{j : U j € B%, 1 < J < «} < jfr + " !n,
»1-+00 n—mn+2 6\ J $f/ ( n + l ) J N J &\ J
where //(•) is the Lebesgue measure. An application of Cauchy-Schwarz inequality gives
sdrnlf h
To obtain an upper bound for the second factor on the right hand side above, the notation of the proof of T heorem 1 will be used. Let e\ > 0 be arbitrary. Then
?p ("«-»-M -2E M > «)
(n-m„+2 § ( a1 Yl—TTln+l j, „ - ßn)2- 2 E [ a ln] > £\ 1j
* ? ' * - £ K"] > t) + ? ' K f ) • «
That the second sum on the right hand side is finite is shown in the proof of The
orem 1. Since {ctj.n} is an (m„ — Independent sequence it follows, using the ap
proach used in the proof of Theorem 1, that for some constant K and if mn < n1/4,
we ha ve P ((n—mn+2) 1Hj=^ln+1 o£n — 25[aQin]>£i/4) < K/n3^2. Furthermore since 2£[c*Qfn] —> 0 as mn -* oo it follows from Lemma 1 that if mn > n1^4 and if n is large enough, then P ((n-mn+2) 1Ei=o>n+1 aj,n~ £[ao,d >^i/4) <2/n2.
Recall that //(Bjv) ^ ^ Then, since the sums in (7) are finite, an application of the Borel-Cantelli lemma yields almost surely that
fesdhslf 5 ^
This inequality, together with (4)-(6) yields the desired result in the limit as
£ , S —y 0 . O
Corollary 1 Under the assumptions of The orem 3, if g(x) is bounded away from zero a.e., i.e. g(x) > p > 0 a.e.,
lim —L_^ E K (
n->oon-mn+2£B Uu+mn)-U{j) J Jb
Proof Using the notation of t he proof of Theorem 3, for n large enough and for
n-¥oo n—mn+2 J=0
implying
E '»S1^ (fo+~)-Oo))) 2£ KJ S 6C'
e < p ,
By letting e and S tend to zero we obtain the desired result. •
( G ( Uu + m n )) - G ( U( j ])
* \ U(j+mn ) — U{j) Z ) l o g P
By combining Theorems 1 and 3 gives the following corollary for
n — m + L
Corollary 2 Under the assumptions of Theorem 3,
lim (Lj,mn)(G(f7))-0(mn) + logmn) < f logg ( x ) d x ( a .s.).
n->oo v ' J O
A function F will be called a pseudo distribution if and only if t here exists a constant c such that cF is a distribution function.
Corollary 3 L e t X i,X2,...,Xn be i.i.d . random variables with d istribution func
tion Fffo and density function fgo. Let {mn} be a sequence of positive integers such that mn = o(n/ log n). Then, for any measurable set A of real number s and for any pseudo dis tribution Fg, with (pseudo) density fg, almost surely,
i«» ö £ log ( F" /X(Ì+mn)| "F" ^fU )\ ) < f dFtfo(x), n-oon—mn+2^ &\ Feo( xu + m n )) - Feo( XU )) ) - J a
where J a — {j : X(j) G A, 0 < j < n — mn+l}, If the index set is redefined as Jji = {jf : [-^yjï^y+mn)] Q 0<j <rc — mn+l}, the inequality above holds if A is a finite union of intervals.
Proof The proof follows from Theorem 3 with G(x) = Fg(F^1(x))i Fpl(x) = i nf { f : F *o(t) > x } since F g o(X i) i s uniformly d i st ri b ut e d a n d F g ( Xi ) ="
FeiF^iFpiXi))). •
The preceding corollary is used in Ekström (1997), where it plays a significant role in a proof of consistency of a so-called generalized maximum spacing (GMSP) estimator. GMSP-estimators are based on certain approximations of different in
formation measures, e.g. the Kullback-Leibler information, Jeffreys' divergence and the Hellinger distance, and can be regarded as alternatives to the maximum likelihood (ML) estimator. As for the ML-method, an unknown distribution Fgo should be estimated from an i.i.d. sample of random variables
Under the assumption that Fgo belongs to a family T = {Fg : 0 G 0} of d istri
butions, absolutely continuous with respect to the Lebesgue measure, the GMSP estimate of Fgo based on the Kullback-Leibler information is the distribution in T that maximizes
(ft(*o+~>)-Wfo)))•
n i'In T ^ J—q \ "*71 /
The corollary above and Theorem 1, together with Jensen's inequality, imply that the limit of this statistic is maximized if a nd only if F$(x) = i^o(x). Thus, for large samples it can be intuitively expected that those distributions which are most probable to have generated the data should be found when L^rin^(Fg(X)) is maximized.
The (original) maximum spacing method, as introduced by Cheng and Amin (1983) and Ranneby (1984), was based on simple spacings. Ranneby proposed the method as an alternative to the maximum likelihood method, and showed that both these methods can be obtained from approximations of t he Kullback- Leibler information I(fe,fgo) = f fgo(x) log (fgo(x)/fg(x)) dx (see Kullback and
Leibler (1951)). This basic idea, that an estimation method may be obtained from an approximation of the Kullback-Leibler information, is applicable even if high order spacings are used. It may be useful to consider Lj™") w ith m„ > 1, to overcome the difficulties with very small values. Because the effect on of inaccurate measurements of the values = -Xy+i) — ^(j) may be considerable, especially for small values, a small inaccuracy in produces a big error in
i»g MS-
Corollary 3 is not only valuable for the (generalized) maximum spacing method, it c a n a ls o b e i nt e r p r et e d i n t e r m s of go o d ness of fit te sts . S u p po s e X \y X 2 , —, Xn
is an i.i.d. sequence of random varibles with distribution F, belonging to a family T of distributions, absolutely continuous with respect to the Lebesgue measure.
Let the hypothesis to be tested be H0 : F(x) = against the alternative H a - F ( x ) Î F0( x ) . Under the null-hypothesis it follows, for m fixed, that
i / n—m+1 \
K'.T'(fl>W) = ^„_m+2( E '»e(<"+')
4 N ( 0 , a2)
where cr2 = (2m2 — 2m + l)r/>'(m) — 2m +1 (see Cressie (1976) and Hoist (1979)).
Cressie (1976) also showed, for m = m„ growing to infinity such that m — o(n1^3), that
V£'(fi>TO) = ^f""è+1log((n+l)(F0(X0+m))-F„(X0))))-tf.(m)j 4 N(0,1).
Thus, using as a test statistic, critical values can be obtained (for large samples) from a standard normal table. The null-hypothesis can therefore be rejected for large values of |V ^^(^o(^0)|«
Suppose that the true underlying distribution is F* ± F0, F, € F. Then by Corollary 3 and Theorem 1,
n-foo Y /n — m + Z J-00 \ J * (X) J and thus, by Jensen's inequality,
lim \V^\FO( X ) ) \ > lim -\/n-m+2 F log D F * ( x ) = oo (a.s.),
n —• oo ' ' 1 n—too J_QO \j*\X) /
where fo and /* are the densities corresponding to Fo and F*, respectiv ely. The case when m grows to infinity is similar. Consequently, the tests are strongly consistent against all continuous alternatives.
Theorem 4 Let X\,X2>..., Xn be i.i.d. random variables on [0,1] with distribu
tion function F. Suppose that {mn} is a sequence of positive int egers such that mn = o(n/ log n). Then,
lim (l4mn)p0 - V'(mn) + logmn) < f \ o g h ( x ) d x (a.s.),
n—»oo v ' J o
where h(x) = ^F"1(x) a.eF~l(x) = inf{t : F(t) > x}. Moreover, if F has a de ns it y f , t he n log h ( x ) d x = — f ( x ) log f ( x ) d x .
Proof If F is continuous then X\ a= F~1(F(Xi)) and F(Xl) is uniformly dis
tributed, so the inequality in the theorem follows as a direct consequence of Corollary 2 with G(x) = Finally if F has a density /, then ^F"l(x) = 1/f ( F ~l( x ) ) and the equality /• log h{ x ) d x = — f ( x ) log f ( x ) d x follows by a change of va riables. •
Remark. The integral — / f(x) log f ( x )d x is known as the entropy of an absolutely continuous distribution F with density /, see Shannon (1948). For a review of methods, including methods based on spacings, to estimate entropy and their statistical properties, see Dudewicz and Van der Meulen (1987). See also Correa (1995), who introduces a different estimator of entropy based on spacings.
Proof of Theorem 2. The result follows by Theorem 1 and Theorem 4, together with Jensen's inequality. •
By assuming that h ( x ) in Theorem 4 is bounded away from zero, using the ideas of Co rollary 1, Theorem 4 becomes a two-sided strong limit theorem.
Corollary 4 Under the assumptions of Theorem ifh(x) is bounded away from zero a.e., Le. h(x) > p > 0 a.e.,
Jim ( L{™n )(X ) - ip(mn) + logm„) = J log h ( x ) d x (a.s.).
If F has a density f, then the equality holds if f is bounded away from infinity.
I n t h i s case / J lo g h ( x ) d x = — fo f ( x ) lo g f ( x ) d x .
Remark. For mn even, L^^X) is related to Vasicek's (1976) entropy estimator.
Vasicek (1976) shows, by rewriting his estimator using Stieltjes sums, that his entropy estimator is weakly c onsistent. Beirlant and Van Zuijlen (1985) show, by using their Glivenko-Cantelli strong limit theorem for the distribution of uni
form mth order spacings, that Vasicek's entropy estimator is strongly consistent under the assumptions that the variance of X\ is finite and that mn oo a nd mn = 0(nl"e) for some 0 < e < 1. Also, by using Vasicek's (1976) approach,
Van Es (1992) give a strong limit theorem for a statistic that has L^^X) as a special case. However Van Es (1992) imposes more restrictive assumptions on the density /, i.e. that it is bounded away from infinity and zero on its support, which is assumed to be an interval. He considers both the case when mn is fixed and the case when mn/ log n ->• oo and mn/n —> 0.
Remark It should be observed that Theorem 4 and Corollary 4 hold for distribu
tions on intervals other than the [0,1]. In the proofs it is sufficient that F"1 be bounded, and this holds for any distribution F defined on any bounded interval.
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