GOODNESS-OF-FIT TESTS VIA PHI-DIVERGENCES. BY LEAH JAGER 1 AND JON A. WELLNER 2 Grinnell College and University of Washington

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1 The Annals of Statistics 2007, Vol. 35, No. 5, DOI: 0.24/ Institute of Mathematical Statistics, 2007 GOODNESS-OF-FIT TESTS VIA PHI-DIVERGENCES BY LEAH JAGER AND JON A. WELLNER 2 Grinnell College and University of Washington A unified family of goodness-of-fit tests based on φ-divergences is introduced and studied. The new family of test statistics S n s) includes both the remum version of the Anderson Darling statistic and the test statistic of Berk and Jones [Z. Wahrsch. Verw. Gebiete ) 47 59] as special cases s = 2ands =, resp.). We also introduce integral versions of the new statistics. We show that the asymptotic null distribution theory of Berk and Jones [Z. Wahrsch. Verw. Gebiete ) 47 59] and Wellner and Koltchinskii [High Dimensional Probability III 2003) Birkhäuser, Basel] for the Berk Jones statistic applies to the whole family of statistics S n s) with s [, 2]. On the side of power behavior, we study the test statistics under fixed alternatives and give extensions of the Poisson boundary phenomena noted by Berk and Jones for their statistic. We also extend the results of Donoho and Jin [Ann. Statist ) ] by showing that all our new tests for s [, 2] have the same optimal detection boundary for normal shift mixture alternatives as Tukey s higher-criticism statistic and the Berk Jones statistic.. Introduction. In this paper we introduce and study a new family of goodness-of-fit tests which includes both the remum version of the Anderson Darling statistic or, equivalently, Tukey s higher criticism statistics as discussed by Donoho and Jin [5]) and the test statistic of Berk and Jones [5] as special cases. The new family is based on phi-divergences somewhat analogously to the phi-divergence tests for multinomial families introduced by Cressie and Read [0], and is indexed by a real parameter s R: s = 2 gives the Anderson Darling test statistic, s = gives the Berk Jones test statistic, s = /2 gives a new Hellinger distance type) statistic, s = 0 corresponds to the reversed Berk Jones statistic studied by Jager and Wellner [24] and s = gives a Studentized or empirically weighted) version of the Anderson Darling statistic. We introduce the corresponding integral versions of the new statistics but will study them in detail elsewhere). Having a family of statistics available gives the possibility of better Received March 2006; revised December Supported in part by NSF Grants DMS and DMS VIGRE Grant). 2 Supported in part by NSF Grants DMS and DMS and by NI-AID Grant 2R0 AI AMS 2000 subject classifications. Primary 62G0, 62G20; secondary 62G30. Key words and phrases. Alternatives, combining p-values, confidence bands, goodness-of-fit, Hellinger, large deviations, multiple comparisons, normalized empirical process, phi-divergence, Poisson boundaries. 208

2 GOODNESS-OF-FIT VIA PHI-DIVERGENCES 209 understanding of individual members of the family, as well as the ability to select particular members of the family that have different desirable properties. In Section 2 we introduce the new test statistics. In Section 3 we briefly discuss the null distribution theory of the entire family of statistics, and note that the exact distributions can be handled exactly for sample sizes up to n = 3000 via Noé s recursion formulas and possibly up to n = 0 4 via the recursion of Khmaladze and Shinjikashvili [3]) along the lines explored for the Berk Jones statistic by Owen [36]. We also generalize the asymptotic distribution theory of Jaeschke [22] and Eicker [7] for the remum version of the Anderson Darling statistic, and of Berk and Jones [5] and Wellner and Koltchinskii [43] for the Berk Jones statistic, by showing that the existing null distribution theory for s = ands = 2 applies to an appropriate version of) the whole family of statistics. We generalize the results of Owen [36] by showing that our family of test statistics provides a corresponding family of confidence bands. In Section 4 we study the behavior of the new family of test statistics under fixed alternatives. We show that for 0 <s< and fixed alternatives the test statistics always converge almost surely to their corresponding natural parameters. For <s<, we provide necessary and sufficient conditions on the alternative d.f. F for convergence to the corresponding natural parameter to hold, and show that the Poisson boundary phenomena noted by Berk and Jones for their statistic continues to hold for s< and for s<0 by identifying the Poisson boundary distributions explicitly. We also briefly discuss further large deviation results and connections between the work of Berk and Jones [5] and Groeneboom and Shorack [9]. In Section 5 we extend the results of Donoho and Jin [5] by showing that all our new tests for s [, 2] have the same optimal detection boundary for normal shift mixture alternatives as Tukey s higher-criticism statistic and the Berk Jones statistic. Our new family of test statistics not only provides a unifying framework for the study of a number of existing test statistics as special cases, but also gives the possibility of designing a new test with several different desirable properties. For example, the new statistic S n /2) satisfies both: a) it consistently estimates its natural parameter for every alternative and b) it has the same optimal detection boundary for the two point normal mixture alternatives of Donoho and Jin [5] as the existing test statistics S n 2) and S n ) considered by these authors. 2. The test statistics. Consider the classical goodness-of-fit problem: pose that X,...,X n are i.i.d. F,andletF n x) = n n i= {X i x} be the empirical distribution function of the sample. We want to test H : F = F 0 versus K : F F 0, where F 0 is continuous. By the probability integral transformation, we can, without loss of generality, pose that F 0 is the uniform distribution on [0, ],

3 2020 L. JAGER AND J. A. WELLNER F 0 x) = x ) 0, and that all the distribution functions F in the alternative K are defined on [0, ]. The basic idea behind our new family of tests is simple. For fixed x 0, ), the interval is divided into two sub-intervals [0,x] and x, ],and we can test the pointwise) null hypothesis H x : Fx)= x versus the pointwise) alternative K x : Fx) x using any of the general phi-divergence test statistics K φ F n x), x) proposed by Csiszár [] see also Csiszár [2] and Ali and Silvey [2]) and studied further in a multinomial context by Cressie and Read [0], where φ is a convex function mapping [0, ) to the extended reals R { }cf. Liese and Vajda [32], pages 0 and 22, and Vajda [39]). Then our proposed test statistics are of the form or S n φ) x K φ F n x), x) T n φ) K φ F n x), x) dx, 0 where the remum and/or integral over x may require some restriction depending on the choice of φ. In our particular case, we define φ = φ s for s R by [ s + sx x s ]/[s s)], s 0,, φ s x) xlog x ) + hx), s =, log/x) + x hx), s = 0 cf. Liese and Vajda [32], page 34), so that ) K s u, v) = vφ s u/v) + v)φ s u)/ v) = s s) { us v s u) s v) s }, s 0,. Note that this definition makes φ s continuous in s for all x in 0, ), and hence, K s is continuous in s for all u, v) 0, ) 2. Also note that K λ+ p, q) = I2 λp : q),wherep = p, p), q = q, q),andi 2 λ p : q) is as defined in 5.), Cressie and Read [0], page 456. Then our proposed test statistics S n s) and T n s) for s R are defined by K s F n x), x), if s, 0<x< ) S n s) K s F n x), x), if s< X ) x<x n) and 2) K s F n x), x) dx, if s>0, 0 T n s) Xn) K s F n x), x) dx, if s 0. X )

4 GOODNESS-OF-FIT VIA PHI-DIVERGENCES 202 The reasons for changing the definitions of the statistics by restricting the remum or integral for different values of s will be explained in Section 3; basically, the restrictions must be imposed for some appropriate value of s in order to maintain the same null distribution theory for all values of s in [, 2]. The most notable special cases of these statistics are s {, 0, /2,, 2}: it is easily checked that K 2 u, v) = u v) 2 2 v v), ) ) u u K u, v) = u log + u) log, v v K /2 u, v) = 4 { uv u) v) } = 2 { u v ) 2 ) 2 } + u v, ) v v K 0 u, v) = K v, u) = v log + v)log, u) u K u, v) = K 2 v, u) = u v) 2 2 u u). It follows that: a) S n 2) is /2 times) the square of the remum form of the Anderson Darling statistic or, in its one-sided form, Tukey s higher criticism statistic ; see Donoho and Jin [5] and Section 5). b) S n ) is the statistic studied by Berk and Jones [5]. c) S n /2) is 4 times) the remum of the pointwise Hellinger divergences between BernoulliF n x)) and BernoulliF 0 x)); as far as we know, this is a new goodness-of-fit statistic [as are all the statistics S n s) for s/ {, 0,, 2}]. d) S n 0) is the reversed Berk Jones statistic introduced by Jager and Wellner [24]. e) S n ) is /2 times) a Studentized version of the remum form of the Anderson Darling statistic; see, for example, Eicker [7], page 6. f) T n ) is the integral form of the Berk Jones statistic introduced by Einmahl and McKeague [8]. g) T n 2) is the classical integral form of) the Anderson Darling statistic introduced by Anderson and Darling [3]. REMARK 2.. Note that K /2 r u, v) = K /2+r v, u) for r R and u, v 0, ), so the families of statistics S n s) and T n s) have a natural symmetry about s = /2. We will continue to use the s-parametrization of these families for reasons of notational simplicity.

5 2022 L. JAGER AND J. A. WELLNER 3. Distributions under the null hypothesis. 3.. Finite sample critical points via Noé s recursion. Owen [36]showedhow to use the recursions of Noé [35] to obtain finite sample critical points of the Berk Jones statistic R n = S n ) for values of n up to 000. See Shorack and Wellner [38], pages for an exposition of Noé s methods.) Jager and Wellner [24] pointed out a minor error in the derivations of Owen [36] and extended his results to the reversed Berk Jones statistic S n 0). Jager [23] gives exact finite sample computations for the whole family of statistics via Noé s recursions for values of n up to The C and R programs are available at the second author s website.) We will not give details of the finite-sample computations here but refer the interested reader to Jager and Wellner [24] and Jager [23]. See Jager and Wellner [24] and Jager [23] for plots of finite sample critical points and several finite sample approximations based on the asymptotic theory given here. During the revision of this paper we learned of an alternative finite-sample recursion for calculation of the null distribution of S n 2) proposed by Khmaladze and Shinjikashvili [3] which apparently works for n 0 4. Presumably this alternative recursion could be used for our entire family of statistics, but this has not yet been carried out Asymptotic distribution theory for S n s) under the null hypothesis. Limit distribution theory for S n 2) and S n ) under the null hypothesis follows from the work of Jaeschke [22] and Eicker [7]; see Shorack and Wellner [38], Chapter 6, pages for an exposition. These results are closely related to the classical results of Darling and Erdös [4]. Berk and Jones [5] stated the asymptotic distribution of their statistic R n = S n ). For details of the proof, see Wellner and Koltchinskii [43], with a minor correction as noted at the end of the proof here. Here we show that the limit distribution of ns n s) r n is the same doubleexponential extreme value distribution for all s 2, where r n = log 2 n + 2 log 3 n 2 log4π), with log 2 n loglog n) and log 3 n loglog 2 n). THEOREM 3. Limit distribution under null hypothesis). Suppose that the null hypothesis H holds so that F is the uniform distribution on [0, ]. Then for s 2 it follows that ns n s) r n d Y4 E 4 v, where E 4 v x) = exp 4exp x)) = PY 4 x).

6 GOODNESS-OF-FIT VIA PHI-DIVERGENCES 2023 Define b n = 2log 2 n, c n = 2log 2 n + 2 log 3 n 2 log4π), d n = n log n) 5, Z n d n x d n n Fn x) x x x). As will be seen, the proof involves the following four facts: Fact. Z n /b n p. Fact 2. b n Z n c n d Y4 E 4 v. Fact 3. /2)c2 n /b2 n = r n + o). Fact 4. ns n s) = /2)Z 2 n + o p). In the ranges s>2ands< we do not know a theorem describing the behavior of the statistics S n s) under the null hypothesis Confidence bands. Owen [36] showed how the Berk Jones statistic R n = S n ) can be inverted to obtain confidence bands for an unknown distribution function F. Similarly, the family of statistics S n s) yields a new family of confidence bands for F as follows: given a continuous d.f. F on R,define K s F n x), F x)), if s, <x< S n s, F ) K s F n x), F x)), if s<. X ) x<x n) By the inverse) probability integral transformation, P F S n s, F ) t) = P F0 S n s) t) for all t where S n s) is as defined in ) andf 0 is the uniform distribution on [0, ]. Hence, with q n s, α) denoting the upper α quantile of the distribution of S n s) under F 0 which is computable via Noé s recursion as discussed in Section 3. or can be approximated for large n via Theorem 3.), it follows that P F Sn s, F ) q n s, α) ) = P F0 Sn s) q n s, α) ) = α for each fixed α 0, ) and n. Hence, {F : S n s, F ) q n s, α)}={f : L n x; s,α) Fx) U n s; s,α) for all x R} yields a family of α confidence bands for F.HereL n x; s,α) and U n x; s,α) are random functions determined by s, α, n and the data in a straightforward way; see Owen [36], Jager and Wellner [24] and Jager [23] for details Asymptotic distribution theory for T n s) under the null hypothesis. Limit distribution theory for T n 2) was established by Anderson and Darling [3]. Einmahl and McKeague [8] noted that this carries over to T n ) for a proof, see Wellner and Koltchinskii [43]) and extended T n ) to other testing problems. Here

7 2024 L. JAGER AND J. A. WELLNER we show that the limit distribution of nt n s) is /2 times) the Anderson Darling limit distribution for all s [, 2], namely, the distribution of 3) A 2 0 [Ut)] 2 t t) dt = d j= Z 2 j jj + ), where U is a standard Brownian bridge process on [0, ] and Z,Z 2,... are i.i.d. N0, ); see, for example, Shorack and Wellner [38], pages THEOREM 3.2 Limit distribution of T n s) under the null hypothesis). Suppose that the null hypothesis H holds so that F is the uniform distribution on [0, ]. Then for <s 2 it follows that nt n s) d A 2 /2, where A 2 is the Anderson Darling limit defined in 3). We will not study the statistics T n s) further in this paper, but intend to continue their study elsewhere. 4. Limit theory under alternatives. The power behavior of individual members of our new family of statistics has previously been studied separately and somewhat in isolation: see, for example, Berk and Jones [5] for the Berk Jones statistic), Durbin, Knott and Taylor [6] and D Agostino and Stephens [3] for T n 2) compared to other integral goodness-of-fit statistics and Nikitin [34] for treatment of Bahadur efficiencies for many goodness-of-fit statistics. Interest in these test statistics has received new impetus via the use of appropriate one-sided versions of the test statistic S n 2) in the context of multiple testing problems; see, for example, Donoho and Jin [5], Jin [28] and Meinshausen and Rice [33]. See Cayón, Jin and Treaster [8] for an interesting application to detection of non- Gaussianity in the cosmic microwave background data gathered by the Wilkinson Microwave Anisotropy Probe WMAP) satellite, and see Cai, Jin and Low [7] for further work on estimation aspects of the problem in connection with the developments in Meinshausen and Rice [33].The work of Owen [36] on confidence bands derived from the Berk Jones statistic S n ) was apparently motivated in large part by the Bahadur efficiency results of Berk and Jones [4]. It is clear from the results of Donoho and Jin [5] and earlier efforts by Révész [37] to combine the strengths of the Kolmogorov and Jaeschke Eicker statistics that tests based on any of the statistics S n s) will do best against alternatives in the tails. As suggested by one of the referees of our paper, this may well be the Achilles heel of such test statistics, since the results of Révész [37] suggest that our statistics will have no asymptotic power for a large class of contiguous alternatives with departures from the null hypothesis in the middle of the distribution). On the other hand, having a family of statistics such as {S n s) : s R} available gives the possibility of choosing or designing ) a test with several desirable properties. We will return to this briefly in Section 6.

8 GOODNESS-OF-FIT VIA PHI-DIVERGENCES 2025 Here we study convergence of the family of statistics to their natural parameters under fixed alternatives, comment briefly on the Bahadur efficiency results of Berk and Jones [5] in light of the results of Groeneboom and Shorack [9], and show that the optimal detection boundary results of Donoho and Jin [5]extend to the whole family of statistics S n s) for s [, 2]. In spite of the negative results of Janssen [27] for goodness-of-fit statistics in general, much remains to be learned about the power behavior of the family {S n s)}. 4.. Almost sure convergence to natural parameter. Let F 0 be the Uniform0, ) distribution function as in Section 2. The Kolmogorov statistic D n F n F 0 has the property that for any distribution function F,ifX,...,X n are i.i.d. F, then a.s. D n F F 0 df). We call df) = F F 0 the natural parameter for the Kolmogorov statistic D n. As Berk and Jones [5] pointed out for their statistic R n = S n ), under alternatives F the convergence S n ) = R n = K F n x), x) a.s. K F x), x) rf) 0<x< 0<x< holds only under some condition on F the exact condition will be given below), and for a slightly more extreme F, namely, what we call the Poisson boundary distribution function, the behavior changes to convergence in distribution to a functional of a Poisson process rather than convergence to a natural parameter. Thus, Berk and Jones [5] showed that if Fx)= / + log/x)), then 4) d Nt) d S n ) = R n = t>0 t U, where N is a standard Poisson process and U Uniform0, ). It turns out that in the range 0 <s< the statistics S n s) behave analogously to the Kolmogorov statistic D n under fixed alternatives. Namely, we show that in this range the statistics converge almost surely to their natural parameter for all d.f. s F. PROPOSITION 4.. Suppose that X,...,X n are i.i.d. F and that 0 <s<. Then S n s) a.s. 0<x< K s F x), x) S s, F ). On the other hand, in the range s> we have the following criterion for almost sure convergence of the statistics S n s) to their natural parameters: PROPOSITION 4.2. Suppose that X,...,X n are i.i.d. F and that s>. Then S n s) a.s. 0<x< K s F x), x) S s, F ) if and only if F satisfies F u) F du <. u)))s )/s 0

9 2026 L. JAGER AND J. A. WELLNER By the inverse) probability integral transformation, the convergence in the last display is equivalent to E F [X X)] /s) <. As Berk and Jones [5] show, if for some γ>0, the distribution function F satisfies and Fx) {log/x)log 2 /x)) +γ }, x γ, Fx) { log / x) ) log 2 / x) )) +γ }, x γ, then R n S n ) a.s. 0<x< K F x), x) S,F) rf) <. It can be shown that this convergence holds if and only if 0 [x x)] Fx) Fx))dx < ; seejager[23] for details. We do not yet know sharp conditions for S n s) a.s. S s, F ) when s Poisson boundaries for s and s < 0. As noted in the previous subsection, the statistic R n = S n ) has a Poisson boundary d.f. F for which R n = S n ) d /U rather than R n = S n ) a.s. rf) S,F).Herewenote that this behavior persists for the entire range s andfors<0. For each fixed s [0, ) c, define the distribution function F s on [0, ] by F s 0) = 0 and, for 0 <x, by + x s ) /s, <s<, s 5) F s x) = ), + log/x) s =, sx s ) ) /s, s <0. Note that F s x) F x) as s for0 x. The following proposition includes the result 4) of Berk and Jones [5] when s =, and it agrees with the case b = /2 of Theorem 2 of Jager and Wellner [25]when s = 2. As in 4), let N be a standard Poisson process, and let U Uniform0, ). PROPOSITION 4.3 Poisson boundaries for s ands<0). i) Fix s and pose that X,...,X n are i.i.d. F s given in 5). Then S n s) d ) Nt) s d= s t>0 t su s. ii) Fix s<0and pose that X,...,X n are i.i.d. F s given in 5). Then S n s) d ) t s 6), s) t S Nt) where S = E is the first jump point of N.

10 GOODNESS-OF-FIT VIA PHI-DIVERGENCES 2027 REMARK 4.. The distribution of t S t/nt)), which is also the limiting distribution of {t/g n t)) : t ξ ) } where G n is the empirical distribution function of n i.i.d. Uniform0, ) random variables ξ,...,ξ n,isgivenby ) P t/nt)) > x t S k ) k = exp x) + x k exp kx), x > ; k! k= see Wellner [40], pages and Shorack and Wellner [38], page 42. This yields an explicit formula for the distribution of the random variable on the righthand side of 6). REMARK 4.2. Although the family of distributions F s satisfies F s x) exp /x )) F 0 x) as s 0, it appears that the natural limit in distribution under F 0 is S n 0) d = 0<x< K 0 F 0 x), x) in this case, so apparently convergence to the natural parameter continues to hold under F 0. We do not know if there is a more extreme) d.f. F 0 for which S n0) d gn) for a nondegenerate functional g of a standard Poisson process N. REMARK 4.3. If F K has Poisson boundary behavior at both 0 and, then natural generalizations of Proposition 4.3 involving two independent Poisson processes can easily be proved. For example, if F is the standard arcsin law with density π u /2 u) /2 0,) u), thens n s) d 2/π 2 ) max{ t>0 Nt)/ t)) 2, t>0 Ñt)/t))2 },wheren, Ñ are independent standard Poisson processes Bahadur efficiency comparisons. Berk and Jones [5] studied the Bahadur efficiency of their statistic S n ) = R n relative to weighted Kolmogorov statistics based on the work of Abrahamson []. As pointed out by Groeneboom and Shorack [9], however, the Bahadur efficacies of the weighted Kolmogorov statistics are 0 for weights heavier than the quite light) logarithmic weight function ψx) = logx x)) because the null distribution large-deviation result is degenerate for heavier weights. A second difficulty for Bahadur efficiency comparisons is that both the weighted Kolmogorov statistics and the Berk Jones statistic fail to converge almost surely to their natural parameters for sufficiently extreme alternative d.f. s F, and as noted by Berk and Jones [5] for the Berk Jones statistic and by Jager and Wellner [25] for the weighted Kolmogorov statistics, there is a certain Poisson boundary d.f. F for which the statistics converge in distribution to a functional of a Poisson process. Thus, comparisons of goodness-of-fit statistics of the remum type via Bahadur efficiency are rendered difficult by breakdowns in both the large-deviation theory under the null hypothesis and by failure of the statistics to converge almost surely under fixed alternatives. Nevertheless, it would be interesting to be able to make comparisons where possible.

11 2028 L. JAGER AND J. A. WELLNER To this end, we consider variants of our statistics in the range 0 <s< with the remum unrestricted as follows: Sn ur,+ s) = K s + F nx), x), 0<x< Sn ur s) = K s F n x), x), 0<x< Sn ur, s) = Ks F nx), x), 0<x< where K s u, v), if 0 <v<u<, K s + u, v) = 0, if 0 u v,, otherwise, K s u, v), if 0 <u<v<, Ks u, v) = 0, if 0 v u,, otherwise. Also, set Ku,v) = K u, v) = u logu/v) + u log u)/ v)) for u, v) 0, ) 2 and K + u, v) = K + u, v). Although we do not yet have large deviation results for the statistics S n s) or S n + s) X ) x<x n) K s + F nx), x),we can establish the following large deviation results for Sn ur,+ s) and Sn ur, s). THEOREM 4.4. Suppose that X,...,X n are i.i.d. with continuous d.f. F 0, the uniform distribution on 0, ). Fix s 0, ). Then n log P 0 S ur,+ n s) a ) inf 0<x< K+ τ s + x, a), x) 7) log[ s s)a] = g s + s a) for each 0 a</[s s)], where τ s + x, a) = inf{t : K+ s t, x) a}. Furthermore, n log P 0 S ur, n s) a ) inf 0<x< K τs x, a), x) log[ s s)a] = gs s a) for each a 0, where τs x, a) = {t : K s t, x) a}. Combining Theorem 4.4 with Proposition 4., we have the following corollary for the Bahadur efficacies of the statistics Sn ur,+ s) and Sn ur, s) with 0 <s<. COROLLARY 4.5. Let F be a continuous distribution function on [0, ]. Then the Bahadur efficacy of Sn ur,+ s) at the alternative F is ɛ s ± F ) = g± s S± s, F )) = g+ s S± s, F )), where g s + is defined in 7) and S ± s, F ) 0<x< K± s F x), x).

12 GOODNESS-OF-FIT VIA PHI-DIVERGENCES 2029 REMARK. Note that lim s g s + page 50, of Berk and Jones [5]. a) = a, in agreement with Theorem 2.2, REMARK. Since g s + a) = g s a) sa as s 0, the Bahadur efficacies of the statistics Sn ur,± s) tend to be smaller than the efficacies of the Berk Jones statistic S +,F)= rf) when the latter exists), and especially so for small s. This,together with extensive numerical computations of Jager [23], strengthens the case in favor of the statistics S n + s) = X ) x<x n) K s + F nx), x) with restricted remum. Unfortunately we do not yet know the large deviation behavior of these statistics with restricted remum. 5. Attainment of the Ingster Donoho Jin optimal detection boundary. Jin [28] and Donoho and Jin [5] consider testing in a sparse heterogenous mixture problem defined as follows: Suppose that Y,...,Y n are i.i.d. G on R and consider testing H 0 : G =, the standard N0, ) distribution function versus H : G = ɛ) + ɛ µ) for some ɛ 0, ), µ > 0. In particular, they consider the n-dependent alternatives H n) given by 8) H n) : G n = ɛ n ) + ɛ n µ n ) for ɛ n = n β,µ n = 2r log n, where /2 <β<and0<r<. By transforming to X i Y i ) i.i.d. F = G )) with the X i s taking values in [0, ]), the testing problem becomes test H 0 : F = F 0, the Uniform0, ) distribution function versus H : F = F 0 u) + ɛ { u) u) µ )} >F 0 u). [The corresponding n-dependent sequence is F n u) = u+ɛ n { u) u) µ n )} with the same choice of ɛ n and µ n as in 8).] Donoho and Jin [5] consider several different test statistics, among which the principal contenders are Tukey s higher criticism statistic HCn defined by HC n X ) x<x [α0 n]) nfn x) x) x x) for some α 0 > 0 they seem to usually take α 0 = /2), and a one-sided version of the Berk Jones statistic BJ n + n X ) x</2 K+ F nx), x),wherek s + u, v) K s u, v){0 <v<u<}.

13 2030 L. JAGER AND J. A. WELLNER Jin [28] see also Ingster [20, 2]) showed that the likelihood ratio test of H 0 versus H n) has a detection boundary defined in terms of the parameters β /2, ) and r 0, ) involved in 8) which is described as follows: set { β /2, /2 <β 3/4, ρ β) = ) 2, β 3/4 <β<. Then for r>ρ β), the likelihood ratio test which makes use of knowledge of β and r) is size and power consistent against H n) as n. Donoho and Jin [5] show that the tests of H 0 versus H n) based on HCn and BJ+ n are also size and power consistent as n and that both of these tests dominate several other tests based on multiple comparison procedures such as the sample range, sample maximum, FDR False Discovery Rate) and Fisher s method; see, for example, Figure of Donoho and Jin [5] and their Theorems.4 and.5. We show here that the tests based on appropriate one-sided versions of the statistics S n s), namely, ns + n s) n X ) x /2 K + s F nx), x), have the same detection boundary for testing H 0 versus H n) as the statistics HC n and BJ + n. More formally, define a function ρ sβ) such that if µ n = 2r log n and if we use a sequence of levels α n 0 slowly enough [slowly enough so that with q n s, α) as defined in Section 3.2, α n satisfies nq n s, α n ) = + o)) log log n], then for r>ρ s β), the resulting sequence of tests has power tending to as n, while for r<ρ s β), the sequence of tests has power tending to zero. In terms of the functions ρ s β), our theorem is as follows. THEOREM 5.. For each s [, 2], ρ s β) = ρ β) for /2 <β<. While this may not be too rising for s 2 in view of the Donoho Jin results for ns n + ) = BJ+ n and ns+ n 2) = /2)H C n )2, it seems new and interesting for s [, ). Figure gives smoothed histograms of the values of the statistics ns n s) r n under the null hypothesis H 0 solid line) and under the alternative hypothesis H n) dotted line) for n = , r = 0.5 and β = /2. This should be compared with Figure 2 on page 978 of Donoho and Jin [5] showing values of HCn and HC+ n, corresponding to our s = 2; their HC n + /n x /2 nfn x) x) + / x x). 6. Discussion and some further problems. In Section 4 we have shown that the statistics {S n s) : s R} behave quite differently for 0 <s<, for s 0and s. In particular, for 0 <s<, the statistics S n s) converge almost surely to their natural parameters for fixed F F 0. Moreover, the different Poisson boundary behaviors for s<0ands suggest that the statistics S n s) with

14 GOODNESS-OF-FIT VIA PHI-DIVERGENCES 203 FIG.. Smoothed histograms of reps) values of the statistics S + n s) under the null hypothesis H 0 solid line) and alternative hypothesis H n) dotted line) with r = 0.5, β = /2 for s = 0, 0.2, 0.5,. s are geared toward heavy tails, while the statistics S n s) with s 0are geared more toward light tails. This also becomes apparent from plots of the functions K F x), x), K F 0 x), x), and of K 0 F x), x) and K 0 F 0 x), x), where F x) = / + log/x)) and F 0 x) = exp /x )); see, for example, Jager [23], page. In Section 5 we have shown that all of the statistics {S n s) : s 2} have the same optimal detection boundary for the two-point normal mixture testing problem considered by Donoho and Jin [5]. Thus, we have some flexibility in designing a test to detect these subtle tail alternatives, and yet behaving very stably under fixed alternatives in the sense of always consistently estimating a natural parameter).

15 2032 L. JAGER AND J. A. WELLNER Thus, it seems that the Hellinger-type) statistic S n /2) may be a very reasonable compromise test statistic. Here is a brief listing of some of the remaining open problems: Problem : What is the limit distribution of S n s) under the null hypothesis when s< ors>2? Problem 2: What are necessary and sufficient conditions for S n s) to converge to its natural parameter under fixed alternatives F for s 0? Problem 3: What is the large deviation behavior of S n s) under the null hypothesis for 0 <s<? Problem 4: Is there an appropriate contiguity theory for the statistics S n s)? The only example involving something similar of which we are aware is Theorem A of Bickel and Rosenblatt [6], but their results do not seem to apply to the statistics S n s).) It is fairly easy to construct versions of our statistics S n s) in more general settings by replacing the intervals [0,x] and x, ] with sets C and C c for C in some class of sets C. Then for testing H 0 : P = P 0 versus H : P P 0,anatural generalization of the statistics S n s) is S n s, C) = K s P n C), P 0 C)), C C where P n is the empirical measure of X,...,X n i.i.d. P. Problem 5: Do the statistics S n s, C) have reasonable power behavior for some of the chimeric alternatives of Khmaladze [30] for some choice of C? 7. Proofs. 7.. Proofs for Section 3. PROOF OF THEOREM 3.. We first carry out the proof for s<, and then indicate the changes that are necessary for s 2. Fix s [, ). Note that ) ) u K su, v) u u = φ s φ s = φ s ) φ s ) = 0 u=v v v u=v and 2 ) u s 2 ) u 2 Ku,v) = u s 2 v v + v v D su, v). Hence, it follows by Taylor expansion of u K s u, v) about u = v that K s u, v) = K s v, v) + u K su, v) u v) + 2 u=v 2 u 2 K su, v) u v) 2 u=u = u v)2 D s u,v)= 2 u v)2 D s u,v)

16 GOODNESS-OF-FIT VIA PHI-DIVERGENCES 2033 for some u satisfying u v u v. This yields 9) K s F n x), x) = 2 Fn x) x ) 2 Ds F n x), x) for 0 <x<, where F n x) x F nx) x ;thatis,x F n x) F nx) on the event x F n x) and F n x) F n x) x on the event F nx) x. We can write 9) as 0) where K s F n x), x) = F n x) x) 2 { + x x)d s F n x), x) }, 2 x x) Rem n x) x x)d s F n x), x) { = F ) x x) n x) s 2 F ) x x + n x) s 2 } x x ) = x 2 s ) x 2 s x) F n x) + x F n x) x) x ) x) x 2 s ) F n x) x 2 s + x F n x). Fix δ 0, /2). Nowforx [δ, δ], F n x) [δ/2, δ/2] a.s. for n N ω, so, much as in Wellner and Koltchinskii [43], Rem n x) =O p n /2 ). δ x δ For 0 <v /2 and s 2, the function u D s u, v) is monotone for u 0, /2], while for 0 <v /2 and s, the function u D s u, v) is monotone for u 0,bv,s)],wherebv,s) /+cv,s)) with cv,s) v)/v) s)/3 s),so x x)d s F n x), x) x x){d sx, x) D s F n x), x)} on the set {F n x) < /2 bx,s)}. SincePF n δ) /2 bδ,s)) 0forδ< /2 ands [, 2],weget ) ) Rem n x) x 2 s X ) x δ X ) x δ F n x) ) + x 2 s X ) x δ F n x) = O p ) + o p ) = O p )

17 2034 L. JAGER AND J. A. WELLNER by Shorack and Wellner [38], inequality, page 45, and inequality 2, 0.3.6), page 46. Alternatively, see Wellner [42], Lemma 2, page 75, and Remark ii). Similarly, ) Rem n x) x 2 s δ x<x n) F n x) 2) Now we write where S n s, I) S n s, II) δ x δ X ) x δ δ x<x n) + δ x<x n) = o p ) + O p ) = O p ). ) x 2 s F n x) S n s) = S n s, I) S n s, II) S n s, III), K s F n x), x) = K s F n x), x) = 2 δ x δ X ) x δ 2 Fn x) x ) 2 Ds F n x), x), Fn x) x ) 2 Ds F n x), x), S n s, III) K s F n x), x) = 2 Fn x) x ) 2 Ds F n x), x). δ x<x n) δ x<x n) By the monotonicity of u D s u, v) for u /2 again, with probability tending to, S n s, II) 2 Fn x) x ) 2 {Ds F n x), x) D s x, x)} X ) x δ Fn x) x ) 2 {Ds F n x), x) D s x, x)}, 2 X ) x δ and similarly, by the monotonicity of u D s u, v) for /2 u<, For S n s, I), S n s, I) = 2 S n s, III) δ x δ Fn x) x ) 2 {Ds F n x), x) D s x, x)} δ x<x n) Fn x) x ) 2 {Ds F n x), x) D s x, x)}. δ x<x n) F n x) x) 2 { + O p n /2 )} x x) Fn x) x ) 2 {Ds F n x), x) D s x, x)}{ + O p n /2 )}. δ x δ

18 GOODNESS-OF-FIT VIA PHI-DIVERGENCES 2035 In the second region the argument above leading to ) yields S n s, II) 2 Fn x) x ) 2 {Ds F n x), x) D s x, x)} 2 X ) x δ X ) x δ and similarly for S n s, III). It follows that S n s) 2 Fn x) x ) 2 X ) x<x n) Fn x) x ) 2 {Ds F n x), x) D s x, x)}, 3) = 2 {D s F n x), x) D s x, x)}{ + O p n /2 )} { Fn x) x) 2 } { x x)d s F n x), x)} X ) x<x n) x x) and, on the other hand, S n s) 2 { + O p n /2 )} Fn x) x ) 2 {Ds F n x), x) D s x, x)} X ) x<x n) 4) = 2 { + O p n /2 )} { Fn x) x) 2 } { x x)d s F n x), x)} X ) x<x n) x x) { + O p n /2 )}. Now we break the remum into the regions [X ),d n ], [d n, d n ] and [ d n,x n) ) with d n = log n) k /n for any k. Then we have F n x) x) 2 n = o p bn 2 X ) x d n x x) ), where b n = 2log 2 n; see Shorack and Wellner [38], 26), page 602. Moreover, so x x)d s F n x), x) =O p ), X ) x d n F n x) x) 2 5) n #x x)ds F n x), x) ) = o p bn 2 X ) x d n x x) ) for # = or # =, and similarly for the region [ d n,x n) ] by a symmetric argument. On the other hand, if we define n Fn x) x Z n, x x) d n x d n

19 2036 L. JAGER AND J. A. WELLNER then, for k 5, Z n p 6) b n and 7) d b n Z n c n Y4 Ev 4, where c n = 2log 2 n + /2) log 3 n /2) log4π) see, e.g., Shorack and Wellner [38], page 600, 6..20) and 6..7)). [Note that for the middle bracket in 5)wehave ) x 2 s ) x 2 s x x)d s F n x), x) = x) + x, F n x) F n x) so ) x 2 s x x)d s F n x), x) + o p ), X ) x X ) F n x) X ) x d n so by Wellner [42], Remark, the probability of large values of the main term can be bounded by ) x 2 s ) x P >λ) = P X ) x d n F n x) X ) x d n F n x) >λ/2 s) ) x P X ) x F n x) >λ/2 s) e λ /2 s) exp λ /2 s)) ]. Furthermore, 8) almost surely where F n x) x x = Oa n ) d n an 2 log 2 n = log 2 n nd n log n) k 0; see Shorack and Wellner [38], page 424, 4.5.0) and 4.5.). It follows from 3), 4) and5) 8) that ns n s) = { nf n x) x) 2 + Op a n ) ) } o p bn 2 2 d n x d n x x) ) 9) { + O p n /2 )} = 2 {Z2 n o pb 2 n )}+o p).

20 GOODNESS-OF-FIT VIA PHI-DIVERGENCES 2037 Hence, we can write 2 Z2 n = 2 Z n c n /b n )Z n + c n /b n ) + 2 cn 2 bn 2 It follows that = 2 b nz n c n /b n ) Z n + c n /b n + b n 2 cn 2 bn 2. 20) ns n s) 2 cn 2 bn 2 = b n Z n c n /b n ) Z n + c n /b n 2b n = b n Z n c n /b n ) Z n/b n + c n /bn 2 2 d + Y 4 { }=Y 4 ; 2 o p bn 2 ) 2 here we used cn 2/b2 n b2 n in the second equality. Since 2 this yields 2) cn 2 bn 2 cn 2 bn 2 ) + o p ) o p ) /2 ) b 2 n + o p) = log 2 n + /2) log 3 n /2) log4π)+ o) = r n + o), P ns n s) r n x ) exp 4exp x)), and completes the proof of Theorem 3.. Note that the centering cn 2/2b2 n ) emerges naturally in the course of this proof. This completes the proof for the case s [, ). For s 2, there are two additional terms that enter, and both of these are o p bn 2 ) from the arguments in the previous section. The case s = 2 is easy since in this case v v)d s u, v) = forallu, while the result was stated for the case s = by Berk and Jones [4] and proved in Wellner and Koltchinskii [43]. Wellner and Koltchinskii [43] incorrectly claim page 324) that KF n x), x) = 0 if x<x ) ; in fact, the remum over this region is stochastically bounded and, hence, can be neglected.) PROOF OF THEOREM 3.2. The fact that nt n 2) = A 2 n /2 d A 2 /2 is classical; see Shorack and Wellner [38], page 48. That nt n ) d A 2 /2 was noted by Einmahl and McKeague [8] and proved by Wellner and Koltchinskii [43]. The proof for s, 2 proceeds along the same lines as the proof in Wellner and Koltchinskii [43] for the case s =, and hence will not be given here. For details, see Jager [23] or Jager and Wellner [26].

21 2038 L. JAGER AND J. A. WELLNER 7.2. Proofs for Section 4. PROOF OF PROPOSITION 4.. We first prove the claim for the unrestricted version of the statistics Sn ur s) defined by Sur n s) 0<x< K sf n x), x), and then show that the difference between S n s) and Sn ur s) is negligible. Now for s 0, ) and C s /s s)),wehave S ur n s) S s, F ) C s 0<x< { F n x) s x s F n x) ) s x) s } { Fx) s x s Fx) ) s x) s } C s { x Fn x) s Fx) s) x s + x { Fn x) ) s Fx) ) s } x s C s { F n x) s Fx) s + Fn x) ) s ) s Fx) x x a.s. 0. Thus, the proposition will be proved if we show that 22) Sn ur s) S ns) a.s. 0. Now write Sn 0s) = max{r n,m n,l n },where and M n K s F n x), x) = S n s), X ) x<x n) L n K s F n x), x) x<x ) R n x Xn) K s F n x), x). Note that 0, if M n L n R n, Sn 0 s) S ns) = max{l n,m n,r n } M n = L n M n, if L n >M n R n, R n M n, if R n >M n L n. Now set α 0 α 0 F ) = {x : Fx)= 0} 0, α α F ) = inf{x : Fx)= }. } }

22 Note that and, on the other hand, Similarly, while GOODNESS-OF-FIT VIA PHI-DIVERGENCES 2039 { ) s } L n = K s F n x), x) = X) x<x ) s s) a.s. s s) { α 0) s } l 0 s, F ), ) ) ) M n = K s F n x), x) K s Fn X),X) = Ks /n,x) X ) x<x n) = { /n) s X) s /n) s ) s } X ) L 0 s s) n a.s. s s) { α 0) s }=l 0 s, F ). R n = x X n) K s F n x), x) = s s) a.s. { α s } r 0 s, F ), s s) { X s} n) M n = K s F n x), x) K s Fn Xn) ) ) ),X n) = Ks /n,xn) X ) x<x n) = { /n) s Xn) s /n) s ) s } X n) R 0 s s) n a.s. { α s }=r 0 s, F ). s s) By combining these pieces, it follows that 0 Sn 0 s) S ns) = max{l n,m n,r n } M n 0, if M n L n R n, 0, if M n L n R n, = L n M n, if L n >M n R n, L n L 0 n, if L n >L 0 n R n, R n M n, if R n >M n L n R n Rn 0, if R n >Rn 0 L n a.s. 0. This shows that 22) holds and completes the proof. PROOF OF PROPOSITION 4.2. Recall that when s = 2 such a condition follows from Theorem 3 in Jager and Wellner [25]: taking b = /2 and applying

23 2040 L. JAGER AND J. A. WELLNER continuous mapping, we conclude that S n 2) a.s. K 2 F x), x) if and only if E{[X X)] /2 } <. 0<x< Similarly, for <s<, { S n s) = Fn x) s x s + F n x) ) s x) s } 0<x< ss ) a.s. { Fx) s x s + Fx) ) s x) s } 0<x< ss ) if and only if F n x) s x s Fx) s x s a.s. 0 and F n x) ) s x) s Fx) ) s x) s a.s. 0, if and only if ) Fn x) s ) Fx) s a.s. x s )/s x s )/s 0 and ) Fn x) s ) Fx) s a.s. x) s )/s x) s )/s 0, if and only if ) Gn F x)) s ) Fx) s a.s. x s )/s x s )/s 0 and ) Gn F x)) s ) Fx) s a.s. x) s )/s x) s )/s 0. Since gu) = u s is uniformly continuous on bounded sets, these last two convergences occur if and only G n F x)) x s )/s Fx) x s )/s a.s. 0 and G n F x)) Fx) x) s )/s x) s )/s a.s. 0. These, in turn, hold if and only if G n u) u F u) s )/s a.s. G 0 and n u) u) F u)) s )/s a.s. 0.

24 GOODNESS-OF-FIT VIA PHI-DIVERGENCES 204 But in view of Wellner [4], these convergences hold if and only if F satisfies 0 F u) F du <. u)))s )/s By the inverse) probability integral transformation, the convergence in the last display is equivalent to E[X X)] s)/s <. This completes the proof of the claimed equivalences. PROOF OF PROPOSITION 4.3. For s =, this follows from Berk and Jones [5], pages Thus, it suffices to prove the claimed convergences for s>ands<0. For s>, fix α 0, ). We begin by breaking the remum over 0, ) into the regions 0 <F s x) < n α, n α F s x) n α and n α <F s x) < : { S n s) = Fn x) s x s + F n x) ) x x) s } 0<x< ss ) { = Fn x) s x s + F n x) ) x x) s } x:0<f s x)<n α ss ) { Fn x) s x s + F n x) ) x x) s } x:n α <F s x)< n α ss ) x: n α <F s x)< I n s) II n s) III n s). { Fn x) s x s + F n x) ) x x) s } ss ) For the main term, I n s), letg n be the empirical d.f. of n i.i.d. Uniform0, ) d random variables and use F n = Gn F s ) to write { ) ss )I n s) = d Gn F s x)) s F s x) s x s 0<F s x)<n α F s x) ) Gn F s x)) s + Fs x) ) s x) s }, F s x) where ) Gn F s x)) s ) Gn t) s ) d Nt) s =, x:f s x)<n α F s x) 0<t<n α t t>0 t F s x) s x s = uniformly in x [0,n α ], while G n F s x)) F s x) x:f s x)<n α x s + x s )/s ) s 0 a.s.

25 2042 L. JAGER AND J. A. WELLNER and Fs x) ) s x) s 0. x:f s x)<n α Combining these last five displays shows that I n s) d s t>0 Nt)/t) s ; note that the limit variable is /s almost surely. To handle the term II n s), write { ) II n s) = d Gn F s x)) s F s x) s x s n α <F s x)< n α F s x) ) Gn F s x)) s + F s x) F s x) ) } s x) s ss ), where now the two terms involving the ratio of the empirical d.f. to the true d.f. F s converge almost surely to. Hence, we conclude that II n s) a.s. K s F s x), x) = 0<x< s, where the equality follows after some calculation. Finally, it is easily shown that III n s) a.s. 0. For s<0, fix α 0, ). We begin by breaking the remum over 0, ) into the regions X ) x<fs n α ), Fs n α ) x Fs n α ) and Fs n α )<x<x n) : { S n s) = Fn x) s x s + F n x) ) x x) s } X ) x<x n) ss ) { = Fn x) s x s + F n x) ) x x) s } x:x ) x<fs n α ss ) ) { Fn x) s x s x:fs n α ) x Fs n α ) x:fs n α )<x<x n) + F n x) ) x x) s } ss ) { Fn x) s x s + F n x) ) x x) s } ss ) I n s) II n s) III n s).

26 GOODNESS-OF-FIT VIA PHI-DIVERGENCES 2043 For the main term, I n s), letg n be the empirical d.f. of n i.i.d. Uniform0, ) d random variables and use F n = Gn F s ) to write { ) ss )I n s) = d Fs x) s F s x) s x s x:x ) x<fs n α G ) n F s x)) ) Fs x) s + G n F s x)) F s x) ) } s x) s, where ) Fs x) s ) t s ) d t s =, x:x ) x<fs n α G ) n F s x)) ξ ) t<n α G n t) t S Nt) F s x) s x s = x s s x s) )) s uniformly in x [0,F n α )], while F s x) x:f s x)<n α G n F s x)) a.s. 0 and Fs x) ) s x) s 0. x:f s x)<n α Combining these last five displays shows that I n s) d s) t S t/ Nt)) s ; note that the limit variable is / s) almost surely. To handle the term II n s), write { ) II n s) = d Fs x) s F s x) s x s G n F s x)) n α <F s x)< n α ) Fs x) s + G n F s x)) F s x) ) } s x) s ss ), where now the two terms involving the ratio of the empirical d.f. to the true d.f. F s converge almost surely to. Hence, we conclude that II n s) a.s. K s F s x), x) = 0<x< s, where the equality follows after some calculation. Finally, it is easily shown that III n s) a.s. 0.

27 2044 L. JAGER AND J. A. WELLNER To prove Theorem 4.4 and its corollary, we will use the following lemma from Chernoff [9]. LEMMA 7.. Let X,X 2,...be i.i.d. with continuous distribution F 0. a) If t<f 0 x), then n log PF n x) t) K t, F 0 x)). b) If t>f 0 x), then n log PF n x) t) K + t, F 0 x)). In both cases, the convergence is from below. PROOF. This follows from Theorem of Chernoff [9]. PROOF OF THEOREM 4.4. We first prove the theorem for Sn ur,+ s). Since K s + t, x) is continuous in t and strictly increasing on x, ), then for 0 <a< x s )/[s s)], there is a unique τ = τx) in x, ) for which K s + τ, x) = a and {t : K s + t, x) a}=[τ, ).Ifa x s )/[s s)],thenτ = necessarily. For any fixed x 0, ),wehave by Lemma 7.. So n log P S ur,+ n s) a ) n log P K + s F nx), x) a ) = n log P F n x) τ ) K + τ, x) 23) lim inf n n log P Sn ur,+ s) a ) inf 0<x< K+ τx), x). Now consider the reverse inequality. Let x i denote the remum for X i) x<x i+).sincef n x) = i/n on this range, we have K s + F nx), x) = K + ) s i/n,xi) K + ) s i/n,xi+) = K + ) s i/n,xi). x i Note that for x<x ),wehavef n x) = 0, and so K s + F nx), x) = 0also.So we can write Sn ur,+ s) = max i n {K s + i/n, X i))} =max i n {K s + F nx i) ), X i) )}. Now, using monotonicity of τ, n log P Sn ur,+ s) a ) = ) n log P { max K + ) )} s Fn Xi),Xi) a i n n n log P K s + ) ) ) Fn Xi),Xi) a i= = n n log P ) )) F n Xi) τ Xi) i=

28 GOODNESS-OF-FIT VIA PHI-DIVERGENCES 2045 = n n log P i/n τ )) n X i) = n log P τ ) i/n) X i) i= i= n n log P F n τ )) i/n)) F n Xi) i= = n n log P F n τ i/n)) i/n ) i= n log n e nk+ i/n,τ i/n)) i= [by Lemma 7.b)] n n log e n min i n K + i/n,τ i/n)) n n log i= e n inf 0<x< K + x,τ x)) i= = n n log e n inf 0<x< K + τx),x) = n log[ ne n inf 0<x< K + τx),x) ] i= = inf 0<x< K+ τx), x) + log n n. So we conclude that 24) lim n n log P S ur,+ n s) a ) inf 0<x< K+ τx), x). Combining this last display with 23) yields the convergence part of 7). To prove the explicit formula for g s +, note that K+ τ + x), x) is a decreasing function of x until x =[ s s)a] / s),wherek + τ + x), x) =. Thus, inf 0<x< K+ τ + x), x) = lim K + τ + [ s s)a] / s) ɛ ), [ s s)a] / s) ɛ ) ɛ 0 = log[ s s)a]/ s), so the given formula for the infimum in 7) holds. This completes the proof for Sn ur,+ s). The proof for Sn ur, s) is analogous using Lemma 7.a) Proofs for Section 5. The following lemma extends Lemma A.4, page 988, Donoho and Jin [5]. LEMMA 7.2. i) For 0 <v u /2, and s 2, 25) K s + u, v) u v) 2 2 v v) K 2u, v).

29 2046 L. JAGER AND J. A. WELLNER ii) Let <s 2 and v = vu) satisfy 0 <v u<. Then, as u 0, K 2 u, v) [ + O v/u) 2 s) O v)/ u) ) 2 s )], K s u, v) = if u/v, vφ s u/v) + o) ) = u{u/v) s s} + o) ) / ss ) ), if u/v. iii) Let s = and v = vu) satisfy 0 <v u<. Then, as u 0, { K2 u, v) [ + O u + u/v) )], if u/v, K u, v) = u logu/v) + o) ), if u/v. iv) Let s [, ) \{0} and v = vu) satisfy 0 <v u<. Then, as u 0, K 2 u, v) [ + O v/u) 2 s) O v)/ u) ) 2 s )], K s u, v) = if u/v, s u + o) ), if u/v. v) Let s = 0 and v = vu) satisfy 0 <v u<. Then, as u 0, K 2 u, v) [ + O v/u) 2) O v)/ u) ) 2 )], K 0 u, v) = if u/v, u + o) ), if u/v. REMARK. Note that for <s 2, as u 0andu/v, { vφ s u/v) = v s) + s u ) u s } v v s s) ) u u s = + s ) ss ){ v } v u s ) u u s } ) s + o), ss ){ v where the right-hand side converges to u logu/v) + o)) as s. PROOF OF LEMMA 7.2. i) Letting u = tv, it suffices to show that for 0 < v /2 and t /2v), K s tv,v) t ) 2 2 v v or, equivalently, since K s tv,v) = vφ s t) + v)φ s tv)/ t)), ) φ s t) + v φ s tv v ) t ) 2 2 v.

30 Let GOODNESS-OF-FIT VIA PHI-DIVERGENCES 2047 ) f s t) φ s t) + v φ s tv v ) t ) 2 2 v. Now by direct calculation, f s ) = 0, and ) ) tv v f s t) = φ s )φ t) + v s t v v v ) tv = φ s t) φ s t v v, so that f s ) = 0. Furthermore, f s t) = φ s t) + = { v = v v v v φ s v)φ s ) tv v v tv t) + vφ s { ) 2 s v) + v t { v) t + v v tv ) } v ) v 2 s } tv )) 2 s } since x 2 s is concave for s 2) = { ) v 2 s } 0 v t tv) using the fact that v v) vt vt) for 0 v vt /2 implies v)/t tv)). Here we have used φ s s sxs x) = s s) = xs s, φ s x) = xs 2. Since t /2v), it follows that v vt /2 <, v vt /2 > 0 and vt)/ v) /2 v)). Whens<, we calculate f s t) = φ s t) v v ) 2 φ s ) tv v and note that f s )<0, while f s t) = 0 has a unique root, so to show f s t) 0, it suffices to show f s /2v)) 0for0 v /2. By a straightforward calculation, we get f s /2v)) = 2v)2 s + v v ) v 2 s /2 v,

31 2048 L. JAGER AND J. A. WELLNER which is 0for0 v /2 ifs. This shows that f s t) 0 in the range s<, and completes the proof of i). ii) By expanding K s u, v) as a function of u as in 0), K s u, v) = K 2 u, v){ + v v)d s u,v) } with u v u v ; since0<v u, we necessarily have 0 <v u u. Here ) v 2 s } { ) v 2 s v v)d s u,v) = v){ u + v u } I + II. Now 0 <v u u implies u /v u/v, sov/u v/u and v u u implies v)/ u) v)/ u ). Thus, I 0and II 0. It follows that { ) v 2 s v v)d s u,v) v }. u Similarly, { ) v 2 s v v)d s u,v) v) } u in this range, and the claimed bound in the first part of ii) follows. To prove the second part of ii), note that when u/v, we can write K s u, v) vφ s u/v) = u/v)s v u)/ v)) s v) v{ s + su/v) u/v) s } where, for <s 2, and = u/v)s v + u)/ v)) s v) u/v) s v v s) su = +[ u)/ v))s v) ]/[u/v) s v] [v s) + su]/[u/v) s v] Bu,v) = + Au, v) Bu,v), v s) + su u/v) s v Au, v) = u)/ v))s v) u/v) s v = s u/v) s + sv/u)s = o) = u)/ v))s [ v) ]+ u)/ v)) s u/v) s v

32 GOODNESS-OF-FIT VIA PHI-DIVERGENCES 2049 u)/ v))s = u/v) s + u)/ v))s u/v) s v su + sv = o) + u/v) s + o) v = o) + s u/v) u/v) s = o) sv/u) s = o). Thus, the second part of ii) holds. The first part of iv) is proved exactly as in ii). To prove the second part of iv), we write { K s u, v) = u/v) s v u)/ v) ) s } v) s s) = s s) = = s s) { ) u s ) v + v) v { ) u s v s ) u + v u) + ) u s )} v } v)su v) + ou) + ov)) sv + ov) { ) ) su v/u) + o) uv/u) s { v/u) s } } s s) = s u + o) ). v) The proof of v) is similar to the proof of iv). LEMMA 7.3. Suppose that X,...,X n are i.i.d. F n with 0 <ρ β) < r < β/3. Then r</4and for any 0 <r 0 <r, F n x) p 26) x 0. n 4r <x<n 4r 0 PROOF. Note that F n ) = d G n F n )),whereg n is the empirical d.f. of n i.i.d. U0, ) random variables ξ,...,ξ n and { F n x) = x + ɛ n x) )} x) µ n x. Thus, with b a a t b ft), F n x) x 27) n 4r <x<n 4r 0 = Fn x) x ) + n 4r 0 n 4r F nx) x ) + n 4r 0 n 4r

33 2050 L. JAGER AND J. A. WELLNER ) d Gn F = n x)) + n 4r 0 x G ) nf n x)) + n 4r 0 28). n 4r x n 4r The second term in this last display converges to 0 in probability easily since F n x)/x implies that it is bounded by G ) nf n x)) + F n x) G ) + nt) p 0 n 4r t n 4r by Theorem 0 of Wellner [42]. On the other hand, G n F n x)) = G nf n x)) F n x) x F n x) x ) ) Gn F n x)) Fn x) Fn x) = +, F n x) x x so again by Theorem 0 of Wellner [42], the first term of 27) convergesto0in probability if ) lim F n x)/x Fn x) n n 4r < and 0. n 4r <x<n 4r 0 x But this holds by a straightforward analysis using the asymptotics of when r<β/3. Now we have the tools in place to prove our extension of the results of Donoho and Jin [5]. PROOF OF THEOREM 5.. First consider <s<2. As in Donoho and Jin [5], we first consider the case r<β/3. Then r</4 and we can choose 0 < r 0 <r</4. From Lemma 7.3 the convergence 26) holds. Thus, by part ii) of Lemma 7.2, it follows that for n 4r <x<n 4r 0,wehave nk s + F nx), x) = Fn x) x) + ) 2 + op ) ), 2 x x) and hence, ns + n s) n 4r <x<n 4r 0 nk + s F nx), x) 2 HC 2 n,r,r 0 + op ) ). Thus, ns + n s) separates H 0 and H n) for s, 2) and r<β/3. Now pose that r> β) 2 and still <s<2). Since r + β)/2 r) <, we can pick a constant q< such that r + β) 2 r r< q<.