The test is mimicked in the con- text of nonparametric density estimation, nonparametric regression and interval-censored data

1998, Vol. 26, No. 1, 288–314

NEW GOODNESS-OF-FIT TESTS AND THEIR APPLICATION TO NONPARAMETRIC CONFIDENCE SETS¹

By Lutz D ¨umbgen

Universit¨at Heidelberg and Medizinische Universit¨at zu L ¨ubeck

Suppose one observes a processVon the unit interval, wheredV= f_o+dWwith an unknown parameterf_o∈L₁01and standard Brownian motionW. We propose a particular test of one-point hypotheses aboutf_o which is based on suitably standardized increments ofV. This test is shown to have desirable consistency properties if, for instance,f_o is restricted to various H¨older classes of functions. The test is mimicked in the con- text of nonparametric density estimation, nonparametric regression and interval-censored data. Under shape restrictions on the parameter, such as monotonicity or convexity, we obtain confidence sets forf_oadapting to its unknown smoothness.

1. Introduction. Suppose one observes a stochastic process V_n =F_n+ n^−1/2Won01, whereF_nis an unknown parameter in⺓01withF_n0 = 0,Wis standard Brownian motion on01, andn >1 is a known scale param- eter. Estimation within this model is closely related to estimation of regression functions or densities based on samples of sizen; see Brown and Low (1996) and Nussbaum (1996). LetC_nV_n αbe a confidence set forF_nwith coverage probability 1−α ∈ 01. Given a model ⺝ ⊂ ⺓01 for F_n and any func- tion φ on⺝, the set φC_nV_n α ∩⺝is obviously a 1−α-confidence set forφF_n. Numerous applications of this type are described, for instance, by Donoho (1988), Davies (1995) and Hengartner and Stark (1995). The first two authors investigate setsC_nV_n αbased on standard goodness-of-fit tests such as the Kolmogorov–Smirnov test. In the context of density estimation, Hen- gartner and Stark (1995) utilize a special test criterion which may, but need not, give optimal confidence bands. The present paper introduces a new type of goodness-of-fit test such that the resulting confidence setsφC_nV_n α∩⺝ have optimal size in terms of rates of convergence simultaneously for various classes⺝ and functionalsφ.

Suppose we want to test the hypothesis “F_n = 0” versus “F_n = 0.” For fixed numbers 0 ≤s < t ≤ 1 andr ∈ R\0 consider the special alternative

“F_n· = ±rLebs t ∩ 0·.” Then an optimal test rejects for large values of nV_ns t²/t−s; generally hs t stands for the increment ht −hs of a functionhon the line. Now we combine these special test statistics. Suppose that the tripletr s tabove has an improper prior distribution with Lebesgue

Received May 1996; revised October 1997.

1Research supported in part by European Union Human Capital and Mobility Program ERB CHRX-CT 940693.

AMS1991subject classifications. 62G07, 62G15

Key words and phrases. Adaptivity, conditional median, convexity, distribution-free, interval censoring, modality, monotonicity, signs of residuals, spacings.

288

density 1s t ∈ δt−s^1/2, where δ = s t ∈ 01²0 < t−s ≤ δ and 0 < δ ≤ 1. Then the corresponding Bayes test statistic is given by Tn^1/2V_n, where

Th =

δexp

hs t² 2t−s

ds dt

This statistic is reminiscent of a goodness-of-fit statistic proposed by Shorack and Wellner (1982). The main difference is the exponential function in the inte- grand, which is essential for our results. It is true, but not obvious, thatTW is finite almost surely with continuous distribution; see Section 5. (Note that ETW = ∞.) Thus we reject the hypothesis “F_n =0” at levelα ifTn^1/2V_n exceeds cα, the 1−α-quantile of ⺜TW. The corresponding 1−α- confidence set forF_n is given by

C_nV_n α =

F∈⺓01F0 =0 Tn^1/2V_n−F ≤cα

As for the power of this test and the size of C_nV_n α, note that T· is convex on⺓01. Hence

2Tn^1/2G−F/2 ≤Tn^1/2V_n−G +Tn^1/2V_n−F (1)

for arbitrary F G∈ ⺓01. In particular, letting G=0 andF=F_n shows that Tn^1/2V_n → ∞ in probability whenever Tn^1/2F_n/2 tends to infinity.

For any fixed F_o = 0, it follows from Fatou’s lemma that Tn^1/2F_o/2 →

∞. HenceT· implies an omnibus test. Unless stated otherwise, asymptotic statements refer ton→ ∞.

In order to investigate the power ofT· more thoroughly, we consider the set

C¹n V_n α =

f∈L₁01

0·fxdx∈C_nV_n α

This is certainly a 1−α-confidence set for the L₁-derivative f_n of F_n (if existent). We consider the intersection ofC¹n V_n αwith H¨older smoothness classes: letI⊂Rbe an interval andβ=k+γ with a nonnegative integerk and 0 < γ≤ 1. Then ⺖_{β L}Istands for the set of all real functions fthat arek times differentiable onIsuch that

f^kx −f^ky ≤Lx−y^γ for allx y∈I

heref^k denotes thekth derivative off(where f⁰=f). Further we define the supremum normf_I=sup_x∈Ifx. All subsequent consistency results are formulated in terms of

ρ_n=logn/n

Theorem 1.1. For arbitrary fixed β L >0, let I_n ⊂ 01 be an interval with length

LebI_n ≥ρ^1/2β+1n

Then there exists a constantR=Rβ Lsuch that sup

f−g_Inf g∈C¹n V_n α f−g∈⺖_{β L}I_n

≤Rρ^β/2β+1n

for any fixedα∈ 01and sufficiently largen.

This result has two straightforward consequences. Suppose that F_n is dif- ferentiable with derivativef_n∈⺖_{β L}01. When testing “f_n=0” versus

“f_n∈

f∈⺖_{β L}01 f₀₁≥ε_n

”

at fixed levelα, it was shown by Ingster (1993) that the maximin power con- verges to one or α as ρ^−β/2β+1n ε_n tends to infinity or zero, respectively. In fact,

Tn^1/2V_n →_p∞ provided that f_n01 ρ^β/2β+1n

→ ∞

Thus our test 1Tn^1/2V_n ≥cαis asymptotically optimal in terms of rates of consistency for arbitrary H¨older classes. Another interesting reference in the context of nonparametric testing is Spokoiny (1996).

A second implication is that sup

f−f_n01f∈C¹n V_n α ∩⺖_{β L}01

≤O_p

ρ^β/2β+1n Note that the confidence set C¹n V_n α ∩⺖_{β L}01 may be empty, where sup⵰ = −∞. In that case⺖_{β L}01 is regarded as a questionable model forf_n. The rateO_pρ^β/2β+1n was shown by Khas’minskii (1978) to be optimal for estimatingf_n∈⺖_{β L}01under sup-norm loss.

Smoothness assumptions such as “f_n ∈⺖_{β L}01” are difficult to justify in practice. It would be desirable to have a1−α-confidence set forf_nwhose size is automatically of the right order of magnitude, depending on the un- known smoothness of f_n. As pointed out by Low (1997), this is essentially impossible. However, some adaptivity is possible iff_n satisfies shape restric- tions such as monotonicity. Restrictions of this type are indeed plausible in many applications. Precisely, we shall investigate the classes

⺖_↑I = fnondecreasing onI and ⺖_↓I = −⺖_↑I

⺖_convI = fconvex onI

⺖_ccI = fconvex–concave or concave–convex onI

Rather than doing so in the present white noise model, we propose and analyze modifications ofTfor three different models.

Section 2 investigates tests for distribution functions on the line and their application to density estimation. Let X_n = X_1n X_2n X_nn be the or- der statistic of n independent random variables with unknown distribution functionF_nin⺠, the set of all continuous distribution functions on the line.

Recall that F_nX_in1≤i≤n is distributed as the order statistic of nindepen- dent random variables with uniform distribution on 01 [cf. Shorack and Wellner (1986), Chapter 1]. Thus

F∈⺠ FX_in1≤i≤n∈B_n

defines a confidence set forF_nwhose coverage probability depends only on the set B_n ⊂ 01ⁿ. Hengartner and Stark (1995) constructed confidence bands for shape-restricted densities (monotonicity or unimodality) with the help of simultaneous confidence bounds forF_nX_{i−1K n} X_{iK n}, 1≤i≤n/K, where X_0n = −∞, X_{n+1 n} = ∞and K =K_n is a bandwidth parameter. One can get rid of the tuning parameter K by considering (essentially) all spacings X_jn X_kn, 0≤j < k≤n+1, in a suitable way. Our particular modification results in greater computational complexity involving convex rather than lin- ear progamming, the reward being (almost) optimal rates of convergence for several functions ofF_n.

Section 3 is concerned with nonparametric regression. Suppose that one observes Y_in = f_nt_in +E_in, 1 ≤ i ≤ n, with an unknown function f_n on R^d, fixed design points t_in ∈ R^d and independent errors E_in with median zero. Davies (1995) obtained tests and confidence sets for (functions of)f_nvia inversion of the runs test, applied to the random vector

signY_n f = signY_in−ft_in1≤i≤n

wherefis a candidate forf_n. For a different application of sign tests in non- parametric regression, see M ¨uller (1991). We propose a test criterion, also based on signY_n f, that yields adaptively optimal confidence bands for f_n. These results complement the literature on point estimation under shape re- strictions [cf. Mammen (1991) and the references therein]. Some numerical examples for our confidence bands are given.

A possible application of the present methods to interval-censored obser- vations is discussed briefly in Section 4. For a detailed treatment of efficient estimation within this model, see Groeneboom and Wellner (1992).

All proofs are deferred to Section 5.

2. Distribution functions and density estimation. The idea is to re- place the processn^1/2V_n−Fin Section 1 with the process

t→n^1/2FX_{n+1t n} −t

Let D¯_n denote the set of pairsj kof integers such that 0≤j < k≤n+1.

Note that F_nX_jn X_kn has a Beta-distribution with parametersk−j and n+1−k+j [cf. Shorack and Wellner (1986), Chapter 3.1]. We utilize the following bounds for tail probabilities of the Beta-distribution.

Proposition 2.1. For0< p <1, define .x p =plogp

x + 1−plog1−p 1−x

if x ∈ 01, and .x p = ∞ otherwise. Let B be a random variable with distribution betamp m1−p,m >0. Then

PB≥x ≤exp−m .x p forx≥p PB≤x ≤exp−m .x p forx≤p

The function .· p is strictly convex on01 with minimum .p p = 0.

For anyc≥0, .x p ≤cimplies that

−

2p1−pc− 1−2p⁻c≤x−p≤

2p1−pc+ 1−2p⁺c Withδ_jkn = k−j/n+1the precise definition of our test statistic is

T_nX_n F =n⁻²

j k∈D_n

expn.FX_jn X_kn δ_jkn

where D_n = j k ∈ ¯D_n δ_{min n} ≤ δ_jkn ≤ δ_n is a nonvoid subset of D¯_n determined by numbers 0 < δ_{min n} ≤ δ_n ≤ 1. A possible reason for using a lower bound δ_{min n} >1/n+1 for δ_jkn are discretization errors in the data X_in. It is assumed throughout that

δ_{min n}=Oρ_n and δ_n→δ

Using an upper boundδ_n<1 reduces computational complexity and empha- sizes smaller intervals. With the 1−α-quantile b_nα of T_nX_n F_n, the set

C_nX_n α = F∈⺠TX_n F ≤b_nα

is a 1−α-confidence set for F_n. The next proposition summarizes some properties ofT_nX_n F_nandC_nX_n α.

Proposition 2.2.

a T_nX_n F_n →_⺜

δexp

Bs t² 2t−s1−t+s

ds dt whereBis a Brownian bridge on01.

(b) There is a constantK_odepending only onD_nn such that the following inequalities hold for anyα∈ 01andngreater than some integern_oα ≥2:

FJ −GJ ≤

K_oρ_nFJ +K_oρ_n for arbitraryF,G∈C_nX_n αand intervalsJ⊂R.

Part (b) is the key to various consistency results. One particular application are confidence bands for monotone densities. Similarly to Section 1, we define C¹n X_n αto be the set of all probability densities on the line whose distribu- tion function belongs toC_nX_n α. A possible notion of consistency is in terms of Hausdorff distance between graphs [cf. Marron and Tsybakov (1995)]. In

case of monotone functions this is essentially equivalent to considering a L´evy distance: for functionsf g∈⺖_↓I, define

df gI =inf

ε >0 fx+ε −gx ∨ gx+ε −fx ≤ε wheneverx x+ε∈I

Theorem 2.3(Monotone densities). Let f and g be arbitrary probability densities inC¹n X_n α ∩⺖_↓Ifor some intervalI⊂R. WithK_oandn_oαas in Proposition 2.2(b),

df gI∩ a∞ ≤ K_ofaρ_n^1/3+ K_oρ_n^1/2 for alla∈I provided that n≥n_oα.

In addition suppose thatf∈⺖_{β L}Ifor someβ∈ 01. Then there exists a constantK₁=K₁K_o β Lsuch that

gx −fx ≤ K₁fxρ_n^β/2β+1+ K₁ρ_n^β/β+1

forx∈Iwithx− fxρ_n^1/2β+1−ρ^1/β+1n ∈I fx −gx ≤ K₁fxρ_n^β/2β+1+ K₁ρ_n^β/β+1

forx∈Iwithx+

infy∈Ifyρ_n1/2β+1

∈I provided that n≥n_oα.

Analogous inequalities hold in case of f g being nondecreasing on some interval. Theorem 2.3 also applies to unimodal or piecewise monotone densi- ties. For instance, let⺠_unibe the class of unimodal distributions. That means, F ∈ ⺠_uni if it has a density which is nondecreasing on − ∞ mF and nonincreasing on mF∞ for some real number mF, a mode of F. Let F_n=F_o∈⺠_uni with unique modemF_oand densityf_o. Theorem 2.4 below shows that for any fixed neighborhood s t of mF_o, with high asymptotic probability the modemFof anyF∈C_nX_n α ∩⺠_uni is contained ins t.

In that case Theorem 2.3 applies to the two intervals − ∞ s and t∞, respectively, so thatC_nX_n α ∩⺠_unigives nontrivial confidence bands forf_o. Theorem 2.4(Inference about the mode). Suppose that F_n = F_o ∈ ⺠_uni with unique modemF_o. Then for anyα∈ 01,

sup

mF −mF_o F∈C_nX_n α ∩⺠_uni

→_p0 In particular, suppose that the densityf_oof F_o satisfies

x→mFlimo

f_omF_o −f_ox

mF_o −x² =γ >0 (2)

Then

sup

mF −mF_oF∈C_nX_n α ∩⺠_uni

=O_pρ^1/5_n

The rateOρ^1/5n for estimatingmF_ois close to the optimal rateOn^−1/5 [cf. Khas’minskii (1979) and Romano (1988)].

3. Confidence sets for regression functions. Given an index set⺤_n= t_1n t_2n t_nn of n points in R^d, let Y_n be a random vector in Rⁿ with components

Y_in=f_nt_in +E_in

for some unknown function f_n on R^d and a random error E_n ∈ Rⁿ having independent components E_in with median zero. That means PE_in ≥ 0 ∧ PE_in≤0 ≥1/2. For a functionfonR^d, define

signY_n f =

s∈ −11ⁿ signY_in−ft_in ∈ 0 s_i for 1≤i≤n This somewhat unusual definition is made in order to deal with possibly discrete error distributions. Let 3_n be uniformly distributed on −11ⁿ. If all components of Y_n have a continuous distribution, then signY_n f_n = signE_n0consists almost surely of one point whose distribution is ⺜3_n. In general, one can easily couple the random vectors3_nandE_nin such a way that

3_n∈signY_n f_n almost surely (3)

Now we define the test statistic T_nY_n f =min

τ_nss∈signY_n f τ_ns = #⺑_n⁻¹

A∈⺑n

exp _n

i=11t_in∈As_i ² 2#A

where ⺑_n is a family of nonvoid subsets of ⺤_n. A corresponding 1−α- confidence set forf_nis given by

C¹n Y_n α =

fT_nY_n f ≤c_nα

withc_nαdenoting the1−α-quantile of⺜τ_n3_n. Note thatTY_n f_n ≤ τ_n3_nalmost surely if (3) holds.

Example. Letn=m^dfor some integersm d >0. Then define

⺤_n^d= 1/m2/m 1^d

⺑_n^dδ_n =

x y ∩⺤_n^dx y∈⺤_n^dwith 0< y_i−x_i≤δ_n for alli where x y = _d

i=1x_i y_i and δ_n → δ. Here #⺑_n ≤ mm−1/2^d ≤ n². Table 1 gives some Monte Carlo estimates forc_nαin dimension one.

Here is the analogue to Proposition 2.2.

Proposition 3.1. Suppose that#⺑_n→ ∞.

(a) In general,

τ_n3_n =O_plog#⺑_n

Table 1

Estimated quantilescnαofτn3nfor⺤n¹⺑n¹δn(20000simulations)

␦nⴝ05 ␦nⴝ025

n cn05 cn01 cn005 cn05 cn01 cn005

50 1.85 3.85 5.56 1.97 3.51 4.47

100 1.87 4.13 6.17 2.02 3.96 5.43

150 1.92 4.29 6.53 2.05 4.12 5.77

200 1.93 4.44 6.67 2.08 4.22 5.82

250 1.95 4.50 6.73 2.08 4.34 6.37

300 1.96 4.53 7.03 2.10 4.35 6.43

400 1.96 4.64 7.15 2.12 4.55 6.60

500 1.98 4.65 7.19 2.13 4.55 6.66

Suppose that ⺑_n⊂ D∩⺤_nD∈⺔ for some Vapnik–Cervonenkis class⺔ of subsets ofR^d, and let

lim sup

n→∞ #⺑_n⁻¹

A∈⺑n

logn/#A<∞

Thenτ_n3_n =O_p1. In particular, for ⺤_n⺑_n = ⺤n¹⺑n¹δ_n, δ−δ²/2τ_n3_n →_⺜ TW

(b) Let H 01 → 0∞ be a fixed nondecreasing function such that all random vectorsE_nsatisfy

PE_in≤Hu ∧PE_in≥ −Hu ≥ 1+u

2 for1≤i≤n u∈ 01 Then for any fixedα∈ 01, the probability of

A∈⺑n

sup

f∈C¹n Ynα

mint∈Aft −f_nt ∨min

t∈Af_nt −ft> H

3

log#⺑_n

#A

tends to zero, whereHu = ∞foru >1.

As for part (b), if all components of E_n are Gaussian with variance not greater than τ, one may take Hu = τ8⁻¹1+u/2 with the standard normal quantile function8⁻¹. A key condition on His

lim sup

u↓0

Hu u <∞ (4)

Theorem 3.2(Isotonic regression). Let allf_nbelong to the class⺖_↑01^d of functionsfon01^d such thatfs ≤ft whenevers≤t componentwise.

Forf g∈⺖_↑01^d, define a L´evy distance df g =inf

ε >0lfs −gs+ε1 ∨ gs −fs+ε1 ≤ε whenevers s+ε1∈ 01^d

where1= 11 1^! ∈R^d. Let⺤_n⺑_n = ⺤n^d⺑n^dδ_n as above. Sup- pose that the assumption of Proposition 3.1(b) holds with a functionHsatis- fying (4). Then

sup

df f_nf∈C¹n Y_n α ∩⺖_↑01^d

≤O_p

ρ^1/2+dn

Theorem 3.3(Convex–concave regression). Let⺤_n⺑_n = ⺤_!n¹⺑n¹δ_n Suppose that allf_nbelong to⺖_cc∩⺖_{β L}a bfor some0≤a < b≤1,β∈ 02 and L > 0. Further suppose that the assumption of Proposition 3.1(b) holds with a functionHsatisfying (4). Then

supf−f_n

a+ρ^1/2β+1n b−ρ^1/2β+1n

f∈C¹n Y_n α ∩⺖_cc

=O_p

ρ^β/2β+1n Numerical examples. In all subsequent examples we consider the pair

⺤n¹⺑n¹025. We simulated dataY_in(shown as dots) having logistic dis- tribution with mean f_ni/n (shown as dotted line) and standard deviation v_in. Point estimators and confidence bands are shown as solid lines.

Figure 1 shows two data vectorsY_n withn=200 andv_in=04. We mini- mizedT_nY_n fover allf∈⺖_conv01. In both examples the minimum turned out to be unique, although this is not necessarily the case. This led to a sign vectors_min, and the solid lines represent the functions

t→

min

ftf∈⺖_conv01 s_min∈signY_n f max

ftf∈⺖_conv01 s_min∈signY_n f The regression functionf_nwas taken to be

f_nt = 1−5t/2 ∨ 5t/3−2/3² and

f_nt = 1−5t/2 ∨ 5t/3−2/3²+12/5< t <4/5sin5πt/2 respectively. The corresponding observed p-value Pτ_n3_n ≥ τ_ns_min was estimated in 40000 Monte Carlo simulations. In the first example it turned out greater than 0.99. In fact, the distance between estimator andf_nis small in comparison with the noise levelv_in. In the second example the Monte Carlo p-value was 0027, so that the nonconvexity off_n is detected at level 0.05.

Figures 2, 3 and 4 depict examples of confidence bands, that is, the envelope functions

t→



 min

ftf∈C¹n Y_n01 ∩⺖ max

ftf∈C¹n Y_n01 ∩⺖

Fig. 1. Point estimators forfn∈⺖conv01.

Precisely, in Figure 2 the parameters aren=250,v_in=05 and f_nt =1t≥1/2 ⺖ =⺖_↑01 In Figure 3 we haven=250,v_in=03 and

f_nt = 1−3t ∨ 3t/2−1/2² ⺖ =⺖_conv01

Fig. 2. Envelope ofC¹n Y_n01 ∩⺖_↑01.

Fig. 3. Envelope ofC¹n Y_n01 ∩⺖_conv01.

Fig. 4. Envelopes ofC¹n Yn01 ∩⺖↑01andC¹n Yn01 ∩⺖↑01 ∩⺖cc01.

Figure 4 depicts heteroscedastic dataY_in withn=200 and f_nt = 3x−1⁺∧1 v_in= 1+f_ni/n/4

The two plots show the envelopes ofC¹n Y_n01 ∩⺖_↑01andC¹n Y_n01∩

⺖_↑01 ∩⺖_cc01, respectively. Here the additional constraint “f_n∈⺖_cc01”

led to considerably smaller confidence bands.

4. Interval censoring. Let X˜_1n, X˜_2n X˜_nn be independent, iden- tically distributed random variables with distribution function F_n. Rather than X˜_in, one only observes Z_in = 1 ˜X_in ≤ r_in, 1 ≤ i ≤ n, where r_1n ≤ r_2n≤ · · · ≤r_nnare given censoring times (viewed as fixed). Given a hypothet- ical distribution functionF, let

Z_ntF =2n^−1/2

i≤nt

Z_in−Fr_in t∈ 01 Then our test statistic for “F_n=F” isT_nZ_n· F, where

T_nh =n⁻²

s t∈nδn

exp

hs t² 2t−s

with _nδ_n = δ_n ∩ 01/n2/n 1² and δ_n → δ. Unfortunately the 1−α-quantile d_nαF_n of the distribution of T_nZ_n· F_n depends on the unknown function F_n. However, the caseF_nr_1n =F_nr_nn =1/2 is the worst case asymptotically. The corresponding quantile is denoted by d_nα.

We define C_nZ_n α to be the set of all distribution functions F such that T_nZ_n· F ≤d_nα.

Proposition 4.1. (a) For any fixedα∈ 01,

n→∞lim d_nα =cα and lim sup

n→∞ PT_nZ_n· F_n ≥d_nα ≤α (b) For any fixedα∈ 01,

n→∞lim P

F∈Csup_nZ_n αFs−F_nt∨F_ns−Ft ≥

3δ⁻¹ρ_n

µ_ns t for somes < t

=0 whereµ_n· =n⁻¹_n

i=11r_in∈ ·.

Part (b) implies consistency of C_nZ_n α in various senses under certain conditions on the sequence of distributions µ_n. We mention only two simple consequences:

Theorem 4.2. (a) Suppose that F_n =F_o is continuous and thatµ_n con- verges weakly to a probability measureµsuch thatsupportµ ⊃supportF_o. Then

sup

F−F_oRF∈C_nZ_n α

→_p0

(b) Suppose that that F_n0 =0 andF_n∈ ⺖_{β L}0∞for some β∈ 01 andL >0. Further letr_1n,r_2n r_nnbe the order statistic of independent, identically distributed random variables R₁, R₂ R_n with distribution µ having continuous densityµ¹ on0∞. Then

sup

F−F_nIF∈C_nZ_n α

=O_p

ρ^β/2β+1n

for any compact subsetIof t≥0µ¹t>0.

5. Proofs. In order to verify the finiteness ofTW, let TW =ˇ sup

s t∈1

Ws t²

2t−sloge/t−s

According to L´evy’s theorem on W’s modulus of continuity, 1 ≤ ˇTW < ∞ almost surely [cf. Shorack and Wellner (1986), Theorem 14.1.1]. Now the key point is that

E1 ˇTW ≤MTW

≤

δE1

Ws t²

2t−s ≤Mlog e

t−s

exp

Ws t² 2t−s

ds dt

≤

δ

1+2Mlog e

t−s

ds dt

<∞

for any constantM >1; see Lemma 5.1. Continuity of⺜TWfollows from the fact that T is strictly convex on the set G ∈ ⺓01 G0 = 0. For Wt = Bt +ξt, 0 ≤ t ≤ 1, with a Brownian bridge B and a standard Gaussian variableξsuch thatBandξare independent. Thus conditional onB, the test statisticTWis a strictly convex function ofξ, whence continuously distributed. This consideration shows in addition that the support of⺜TW is connected.

Lemma 5.1. LetXbe a nonnegative random variable such thatPX≥r ≤ 2 exp−rfor allr≥0(e.g.,X=Z²/2withZ∼⺞01). Then for allγ,l >0,

E1X≤lexpγX ≤

1+2l ifγ=1 1+2expγ−1l −1/1−1/γ ifγ=1 Proof. The expectation of 1X≤lexpγXequals

∞

0 PX≤land expγX> rdr

≤1+_expγl

1 PexpγX> rdr

≤1+2_expγl

1 r^−1/γdr

=

1+2l ifγ=1

1+2expγ−1l −1/1−1/γ ifγ=1 ✷ The proofs of Theorems 1.1 and 3.3 are based on a lemma on H¨older classes of functions.

Lemma 5.2. For β L > 0 there is a universal constantK_{β L} >0 such that for arbitrary compact intervalsI⊂Rand anyf∈⺖_{β L}Ithe following hold.

(a) There is an intervalJ_f⊂Isuch that

f ≥K_{β L}f_I onJ_f LebJ_f ≥K_{β L}

f^1/β_I ∧LebI

(b) Ifβ≤2, then for arbitrary g∈⺖_ccIthere is an interval J_fg ⊂I such that

g−f ≥ gx_o −fx_o/4 onJ_fg LebJ_fg ≥K_{β L}

gx_o −fx_o^1/β∧LebI wherex_odenotes the midpoint ofI.

Proof of Lemma 5.2(a). Let x₁ ∈ I with fx₁ = f_I, and define γ= f^1/β_I ∧LebI.

If 0< β ≤ 1, thenfx ≥ f_I−Lx−x₁^β ≥ f_I/2 for any pointx in J_f= x₁− 2L^−1/βγ x₁+ 2L^−1/βγ ∩I, where LebJ_f ≥ 2L^−1/β∧2⁻¹γ.

For β > 1 we use induction on k = kβ. Suppose the assertion is true for β−1 L in place of β L. If fx ≥ f_I/2 for all x ∈ J^!_f = x₁− γ/2 x₁+γ/2∩I, the assertion would be true withJ_f=J^!_fandK_{β L}=1/2.

Otherwise letx₂∈J^!_fwithfx₂ ≤ f_I/2. Then

f¹_I≥ fx₁ −fx₂/x₁−x₂ ≥ f_I/γ

By assumption, sincef¹∈⺖_{β−1 L}I, there is an intervalJ^!!_f⊂Isuch that f¹ ≥K_{β−1 L}f_I/γonJ^!!_f

LebJ^!!_f ≥K_{β−1 L}f_I/γ^1/β−1∧LebI =K_{β−1 L}γ Hence, ifa₀=infJ^!!_fanda_i=a₀+ i/4LebJ^!!_f, then

fa_i −fa_i−1 ≥4⁻¹K²_{β−1 L}f_I for 1≤i≤4

In addition,fis strictly monotone onJ^!!_fby continuity off¹. Hence one easily verifies thatf ≥4⁻¹K²_{β−1 L}f_Iona₀ a₁ ora₃ a₄. ✷

Proof of Lemma 5.2(b). At first we consider the special case, where I=

−44, f ≡ 0, g0 = 1 and g is convex–concave on −44. Under these assumptions there is an intervalJ⊂ −44 such that

g ≥1/2 onJ and LebJ ≥1

Obviously, this is true if g >1/2 on −10or on 01. Otherwise, let −1≤ x₁<0< x₂≤1 withgx₁ ∨gx₂ ≤1/2. Convex-concavity ofg implies that

g is concave onx_∗4 and convex on −4 x_∗for somex_∗ ∈ −41, because otherwisegwould be convex on−11. Ifx_∗ ≤0, then theL₁-derivativeg¹ of gsatisfies

g¹x ≤g¹x₂ ≤ gx₂ −1/x₂≤ −1/2 forx₂≤x <4

If x_∗ >0, then convexity of g on−4 x_∗and gx₁<1 together imply that gx_∗>1, whence

g¹x ≤g¹x₂ ≤ gx₂ −gx_∗/x₂−x_∗<−1/2 forx₂≤x <4 Thus

gx ≤gx₂ +x x2

g¹rdr≤1/2− x−x₂/2≤ −1/2 for 3≤x <4 With the help of affine transformations, one can deduce that in the general case for any 0< γ≤LebI/2 there is an intervalJ_fgγ ⊂ x_o−γ x_o+γ ⊂I such that

G ≥ gx_o −fx_o/2 onJ_fgγ and LebJ_fgγ ≥γ/4 where

Gx =

gx −fx_o if 0< β≤1 gx −fx_o −f¹x_ox−x_o if 1< β≤2 is also in⺖_ccI. But forx∈ x_o−γ x_o+γ,

Gx − g−fx =

fx −fx_o if 0< β≤1 _x

x_of¹r −f¹x_odr if 1< β≤2

≤Lγ^β≤ gx_o −fx_o/4 provided thatγ≤ 4L^−1/βgx_o −fx_o^1/β ✷

Proof of Theorem 1.1. Let F G be continuous functions on 01 with L₁-derivatives f g ∈ C¹n V_n α, respectively, such that f−g ∈ ⺖_{β L}I_n. Then by (1),

Tn^1/2F−G/2 ≤2⁻¹Tn^1/2V_n−F +Tn^1/2V_n−G ≤cα Given any fixed number R ≥ 1, suppose that fx −gx ≥ Rρ^β/2β+1n for somex∈I_n. According to Lemma 5.2 there exists an intervalJ=J_fgn⊂I_n such that

f−g≥K_∗Rρ^β/2β+1n onJ LebJ ≥K_∗

f−g^1/β_In∧LebI_n ≥K_∗ρ^1/2β+1n

where K_∗ denotes a generic positive constant depending only on β L but possibly different in various places. If J¹ andJ³ denote the left and right third ofJ, respectively, then fors∈J¹ andt∈J³,

t−s≥K_∗ρ^1/2β+1n F−Gs t²

2t−s ≥K_∗R²ρ^2β/2β+1n t−s ≥K_∗R²ρ_n Thus

Tn^1/2F−G/2 ≥

1s∈J¹1t∈J³exp

K_∗R²nρ_n ds dt

≥K_∗n^{K∗R2−2/2β+1}

For n and R sufficiently large, the latter bound exceeds cα. In that case f−g_In is necessarily smaller thanRρ^β/2β+1n . ✷

Proof of Proposition 2.1. LetGandG^!be independent Gamma-distrib- uted random variables with meanmp andm1−p, respectively. ThenB= G/G+G^!has the desired Beta-distribution, and forp < x <1,

PB≥x =P1−xG−xG^!≥0

≤inf

r>0Eexpr1−xG−rxG^!

= inf

0<r<1/1−xexp−mplog1−r1−x −m1−plog1+rx

=exp−m .x p

wherer_min= x−p/x1−x. Withκ=p1−pandγ=1−2pone can write

.x p =_x−p

0

r

κ+γr−r²dr

≥_x−p

0

r κ+γrdr

≥κ⁻¹x−p²/2 ifp≥1/2

=κγ⁻²Hκ⁻¹γx−p ifp <1/2 where Hy =y−log1+yis strictly increasing iny≥0. It follows easily from the series expansion of exp·thatH⁻¹y ≤ 2y^1/2+y. Thus.x p ≤c implies that

x−p≤

2κc^1/2 ifp≥1/2 κγ⁻¹H⁻¹κ⁻¹γ²c ≤ 2κc^1/2+γc ifp <1/2

For 0< x < pthe assertions follow from the fact that 1−B∼Betam1− p mpand.x p =.1−x1−p. ✷

Here is a modified version of Lemma VII.9 of Pollard (1984), which is con- venient for our purposes. The proof is essentially the same.

Lemma 5.3 (Chaining). Let S = St_t∈⺤ be a stochastic process on a to- tally bounded metric space⺤ ρhaving continuous sample paths. LetQ be a measurable, nonnegative function on 0∞² such that for allη δ >0 and s t∈⺤,

PSs −St ≥ρs tQη δ ≤2 exp−η ifρs t ≥δ Then

PSs −St>12Jρs t afor somes t∈⺤ withρs t ≤δ ≤2δ/a for arbitrarya δ >0, where

Jε a =_ε

0 QlogaDu²/u udu Du =sup

#⺤_o⺤_o⊂⺤ ρs t> ufor differents t∈⺤_o

✷

Proof of Proposition 2.2(a). At first it is shown that T˜_n= max

j k∈ ¯Dn

n.F_nX_jn X_kn δ_jkn log

δ⁻¹_jkn1−δ_jkn⁻¹ =O_p1 It follows from Proposition 2.1 that forη_n>0,

P

j k∈ ¯maxD_nn.F_nX_jn X_kn δ_jkn ≥η_n

≤2 n+2

2

exp−η_n If η_n = 3 logn, the latter bound tends to zero. Thus for arbitrary fixed 0< γ <1/2,

j k∈ ¯D_nδmax_jkn1−δ_jkn≤n^−γ

n.F_nX_jn X_kn δ_jkn log

δ⁻¹_jkn1−δ_jkn⁻¹ =O_p1 On the other hand,

j k∈ ¯Dnδmaxjkn1−δjkn≥n^−γ

n.F˜ _nX_jn X_kn δ_jkn log

δ⁻¹_jkn1−δ_jkn⁻¹ =O_p1 (5)

where .x p = 2p1˜ −p⁻¹x−p². This follows, for instance, from the Chaining Lemma 5.3 applied to the uniform quantile processSj/n+1 = n+1^1/2F_nX_jnon⺤ = j/n+10≤j≤n+1equipped withρs t = VarBs t^1/2. For elementary calculations, show that Du ≤ 2/u² for 0 <

u≤1. Further, one can easily deduce from Proposition 2.1 that Qη δ = 2η^1/2+max

n+1^1/2δ1₋₁ η

satisfies the assertion of Lemma 5.3. Then elementary calculations show that Jε a ≤Ka

εlog1/ε^1/2+n^−1/2logn²

for allε∈ 01/2and some constantKanot depending onn. Alternatively one may deduce (5) from the Hungarian approximation [cf. Shorack and Well- ner (1986), Chapter 12.2]. But (5) implies that

j k∈ ¯Dnδmaxjkn1−δjkn≥n^−γlogitF_nX_jn X_kn −logitδ_jkn →_p0

where logitx =logx/1−x. Elementary calculations show that.x p/

.x p →˜ 1 as logitx −logitp →0. Thus one may replace.˜ in (5) with. and obtains thatT˜_n=O_p1.

Analogously one can show that TB =˜ sup

s t∈1

Bs t² 2ρs t²logρs t⁻² is finite almost surely.

Now it follows from Lemma 5.1 that for arbitraryε,M >0, E1 ˜T_n≤Mn⁻²

j k∈D_nδ_jkn1−δ_jkn≤ε

expn.F_nX_jn X_kn δ_jkn

≤n⁻²

j k∈Dnδjkn1−δjkn≤ε

1+2Mlogδ⁻¹_jkn1−δ_jkn⁻¹

→

s t∈δρs t²≤ε1+2Mlogρs t⁻²ds dt E1 ˜TB ≤M

s t∈δρs t²≤εexp

Bs t² 2ρs t²

ds dt

≤

s t∈δρs t²≤ε1+2Mlogρs t⁻²ds dt

This bound tends to zero asε↓0. Moreover,S· =S_n·converges in distri- bution toBif it is suitably extended toS_n∈⺓01, whence

n⁻²

j k∈Dnδjkn1−δjkn≥ε

expn.F_nX_jn X_kn δ_jkn

→_⺜

s t∈δρs t²≥εexp

Bs t² 2ρs t²

ds dt

[Here we applied Rubin’s extended continuous mapping theorem; see Billings- ley (1968), Theorem 5.5.] These two facts together imply the asserted conver- gence in distribution ofT_nX_n F_n. ✷

Proof of Proposition 2.2(b). LetK_∗ be a generic real constant depend- ing only onD_nnand possibly different in various (in)equalities. LetFbe an arbitrary element ofC_nX_n α. It follows straightforwardly from part (a) that

.FX_jn X_kn δ_jkn ≤3ρ_n for allj k ∈D_n (6)

provided thatn≥n_oα ≥2. This implies that

FX_jn X_kn −δ_jkn ≤ K_∗δ_jknρ_n^1/2+K_∗ρ_n for allj k ∈ ¯D_n (7)

It follows from Proposition 2.1 and (6) that (7) holds for D_n in place of D¯_n with K_∗ =6. Then elementary considerations show that D_n can be replaced withD¯_nifK_∗ is adjusted properly.

Now let G be another element of C_nX_n α and J ⊂R an interval with FJ< GJ. Define

j=jJ =maxlX_ln≤infJ k=kJ =minlX_ln≥supJ Then (7) implies that

GJ ≤GX_jn X_kn ≤δ_jkn+ K_∗δ_jknρ_n^1/2+K_∗ρ_n (8)

FJ ≥FX_{j+1 n} X_{k−1 n} ≥δ_jkn−2/n− K_∗δ_jknρ_n^1/2−K_∗ρ_n (9)

and one easily deduces from (9) that

δ_jkn≤2FJ +K_∗ρ_n (10)

Now subtracting (9) from (8) and plugging in (10) yields

GJ −FJ ≤ K_∗FJρ_n^1/2+K_∗ρ_n ✷ Proof of Theorem 2.3. LetFandGbe the distribution function offand g, respectively. For arbitrarya x y∈I with a≤x < y, the monotonicity of fandg, together with Proposition 2.2(b), implies that

gy −fx ∨ fy −gx ≤ Gx y −Fx y/y−x

≤

K_oFx yρ_n/y−x^{2 1/2}+K_oρ_n/y−x

≤ K_ofaρ_n/y−x^1/2+K_oρ_n/y−x where we assume throughout thatn≥n_oα. Ify−xis greater than

κ_na = K_ofaρ_n^1/3+ K_oρ_n^1/2

thenK_ofaρ_n/y−x^1/2+K_oρ_n/y−x ≤κ_na. Hencedf gI∩a∞ ≤ κ_na.

Now suppose in addition thatf∈⺖_{β L}I. Then gy −fy ∨ fx −gx

≤Ly−x^β+ K_ofxρ_n/y−x^1/2+K_oρ_n/y−x (11)

Letx=y− fyρ_n^1/2β+1−ρ^1/β+1n , assuming that this point is also inI.

Iffyρ_n^1/2β+1≥ρ^1/β+1n , which is equivalent toρ_n≤fy^β+1/β, then fx ≤fy +Ly−x^β ≤fy +L2^βfyρ_n^β/2β+1≤ 1+L2^βfy

and (11) yields

gy −fy ≤

L2^β+

K_o1+L2^{β 1/2}+K_o fyρ_n^β/2β+1

On the other hand, fyρ_n^1/2β+1≤ρ^1/β+1n is equivalent tofy ≤ρ^β/β+1n

and implies that

fx ≤fy +L2^βρ^β/β+1n ≤ 1+L2^βρ^β/β+1n Thus (11) leads to

gy −fy ≤

L2^β+ K_o1+L2^β1/2+K_o ρ^β/β+1n

As for the lower bound, let γ = inf_y∈Ify, and suppose that x + γρ_n^1/2β+1 ∈ I. Since fx −gx ≤ fx, we may assume that fx ≥ Kρ^β/β+1n and define y=x+ fxρ_n/K^1/2β+1 for some constant K≥1 to be specified later. This definition implies that

ρ^1/β+1n ≤y−x≤ fx/K^1/β Ify∈I, one can easily deduce from (11) that

fx −gx ≤

LK^−β/2β+1+K^1/2_o K^1/4β+2 fxρ_n^β/2β+1+K_oρ^β/β+1n It remains to be shown that y∈ I for suitable K=Kβ L. If fx ≤Kγ, theny−x≤ γρ_n^1/2β+1. Otherwise,

Lz−x^β≥ 1−fz/fxfx ≥ 1−1/Kfx

for somez∈I,z > x. Thus ifK=L+1, thenz−x≥ fx/K^1/β≥y−x. ✷ Proof of Theorem 2.4. According to Proposition 2.2(b), it suffices to show that mG_n → mF_o and, in case of (2), mG_n =mF_o +Oρ^1/5n , where G_nn is an arbitrary sequence of distribution functions in⺠_uni with

G_nJ −F_oJ ≤ K_oρ_nF_oJ^1/2+K_oρ_n

for all intervals J ⊂ R and anyn > 1. For fixed ε > 0 there are bounded, nondegenerate intervalsJ₁≤J₂≤J₃(in a pointwise sense) such that

maxl=13F_oJ_l/LebJ_l< F_oJ₂/LebJ₂ J₁∪J₂∪J₃⊂ mF_o −ε mF_o +ε

It follows from Proposition 2.2(b), that there is an integer n₁ such that maxl=13G_nJ_l/LebJ_l< G_nJ₂/LebJ₂ ifn≥n₁ But these two inequalities forG_n imply thatmG_n ⊂J₁∪J₂∪J₃.

Suppose that f_o satisfies the regularity condition (2), where mF_o = 0 without loss of generality. We define

J_n1= −2κ_n−κ_n J_n2= −κ_n κ_n J_n3= κ_n2κ_n