TESTING FOR A GENERAL CLASS OF FUNCTIONAL INEQUALITIES

Transcription

1 TESTING FOR A GENERAL CLASS OF FUNCTIONAL INEQUALITIES SOKBAE LEE 1,2, KYUNGCHUL SONG 3, AND YOON-JAE WHANG 4 Abstract. In tis paper, we propose a general metod for testing inequality restrictions on nonparametric functions. Our framework includes many nonparametric testing problems in a unified framework, wit a number of possible applications in auction models, game teoretic models, wage inequality, and revealed preferences. Our test involves a one-sided version of L p functionals of kernel-type estimators 1 p < ) and is easy to implement in general, mainly due to its recourse to te bootstrap metod. Te bootstrap procedure is based on te nonparametric bootstrap applied to kernel-based test statistics, wit an option of estimating contact sets. We provide regularity conditions under wic te bootstrap test is asymptotically valid uniformly over a large class of distributions, including cases were te limiting distribution of te test statistic is degenerate. Our bootstrap test is sown to exibit good power properties in Monte Carlo experiments, and we provide a general form of te local power function. As an illustration, we consider testing implications from auction teory, provide primitive conditions for our test, and demonstrate its usefulness by applying our test to real data. We plement tis example wit te second empirical illustration in te context of wage inequality. Key words. Bootstrap, conditional moment inequalities, kernel estimation, local polynomial estimation, L p norm, nonparametric testing, partial identification, Poissonization, quantile regression, uniform asymptotics JEL Subject Classification. C12, C14. 1 Department of Economics, Columbia University, 1022 International Affairs Building 420 West 118t Street, New York, NY 10027, USA. 2 Centre for Microdata Metods and Practice, Institute for Fiscal Studies, 7 Ridgmount Street, London, WC1E 7AE, UK. 3 Vancouver Scool of Economics, University of Britis Columbia, East Mall, Vancouver, BC, V6T 1Z1, Canada 4 Department of Economics, Seoul National University, 1 Gwanak-ro, Gwanak-gu, Seoul, , Republic of Korea. addresses: sl3841@columbia.edu, kysong@mail.ubc.ca, wang@snu.ac.kr. Date: 5 February We would like to tank Editor, Peter C.B. Pillips, Co-Editor, Liangjun Su, tree anonymous referees, Emmanuel Guerre and participants at numerous seminars and conferences for teir elpful comments. We also tank Kyeongbae Kim, Kooyun Kwon and Jaewon Lee for capable researc assistance. Lee s work was ported by te European Researc Council ERC-2014-CoG ROMIA). Song acknowledges te financial port of Social Sciences and Humanities Researc Council of Canada. Wang s work was ported by te National Researc Foundation of Korea Grant funded by te Korean Government NRF B00004) and te SNU Creative Leading Researcer Grant. 1

2 2 LEE, SONG, AND WHANG 1. Introduction In tis paper, we propose a general metod for testing inequality restrictions on nonparametric functions. To describe our testing problem, let v τ,1,..., v τ,j denote nonparametric real-valued functions on R d for eac index τ T, were T is a subset of a finite dimensional space. We focus on testing H 0 : max{v τ,1 x),, v τ,j x)} 0 for all x, τ) X T, against 1.1) H 1 : max{v τ,1 x),, v τ,j x)} > 0 for some x, τ) X T, were X T is a domain of interest. We propose a one-sided L p integrated test statistic based on nonparametric estimators of v τ,1,..., v τ,j. We provide general asymptotic teory for te test statistic and suggest a bootstrap procedure to compute critical values. We establis tat our test as correct uniform asymptotic size and is not conservative. We also determine te asymptotic power of our test under fixed alternatives and some local alternatives. We allow for a general class of nonparametric functions, including, as special cases, conditional mean, quantile, azard, and distribution functions and teir derivatives. For example, v τ,j x) = P Y j τ X = x) can be te conditional distribution function of Y j given X = x, or v τ,j x) can be te τ-t quantile of Y j conditional on X = x. We also allow for transformations of tese functions satisfying some regularity conditions. Te nonparametric estimators we consider are mainly kernel-type estimators but can be allowed to be more general, provided tat tey satisfy certain Baadur-type linear expansions. Inequality restrictions on nonparametric functions arise often as testable implications from economic teory. For example, in first-price auctions, Guerre, Perrigne, and Vuong 2009) sow tat te quantiles of te observed equilibrium bid distributions wit different numbers of bidders sould satisfy a set of inequality restrictions Equation 5) of Guerre, Perrigne, and Vuong 2009)). If te auctions are eterogeneous so tat te private values are affected by observed caracteristics, we may consider conditionally exogenous participation wit a conditional version of te restrictions see Section 3.2 of Guerre, Perrigne, and Vuong 2009)). Suc restrictions are in te form of multiple inequalities for linear combinations of nonparametric conditional quantile functions. Our test ten can be used to test weter te restrictions old jointly uniformly over quantiles and observed caracteristics in a certain range. In tis paper, we use tis auction example to illustrate te usefulness of our general framework. To te best of our knowledge, tere does not exist an alternative test available in te literature for tis kind of example. In addition to Guerre, Perrigne, and Vuong 2009, GPV ereafter), a large number of auction models are associated wit some forms of functional inequalities. See, for example, Haile and Tamer 2003), Haile, Hong, and Sum 2003), Aradillas-López, Gandi, and Quint

3 GENERAL FUNCTIONAL INEQUALITIES ), and Krasnokutskaya, Song, and Tang 2016) among oters. Our metod can be used to make inference in teir setups, wile allowing for continuous covariates. See Online Appendix E for furter discussions on potential applications in economics, suc as inference in models of games, tests of revealed preferences, and inference in partial identification. Our framework as several distinctive merits. First, our proposal is easy to implement in general, mainly due to its recourse to te bootstrap metod. Te bootstrap procedure is based on nonparametric bootstrap applied to kernel-based test statistics. We establis te general asymptotic uniform) validity of te bootstrap procedure. Second, our proposed test is sown to exibit good power properties bot in finite and large samples. Good power properties can be acieved by te use of critical values tat adapt to te binding restrictions of functional inequalities. Tis could be done in various ways; in tis paper, we follow te contact set approac of Linton, Song, and Wang 2010) and propose bootstrap critical values. As is sown in tis paper, te bootstrap critical values yield significant power improvements. Furtermore, we find troug our local power analysis tat tis class of tests exibits dual convergence rates depending on Pitman directions, and in many cases, te faster of te two rates acieves a parametric rate of n, despite te use of kernel-type test statistics. Tird, we establis te asymptotic validity of te proposed test uniformly over a large class of distributions, witout imposing restrictions on te covariance structure among nonparametric estimates of v τ,j ), tereby allowing for degenerate cases. Suc a uniformity result is crucial for ensuring good finite sample properties for tests wose pointwise) limiting distribution under te null ypotesis exibits various forms of discontinuity. Te discontinuity in te context of tis paper is igly complex, as te null ypotesis involves inequality restrictions on a multiple number of or even a continuum of) nonparametric functions. We establis te uniform validity of te test in a way tat covers tese various incidences of discontinuity. Our new uniform asymptotics may be of independent interest in many oter contexts. Muc of te recent literature on testing inequality restrictions focuses on conditional moment inequalities. Researc on conditional moment inequalities includes Andrews and Si 2013), Andrews and Si 2014), Andrews and Si 2017), Aradillas-López, Gandi, and Quint 2016), Armstrong 2015), Armstrong and Can 2016), Cernozukov, Lee, and Rosen 2013), Cetverikov 2011), Fan and Park 2014), Kan and Tamer 2009), Lee, Song, and Wang 2013), and Menzel 2014) among oters. In contrast, tis paper s approac naturally covers a wide class of inequality restrictions among nonparametric functions tat te moment inequality framework does not or at least is cumbersome to) apply. Suc examples include testing multiple inequalities tat are defined by differences in conditional quantile

4 4 LEE, SONG, AND WHANG functions uniformly over covariates and quantiles. As tis paper demonstrates troug an empirical application, suc a testing problem can arise in empirical researc see Section 6). One important class tat is covered by our general approac is to test conditional stocastic dominance, wic is a generalization of stocastic dominance to conditional distributions. See, for example, Cang, Lee, and Wang 2015), Delgado and Escanciano 2013) and Hsu 2016) among oters. Our test can also be used as an alternative to testing monotonicity of mean regression e.g. Cetverikov 2012) and Gosal, Sen, and van der Vaart 2000)) as well as testing stocastic monotonicity e.g. Lee, Linton, and Wang 2009) and Delgado and Escanciano 2012)). Our paper is also related to testing affiliation Jun, Pinkse, and Wan 2010) and testing density ratio ordering Beare and Moon 2015). More generally speaking, our framework is related to testing qualitative nonparametric ypoteses suc as positivity, monotonicity and convexity in nonparametric models. See, for example, Dümbgen and Spokoiny 2001), Juditsky and Nemirovski 2002) and Baraud, Huet, and Laurent 2005) among oters. However, tere are some important differences. First, te existing statistics literature mainly focuses on te ypotesis on te mean regression function in te ideal Gaussian wite noise model, wile we do not impose any parametric assumption suc as Gaussianity and allow for more general functional inequalities among nonparametric functions oter tan te mean regression function. Second, te tecnical details of te existing positivity tests are quite different from ours. Our current work is substantially different from our previous paper, Lee, Song, and Wang 2013). Te latter considered only conditional moment inequalities under pointwise asymptotics. Te test in Lee, Song, and Wang 2013) is based on asymptotic normality under te least favorable case. Te current paper goes muc beyond conditional moment inequalities and provides uniform validity of te bootstrap test wit contact set estimation. Te remainder of te paper is as follows. Section 2 gives an informal description of our general framework by introducing test statistics and critical values and by providing intuition beind our approac. In Section 3, we establis te uniform asymptotic validity of our bootstrap test. We also provide a class of distributions for wic te asymptotic size is exact. In Section 4, we establis consistency of our test and its local power properties. In Section 5, we report results of some Monte Carlo experiments. In Sections 6 and 7, we give two empirical examples. Te first empirical example in Section 6 is on testing auction models following GPV, and te second one in Section 7 is about testing functional inequalities via differences-in-differences in conditional quantiles, inspired by Acemoglu and Autor 2011). Te empirical examples given in tis section are not covered easily by existing inference metods; owever, tey are all special cases of our general framework. Section 8 concludes. Online Appendices provide all te proofs of teorems wit a roadmap of te proofs to elp readers.

5 GENERAL FUNCTIONAL INEQUALITIES 5 2. General Overview 2.1. Test Statistics. We present a general overview of tis paper s framework by introducing test statistics and critical values. To ease te exposition, we confine our attention to te case of J = 2 ere. Te definitions and formal results for general J are given later in Section 3. Trougout tis paper, we assume tat T is a compact subset of a Euclidean space. Tis does not lose muc generality because wen T is a finite set, we can redefine our test statistic by taking T as part of te finite index j indexing te nonparametric functions. For j = 1, 2, let ˆv τ,j x) be a kernel-based nonparametric estimator of v τ,j x) and let its appropriately scaled version be û τ,j x) r n,jˆv τ,j x), ˆσ τ,j x) were r n,j is an appropriate normalizing sequence tat diverges to infinity, 1 and ˆσ τ,j x) is an appropriate possibly data-dependent) scale normalization. 2 Ten inference is based on te following statistic: 2.1) ˆθ T X T X max {û τ,1 x), û τ,2 x), 0} p dxdτ max {û τ,1 x), û τ,2 x), 0} p dqx, τ), were Q is Lebesgue measure on X T. In tis overview section, we focus on te case of using te max function under te integral in 2.1). Our general teory permits an alternative form 2 j=1 max {û τ,jx), 0} p in place of te max function see 3.1)) Bootstrap Critical Values. As we sall see later, te asymptotic distribution of te test statistic exibits complex ways of discontinuities as one perturbs te data generating processes. Tis suggests tat te finite sample properties of te asymptotic critical values may not be stable. Furtermore, te location-scale normalization requires nonparametric estimation and tus a furter coice of tuning parameters. Tis can worsen te finite sample properties of te critical values furter. To address tese issues, tis paper develops a bootstrap procedure. In te following, we let ˆv τ,jx) and ˆσ τ,jx), j = 1, 2, denote te bootstrap counterparts of ˆv τ,j x) and ˆσ τ,j x), j = 1, 2. Let te bootstrap counterparts be constructed in te same way as te nonparametric estimators ˆv τ,j x) and ˆσ τ,j x), j = 1, 2, wit te bootstrap sample independently drawn wit replacement from te empirical distribution of te original sample.

6 6 LEE, SONG, AND WHANG We let 2.2) ŝ τ,jx) r n,j{ˆv τ,jx) ˆv τ,j x)} ˆσ τ,j x), j = 1, 2. Note tat ŝ τ,jx) is a centered and scale normalized version of te bootstrap quantity ˆv τ,jx). Using tese bootstrap quantities, we consider two versions of bootstrap critical values: one based on te least favorable case and te oter based on estimating a contact set Te Least Favorable Case. Under te least favorable configuration LFC), we construct a bootstrap version of te rigt and side of 2.1) as ˆθ LFC max { ŝ τ,1x), ŝ τ,2x), 0 } p dqx, τ). Under regularity conditions, bootstrap critical values based on te LFC can be sown to yield tests tat are asymptotically valid uniformly in P. However, tey are often too conservative in practice. As an alternative to te LFC-based bootstrap critical value, we propose a bootstrap critical value tat can be less conservative but at te expense of introducing an additional tuning parameter Estimating a Contact Set. As we sall sow formally in a more general form in Lemma 1 in Section 3 below, it is satisfied tat under H 0, for eac sequence c n suc tat log n/cn 0 as n, 2.3) ˆθ = B n,{1} c n) + + B n,{2} c n) B n,{1,2} c n) max {û τ,1 x), 0} p dqx, τ) max {û τ,2 x), 0} p dqx, τ) max {û τ,1 x), û τ,2 x), 0} p dqx, τ), wit probability approacing one, were, letting u n,τ,j x) r n,j v n,τ,j x)/σ n,τ,j x), i.e., a population version of û τ,j x), 3 we define B n,{1} c n ) {x, τ) X T : u n,τ,1 x) c n and u n,τ,2 x) < c n }, B n,{2} c n ) {x, τ) X T : u n,τ,2 x) c n and u n,τ,1 x) < c n } and B n,{1,2} c n ) {x, τ) X T : u n,τ,1 x) c n and u n,τ,2 x) c n }. For example, te set B n,{1} c n ) is a set of points x, τ) suc tat v n,τ,1 x)/σ n,τ,1 x) is close to zero, and v n,τ,2 x)/σ n,τ,2 x) is negative and away from zero. We call contact sets suc sets as B n,{1} c n ), B n,{2} c n ), and B n,{1,2} c n ).

7 GENERAL FUNCTIONAL INEQUALITIES 7 Now, comparing 2.3) wit 2.1) reveals tat te limiting distribution of ˆθ under te null ypotesis will not depend on points outside te union of te contact sets. Tus it is natural to base te bootstrap critical values on te quantity on te rigt and side of 2.3) instead of tat on te last integral in 2.1). As we will explain sortly in te next subsection, tis leads to a test tat is uniformly valid and exibits substantial improvement in power. Figure 1. Contact Set Estimation c n B 1 c n B 1,2 c n B 2 c n c n u 1 x) u 2 x) Note: Tis figure illustrates estimated contact sets wen J = 2. Te tick line segments on te x-axis represent estimated contact sets. To construct bootstrap critical values, we introduce sample versions of te contact sets: ˆB {1} c n ) {x, τ) X T : û τ,1 x) c n and û τ,2 x) < c n }, ˆB {2} c n ) {x, τ) X T : û τ,2 x) c n and û τ,1 x) < c n } and ˆB {1,2} c n ) {x, τ) X T : û τ,1 x) c n and û τ,2 x) c n }. See Figure 1 for illustration of estimation of contact sets wen J = 2. Given te contact sets, we construct a bootstrap version of te rigt and side of 2.3) as ˆθ max { ŝ τ,1x), 0 } p 2.4) dqx, τ) ˆB {1} ĉ n) + max { ŝ τ,2x), 0 } p dqx, τ) ˆB {2} ĉ n) + max { ŝ τ,1x), ŝ τ,2x), 0 } p dqx, τ), ˆB {1,2} ĉ n)

8 8 LEE, SONG, AND WHANG were ĉ n is a data dependent version of c n. We will discuss a way to construct ĉ n sortly. We also define â E ˆθ, were E denotes te expectation under te bootstrap distribution. Let c α be te 1 α)- t quantile from te bootstrap distribution of ˆθ. In practice, bot quantities â and c α are approximated by te sample mean and sample 1 α)-t quantile, respectively, from a large number of bootstrap repetitions. Ten for a small constant η 10 3, we take c α,η max{c α, d/2 η + â } as te critical value to form te following test: 2.5) Reject H 0 if and only if ˆθ > c α,η. Ten it is sown later tat te test as asymptotically correct size, i.e., 2.6) lim n P {ˆθ > c α,η} α, 0 were P 0 is te collection of potential distributions tat satisfy te null ypotesis. implement te test, tere are two important tuning parameters, namely te bandwidt used for nonparametric estimation and te constant ĉ n for contact set estimation. 4 We discuss ow to obtain te latter in te context of our Monte Carlo experiments in Section 5.1. To 2.3. Discontinuity, Uniformity, and Power. Many tests of inequality restrictions exibit discontinuity in its limiting distribution under te null ypotesis. Wen te inequality restrictions involve nonparametric functions, tis discontinuity takes a complex form, as empasized in Section 5 of Andrews and Si 2013). To see te discontinuity problem in our context, let {Y i, X i ) } n i=1 be i.i.d. copies from an observable bivariate random vector, Y, X) R R, were X i is a continuous random variable wit density f. We consider a simple testing example: 2.7) H 0 : E[Y X = x] 0 for all x X vs. H 1 : E[Y X = x] > 0 for some x X. Here, wit te subscript τ pressed, we set J = 1, r n,1 = n, p = d = 1, and define [v] + max{v, 0}. Let 2.8) ˆv 1 x) = 1 n n ) Xi x Y i K i=1 and ˆσ 2 1x) = 1 n n i=1 ) Yi 2 K 2 Xi x, were K is a nonnegative, univariate kernel function wit compact port and is a bandwidt.

9 GENERAL FUNCTIONAL INEQUALITIES 9 Assume tat te density of X is strictly positive on X. Ten, in tis example, v n,1 x) Eˆv 1 x) 0 for almost every x in X wenever te null ypotesis is true. Define Z n,1 x) = { } ˆv1 x) v n,1 x) { n and B n,1 0) = x X : } nv n,1 x) = 0. ˆσ 1 x) We analyze te asymptotic properties of ˆθ as follows. We first write { } 2.9) 1/2 ˆθ a n,1 ) = 1/2 [Z n,1 x)] + dx a n,1 B n,1 0) [ ] + 1/2 nvn,1 x) Z n,1 x) + ˆσ 1 x) were X \B n,1 0) [ ] a n,1 = E [Z n,1 x)] + dx. B n,1 0) Wen liminf n Q B n,1 0)) > 0 wit QB n,1 0)) denoting Lebesgue measure of B n,1 0), we can sow tat te leading term on te rigt and side in 2.9) becomes asymptotically N0, σ 2 0) for some σ 2 0 > 0. On te oter and, te second term vanises in probability as n under H 0 because for eac x X \B n,1 0), 0 > nv n,1 x) as n under H 0. Tus we conclude tat wen liminf n Q B n,1 0)) > 0 under H 0, + dx, { } 2.10) 1/2 ˆθ a n,1 ) 1/2 [Z n,1 x)] + dx a n,1 d N0, σ0). 2 B n,1 0) Tis asymptotic teory is pointwise in P wit P fixed and letting n ), and may not be adequate for finite sample approximation. Tere are two sources of discontinuity. First, te pointwise asymptotic teory essentially regards te drift component nv n,1 x) as, wereas in finite samples, te component can be very negative, but not. Second, even if te nonparametric function nv n,1 x) canges continuously, te contact set B n,1 0) may cange discontinuously in response. 5 Wile tere is no discontinuity in te finite sample distribution of te test statistic, tere may arise discontinuity in its pointwise asymptotic distribution. Furtermore, te complexity of te discontinuity makes it arder to trace its source, wen we ave J > 2. As a result, te asymptotic validity of te test tat is establised pointwise in P is not a good justification of te test. We need to establis te asymptotic validity tat is uniform in P over a reasonable class of probabilities.

10 10 LEE, SONG, AND WHANG Recall tat bootstrap critical values based on te least favorable configuration use a bootstrap quantity suc as 2.11) ˆθ LFC X [ŝ x)] + dx, were ŝ x) = { } ˆv n 1 x) ˆv 1 x), ˆσ 1x) wic can yield tests tat are asymptotically valid uniformly in P. However, using a critical value based on ˆθ 1 [ŝ x)] + dx ˆB {1} c n) also yields an asymptotically valid test, and yet ˆθ LFC > ˆθ 1 in general. Tus te bootstrap tests tat use te contact set ave better power properties tan tose tat do not. Te power improvement is substantial in many simulation designs and can be important in real-data applications. Now, let us see ow te coice of c α,η max{c α, 1/2 η + â } wit d = 1 ere) leads to bootstrap inference tat is valid even wen te test statistic becomes degenerate under te null ypotesis. Te degeneracy arises wen te inequality restrictions old wit large slackness, so tat te convergence in 2.10) olds wit σ 2 0 = 0, and ence For te bootstrap counterpart, note tat 1/2 ˆθ a n,1 ) = o P 1). 1/2 c α,η a n,1 ) = 1/2 max{c α a n,1, 1/2 η + â a n,1 } η + 1/2 â a n,1 ), were it can be sown tat 1/2 â a n,1 ) = o P 1). Terefore, te bootstrap inference is designed to be asymptotically valid even wen te test statistic becomes degenerate. Note tat for te sake of validity only, one may replace 1/2 η by a fixed constant, say η > 0. However, tis coice would render te test asymptotically too conservative. Te coice of 1/2 η in tis paper makes te test asymptotically exact for a wide class of probabilities, wile preserving te uniform validity in bot te cases of degeneracy and nondegeneracy. 6 precise class of probabilities under wic te test becomes asymptotically exact is presented in Section 3. Tere are two remarkable aspects of te local power beavior of our bootstrap test. First, te test exibits two different kinds of convergence rates along different directions of Pitman local alternatives. Second, despite te fact tat te test uses te approac of local smooting by kernel as in Härdle and Mammen 1993), te faster of te two convergence rates acieves a parametric rate of n. To see tis more closely, let us return to te simple example in Te

11 2.7), and consider te following local alternatives: GENERAL FUNCTIONAL INEQUALITIES ) v n x) = v 0 x) + δx) b n, were v 0 x) 0 for all x X and δx) > 0 for some x X, and b n as n suc tat v n x) > 0 for some x X. Te function δ ) represents a Pitman direction of te local alternatives. As we sow later, tere exist two types of convergence rates of our test, depending on te coice of δx). Let B 0 0) {x X : v 0 x) = 0} and σ1x) 2 E[Yi 2 X i = x]fx) K 2 u)du. Wen δ ) is suc tat δx) dx > 0, σ 1 x) B 0 0) B 0 0) te test acieves a parametric rate b n = n. On te oter and, wen δ ) is suc tat δx) δ 2 x) dx = 0 and dx > 0, σ 1 x) σ1x) 2 te test acieves a slower rate b n = n 1/4. See Section 4.2 for euristics beind te results. B 0 0) In Section 4.3, te general form of local power functions is derived. 3. Uniform Asymptotics under General Conditions In tis section, we establis uniform asymptotic validity of our bootstrap test. 7 We also provide a class of distributions for wic te asymptotic size is exact. We first define te set of distributions we consider. Definition 1. Let P denote te collection of te potential joint distributions of te observed random vectors tat satisfy Assumptions A1-A6, and B1-B4 given below. Let P 0 P be te sub-collection of potential distributions tat satisfy te null ypotesis. Let denote te Euclidean norm trougout te paper. For any given sequence of subcollections P n P, any sequence of real numbers b n > 0, and any sequence of random vectors Z n, we say tat Z n /b n P 0, P n -uniformly, or Z n = o P b n ), P n -uniformly, if for any a > 0, lim P { Z n > ab n } = 0. n n Similarly, we say tat Z n = O P b n ), P n -uniformly, if for any a > 0, tere exists M > 0 suc tat lim P { Z n > Mb n } < a. n n We also define teir bootstrap counterparts. Let P denote te probability under te bootstrap distribution. For any given sequence of subcollections P n P, any sequence of real

12 12 LEE, SONG, AND WHANG numbers b n > 0, and any sequence of random vectors Z n, we say tat Z n/b n P 0, P n - uniformly, or Z n = o P b n ), P n -uniformly, if for any a > 0, lim n P {P { Zn > ab n } > a} = 0. n Similarly, we say tat Z n = O P b n ), P n -uniformly, if for any a > 0, tere exists M > 0 suc tat lim P {P { Zn > Mb n } > a} < a. n n In particular, wen we say Z n = o P b n ) or O P b n ), P-uniformly, it means tat te convergence olds uniformly over P P, and wen we say Z n = o P b n ) or O P b n ), P 0 -uniformly, it means tat te convergence olds uniformly over all te probabilities in P tat satisfy te null ypotesis Test Statistics and Critical Values in General Form. First, let us extend te test statistics and te bootstrap procedure to te general case of J 1. Let Λ p : R J [0, ) be a nonnegative, increasing function indexed by p suc tat 1 p <. Wile te teory of tis paper can be extended to various general forms of map Λ p, we focus on te following type: 3.1) Λ p v 1,, v J ) = max{[v 1 ] +,, [v J ] + }) p or Λ p v 1,, v J ) = were for a R, [a] + = max{a, 0}. Te test statistic is defined as ˆθ = Λ p û τ,1 x),, û τ,j x)) dqx, τ). X T J [v j ] p +, To motivate our bootstrap procedure, it is convenient to begin wit te following lemma. Let us introduce some notation. Define N J 2 N J \{ }, i.e., te collection of all te nonempty subsets of N J {1, 2,, J}. For any A N J and v = v 1,, v J ) R J, we define v A to be v except tat for eac j N J \A, te j-t entry of v A is zero, and let 3.2) Λ A,p v) Λ p v A ). Tat is, Λ A,p v) is a censoring of Λ p v) outside te index set A. Now, we define a general version of contact sets: for A N J and for c n,1, c n,2 > 0, 3.3) B n,a c n,1, c n,2 ) { x, τ) X T : j=1 r n,j v n,τ,j x)/σ n,τ,j x) c n,1, for all j A r n,j v n,τ,j x)/σ n,τ,j x) < c n,2, for all j N J /A were σ n,τ,j x) is a population version of ˆσ τ,j x) see e.g. Assumption A5 below.) Wen c n,1 = c n,2 = c n for some c n > 0, we write B n,a c n ) = B n,a c n,1, c n,2 ). },

13 GENERAL FUNCTIONAL INEQUALITIES 13 Lemma 1. Suppose tat Assumptions A1-A3 and A4i) in Section 3.2 old. Suppose furter tat c n,1 > 0 and c n,2 > 0 are sequences suc tat log n{c 1 n,1 + c 1 n,2} 0, as n. Ten as n, { inf P ˆθ = 0 B n,a c n,1,c n,2 ) Λ A,p û τ,1 x),, û τ,j x))dqx, τ) } 1, were P 0 is te set of potential distributions of te observed random vector under te null ypotesis. Te lemma above sows tat te test statistic ˆθ is uniformly approximated by te integral wit domain restricted to te contact sets B n,a c n,1, c n,2 ) in large samples. Note tat te result of Lemma 1 implies tat te approximation error between ˆθ and te expression on te rigt-and side is o P ε n ) for any ε n 0, tereby suggesting tat one may consider a bootstrap procedure tat mimics te representation of ˆθ in Lemma 1. We begin by introducing a sample version of te contact sets. For A N J, { } r n,jˆv τ,j x)/ˆσ τ,j x) ĉ n, for all j A ˆB A ĉ n ) x, τ) X T :. r n,jˆv τ,j x)/ˆσ τ,j x) < ĉ n, for all j N J \A Te explicit condition for ĉ n is found in Assumption A4 below. Given te bootstrap counterparts, {[ˆv τ,jx), ˆσ τ,jx)] : j N J }, of {[ˆv τ,j x), ˆσ τ,j x)] : j N J }, we define our bootstrap test statistic as follows: ˆθ Λ A,p ŝ τ,1x),, ŝ τ,jx))dqx, τ), ˆB A ĉ n) were for j N J, ŝ τ,jx) r n,j ˆv τ,jx) ˆv τ,j x))/ˆσ τ,jx). We also define â E Λ A,p ŝ τ,1x),, ŝ τ,jx))dqx, τ). ˆB A ĉ n) Let c α be te 1 α)-t quantile from te bootstrap distribution of ˆθ and take c α,η = max{c α, d/2 η + â } as our critical value, were η 10 3 is a small fixed number. One of te main tecnical contributions of tis paper is to present precise conditions under wic tis proposal of bootstrap test works. We present and discuss tem in subsequent sections.

14 14 LEE, SONG, AND WHANG To see te intuition for te bootstrap validity, first note tat te uniform convergence of r n,j {ˆv τ,j x) v n,τ,j x)}/ˆσ τ,j x) over x, τ) implies tat 3.4) B n,a c n,l, c n,u ) ˆB A ĉ n ) B n,a c n,u, c n,l ) wit probability approacing one, wenever P {c n,l ĉ n c n,u } 1. Terefore, if log n/c n,l 0, ten, letting ŝ τ,j r n,j ˆv τ,j x) v n,τ,j x))/ˆσ τ,j x)), we ave 3.5) ˆθ Λ A,p ŝ τ,1 x),, ŝ τ,j x)) dqx, τ), B n,a c n,l,c n,u ) wit probability approacing one, by Lemma 1 and te null ypotesis. Wen te last sum as a nondegenerate limit, we can approximate its distribution by te bootstrap distribution ŝ Λ A,p τ,1 x),, ŝ τ,jx) ) dqx, τ) B n,a c n,l,c n,u ) were te inequality follows from 3.4). 8 distribution of ˆθ. ŝ Λ A,p τ,1 x),, ŝ τ,jx) ) dqx, τ) ˆθ, ˆB A ĉ n) Tus te critical value is read from te bootstrap On te oter and, if te last sum in 3.5) as limiting distribution degenerate at zero, we simply take a small positive number η to control te size of te test. Tis results in our coice of c α,η = max{c α, d/2 η + â } Assumptions. In tis section, we provide assumptions needed to develop general results. We assume tat S X T is a compact subset of a Euclidean space. We begin wit te following assumption. Assumption A1. Asymptotic Linear Representation) For eac j N J {1,, J}, tere exists a nonstocastic function v n,τ,j ) : R d R suc tat a) v n,τ,j x) 0 for all x, τ) S under te null ypotesis, and b) as n, 3.6) x,τ) S { } r ˆvτ,j x) v n,τ,j x) n,j n ˆσ τ,j x) d {ĝ τ,j x) Eĝ τ,j x)} = o P d ), P-uniformly, were, wit {Yi, Xi )} n i=1 being a random sample suc tat Y i = Yi1,..., YiJ ) R J L, Y ij R L, X i R d, and te distribution of X i is absolutely continuous wit respect to Lebesgue measure, 9 we define ĝ τ,j x) 1 n d n i=1 β n,x,τ,j Y ij, X ) i x, and β n,x,τ,j : R L R d R is a function wic may depend on n 1.

15 GENERAL FUNCTIONAL INEQUALITIES 15 Assumption A1 requires tat tere exist a nonparametric function v n,τ,j x) around wic te asymptotic linear representation olds uniformly in P P, and v n,τ,j x) 0 under te null ypotesis. Te required rate of convergence in 3.6) is o P d/2 ) instead of o P 1). We need tis stronger convergence rate primarily because ˆθ a n is O P d/2 ) for some nonstocastic sequence a n. 10 Wen ˆv τ,j x) is a sample mean of i.i.d. random quantities involving nonnegative kernels and ˆσ n,τ x) = 1, we may take v n,τ,j x) = Eˆv τ,j x), and ten o P d ) is in fact precisely equal to 0. If te original nonparametric function v τ,j ) satisfies some smootness conditions, we may take v n,τ,j x) = v τ,j x), and andle te bias part Eˆv τ,j x) v τ,j x) using te standard arguments to deduce te error rate o P d ). Assumption A1 admits bot set-ups. For instance, consider te simple example in Section 2.3. Te asymptotic linear representation in Assumption A1 can be sown to old wit β n,x,1 Y i, X i x)/) = Y i KX i x)/)/σ n,1 x), were σ 2 n,1x) = E[Y 2 i K 2 X i x)/)]/, if ˆσ n,1 x) is cosen as in 2.8). Te following assumption for β n,x,τ,j essentially defines te scope of tis paper s framework. Assumption A2. Kernel-Type Condition) For some compact K 0 R d tat does not depend on P P or n, it is satisfied tat β n,x,τ,j y, u) = 0 for all u R d \K 0 and all x, τ, y) X T Y j and all j N J, were Y j denotes te port of Y ij. Assumption A2 can be immediately verified wen te asymptotic linear representation in 3.6) is establised. Tis condition is satisfied in particular wen te asymptotic linear representation involves a multivariate kernel function wit bounded port in a multiplicative form. In suc a case, te set K 0 depends only on te coice of te kernel function, not on any model primitives. Assumption A3. Uniform Convergence Rate for Nonparametric Estimators) For all j N J, x,τ) S r n,j ˆv τ,j x) v n,τ,j x) ˆσ τ,j x) = O P log n ), P-uniformly. Assumption A3 in combination wit A5 below) requires tat ˆv τ,j x) v n,τ,j x) ave te uniform convergence rate of O P r 1 n,j log n) uniformly over P P. Lemma 2 in Section 3.4 provides some sufficient conditions for tis convergence. We now introduce conditions for te bandwidt and te tuning parameter c n for te contact sets. Assumption A 4. Rate Conditions for Tuning Parameters) i) As n, 0, log n/rn 0, and n 1/2 d ν 1 0 for some arbitrarily small ν 1 > 0, were r n

16 16 LEE, SONG, AND WHANG min j NJ r n,j. ii) For eac n 1, tere exist nonstocastic sequences c n,l > 0 and c n,u c n,l c n,u, and > 0 suc tat as n. inf P {c n,l ĉ n c n,u } 1, and log n/c n,l + c n,u /r n 0, Te requirement tat log n/r n 0 is satisfied easily for most cases were r n increases at a polynomial order in n. Assumption A4ii) requires tat ĉ n increase faster tan log n but slower tan r n wit probability approacing one. Assumption A5. Regularity Conditions for ˆσ τ,j x)) For eac τ, j) T N J, tere exists σ n,τ,j ) : X 0, ) suc tat lim inf n inf x,τ) S inf σ n,τ,j x) > 0, and ˆσ τ,j x) σ n,τ,j x) = o P 1), P-uniformly. x,τ) S Assumption A5 requires tat te scale normalization ˆσ τ,j x) sould be asymptotically well defined. Te condition precludes te case were estimator ˆσ τ,j x) converges to a map tat becomes zero at some point x, τ) in S. appropriate coice of ˆσ τ,j x). Assumption A5 is usually satisfied by an Wen one cooses ˆσ τ,j x) = 1, wic is permitted in our framework, Assumption A5 is immediately satisfied wit σ n,τ,j x) = 1. Again, if we go back to te simple example considered in Section 2.3, it is straigtforward to see tat under regularity conditions, wit te subscript τ pressed, ˆσ 2 1x) = σ 2 n,1x)+o P 1) and σ 2 n,1x) = σ 2 1x) + o1), were σ 2 1x) EY 2 X = x)fx) K 2 u)du, as n. Te convergence can be strengtened to a uniform convergence wen σ 2 1x) is bounded away from zero uniformly over x X and P P, so tat Assumption A5 olds. We introduce assumptions about te moment conditions for β n,x,τ,j, ) and oter regularity conditions. For τ T and ε 1 > 0, let S τ ε 1 ) {x + a : x S τ, a [ ε 1, ε 1 ] d }, were S τ {x X : x, τ) S} for eac τ T. Let U K 0 + K 0 suc tat U contains {0} in its interior and K 0 is te same as Assumption A2. Here, + denotes te Minkowski sum of sets. Assumption A6. i) Tere exist M 2p + 2), C > 0, and ε 1 > 0 suc tat E[ β n,x,τ,j Y ij, u) M X i = x]fx) C, for all x, u) S τ ε 1 ) U, τ T, j N J, n 1, and P P, were f ) is te density of X i. 11 ii) For eac a 0, 1/2), tere exists a compact set C a R d suc tat 0 < inf P {X i R d \C a } P {X i R d \C a } < a.

17 GENERAL FUNCTIONAL INEQUALITIES 17 Assumption A6i) requires tat conditional moments of β n,x,τ,j Y ij, z) be bounded. Assumption A6ii) is a tecnical condition for te distribution of X i. Te tird inequality in Assumption A6ii) is satisfied if te distribution of X i is uniformly tigt in P, and follows, for example, if E X i <. Te first inequality in Assumption A6ii) requires tat tere be a common compact set outside wic te distribution of X i still as positive probability mass uniformly over P P. Te main trust of Assumption A6ii) lies in te requirement tat suc a compact set be independent of P P. Wile it is necessary to make tis tecnical condition explicit as stated ere, te condition itself appears very weak. Tis paper s asymptotic analysis adopts te approac of Poissonization see, e.g., Horvát 1991) and Giné, Mason, and Zaitsev 2003)). However, existing metods of Poissonization are not readily applicable to our testing problem, mainly due to te possibility of local or global redundancy among te nonparametric functions. In particular, te conditional covariance matrix of β n,x,τ,j Y ij, u) s across different x, τ, j) s given X i can be singular in te limit. Since te empirical researcer rarely knows a priori te local relations among nonparametric functions, it is important tat te validity of te test is not sensitive to te local relations among tem, i.e., te validity sould be uniform in P. Tis paper deals wit tis callenge in tree steps. First, we introduce a Poissonized version of te test statistic and apply a certain form of regularization to facilitate te derivation of its limiting distribution uniformly in P P, i.e., regardless of singularity or degeneracy in te original test statistic. Second, we use a Berry-Esseen-type bound to compute te finite sample influence of te regularization bias and let te regularization parameter go to zero carefully, so tat te bias disappears in te limit. Tird, we translate tus computed limiting distribution into tat of te original test statistic, using so-called de-poissonization lemma. Tis is ow te uniformity issue in tis complex situation is covered troug te Poissonization metod combined wit te metod of regularization Asymptotic Validity of Bootstrap Procedure. Recall tat E and P denote te expectation and te probability under te bootstrap distribution. We make te following assumptions for ˆv τ,jx). Assumption B1. Bootstrap Asymptotic Linear Representation) For eac j N J, { ˆv } r τ,j x) ˆv τ,j x) n,j ˆσ τ,j x) n d {ĝτ,jx) E ĝτ,jx)} = o P d ), P-uniformly, x,τ) S were ĝ τ,jx) 1 n d n i=1 ) β n,x,τ,j Yij, X i x, and β n,x,τ,j is a real valued function introduced in Assumption A1.

18 18 LEE, SONG, AND WHANG Assumption B2. For all j N J, x,τ) S r n,j ˆv τ,jx) ˆv τ,j x) ˆσ τ,j x) Assumption B3. For all j N J, x,τ) S = O P log n), P-uniformly. ˆσ τ,jx) ˆσ τ,j x) = op 1), P-uniformly. Assumption B1 is te asymptotic linear representation of te bootstrap estimator ˆv τ,jx). Te proof of te asymptotic linear representation can be typically proceeded in a similar way tat one obtains te original asymptotic linear representation in Assumption A1. Assumptions B2 and B3 are te bootstrap versions of Assumptions A3 and A5. Proving Assumption B3 is very similar to te way we prove Assumption A5. One can use similar arguments in te proof of Lemma 2ii) below. See te plemental note for its proof.) Sufficient conditions for Assumption B2 are provided in Lemma 2ii) below. Assumption B4. Bandwidt Condition) n 1/2 3M 4 2M 4)d ν 2 small ν 2 > 0 and for M > 0 tat appears in Assumption A6i). 0 as n, for some Wen β n,x,τ,j Y ij, u) is bounded uniformly over n, x, τ, j), te bandwidt condition in Assumption B4 can be reduced to n 1/2 3d/2 ν 2 0. If Assumption A6i) olds wit M = 6 and p = 1, te bandwidt condition in Assumption B4 is reduced to n 1/2 7d/4 ν 2 0. Note tat Assumption B4 is stronger tan te bandwidt condition in Assumption A4i). Te main reason is tat we need to prove tat for some a > 0, we ave a n = a + o d/2 ) and a n = a + o P d/2 ), P-uniformly, were a n is an appropriate location normalizer of te test statistic, and a n is a bootstrap counterpart of a n. To sow tese, we utilize a Berry-Esseen-type bound for a nonlinear transform of independent sum of random variables. Since te approximation error depends on te moment bounds for te sum, te bandwidt condition in Assumption B4 takes a form tat involves M > 0 in Assumption A6. We now present te result of te uniform validity of our bootstrap test. Teorem 1. Suppose tat Assumptions A1-A6 and B1-B4 old. Ten lim n P {ˆθ > c α,η} α. 0 One migt ask weter te bootstrap test 1{ˆθ > c α,η} can be asymptotically exact, i.e., weter te inequality in Teorem 1 can old as an equality. As we sow below, te answer is affirmative. Te remaining issue is a precise formulation of a subset of P 0 suc tat te rejection probability of te bootstrap test acieves te level α asymptotically, uniformly over te subset.

19 GENERAL FUNCTIONAL INEQUALITIES 19 To see wen te test will ave asymptotically exact size, we apply Lemma 1 to find tat wit probability approacing one, ˆθ = Λ A,p ŝ τ x) + u n,τ x; ˆσ)) dqx, τ), A N B n,a c n,u,c n,l ) J were ŝ τ x) [r n,j {ˆv n,τ,j x) v n,τ,j x)}/ˆσ τ,j x)] J j=1, and u n,τx; ˆσ) [r n,j v n,τ,j x)/ˆσ τ,j x)] J j=1, and c n,u > 0 and c n,l > 0 are nonstocastic sequences tat satisfy Assumption A4ii). We fix a positive sequence q n 0, and write te rigt and side as 3.7) Λ A,p ŝ τ x) + u n,τ x; ˆσ)) dqx, τ) + B n,a q n) A N B n,a c n,u,c n,l )\B n,a q J n) Λ A,p ŝ τ x) + u n,τ x; ˆσ)) dqx, τ). Under te null ypotesis, we ave v n,τ,j x) 0, and ence te last sum is bounded by Λ A,p ŝ τ x)) dqx, τ), A N B n,a c n,u,c n,l )\B n,a q J n) wit probability approacing one. Using te uniform convergence rate in Assumption A3, we find tat as long as QB n,a c n,u, c n,l )\B n,a q n )) 0, fast enoug, te second term in 3.7) vanises in probability. As for te first integral, since for all x B n,a q n ), we ave r n,j v n,τ,j x)/σ n,τ,j x) q n for all j A, we use te Lipscitz continuity of te map Λ A,p on a compact set, to approximate te leading sum in 3.7) by θ 1,n q n ) Λ A,p ŝ τ x)) dqx, τ). A N B n,a q J n) Tus we let { ) } 3.8) Pn λ n, q n ) P P : Q B n,a c n,u, c n,l )\B n,a q n ) λ n, and find tat ˆθ = θ 1,n q n ) + o P d/2 ), Pn λ n, q n ) P 0 -uniformly, as long as λ n and q n converge to zero fast enoug. We will specify te conditions in Teorem 2 below. Let us deal wit θ 1,n q n ). First, it can be sown tat tere are sequences of nonstocastic numbers a n q n ) R and σ n q n ) > 0 tat depend on q n suc tat 3.9) d/2 { θ 1,n q n ) a n q n )}/σ n q n ) d N0, 1),

20 20 LEE, SONG, AND WHANG if liminf n σ n q n ) > 0. We provide te precise formulae for σ n q n ) and a n q n ) in Section 4.3. Since te distribution of d/2 { θ 1,n q n ) a n q n )}/σ n q n ) is approximated by te bootstrap distribution of d/2 {ˆθ a n q n )}/σ n q n ) in large samples, we find tat d/2 {c α a n q n )} σ n q n ) = Φ 1 1 α) + o P 1). Hence te bootstrap critical value c α will dominate d/2 η + â > 0, if for all n 1, Φ 1 1 α) d/2 { d/2 η + â a n q n )} σ n q n ) = η + d/2 {â a n q n )}. σ n q n ) We can sow tat â a n q n ) = o P d/2 ), wic follows if λ n in 3.8) vanises to zero sufficiently fast. Hence if σ n q n ) η/φ 1 1 α), we ave c α becomes approximately equal to our bootstrap critical value c α,η. Tis leads to te following formulation of probabilities. Definition 2. Define P n λ n, q n ) were P n λ n, q n ) is as defined in 3.8). { P P } n λ n, q n ) : σ n q n ) η/φ 1 1 α), Te following teorem establises te asymptotic exactness of te size of te bootstrap test over P P n λ n, q n ) P 0. Teorem 2. Suppose tat Assumptions A1-A6 and B1-B4 old. Let λ n 0 and q n 0 be positive sequences suc tat 3.10) Ten d/2 log n) p/2 λ n 0 and d/2 q n {log n) p 1)/2 + q p 1 n } 0. lim n nλ n,q n) P 0 P {ˆθ > c α,η} α = 0. Teorem 2 sows tat te rejection probability of our bootstrap test acieves exactly te level α uniformly over te set of probabilities in P n λ n, q n ) P 0. If v n,τ,j x) 0 for eac x, τ) and for eac j te least favorable case, say P LFC ), ten it is obvious tat te distribution P LFC belongs to P n λ n, q n ) for any positive sequences λ n 0 and q n 0. Tis would be te only case of asymptotically exact coverage if bootstrap critical values were obtained as

21 GENERAL FUNCTIONAL INEQUALITIES 21 in 2.11), witout contact set estimation. By estimating te contact sets and obtaining a critical value based on tem, Teorem 2 establises te asymptotically uniform exactness of te bootstrap test for distributions suc tat tey may not satisfy v n,τ,j x) 0 everywere Sufficient Conditions for Uniform Convergences in Assumptions A3 and B2. Tis subsection gives sufficient conditions tat yield Assumptions A3 and B2. Te result is formalized in te following lemma. Lemma 2. i) Suppose tat Assumptions A1-A2 old and tat for eac j N J, tere exist finite constants C, γ j > 0, and a positive sequence δ n,j > 0 suc tat for all n 1, and all x 1, τ 1 ) S, [ 3.11) E ] b n,ij x 1, τ 1 ) b n,ij x 2, τ 2 )) 2 Cδn,jλ 2 γ j, for all λ > 0, x 2,τ 2 ) S: x 1 x 2 + τ 1 τ 2 λ were b n,ij x 1, τ 1 ) β n,x1,τ 1,j Y ij, X i x 1 )/) and lim n E[ x,τ) S b 4 n,ijx, τ)] C and δ n,j = n s 1,j and = n s 2 for some s 1,j, s 2 R. Furtermore, assume tat n 1/2 d ν 0, for some small ν > 0. Ten, Assumption A3 olds. ii) Suppose furter tat Assumptions B1 and B3 old. Ten, Assumption B2 olds. Te condition 3.11) is te local L 2 -continuity condition for β n,x,τ,j Y ij, X i x)/) in x, τ). Te condition corresponds to wat Andrews 1994) called Type IV class. Te condition is satisfied by numerous maps tat are continuous or discontinuous, as long as regularity conditions for te random vector Y i, X i ) are satisfied. 12 Typically, δ n,j diverges to infinity at a polynomial rate in 1. Te constant γ j is 2 or can be smaller tan 2, depending on te smootness of te underlying function b n,ij x, τ). Te value of γ j does not affect te asymptotic teory of tis paper, as long as it is strictly positive. In Section 6.4, we provide primitive sufficient conditions to establis te uniform validity of our bootstrap test for te first empirical example. 4. Power Properties In tis section, we consider te power properties of te bootstrap test. 13 In Section 4.1, we establis te consistency of our test. Section 4.2 provides euristic arguments beind local power properties of our tests, and Section 4.3 presents te local power function in a general form. 14

22 22 LEE, SONG, AND WHANG 4.1. Consistency. First, to sow consistency of our test, we make te following assumption. Assumption C1. For eac j N J and x, τ) S, v n,τ,j x) = v τ,j x) + o1), and 4.1) lim n x,τ) S v n,τ,j x) <. Te pointwise convergence v n,τ,j x) = v τ,j x) + o1) olds typically by an appropriate coice of v n,τ,j x). In many examples, condition 4.1) is often implied by Assumptions A1- A6. If we revisit te simple example considered in Section 2.3, it is straigtforward to see tat under Assumptions A1-A6, wit te subscript τ pressed, v n,1 x) = v 1 x) + o1), were v n,1 x) Eˆv n,1 x) and v 1 x) EY X = x)fx), and 4.1) olds easily. We now establis te consistency of our proposed test as follows. Teorem 3. Suppose tat Assumptions A1-A6, B1-B4, and C1 old and tat we are under a fixed alternative ypotesis suc tat Λ p v τ,1 x),, v τ,j x)) dqx, τ) > 0. Ten as n, P {ˆθ > c α,η} Local Power Analysis: Definitions and Heuristics. In tis section, we investigate te local power properties of our test. For local power analysis, we formally define te space of Pitman directions. Let D be te collection of R J -valued bounded functions on X T suc tat for eac δ = δ 1,, δ J ) D, Q{x, τ) S : δ j x, τ) 0} > 0 for some j = 1,..., J. Tat is, at least one of te components of any δ D is a non-zero function a.e. For eac δ = δ 1,, δ J ) D, we write δ τ,j x) = δ j x, τ), j = 1,, J. For a given vector of sequences b n = b n,1,, b n,j ), suc tat b n,j, and δ D, we consider te following type of local alternatives: 4.2) H δ : v τ,j x) = v 0 τ,jx) + δ τ,jx) b n,j, for all j N J, were v 0 τ,jx) 0 for all x, τ, j) X T N J, δ τ,j x) > 0 for some x, τ, j) X T N J suc tat v τ,j x) > 0 for some x, τ, j) X T N J. Note tat in 4.2), v τ,j x) is a sequence of Pitman local alternatives tat consist of tree components: v 0 τ,jx), b n, and δ τ,j x). Te first component v 0 τ,jx) determines were te sequence of local alternatives converges to. For example, if v 0 τ,jx) 0 for all x, τ, j), ten we ave a sequence of local alternatives tat converges to te least favorable case. We allow for negative values for v 0 τ,jx), so tat we include te local alternatives tat do not converge to te least favorable case as well.

23 GENERAL FUNCTIONAL INEQUALITIES 23 From ere on, we assume te local alternative ypoteses of te form in 4.2). We fix v 0 τ,jx) and identify eac local alternative wit a pair b n, δ) for eac Pitman direction δ D. Te following definitions are useful to explain our local power results. Definition 3. i) Given a Pitman direction δ D, we say tat an α-level test, 1{T > c α }, as nontrivial local power against b n, δ), if under te local alternatives b n, δ), liminf n P {T > c α } > α, and say tat te test as trivial local power against b n, δ), if under te local alternatives b n, δ), lim n P {T > c α } α. ii) Given a collection D, we say tat a test as convergence rate b n against D, if te test as nontrivial local power against b n, δ) for some δ D, and as trivial local power against b n, δ) for all δ D and all b n suc tat b n,j/b n,j as n, for all j = 1,..., J. One of te remarkable aspects of te local power properties is tat our test as two types of convergence rates. More specifically, tere exists a partition D 1, D 2 ) of D, were our test as a rate b n against D 1 and anoter rate b n against D 2. Furtermore, in many nonparametric inequality testing environments, te faster of te two rates b n and b n acieves te parametric rate of n. To see tis closely, let us assume te set-up of testing inequality restrictions on a mean regression function in Section 2.4, and consider te following local alternatives: 4.3) v n,1 x) = v 0 x) + δx) b n, were v 0 x) 0 for all x X, and δ D. First, we set b n = n. Ten under tis local alternative ypotesis b n, δ), we can verify tat wit probability approacing one, { [ ] } 4.4) 1/2 ˆθ a n,0 ) = 1/2 nv0 x) Z n,1 x) + + 1/2 δx) dx a n,0, Bnc 0 n) ˆσ 1 x) ˆσ 1 x) + were Z n,1 x) = { n {ˆv 1 x) v n,1 x)} /ˆσ 1 x), Bnc 0 n ) = x X : } nv 0 x) c n, c n, log n/c n 0, and [ ] a n,0 = E [Z n,1 x)] + dx. Bnc 0 n) Under regularity conditions, te rigt-and side of 4.4) is approximated by 4.5) 1/2 { B 0 0) [ Z n,1 x) + 1/2 δx) σ 1 x) ] } dx a n,δ + 1/2 {a n,δ a n,0 }, +

24 24 LEE, SONG, AND WHANG were B 0 0) = {x X : v 0 x) = 0} and [ a n,δ = E B 0 0) [ ] ] Z n,1 x) + 1/2 δx) dx. σ 1 x) + Te leading term in 4.5) converges in distribution to Z 1 N0, σ 2 0) precisely as in 2.10). Furtermore, we can sow tat a n,δ = a n,0 = B 0 0) B 0 0) [ ] E Z 1 + 1/2 δx) dx + o 1/2 ) and σ 1 x) + E [Z 1 ] + dx + o 1/2 ). Terefore, as for te last term in 4.5), we find tat [ ] 1/2 {a n,δ a n,0 } = E 1/2 Z 1 + 1/2 δx) B 0 0) σ 1 x) δx) = 2φ0) dx + o1), σ 1 x) B 0 0) + E [Z 1 ] + ) dx + o1) were te last equality follows from expanding 1/2 { E [ Z 1 + 1/2 δx)/σ 1 x) ] + E [Z 1] + }. We conclude tat under te local alternatives, we ave 1/2 ˆθ a n,0 ) d Z 1 + 2φ0) B 0 0) δx) σ 1 x) dx. Te magnitude of te last term in te limit determines te local power of te test. Tus under Pitman local alternatives suc tat δx) 4.6) dx > 0, σ 1 x) B 0 0) te test as nontrivial power against n-converging Pitman local alternatives. Note tat te integral in 4.6) is defined on te population contact set B 0 0). Tus, te test as nontrivial power, unless te contact set as Lebesgue measure zero or δ ) is too often negative on te contact set. Wen te integral in 4.6) is zero, we consider te local alternatives b n, δ) wit a slower convergence rate b n = n 1/2 1/4. Following similar arguments as before, we now ave 1/2 ˆθ a n,0 ) d Z 1 + lim n 1/2 {ā n,δ a n,0 }, were ā n,δ = B 0 0) [ ] E Z n,1 x) + 1/4 δx) dx, σ 1 x) +