In this article, we develop a nonparametric estimator for the Hölder constant of a density function. We consider a simulation study to evaluate the performance of the proposal and construct smooth bootstrap confidence intervals. Also, we give a brief review over the impossibility to decide whether a density function is Hölder.Fil: Henry, Guillermo Sebastian. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Rodriguez, Daniela Andrea. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Sued, Raquel Mariela. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentin

Henry, Guillermo Sebastian

Rodriguez, Daniela Andrea

Sued, Raquel Mariela

Communications in Statistics - Simulation and Computation

English

Guillermo Henry

Daniela Rodriguez

Mariela Sued

Crossref

A Note on the Estimation of the Hölder Constant

CONICET Digital

A note on the estimation of the Ho¨lder constantGuillermo Henry, Daniela Rodriguez and Mariela SuedFCEyN, Universidad de Buenos Aires and CONICET, ArgentinaAbstractIn this paper we develop a nonparametric estimator for the Ho¨lder constant of a densityfunction. We consider a simulation study to evaluate the performance of the proposal and con-struct smooth bootstrap confidence intervals. Also, we give a brief review over the impossibilityto decide whether a density function is Ho¨lder.Key words and phrases: Discernibility, Ho¨lder density, Nonparametric estimation.1 IntroductionModern statistics assumes many hypothesis concerning the underlying distribution of the data inorder to obtain good estimators. It is often assumed that the underlying distribution has a densityf that belongs to certain class F . To decide whether f belongs to F based on the data is not alwayspossible. When it is, we say that the class F is discernible. Devroy and Lugossi (2002) discussthis problem in detail. Between many examples considered in their work, they show that there isno rule able to decide whether a density is Lipschitz or not. We generalize this fact proving thatthe class of γ-Ho¨lder densities is not discernable. Given these circumstances, we limit ourselves toestimate the Ho¨lder constat of a density, including the Lipschitz case, whenever it is finite. In otherwords, assuming that the density is γ-Ho¨lder, we will make inference on its constant. Carando, et.al. (2009) studied a maximum likelihood non parametric density estimator under the assumptionthat the density is Lipschitz with a known constant: our proposal can be plugged in their method.This note is organized as follows. In Section 2, we discuss the discernibility of the class of Ho¨lderfunctions. The proposed estimator of the Ho¨lder constant and a simulation study to evaluate itsperformance are included in Section 2 and 3, respectively. Finally, smoothed bootstrap confidenceintervals are defined in Section 4.2 Ho¨lder densities are not discernibleAccording to Devroy and Lugossi (2002), a class of densities F is said to be discernible if for anydensity f we can decide whether it belongs to F or not, based on sample X1, X2, . . . of i.i.d. random1variables with common density f . The decision is made based on a classification rule, which is asequence {Tn} of functions Tn : Rn → {0, 1} that satisfieslimn→∞Tn(X1, . . . , Xn) ={1 a.s. if Xi has denisty f ∈ F ,0 a.s. if Xi has denisty f 6∈ F . (1)The authors also show that the class F of densities supported in [0, 1], such that each f ∈ F hasa Lipschitz version, is not discernible. We will prove that γ- Ho¨lder densities are not discernible,including the Lipschitz case with γ = 1. Recall that, for γ ∈ (0, 1], f is γ-Ho¨lder if|f(x)− f(y)| ≤ wγ(f)|x− y|γ (2)with finite wγ(f), wherewγ(f) = supx6=y|f(x)− f(y)||x− y|γ . (3)We follow the approach by Fraiman and Meloche (1999) used to prove that the class of densitieswith finitely many modes is not discernable. Their presentation is based on the results obtained byLe Cam and Schartz (1960), where necessary and sufficient conditions for the existence of consistentestimates are discussed.We denote by L1[0, 1] the set of integrable Lebesgue functions on [0, 1], and denote the L1norm with ||f ||1 =∫ 10 f(x) dx. Let B ⊂ L1[0, 1] be the set of densities in the interval [0, 1]. Notethat (B, || · ||1) is a closed subset of L1[0, 1]. Let G be a subset of B such that both G and Gc aredense in B. By Lemma 1.1 in Fraiman and Meloche (1999) we can conclude that such G is notdiscernible. As we mentioned, we will take advantage of this fact to prove that Gγ , the class ofdensities supported in [0, 1] such that each f ∈ Gγ has a γ-Ho¨lder version, is not discernible. Wewill see that both Gγ and Gcγ are dense in B, as follows:1. To prove that Gγ is dense in B, we use the fact that the convolution preserves regularity inorder to approximate any density f by a sequence of densities in Gγ . To be more precise, letK be a kernel in Gγ and consider Kh(y) = h−1K(y/h). Given f ∈ B, considerfh(x) =f ∗Kh(x) I[0,1](x)Ch(f), where Ch(f) =∫ 10 f ∗Kh(x)dx .Note that f ∗Kh converges to f in L1[R] and so, since f is supported in [0, 1], we get thatCh → 1. This guarantees that fh converges to f in L1[0, 1]. On the other hand, f ∗Kh is asregular as K, meaning that if K ∈ Gγ , f ∗Kh is γ-Ho¨lder and by consequence fh ∈ Gγ .2. In order to prove that Gcγ is dense in L1[0, 1], we need to prove that any element f ∈ Gγ canbe approximated by a sequence in Gcγ . Take g ∈ Gcγ and definefh = (1− h)f + hg .Then, fh ∈ Gcγ and converges to f in L1[0, 1], when h ↓ 0.23 Estimation of the Ho¨lder constant.Let X be a random variable with density f γ-Ho¨lder; i.e. f satisfies condition (2). We want toremark that this condition has a nice geometric interpretation. In the Lipschitz case, i.e. Ho¨lderfunction with γ = 1, there is a double cone whose vertex can be translated along the graph in sucha way that the graph of the density remains always entirely inside the cone. When the densityis γ-Ho¨lder, the situation is analogous, but considering a double parabolic cone instead of linearcone, as we can see in Figure 1. Note that given a double cone with slope m, any cone with slopem′ ≥ m satisfies also this property. The Ho¨lder constant agree with the smallest of these slopes.a) b)−2 −1 0 1 20.10.20.30.40.5−0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.20.00.20.40.60.81.0Figure 1: a) Lipschitz’s cone. b) Ho¨lder’s cone.Let X1, ..., Xn be independent observations of a random variable X with unknown probabilitydensity function f γ-Ho¨lder, with known γ. In order to define a nonparametric estimator of theHo¨lder constant wγ(f), we consider the following result whose proof can be obtained straightfor-wardly by Borel–Cantelli’s Lemma.Lemma: Let G ⊂ Rk be an open set and h : G → R a bounded and continuous function. Let(Yi)i≥1 be independent and identically distributed random variables taking values on Rk, withdensity function f , such that f(y) > 0 for all y ∈ G and P (Y1 ∈ G) = 1. ThenMn = max1≤i≤nh(Yi)→ supy∈Gh(y) =M a.s.. (4)Note that if we apply the Lemma to the function h(x, y) = |f(x)−f(y)||x−y| and the variables Wj =(X2j−1, X2j), for 1 ≤ j, in the open set that excludes the diagonal, then a natural approach is toconsiderŵγ(f) = max1≤i,j≤n; i 6=j|f(Xi)− f(Xj)||Xi −Xj |γ .But, since f is unknown, we will estimate f trough a nonparametric estimator fn. Thereforethe nonparametric estimator of the Ho¨lder constant wγ(f) is defined as ŵγ(fn).3In this work, we consider the class of kernel type estimators of f which has been widely studiedsince the work of Rosenblatt (1956) and Parzen (1962). These estimators are defined asfn(x) =1nhn∑j=1K(x−Xjh), (5)where K(u) is a bounded density on R and h is a sequence of positive number such that h→ 0 andnh→∞ as n→∞.We note that the estimator ŵγ(fn) has also a geometric interpretation. If we consider theLipschitz case, the estimator can be defined as follow: for each point of the random sample of sizen, for exampleX1, we consider the n−1 segments that join the points (X1, fn(X1)) and (Xi, fn(Xi))for i = 2, . . . n, and denote by mX1(Xi) the slopes of these segments. Figure 2 a) shows some slopesfor two points on the random sample. Therefore for each point of the sample, we can estimatea local double cone as the cone given by the maximum of these slopes. Finally, we consider aestimator of the Lipschitz constant as the maximum slope of the local cones given the sample.In the Ho¨lder case, the procedure is the same but instead of consider straights we have toconsider curves defined by the graph of xγ . In Figure 2 b), we show this situation.a) b)−2 −1 0 1 20.00.10.20.30.40.5−0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.20.00.20.40.60.8Figure 2: a) Lipschitz’s cone. The lines in red correspond to the cone with the estimated constant. Thesolid lines in blue correspond to the local cone estimator for a particular sample point and the dashed linescorrespond to the straight lines for this point with some of the others points in the sample. In green thesame kind of plots are shown, constructed at a different point of the sample. b) Ho¨lder’s cone. The figure isanalogous to a) but with an appropriate curve.4 MonteCarlo studyThis section contains the results of a simulation study designed to evaluate the performance of theestimator defined in Section 3. To perform this study, we considered four models. For each model,we generated X1, . . . , Xn independent samples of size n = 100, 500, 1000 according with a density4f . In each model, we performed N = 1000 replications. The different models were generated fromthe following density functions.Model 1 : fX(x) = 1√2pi(2φ(2)−1) exp{−12x2}I(−2,2)(x) in this case the Lipschitz constantw1(f) = e−1/2(√2pi(2φ(2)− 1))−1 = 0.2535053 where φ denotes the standar normal distribu-tion function.Model 2 : fX(x) = (x + 1)I(−1,0)(x) + (1 − x)I(0,1) in this case the Lipschitz constantw1(f) = 1.Model 3 : fX(x) = 12((x+ 1)I(−1,0)(x) + (1− x)I(0,1)(x) + (x− 1)I(1,2)(x) + (3− x)I(2,3)(x))in this case the Lipschitz constant w1(f) = 12 .Model 4 : fX(x) = 23√xI(0,1)(x) in this case the Ho¨lder constant w 12(f) = 23 .Figure 1 shows the densities considered in models 1 and 4 while Figure 3 shows the densityfunctions in the models 2 and 3.a) b)−1.0 −0.5 0.0 0.5 1.00.00.20.40.60.81.01.2−1 0 1 2 30.00.20.40.60.81.0Figure 3: a) Density function of Model 2. b) Density function of Model 3.In all cases, for smoothing procedure, we take the gaussian kernel and we consider an auto-matic bandwidth selector given by Scott (1992). To study the performance of the estimators ofwγ(f), denoted by ŵγ(fn), we have considered the mean square error (MSE) calculated over the Nreplications, i.e.MSE(ŵγ(fn)) =1NN∑i=1[ŵγ(fn)i − wγ(f)]2where ŵγ(fn)i denotes the estimator of the constant given by the replication i.Table 1 gives the values of the MSE of ŵγ(fn) for the four considered models. The simulationstudy confirms the good behavior of constant estimators, under the different models. Model 1corresponds to a infinitely differential function and as we expected the estimator of the constant5has a good performance. The densities considered in Models 2 and 3 correspond to non differentiablefunctions however the behavior is comparable. Finally, Model 4 is an example of the Ho¨lder casewhere we also obtained a good performance.Model 1 Model 2 Model 3 Model 4n = 100 0.0101 0.0113 0.0281 0.0174n = 500 0.0050 0.0084 0.0051 0.0062n = 1000 0.0037 0.0096 0.0016 0.0047Table 1: MSE for the different models.5 Smoothed bootstrap confidence intervalsIn this section, we develop a bootstrap procedure to implement confidence intervals for the Ho¨lderconstant. One of the main motivations to construct confidence intervals is to be able to answerquestions about the true value of the Holder constant. For example, if a 100(1 − α)% confidenceintervals for wγ(f) does not contain the value c0, this is evidence against the null hypothesis thatwγ(f) is c0. In this way, as usual, the confidence intervals can provide a test for such null hypothesis.Moreover, confidence intervals give an idea about the global variability of the estimator.The bootstrap method introduced by Efron (1979) is a very useful tool used primarily to estimatethe sampling distribution of some statistic of interest. Given a set of independent and identicallydistributed observations X1, . . . , Xn, the estimator of the Ho¨lder constant ŵγ(fn) can be definedas some function T (X1, . . . , Xn). The smoothed bootstrap technic (Efron (1979, 1982); Silvermanand Young, (1987)) estimates the sampling distribution of interest by that of T (X∗1 , . . . , X∗n), forbootstrap samples X∗1 , . . . , X∗n of size n obtained by independent sampling from the distributionFn with density fn defined in (5).In order to construct bootstrap confidence intervals, we consider the smoothed bootstrapmethod. Thus, the bootstrap confidence intervals can be described as follows.Step 1 Based on X1, . . . , Xn, we estimate f through fn(x) = (nh)−1∑nj=1K(x−Xjh)and computeŵγ(fn).Step 2 A bootstrap sample X∗1 , . . . , X∗n is generated as X∗j = Xij + εj , with i1, . . . , in independentand uniformly distributed integers from {1, . . . , n} and ε1, . . . , εn independent and identicallydistributed with density Kh(t) = h−1K(t/h). The bootstrap estimate ŵ∗γ(f∗n) of wγ(f) iscalculated based on X∗1 , . . . , X∗n.Step 3 Repeat Step 2, B times, to get B values of ŵ∗γ(f∗n)r for 1 ≤ r ≤ B.Step 4 A bootstrap estimator of the variance of ŵγ(fn) is computed asvarB(ŵγ(fn)) =1B − 1B∑r=1(ŵ∗γ(f∗n)r − ŵ∗γ(f∗n))26where ŵ∗γ(f∗n) =1BB∑r=1ŵ∗γ(f∗n)r.Step 5 The 1− α confidence interval of wγ(f) is defined asIC(wγ(f)) =(ŵγ(fn)− q∗(1− α2)√varB(ŵγ(fn)), ŵγ(fn)− q∗(α2)√varB(ŵγ(fn)))where q∗(α) is the α−quantile of the empirical bootstrap distribution of T ∗r = ŵ∗γ(f∗n)r−ŵγ(fn)√varB(ŵγ(fn))for 1 ≤ r ≤ B.To evaluate the performance of the bootstrap intervals described above, we implement this proce-dure to the models considered in the previous Section. The following Table summarizes the resultsof the empirical coverage, defined as the proportion of the bootstrap intervals containing the trueparameter over the N = 1000 replications. For each model, we consider B = 1000 bootstrap repli-cations and in all cases, we compute confidence intervals with level 0.95. Note that, as expected, thebootstrap intervals have a good performance since the empirical coverages are near to the nominallevel 0.95.Model 1 Model 2 Model 3 Model 4n = 100 94.1 95 97 96.5Table 2: Empirical coverage of nominal 95% confidence intervals. The empirical coverages are inpercent.AcknowledgmentsWe wish to thank the anonymous referee for his/her valuable comments which led to an improvedversion of the original article. We would like to thank to Lucio Pantazis for his careful reading ofthe manuscript and for useful suggestions. This research was partially supported by Grants pip112-200801-00216 and 0592 from conicet and UBACYT 20020100300057 from the Universidadde Buenos Aires.References[1] Carando, D.; Fraiman, R. and Groisman, P. (2009). Nonparametric likelihood based estimationfor a multivariate Lipschitz density. Journal of Multivariate Analysis. 100. 981-992.[2] Devroye, L and Lugosi, G. (2002). Almost sure classification of densities. Journal of Nonpara-metric Statistics. 14, 675–698.[3] Efron, B. (1979) Bootstrap methods-another look at the jack-knife. Ann. Statist., 7, 1-26.7[4] Efron, B. (1982) The Jackknife, the Bootstrap and Other Resampling Plans. Philadelphia:Society for Industrial and Applied Mathematics.[5] Fraiman, R. an Meloche, J. (1999). Counting Bumps. Annals of the Institute of StatisticalMathematics, 51, 1–29.[6] Le Cam. L. and Schwartz, L. (1960) A necessary and sufficient condition for the existence ofconsistent estimates. Annals of Statistics. 31, 140–150.[7] Parzen, E. (1962). On estimation of a probability density function and mode. Ann. Math.Statist. 33, 1065–1076.[8] Rosenblatt, M. (1956). Remarks on some nonparametric estimates of a density function. Ann.Math. Statist.27, 832–837.[9] Scott, D. W. (1992) Multivariate Density Estimation: Theory, Practice, and Visualization.Wiley.[10] Silverman, B. W. and Young, G. A. (1987) The bootstrap: to smooth or not to smooth?Biometrika, 74, 469-479.8

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A note on the estimation of the Hölder constantGuillermo Henry, Daniela Rodriguez and Mariela SuedFCEyN, Universidad de Buenos Aires and CONICET, ArgentinaAbstractIn this paper we develop a nonparametric estimator for the Hölder constant of a densityfunction. We consider a simulation study to evaluate the performance of the proposal and con-struct smooth bootstrap confidence intervals. Also, we give a brief review over the impossibilityto decide whether a density function is Hölder.Key words and phrases: Discernibility, Hölder density, Nonparametric estimation.1 IntroductionModern statistics assumes many hypothesis concerning the underlying distribution of the data inorder to obtain good estimators. It is often assumed that the underlying distribution has a densityf that belongs to certain class F . To decide whether f belongs to F based on the data is not alwayspossible. When it is, we say that the class F is discernible. Devroy and Lugossi (2002) discussthis problem in detail. Between many examples considered in their work, they show that there isno rule able to decide whether a density is Lipschitz or not. We generalize this fact proving thatthe class of γ-Hölder densities is not discernable. Given these circumstances, we limit ourselves toestimate the Hölder constat of a density, including the Lipschitz case, whenever it is finite. In otherwords, assuming that the density is γ-Hölder, we will make inference on its constant. Carando, et.al. (2009) studied a maximum likelihood non parametric density estimator under the assumptionthat the density is Lipschitz with a known constant: our proposal can be plugged in their method.This note is organized as follows. In Section 2, we discuss the discernibility of the class of Hölderfunctions. The proposed estimator of the Hölder constant and a simulation study to evaluate itsperformance are included in Section 2 and 3, respectively. Finally, smoothed bootstrap confidenceintervals are defined in Section 4.2 Hölder densities are not discernibleAccording to Devroy and Lugossi (2002), a class of densities F is said to be discernible if for anydensity f we can decide whether it belongs to F or not, based on sample X1, X2, . . . of i.i.d. random1variables with common density f . The decision is made based on a classification rule, which is asequence {Tn} of functions Tn : Rn → {0, 1} that satisfieslimn→∞Tn(X1, . . . , Xn) ={1 a.s. if Xi has denisty f ∈ F ,0 a.s. if Xi has denisty f 6∈ F . (1)The authors also show that the class F of densities supported in [0, 1], such that each f ∈ F hasa Lipschitz version, is not discernible. We will prove that γ- Hölder densities are not discernible,including the Lipschitz case with γ = 1. Recall that, for γ ∈ (0, 1], f is γ-Hölder if|f(x)− f(y)| ≤ wγ(f)|x− y|γ (2)with finite wγ(f), wherewγ(f) = supx6=y|f(x)− f(y)||x− y|γ . (3)We follow the approach by Fraiman and Meloche (1999) used to prove that the class of densitieswith finitely many modes is not discernable. Their presentation is based on the results obtained byLe Cam and Schartz (1960), where necessary and sufficient conditions for the existence of consistentestimates are discussed.We denote by L1[0, 1] the set of integrable Lebesgue functions on [0, 1], and denote the L1norm with ||f ||1 =∫ 10 f(x) dx. Let B ⊂ L1[0, 1] be the set of densities in the interval [0, 1]. Notethat (B, || · ||1) is a closed subset of L1[0, 1]. Let G be a subset of B such that both G and Gc aredense in B. By Lemma 1.1 in Fraiman and Meloche (1999) we can conclude that such G is notdiscernible. As we mentioned, we will take advantage of this fact to prove that Gγ , the class ofdensities supported in [0, 1] such that each f ∈ Gγ has a γ-Hölder version, is not discernible. Wewill see that both Gγ and Gcγ are dense in B, as follows:1. To prove that Gγ is dense in B, we use the fact that the convolution preserves regularity inorder to approximate any density f by a sequence of densities in Gγ . To be more precise, letK be a kernel in Gγ and consider Kh(y) = h−1K(y/h). Given f ∈ B, considerfh(x) =f ∗Kh(x) I[0,1](x)Ch(f), where Ch(f) =∫ 10 f ∗Kh(x)dx .Note that f ∗Kh converges to f in L1[R] and so, since f is supported in [0, 1], we get thatCh → 1. This guarantees that fh converges to f in L1[0, 1]. On the other hand, f ∗Kh is asregular as K, meaning that if K ∈ Gγ , f ∗Kh is γ-Hölder and by consequence fh ∈ Gγ .2. In order to prove that Gcγ is dense in L1[0, 1], we need to prove that any element f ∈ Gγ canbe approximated by a sequence in Gcγ . Take g ∈ Gcγ and definefh = (1− h)f + hg .Then, fh ∈ Gcγ and converges to f in L1[0, 1], when h ↓ 0.23 Estimation of the Hölder constant.Let X be a random variable with density f γ-Hölder; i.e. f satisfies condition (2). We want toremark that this condition has a nice geometric interpretation. In the Lipschitz case, i.e. Hölderfunction with γ = 1, there is a double cone whose vertex can be translated along the graph in sucha way that the graph of the density remains always entirely inside the cone. When the densityis γ-Hölder, the situation is analogous, but considering a double parabolic cone instead of linearcone, as we can see in Figure 1. Note that given a double cone with slope m, any cone with slopem′ ≥ m satisfies also this property. The Hölder constant agree with the smallest of these slopes.a) b)−2 −1 0 1 20.10.20.30.40.5−0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.20.00.20.40.60.81.0Figure 1: a) Lipschitz’s cone. b) Hölder’s cone.Let X1, ..., Xn be independent observations of a random variable X with unknown probabilitydensity function f γ-Hölder, with known γ. In order to define a nonparametric estimator of theHölder constant wγ(f), we consider the following result whose proof can be obtained straightfor-wardly by Borel–Cantelli’s Lemma.Lemma: Let G ⊂ Rk be an open set and h : G → R a bounded and continuous function. Let(Yi)i≥1 be independent and identically distributed random variables taking values on Rk, withdensity function f , such that f(y) > 0 for all y ∈ G and P (Y1 ∈ G) = 1. ThenMn = max1≤i≤nh(Yi) → supy∈Gh(y) = M a.s.. (4)Note that if we apply the Lemma to the function h(x, y) = |f(x)−f(y)||x−y| and the variables Wj =(X2j−1, X2j), for 1 ≤ j, in the open set that excludes the diagonal, then a natural approach is toconsiderŵγ(f) = max1≤i,j≤n; i 6=j|f(Xi)− f(Xj)||Xi −Xj |γ .But, since f is unknown, we will estimate f trough a nonparametric estimator fn. Thereforethe nonparametric estimator of the Hölder constant wγ(f) is defined as ŵγ(fn).3In this work, we consider the class of kernel type estimators of f which has been widely studiedsince the work of Rosenblatt (1956) and Parzen (1962). These estimators are defined asfn(x) =1nhn∑j=1K(x−Xjh), (5)where K(u) is a bounded density on R and h is a sequence of positive number such that h → 0 andnh →∞ as n →∞.We note that the estimator ŵγ(fn) has also a geometric interpretation. If we consider theLipschitz case, the estimator can be defined as follow: for each point of the random sample of sizen, for example X1, we consider the n−1 segments that join the points (X1, fn(X1)) and (Xi, fn(Xi))for i = 2, . . . n, and denote by mX1(Xi) the slopes of these segments. Figure 2 a) shows some slopesfor two points on the random sample. Therefore for each point of the sample, we can estimatea local double cone as the cone given by the maximum of these slopes. Finally, we consider aestimator of the Lipschitz constant as the maximum slope of the local cones given the sample.In the Hölder case, the procedure is the same but instead of consider straights we have toconsider curves defined by the graph of xγ . In Figure 2 b), we show this situation.a) b)−2 −1 0 1 20.00.10.20.30.40.5−0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.20.00.20.40.60.8Figure 2: a) Lipschitz’s cone. The lines in red correspond to the cone with the estimated constant. Thesolid lines in blue correspond to the local cone estimator for a particular sample point and the dashed linescorrespond to the straight lines for this point with some of the others points in the sample. In green thesame kind of plots are shown, constructed at a different point of the sample. b) Hölder’s cone. The figure isanalogous to a) but with an appropriate curve.4 MonteCarlo studyThis section contains the results of a simulation study designed to evaluate the performance of theestimator defined in Section 3. To perform this study, we considered four models. For each model,we generated X1, . . . , Xn independent samples of size n = 100, 500, 1000 according with a density4f . In each model, we performed N = 1000 replications. The different models were generated fromthe following density functions.Model 1 : fX(x) = 1√2π(2φ(2)−1) exp{−12x2}I(−2,2)(x) in this case the Lipschitz constantw1(f) = e−1/2(√2π(2φ(2)− 1))−1 = 0.2535053 where φ denotes the standar normal distribu-tion function.Model 2 : fX(x) = (x + 1)I(−1,0)(x) + (1 − x)I(0,1) in this case the Lipschitz constantw1(f) = 1.Model 3 : fX(x) = 12((x + 1)I(−1,0)(x) + (1− x)I(0,1)(x) + (x− 1)I(1,2)(x) + (3− x)I(2,3)(x))in this case the Lipschitz constant w1(f) = 12 .Model 4 : fX(x) = 23√xI(0,1)(x) in this case the Hölder constant w 12(f) = 23 .Figure 1 shows the densities considered in models 1 and 4 while Figure 3 shows the densityfunctions in the models 2 and 3.a) b)−1.0 −0.5 0.0 0.5 1.00.00.20.40.60.81.01.2−1 0 1 2 30.00.20.40.60.81.0Figure 3: a) Density function of Model 2. b) Density function of Model 3.In all cases, for smoothing procedure, we take the gaussian kernel and we consider an auto-matic bandwidth selector given by Scott (1992). To study the performance of the estimators ofwγ(f), denoted by ŵγ(fn), we have considered the mean square error (MSE) calculated over the Nreplications, i.e.MSE(ŵγ(fn)) =1NN∑i=1[ŵγ(fn)i − wγ(f)]2where ŵγ(fn)i denotes the estimator of the constant given by the replication i.Table 1 gives the values of the MSE of ŵγ(fn) for the four considered models. The simulationstudy confirms the good behavior of constant estimators, under the different models. Model 1corresponds to a infinitely differential function and as we expected the estimator of the constant5has a good performance. The densities considered in Models 2 and 3 correspond to non differentiablefunctions however the behavior is comparable. Finally, Model 4 is an example of the Hölder casewhere we also obtained a good performance.Model 1 Model 2 Model 3 Model 4n = 100 0.0101 0.0113 0.0281 0.0174n = 500 0.0050 0.0084 0.0051 0.0062n = 1000 0.0037 0.0096 0.0016 0.0047Table 1: MSE for the different models.5 Smoothed bootstrap confidence intervalsIn this section, we develop a bootstrap procedure to implement confidence intervals for the Hölderconstant. One of the main motivations to construct confidence intervals is to be able to answerquestions about the true value of the Holder constant. For example, if a 100(1 − α)% confidenceintervals for wγ(f) does not contain the value c0, this is evidence against the null hypothesis thatwγ(f) is c0. In this way, as usual, the confidence intervals can provide a test for such null hypothesis.Moreover, confidence intervals give an idea about the global variability of the estimator.The bootstrap method introduced by Efron (1979) is a very useful tool used primarily to estimatethe sampling distribution of some statistic of interest. Given a set of independent and identicallydistributed observations X1, . . . , Xn, the estimator of the Hölder constant ŵγ(fn) can be definedas some function T (X1, . . . , Xn). The smoothed bootstrap technic (Efron (1979, 1982); Silvermanand Young, (1987)) estimates the sampling distribution of interest by that of T (X∗1 , . . . , X∗n), forbootstrap samples X∗1 , . . . , X∗n of size n obtained by independent sampling from the distributionFn with density fn defined in (5).In order to construct bootstrap confidence intervals, we consider the smoothed bootstrapmethod. Thus, the bootstrap confidence intervals can be described as follows.Step 1 Based on X1, . . . , Xn, we estimate f through fn(x) = (nh)−1∑nj=1 K(x−Xjh)and computeŵγ(fn).Step 2 A bootstrap sample X∗1 , . . . , X∗n is generated as X∗j = Xij + εj , with i1, . . . , in independentand uniformly distributed integers from {1, . . . , n} and ε1, . . . , εn independent and identicallydistributed with density Kh(t) = h−1K(t/h). The bootstrap estimate ŵ∗γ(f∗n) of wγ(f) iscalculated based on X∗1 , . . . , X∗n.Step 3 Repeat Step 2, B times, to get B values of ŵ∗γ(f∗n)r for 1 ≤ r ≤ B.Step 4 A bootstrap estimator of the variance of ŵγ(fn) is computed asvarB(ŵγ(fn)) =1B − 1B∑r=1(ŵ∗γ(f∗n)r − ŵ∗γ(f∗n))26where ŵ∗γ(f∗n) =1BB∑r=1ŵ∗γ(f∗n)r.Step 5 The 1− α confidence interval of wγ(f) is defined asIC(wγ(f)) =(ŵγ(fn)− q∗(1− α2)√varB(ŵγ(fn)), ŵγ(fn)− q∗(α2)√varB(ŵγ(fn)))where q∗(α) is the α−quantile of the empirical bootstrap distribution of T ∗r = ŵ∗γ(f∗n)r−ŵγ(fn)√varB(ŵγ(fn))for 1 ≤ r ≤ B.To evaluate the performance of the bootstrap intervals described above, we implement this proce-dure to the models considered in the previous Section. The following Table summarizes the resultsof the empirical coverage, defined as the proportion of the bootstrap intervals containing the trueparameter over the N = 1000 replications. For each model, we consider B = 1000 bootstrap repli-cations and in all cases, we compute confidence intervals with level 0.95. Note that, as expected, thebootstrap intervals have a good performance since the empirical coverages are near to the nominallevel 0.95.Model 1 Model 2 Model 3 Model 4n = 100 94.1 95 97 96.5Table 2: Empirical coverage of nominal 95% confidence intervals. The empirical coverages are inpercent.AcknowledgmentsWe wish to thank the anonymous referee for his/her valuable comments which led to an improvedversion of the original article. We would like to thank to Lucio Pantazis for his careful reading ofthe manuscript and for useful suggestions. This research was partially supported by Grants pip112-200801-00216 and 0592 from conicet and UBACYT 20020100300057 from the Universidadde Buenos Aires.References[1] Carando, D.; Fraiman, R. and Groisman, P. (2009). Nonparametric likelihood based estimationfor a multivariate Lipschitz density. Journal of Multivariate Analysis. 100. 981-992.[2] Devroye, L and Lugosi, G. (2002). Almost sure classification of densities. Journal of Nonpara-metric Statistics. 14, 675–698.[3] Efron, B. (1979) Bootstrap methods-another look at the jack-knife. Ann. Statist., 7, 1-26.7[4] Efron, B. (1982) The Jackknife, the Bootstrap and Other Resampling Plans. Philadelphia:Society for Industrial and Applied Mathematics.[5] Fraiman, R. an Meloche, J. (1999). Counting Bumps. Annals of the Institute of StatisticalMathematics, 51, 1–29.[6] Le Cam. L. and Schwartz, L. (1960) A necessary and sufficient condition for the existence ofconsistent estimates. Annals of Statistics. 31, 140–150.[7] Parzen, E. (1962). On estimation of a probability density function and mode. Ann. Math.Statist. 33, 1065–1076.[8] Rosenblatt, M. (1956). Remarks on some nonparametric estimates of a density function. Ann.Math. Statist.27, 832–837.[9] Scott, D. W. (1992) Multivariate Density Estimation: Theory, Practice, and Visualization.Wiley.[10] Silverman, B. W. and Young, G. A. (1987) The bootstrap: to smooth or not to smooth?Biometrika, 74, 469-479.8

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A Note on the Estimation of the Hölder Constant

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