Estimation of a k monotone density, part 3: limiting Gaussian versions of the problem; invelopes and envelopes

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1 Estimation of a monotone density, part 3: limiting Gaussian versions of the problem; invelopes and envelopes Fadoua Balabdaoui 1 and Jon A. Wellner University of Washington September, 004 Abstract Let be a positive integer. The limiting distribution of the nonparametric maximum lielihood estimator of a monotone density is given in terms of a smooth stochastic process H described as follows: i H is everywhere above or below Y, the 1 fold integral of two-sided standard Brownian motion plus!/!t when is even or odd. ii H is convex. iii H touches Y at exactly those points where H has changes of slope. We show that H exists if a certain conjecture concerning a particular Hermite interpolation problem holds. The process H 1 is the familiar greatest convex minorant of two-sided Brownian motion plus 1/t, which arises in connection with nonparametric estimation of a monotone 1-monotone function. The process H is the invelope process studied in connection with nonparametric estimation of convex functions up to a scaling of the drift term in Groeneboom, Jongbloed, and Wellner 001a. We therefore refer to H as an invelope process when is even, and as an envelope process when is odd. We establish existence of H for all non-negative integers under the assumption that our ey conjecture holds, and study basic properties of H and its derivatives. Approximate computation of H is possible on finite intervals via the iterative 1 spline algorithm which we use here to illustrate the theoretical results. 1 Research supported in part by National Science Foundation grant DMS Research supported in part by National Science Foundation grants DMS-00330, and NIAID grant R01 AI AMS 000 subject classifications. Primary: 6G05; secondary 60G15, 6E0. Key words and phrases. asymptotic distribution, completely monotone, canonical limit, direct estimation problem, Gaussian, inverse estimation problem, onvex, monotone, mixture model, monotone 1

2 Outline 1. Introduction. The Main Result 3. The processes H c, on [, c] 3.1 Existence and characterization of H,c for even Existence and characterization of H,c for odd Tightness as c 4.1 Existence of points of touch Tightness Completion of the proof of the main theorem Appendix 69-70

3 1 Introduction Consider the following nonparametric estimation problem: X 1,..., X n is a sample from a density g 0 with the property that g 0 is monotone on the support of the distribution of the X i s where 1 is a fixed integer. Before proceeding further, we describe these classes of densities more precisely. For = 1, D 1 is the class of all decreasing nonincreasing densities with respect to Lebesgue measure on R + = 0,. For integers, the class D is the collection of all densities with respect to Lebesgue measure on R + for which j g j x is nonnegative, nonincreasing, and convex for all x R + and j = 0,...,. Another way to describe these shape constrained classes is in terms of scale mixtures of Beta1, densities: it is nown that the classes D correspond exactly to the classes of densities that can be represented as gx = 0 y x y + df y for some distribution function F on R +. For example, when = 1 the class of monotone decreasing densities D 1 corresponds exactly to the class of all scale mixtures of the uniform, or Beta1, 1, density; and when =, the class of convex decreasing densities corresponds to the class of all scale mixtures of the triangular Beta1, density. These correspondences for non-negative functions are due to Williamson 1956 see also Gneiting 1999, and those results carry-over to the class of densities considered here as has been nown for = 1 and =, and as shown in general by Lévy 196 for arbitrary 1. As noted, the case = 1 corresponds to g D 1, the class of monotone decreasing densities. In this case the nonparametric maximum lielihood estimator is the wellnown Grenander estimator, given by the left-continuous slopes of the least concave majorant of the empirical distribution function. It is well nown that if g 0 t 0 < 0 and g 0 is continuous in a neighborhood of t 0, the Grenander estimator ĝ n satisfies n 1/3 ĝ n t 0 g 0 t 0 1 gt 0 g t 0 1/3 d Z H where Z is the slope at zero of the greatest convex minorant H1 of {W t + t : t R} where W is standard two-sided Brownian motion starting at zero; this is due to Praasa Rao 1969, and was reproved by Groeneboom 1985 and Kim and Pollard The results of Groeneboom 1985, 1989 yield methods for computing the exact distribution of Z; see e.g. Groeneboom and Wellner 001. The case =, corresponding to g being a convex and decreasing density, was treated by Groeneboom, Jongbloed, and Wellner 001a, Groeneboom, Jongbloed, and Wellner 001b. Assuming that g 0 t 0 > 0, they showed that the nonparametric maximum lielihood estimator ĝ n of g satisfies n /5 ĝ n t 0 gt 0 n 1/5 ĝ nt 0 g t 0 d 3 c,0 g H 0 3 c,1 g H 0

4 H 3 where 0, H 0 are the second and third derivatives at 0 of the invelope H of Ỹ as described in Theorem.1 of Groeneboom, Jongbloed, and Wellner 001a referred to hereafter as GJW and c,0 g = g t 0 g t 0 /4 1/5, c,1 f = g t 0 3 gt 0 /4 3 1/5. Existence and uniqueness of the process H was established in GJW. Our reason for using a tilde over the H here is to distinguish the process H H of Groeneboom, Jongbloed, and Wellner 001a where the drift term was of the form t 4 at the level of the process Ỹ from the process H here where the drift term is taen to be t 4 /1 for = and of the form!/!t for general in order to mae the problem more stable for large. Our goal in this paper is to establish existence and uniqueness of the corresponding limit processes H for all integers 3. In the companion paper Balabdaoui and Wellner 004c we show that the nonparametric maximum lielihood estimator ĝ n of the monotone density g with g t 0 0 satisfies n /+1 ĝ n t 0 g 0 t 0 n /+1 ĝ 1 n t 0 g 1 t 0 n 1/+1 ĝ n t 0 g t 0 d c,0 gh 0 c,1 gh +1 0 c, gh 0 where H is the invelope or envelope process described in Theorem.1 of the next section and where { } j+1 c,j g = gt 0 j g +1 t +1 0! for j = 0,..., 1. For further bacground concerning the statistical problem, see Balabdaoui and Wellner 004a, and for computational issues, see Balabdaoui and Wellner 004b. Our proof of existence of the processes H on R for 3 proceeds by establishing existence of appropriate processes H c, on [, c] for each c > 0, and then showing that these processes and their first derivatives are tight in C[ K, K] for fixed K > 0 as c. The ey step in proving this tightness is essentially showing that two successive jump points τc and τ c + of two successive jump points of H c, to the left and right of 0 satisfy τ c + τc = O p 1 as. We show in section 4 that this is equivalent to τ + c τ c = O p c / in a re-scaled version of the problem, and that in this re-scaled setting the problem is essentially the same problem arising in the finite-sample problem discussed above and in more detail in Balabdaoui and Wellner 004c. We call the problem 4

5 of showing that 1.1 holds the gap problem. In Section 4, we show that when >, the solution of the gap problem reduces to a conjecture concerning a non-classical Hermite interpolation problem via odd-degree splines. To put the interpolation problem encountered in the section 4 in context, it is useful to review briefly the related complete Hermite interpolation problem for odd-degree splines which is more classical and for which error bounds uniform in the nots are now available. Given a function f C [0, 1] and an increasing sequence 0 = t 0 < t 1 < < t m < t m+1 = 1 where m 1 is an integer, it is well-nown that there exists a unique spline, called the complete spline and denoted here by Cf, of degree 1 with interior nots t 1,, t m that satisfies the + m conditions { Cfti = ft i, i = 1,, m Cf l t 0 = f l t 0, Cf l t m+1 = f l t m+1, l = 0,, 1; see Schoenberg 1963, de Boor 1974, or Nürnberger 1989, page 116, for further discussion. If f C [0, 1], then there exists c > 0 such that sup f Cf c f. 1. 0<t 1 < <t m<1 This uniform in nots bound in the complete interpolation problem was first conjectured by de Boor 1973 in 197 for > 4 as a generalization that goes beyond =, 3 and 4 for which the result was already established see also de Boor However, it too more than 5 years to prove de Boor s conjecture; the proof of 1. is due to Shadrin 001. By a scaling argument, the bound 1. implies that, if f C [a, b], a < b R, the interpolation error in the complete Hermite interpolation problem is uniformly bounded in the nots, and that the bound is of the order of b a. One ey property of the complete spline interpolant Cf is that Cf is the Least Squares approximation of f when f L [0, 1]; i.e., if S t 1,, t m denotes the space of splines of order degree 1 and interior nots t 1,, t m, then 0 Cf f x dx = min S S t 1,,t m 0 Sx f x dx 1.3 see e.g. Schoenberg 1963, de Boor 1974, Nürnberger Consequently, if L denotes the space of bounded functions on [0, 1], then the properly defined map C [0, 1] S t 1,, t m f Cf is the restriction of the orthoprojector, denoted here by P S t, where t = t 1,, t m, from L to L with respect to the inner product g, h = 0 gxhxdx. de Boor 5

6 1974 pointed out that, in order to prove the conjecture, it is enough to prove that sup t P S tg P S t = sup sup t g g is bounded, and this was successfully achieved by Shadrin 001. The Hermite interpolation problem which arises naturally in Section appears to be another variation of Hermite interpolation problems via odd-degree splines, which has not yet been studied in the approximation theory or spline literature. More specifically, if f is some real-valued function in C j [0, 1] for some j, 0 = t 0 < t 1 < < t 4 < t 3 = 1 is a given increasing sequence, then there exists a unique spline Hf of degree 1 and interior nots t 1,, t 4 satisfying the 4 4 conditions Hft i = ft i, and Hf t i = f t i, i = 0,, 3. Note that the spline Hf matches not only the value of the function at the nots but also the value of its first derivative. Thus we should expect the interpolation error to be smaller than in the complete interpolation, and this gives hope that boundedness of the interpolation error also holds in this uncommon problem. Here is our conjecture concerning a uniform in nots error bound for the Hermite interpolant H. Conjecture 1.1 Let a = x 0 < x 1 < < x 3 = b be arbitrary points and 1 r 1. Suppose that f that is a function that is r-times differentiable on [a, b] except for a finite number of points. If Hf denotes the unique interpolating spline of degree 1 that solves the Hermite problem: Hfx j = fx j, and Hf x j = f x j for j = 0,, 3, then there exists a constant C > 0 depending only on such that sup Hft ft Cωf r ; b a b a r t [a,b] where ωf r ; is the modulus of continuity of f r on [a, b]: ωh; δ = sup{ ht ht 1 : t 1, t [a, b], t t 1 δ}. The Main Result Suppose that 1 and let W be a two-sided Brownian motion starting from 0 at 0. Define the Gaussian processes {Y t : t R} by { t s 0 0 s 0 Y t = W s 1ds 1 ds +!! t, t 0, 0 0 t s 0 s W s 1 ds 1 ds +!! t, t < 0, 6

7 and set X t Y t = W t t +1 for t R. Thus dx t = t dt + dw t f,0 tdt + dw t where f,0 is monotone for = 1, convex for =, and, for 3 the -th derivative f,0 t =!/t is convex. Thus we can consider estimation of the function f,0 in Gaussian noise dw t subject to the constraint of convexity of f or monotonicity of f in the case = 1. Here is our main result. Theorem.1 If = 1, or if 3 and Conjecture 1.1 holds, there exists an almost surely uniquely defined stochastic process H characterized by the four following conditions: i H t Y t 0, t R. ii H is -convex; i.e. H iii iv exists and is convex. For any t R, H t = Y t if and only if H changes slope at t; equivalently, H t Y t dh t = 0. If is even, lim t H j t Y j t = 0 for j = 0,, /; if is odd, lim t H t Y t = 0 and lim t H j+1 t Y j+1 t = 0, for j = 0,, 3/. Note that H is below Y for odd and hence is an envelope, while H lies above Y for even and hence is an invelope, a term that was coined by Groeneboom, Jongbloed, and Wellner 001a to describe the situation in the case =. One can view H f as an estimator of f,0, and H +j as estimators of f j,0, j = 1,..., 1. Note that in Balabdaoui and Wellner 004c, section 3, the drift term in the limiting process is equal to!/! t, and hence a slightly different version of Theorem.1 is needed: Corollary.1 Suppose that = 1, or 3 and Conjecture 1.1 holds. If Z is the 1-fold integral of two-sided Brownian motion +!/! t, then there exists an almost surely uniquely defined stochastic process G characterized by the four following conditions: i G t Z t 0, t R. ii G is -convex. 7

8 iii For any t R, G t = Z t if and only if G changes slope at t; equivalently, iv G t Z t dh t = 0. If is even, lim t G j t Z j t = 0 for j = 0,, /; if is odd, lim t G t Z t = 0 and lim t G j+1 t Z j+1 t = 0, for j = 0,, 3/. Proof. Since for all 1, W = d W, it follows that d d Z = Y, or Z = Y. From Theorem.1, it follows that the process G = a.s. H is almost surely uniquely defined by the conditions i-iv of Corollary.1. Remar.1 It follows from the proof that G 0,..., G 0 H 0,..., H 0. Thus these random vectors have the same distribution for all even. For = 1 this gives G = d H = d H since the distribution of H is nown to be symmetric about 0 from Groeneboom [Conjecture: G d 0,..., G 0 = H 0,..., H 0 for all odd as well. ] Our proof of Theorem.1 proceeds along the general lines of the proof for the case = in Groeneboom, Jongbloed, and Wellner 001a. We first establish the existence and give characterizations of processes H c, on [, c], we then show that these processes are tight and converge to the limit process H as c. But there are a number of new difficulties and complications. For example, we have not yet found analogues of the mid-point relations given in Lemma.4 and Corollary. of Groeneboom, Jongbloed, and Wellner 001a. Those arguments are replaced by new results involving perturbations by B-splines. Several of our ey results for the general case involve the theory of splines as given in Nürnberger 1989 and DeVore and Lorentz Some of the arguments setched in Groeneboom, Jongbloed, and Wellner 001a are given in more detail and greater generality here. Throughout the remainder of this paper we assume that Conjecture 1.1 holds. The tightness claims in this paper all depend on the validity of Conjecture 1.1. This paper is organized as follows: In section 3 we establish existence and give characterizations of processes H c, on compact intervals [, c] as solutions of certain minimization problems that can be viewed in terms of estimation of the canonical onvex function t and its derivatives in Gaussian white noise dw t. These problems are slightly different for even and odd due to the different boundary conditions involved, and hence are treated separately for even and odd s. In section 4 we establish tightness of the processes H c, and derivatives H j c, for j {1,..., } 8 d =

9 as c. These arguments rely on the crucial fact that two successive changes of slope τ c + and τc of H c, to the right and left of a fixed point t satisfy τ c + t = O p 1 and t τc = O p 1 as c. In section 5 we combine the results from sections 3 and 4 to complete the proof of Theorem.1. 3 The processes H c, on [, c] To prepare for the proof of Theorem.1, we first consider the problem of minimizing the criterion function Φ c f = 1 f tdt ftdx t 3.1 over the class of -convex functions on [, c] and which satisfy two different sets of boundary conditions depending on the parity of. We will start by considering the case even, >. 3.1 Existence and Characterization of H c, for even Throughout this subsection is assumed to be an even integer, > since the case = is covered by Groeneboom, Jongbloed, and Wellner 001a. Let c > 0 and m 1 and m R l, where = l. Consider the problem of minimizing Φ c over the class of -convex functions satisfying C,m1,m f,, f, f = m 1 and f c,, f c,, fc = m. Proposition 3.1 The functional Φ c admits a unique minimizer in C,m1,m. We preface the proof of the proposition by the following lemma: Lemma 3.1 Let g be a convex function defined on [0, 1] such that g0 = 1 and g1 = where 1 and are arbitrary real constants. If there exists t 0 0, 1 such that gt 0 < M, then gt < M/ on the interval [t L, t U ] where t L = 1 + M/ 1 + M t 0, t U = + M/t 0 + M/. + M Proof. Since g is convex, it is below the chord joining the points 0, 1 and t 0, M and the chord joining the points t 0, M and 1,. We can easily verify that these chords intercept the horizontal line y = M/ at the points t L, M/ and t U, M/ where t L and t U are as defined in the lemma. 9

10 Proof of Proposition 3.1 We first prove that we can restrict ourselves to the class of functions { } C,m1,m,M = f C,m1,m, f > M for some M > 0. Without loss of generality, we assume that f f c; i.e., m 1,1 m 1,. Now, by integrating f twice 4, we have f 4 x = x x sf sds + α 1 x + c + α 0, 3. where α 0 = f 4 = m 1, and α 1 = f 4 c f 4 c sf sds /c = m, m 1, c sf sds /c. Using the change of variable x = t 1c, t [0, 1], and denoting we can write, for all t [0, 1] d t = f t 1c m 1,1 f 4 t 1c t = c t sd sds t 0 t + c m 1,1 t sds t = c t 1 0 t + c m 1,1 t t 0 0 s d sds t 1 sd sds 1 sds + m, m 1, t + m 1, 1 sd sds 0 + m, m 1, t + m 1,. If there exists x 0 [, c] such that 3M/ + m 1,1 < f x 0 < M + m 1,1 for M > 0 large, then 3M/ < d t 0 < M where x 0 = t 0 1c. Let t L and t U be the same numbers defined in Lemma 3.1. Now, since d 0 on [0, 1] recall that it was assumed that f > f c, we have for all 0 t 1 t f 4 t 1c c t m 1,1 + m, m 1, t + m 1, 10 t

11 and in particular, if t [t L, t U ], we have t f 4 t 1c c 1 t t + c t m 1,1 Mc 1 t s d sds 3.3 t L + m, m 1, t + m 1, t t t t L s ds + c m 1,1 + m, m 1, t + m 1, = Mc t 1 tt t 4 L + c t m 1,1 + m, m 1, t + m 1,. Hence, if = 4, this implies that t U t L f t 1c dt is of the order of M. In fact, if M is chosen to be large enough so that the term in 3.3 is positive for all t [t L, t U ], it is easy to establish that, using the fact that 1 t 1 t U and t + t L t L tu t L f t 1c dt α M + α 1 M 1, where α = c 4 1 t U t L t U t L 3 /3, and α 1 = 1 m 1,1 c tu 1 tt t Lt tdt t L tu + m, m 1, t1 tt t Ldt + m t L tu t L 1 tt t Ldt But α does not vanish as M since t L t 0 /, t U t 0 +1/ and t U t L 1/. Therefore, for = 4, if there exists x 0 such that f x 0 < M, then we can find real constants c > 0, c 1 and c 0 such that Φ c f = 1 c tu t L f tdt f t 1c dt c M + c 1 M + c 0, ftdx 4 t ftdx 4 t 3.4 since the second term in 3.4 is of the order of M. Indeed, using integration by parts, we can write ftdx 4 t = X 4 cfc X 4 f f tx 4 tdt 11.

12 where for all t, c Hence, and f t = t f t 3M f sds + t m, m 1, ds + m, m 1, + 3M 6M c + m, m 1, c c sf sds /c. c sds /c ftdx 4 t 1Mc + m, m 1,1 + m 1, + m, sup X 4 t. [,c] This implies that the functions in C,m1,m candidates for the minimization problem. have to be bounded in order to be possible Suppose now that > 4. In order to reach the same conclusion, we are going to show that in this case too, there exist constants c > 0, c 1, and c 0 such that 1 f tdt ftdx t c M + c 1 M + c 0. For this purpose we use induction. Suppose that for j < /, there exists a polynomial P 1,j whose coefficients depend only on c and the first j components of m 1 and m such that we have for all t [0, 1] j f j t 1c P 1,j t, and suppose that there exists a polynomial Q j depending only on t L and c such that Q j > 0 on t L, t U and lastly P,j a polynomial whose coefficients depend on t L, c and the first j components of m 1 and m such that for all t [t L, t U ], we have By integrating f j twice, we have j f j t 1c MQ j t + P,j t. f j x = x x sf j sds + α 1,j x + c + α 0,j, where α 0,j = f j = m 1,j+1 and α 1,j = f j c f j c sf j sds /c = m,j+1 m 1,j+1 c sf j sds /c. 1

13 For j < /, we denote d j t = f j c 1t, for t [0, 1]. By the same change of variable we used before, we can write for all t [0, 1] j f j ct 1 t = c t s j d j sds t 0 + m,j+1 m 1,j+1 t + m 1,j+1 = c t 1 t 0 s j d j sds t + m,j+1 m 1,j+1 t + m 1,j s j d j sds Hence, by using the induction hypothesis, we have for all t [0, 1] j f j t 1c c t 1 which is equivalent to j+1 f j t 1c c 1 t and if t [t L, t U ] j f j t 1c t tl t 0 t 1 s j d j sds sp 1,j sds t + m,j+1 m 1,j+1 t + m 1,j+1 t 0 sp 1,j sds + t t 1 sp 1,j sds m,j+1 m 1,j+1 t m 1,j+1 = P 1,j+1 t, t t 1 sp 1,j sds c t 1 sp 1,j sds + t 1 smq j s + P,j sds 0 t L t 1 sp 1,j sds + m,j+1 m 1,j+1 t + m 1,j+1. This can be rewritten j+1 f j t 1c c M1 t sq j sds + 1 t sp 1,j sds t L 0 t + 1 t P,j sds + t 1 sp 1,j sds t L t m,j+1 m 1,j+1 t m 1,j+1 = MQ j+1 t + P,j+1 t, t tl 13

14 where P 1,j+1, P 1,j+1 and Q j+1 satisfy the same properties assumed in the induction hypothesis. Therefore, there exist two polynomials P and Q such that for all t [t L, t U ], / f t 1c MQt + P t and Q > 0 on t L, t U. Thus, for M chosen large enough tu Φ c f M Q tdt + O p M t L since it can be shown using induction and similar arguments as for the case = 4 that ftdx t = O pm. We conclude that there exists some M > 0 such that we can restrict ourselves to the space C,m1,m,M while searching for the minimizer of Φ c. Let us endow the space C,m1,m,M with the distance dg, h = g h = sup g t h t. t [,c] d is indeed a distance since dg, h = 0 if an only if g and h are equal on [, c] and hence g = h using the boundary conditions; i.e., g p ±c = h p ±c, for p /. Consider a sequence f n n in C,m1,m,M. Denote g n = f n. Since g n n is uniformly bounded and convex on the interval [, c], there exists a subsequence g of g n n and a convex function g such that g = m 1,1, gc = m,1, g M and g converges uniformly to g on [, c] e.g. Roberts and Varberg 1973, pages 17 and 0. Define f as the -fold integral of the limit g that satisfies f 4 = m 1,,, f = m 1, and f 4 c = m,,, fc = m,. Then, f belongs to C,m1,m,M and df, f 0, as. Thus, the space C,m1,m,M, d is compact. It remains to show now that Φ c is continuous with respect to d and that the minimizer is unique. Fix a small ɛ > 0 and consider f and g two elements in C,m1,m,M. Φ c g Φ c f = 1 1 g t f t dt gt ft dx t g t f t c dt + gt ft dx t. Suppose that = 4. By using the expression obtained in 3., we can write gt ft = t t s g s f s ds + α 1 t + c, t [, c] 14

15 where α 1 = c s g s f s ds/c since f±c = g±c and f ±c = g ±c. Therefore, for all t [, c], we have t c sds gt ft t sds df, g + t + cdf, g c t + c = + c t + c df, g c c + c df, g = c df, g. Also, we obtain using the same expression t ft t sds + c sds max m 1,1, m,1, M + m 1, + m, 4 c max m 1,1, m,1, M + m 1, + m, for all t [, c] and the same inequality holds for g. By denoting it follows that 1 K 0 = 4 c max m 1,1, m,1, M + m 1, + m,, g t f t dt 1 gt + ft gt ft dt K 0 gt ft dt ck 0 sup gt ft t [,c] c 3 K 0 df, g. 3.5 Now, using integration by parts and again the fact that f±c = g±c, we can write gt ft dx t = g t f t X tdt 3.6 But, g t f t g f = t g s f s ds

16 for all t [, c]. On the other hand, we obtain using integration by parts c s g s f s ds/c = g f. 3.8 By the triangle inequality, we obtain t g t f t g f + g s f s ds c s g s f s ds/c + t g s f s ds c df, g + t + cdf, g c + c df, g = 3cdf, g. 3.9 Combining 3.5 and 3.9, it follows that Φ c g Φ c f c 3 K 0 + 3c X t dt df, g. Now, let > 4 be an even integer. We have g 4 t f 4 t = t t s g s f s ds + α 1 t + c, t [, c] where α 1 = c s g s f s ds/c we obtain, applying the same techniques used for = 4, that g 4 t f 4 t c df, g, t [, c]. By induction and using the fact that for j = 3,, / g j t f j t = for t [, c] where α 1,j = t t s g j+ s f j+ s ds + α 1,j t + c, c s g j+ s f j+ s ds/c, 16

17 it follows that and in particular sup g j t f j t c j df, g, t [,c] sup gt ft c df, g. t [,c] Now, notice that the identities in 3.6, 3.7, 3.8, and the inequality in 3.9 continue to hold. It follows that there exist constants K j > 0, j =,, / such that for all t [, c] where for j = 3,, / On the other hand, we have f j t, g j t K j K j 4 c K j+ + m,j m 1,j + m 1,j. t g t f t g f + g s f s ds c s g s f s ds/c + t c c 4 df, g + t + cc 4 df, g c 3 + c 3 df, g = 3 c 3 df, g g s f s ds and hence c Φ c g Φ c f c K 0 + 3/c 3 X t dt df, g. We conclude that the functional Φ c admits a minimizer in the class C m1,m,m and hence in C m1,m. This minimizer is unique by the strict convexity of Φ c. The next proposition gives a characterization of the minimizer. 17

18 Proposition 3. The function f c, C,m1,m is the minimizer of Φ c if and only if and where H c, is the -fold integral of f c, satisfying and H c, = Y, H c, H c, c = Y c, H c, H c, t Y t, t [, c], 3.10 H c, t Y t df c, t = 0, 3.11 = Y c = Y,, H c, c,, H c, Our proof of Proposition 3. will use the following lemma. = Y, c = Y c. Lemma 3. Let t 0 [, c]. The probability that there exists a polynomial P of degree such that P t 0 = Y t 0, P t 0 = Y t 0,, P t 0 = Y t and satisfies P Y or P Y in a small neighborhood of t 0 right resp. left neighborhood if t 0 = resp. t 0 = c is equal to 0. Proof. Without loss of generality, we assume that 0 t 0 < c. As a consequence of Blumenthal s 0-1 law and the Marov property of a Brownian motion, the probability that a straight line intercepting a Brownian motion W at the point t 0, W t 0 is above or below W in a neighborhood of t 0 is equal to 0 since W crosses the horizontal line y = W t 0 infinitely many times in such neighborhood with probability 1 see e.g. Durrett 1984, 5, page 14. Suppose that there exist δ > 0 and a polynomial P satisfying the condition in 3.1 and P t Y t for all t [t 0, t 0 + δ] the case P Y can be handled similarly. Denote = P Y. Using the condition in 3.1 and successive integrations by parts, we can establish for all t R the identity P t Y t = t Moreover, we have for all t [t 0, t 0 + δ] t t 0 t 0 t s sds.! t s sds ! 18

19 This implies that there exists a subinterval [t 0 + δ 1, t 0 + δ ] [t 0, t 0 + δ] such that t = P t Y t 0, t [t 0 + δ 1, t 0 + δ ] 3.14 since otherwise, the integral in 3.13 would be strictly negative. But a polynomial P of degree satisfying 3.1 can be written as P t = Y t 0 + Y t 0t t Y t 0 t t 0 1! + P t 0 t t 0,! and therefore, it follows from the inequality in 3.14 that or equivalently Y t 0 + P t 0 t t 0 Y t, t [t 0 + δ 1, t 0 + δ ], W t t P t 0 t t 0 W t t+1, t [t 0 + δ 1, t 0 + δ ]. The latter event occurs with probability 0 since the law of the process {W t + t+1 +1 : t [0, c] is equivalent to the law of the Brownian motion process {W t : t [0, c]}, and the result follows. Proof of Proposition 3.. Let f c, be a function in C,m1,m satisfying 3.10 and To avoid conflicting notations, we replace f c, by f. For an arbitrary function g in C,m1,m, we have and therefore Φ c g Φ c f Using the fact that H j c, and g f = g f + fg f fg f, 3.15 ft gt ft dt gt ft dx t. is the j-fold integral of f for j = 1,,, g i ±c = f i ±c, for i = 0,, / H j j c, ±c = Y ±c, for j = 0,, /, we obtain, using successive integrations by parts, ft gt ft dt gt ft dx t 19

20 = [ H c, = [ = = + ] t Y c t gt ft H c, H c, H c, t Y t g t f t dt g t Y t t f t dt ] t Y c t g t f t f t Y t t f c t dt H c H c, t Y t g t f t dt. = H c, t Y t dg t df t which yields, using the condition in 3.11, = ft gt ft dt H c, t Y t dg t. gt ft dx t Using condition 3.10 and the fact that g is nondecreasing, we conclude that Φ c g Φ c f. Since g was arbitrary, f is the minimizer. In the previous proof, we used implicitly the fact that f and g exist at and c. Hence, we need to chec that such an assumption can be made. First, notice that with probability 1, there exists j {1,, 1} such that H j j c, c Y c. If such a j does not exist, it will follow that there exists a polynomial P of degree such that P i c = Y i c, for i = 0,, 1 and P t Y t, for t in a left neighborhood of c. Indeed, using Taylor expansion of H c, at the point c, we have for some small δ > 0 and u [c δ, c H c, u = H c, c + H c, + ou c H c, c cu c + + 1! u c + H c, c! u c 0

21 = Y c + Y Y c cu c + + 1! u c + H Y u. + ou c Hence, there exists δ 0 > 0 such that the polynomial P given by c, c! u c P u = Y c + Y Y c cu c + + 1! u c + H c, c + 1 u c! satisfies P Y on [c δ 0, c. But by Lemma 3., we now that the probability of the latter event is equal to 0. Consider j 0 the smallest integer in {1,, 1} such that H j 0 c, c Y j 0 c. Notice first that j 0 has to be odd. Besides, since H c, Y, H j 0 c, c Y j 0 c implies H j 0 c, c < Y j 0 c, and by continuity there exists a left neighborhood [c δ, c of c such that H j 0 c, t < Y j 0 t for all t [c δ, c. Hence, if we suppose that g t as t c, where g C,m1,m then u c δ Now, if j 0 = 1 we have c δ = g t and hence lim u c [ g t c δ g t H j 0 c, t Y j 0 t dt as u c. H c, H c, t Y t dt ] t Y c t c δ g th c, t Y tdt = g ch c, c X c c δ g tftdt + g tdx t c δ g c δh c δ X c δ >. c δ c, g tftdt + c δ g tdx t Therefore, when t c, g t converges to a finite limit and we can assume that g c is finite. Using a similar arguments, we can show that lim t g t >. The same conclusion is reached when j 0 < 1. 1

22 Now, suppose that f minimizes Φ c over C,m1,m. Fix a small ɛ > 0 and let t, c. We define the function f t,ɛ on [, c] by f t,ɛ u = u t + u + c fu + ɛ + α 1! 1! u + c 3 + α α 1 u + c 3! = fu + ɛp t u satisfying p i t ±c = 0, for i = 0,, / For this choice of a perturbation function, we have for all u [, c] f t,ɛ u = f u + ɛ u t + + α u + c. Thus, for any ɛ > 0, f t,ɛ is the sum of two convex functions and so it is convex. The condition 3.16 ensures that f t,ɛ remains in the class C,m1,m and the parameters α j, j = 1, 3,, 1 are uniquely determined: c t α = c c 3 α 3 = α 3! c t3 3!. c α 1 = α 1! α c 3 3 3! Since f is the minimizer of Φ c, we have On the other hand, Φ c f ɛ,t Φ c f lim ɛ 0 ɛ = = fup t udu [ H c, [ = H c, u Y u Φ c f ɛ,t Φ c f lim 0. ɛ 0 ɛ p t udx u ] c p t u u Y u p tu ] c + H c, H c, c t. 1! u Y u p tudu u Y u p t udu

23 = H c, u Y u p t udu. = H c, u Y u dp t udu = H c, t Y t, and therefore the condition in 3.10 is satisfied. Similarly, consider the function f ɛ defined as Notice first that, u + c f ɛ u = fu + ɛ fu + β 1! = fu + ɛhu + + β 1 u + c + β 0. f ɛ u = 1 + ɛf u + ɛβ u + c + β u + c! which is convex for ɛ > 0 sufficiently small. In order to have f ɛ in the class C ɛ,m1,m, we choose β, β,, β 0 such that h i ±c = 0, for i = 0,, /. It is easy to chec that the latter conditions determine β,, β 0 uniquely. Thus, we have = fuhudu hudx 0 = lim ɛ 0 Φ c f ɛ Φ c f ɛ = H c, u Y u h udu. = = H c, u Y u dh u H c, u Y u df u and hence condition 3.11 is satisfied. 3

24 3. Existence and Characterization of H c, for odd In the previous section, we proved that the minimization problem for = studied in Groeneboom, Jongbloed, and Wellner 001a can be generalized naturally for any even >. For odd, the problem remains to be formalized. For the particular case = 1, it is very well nown that the stochastic process involved in the limiting distribution of the MLE of a monotone density at a fixed point x 0 under some regularity conditions is determined by the slope at 0 of the greatest convex minorant of the process W t+t, t R. In this case, a switching relationship was exploited as a fundamental tool to derive the asymptotic distribution of the MLE. It is based on the observation that if ĝ n is the MLE the Grenander estimator; i.e., the left derivative of the greatest concave majorant of the empirical distribution G n based on an i.i.d. sample from the true monotone density, then for a fixed a > 0 [ { }] sup s 0 : G n s as is maximal = [ ] ĝ n t a see Groeneboom A similar relationship is currently unnown when > 1. The difficulty is apparent already for = and hence there was a need to formalize the problem differently. As we did for even integers, we need to pose an appropriate minimization problem for odd integers > 1. Wellner 003 revisited the case = 1 and established a necessary and sufficient condition for a function in the class of monotone functions g such that g,[,c] K to be the minimizer of the functional Ψ c g = 1 g tdt gtdw t + t see Theorem 3.1 in Wellner 003. However, the characterization involves two Lagrange parameters which maes the resulting optimizer hard to study. Wellner 003 pointed out that when K = K c, the Lagrange parameters will vanish as c. Here we define the minimization problem differently. Let > 1 be an odd integer, c > 0, m 0 R and m 1 and m R l where = l + 1. Consider the problem of minimizing the same criterion function Φ c introduced in 3.1 over the class C,m0,m 1,m of -convex functions satisfying f,, f 1 = m 1 and f c,, f 1 c = m, and fc = m 0. Proposition 3.3 Φ c C,m0,m 1,m. defined in 3.1 admits a unique minimizer in the class Proof. The proof is very similar to the one we used for even. 4

25 The following proposition gives a characterization for the minimizer. Although the techniques are similar to those developed for even, we prefer to give a detailed proof in order to show clearly the differences between the cases even and odd. Proposition 3.4 The function f c, C,m0,m 1,m if is the minimizer of Φ c if and only H c, t Y t, t [, c] 3.17 and H c, t Y t df c, t = 0, 3.18 where H c, is the -fold integral of f c, satisfying H c, = Y, H c, = Y,, H 3 c, = Y 3, and H c, c = Y c, H c, c = Y c,, H 3 c, H c, = Y. c = Y 3 c, Proof. To avoid conflicting notations, we replace f c, by f. Let f be a function in C,m0,m 1,m satisfying 3.17 and Using the inequality in 3.15, we have for an arbitrary function g in C,m0,m 1,m Φ c g Φ c f ft gt ft dt gt ft dx t. Using the fact that H j c, that is the j-fold integral of f for j = 1,, and the fact gc = fc, H c, = Y, g i+1 ±c = f i+1 ±c, for i = 0,, 3/, and H j j c, ±c = Y ±c, for j = 0,, 3/, 5

26 we obtain by successive integrations by parts ft gt ft dt [ = H c, = [ = =. + = t Y t H c, gt ft dx t ] c gt ft t Y t g t f t dt g H c, t Y t t f t dt H c, t Y t This yields, using the condition in 3.18, g t f t ] c g H c, t Y t t f t dt g H c, t Y t t f t dt H c, t Y t dg t df t. ft gt ft dt gt ft dx t = H c, t Y t dg t. Now, using condition 3.17 and the fact that g is nondecreasing, we conclude that Φ c g Φ c f and that f is the minimizer of Φ c. Conversely, suppose that f minimizes Φ c over the class C,m0,m 1,m. Fix a small ɛ > 0 and let t, c. We define the function f t,ɛ on [, c] by f t,ɛ u = fu + ɛ = fu + ɛp t u u t + 1! + + α u + c u + c + α 1! + α 0! + α 3 u + c 3 3! satisfying p i+1 t ±c = 0, for i = 0,, 3/

27 and p t c = For this choice of a perturbation function, we have for all u [, c] f t,ɛ u = f u + ɛu t + + α u + c. Thus, f t,ɛ is convex for any ɛ > 0 as a sum of two convex functions. The conditions 3.19 and 3.0 ensures that f t,ɛ remains in the class C,m0,m 1,m and the parameters α, α 3,, α 0 are uniquely determined: c t α = c α 3 = 1 c α 3 c 3! + c t3 3!. α = 1 c α c! + + α c c 3!! c α 0 = α 1! + + α c c t +.! 1! Since f is the minimizer of Φ c, we have But Φ c f ɛ Φ c f lim ɛ 0 ɛ = = [ [ =. = fup t udu H c, H c, u Y u u Y u Φ c f ɛ Φ c f lim 0. ɛ 0 ɛ p t udx u ] c p t u p tu ] c H c, u Y u dp t u = H c, t Y t, + H c, H c, u Y u p tudu u Y u p t udu 7

28 and therefore the condition in 3.17 is satisfied. Similarly, consider the function f ɛ defined as f ɛ u = u + c fu + ɛ fu u + c + β + β + + β 1 u + c + β 0 1!! = fu + ɛhu. Notice first that, f ɛ u = 1 + ɛf u + ɛβ u + c which is convex for ɛ small enough. In order to have f ɛ in the class C m0,m 1,m, we choose the coefficients β, β,, β 0 such that h i+1 ±c = 0, for i = 0,, 3/, and hc = 0. It is easy to chec that the previous equations admit a unique solution. Thus, we have = fuhudu hudx u 0 = lim ɛ 0 Φ c f ɛ Φ c f ɛ = H c, u Y u h udu. = = and hence condition 3.18 is satisfied. H c, u Y u dh u H c, u Y u df u, 4 Tightness as c 4.1 Existence of points of touch Although the characterizations given in Propositions 3. and 3.4, indicate that f c, is piecewise linear and the -fold integral of f c, touches Y whenever f c, changes its slope, they do not provide us with any information about the number of the jump points of f c,. It is possible, at least in principle, that f c, does not have any jump point, in which case f c, is a straight line. However, if we tae! m 1 = m =! c,! 4! c4,, c 8

29 when is even, and m 0 = c, m 1 = m =!! c,!! 4! c4,, 1! c when is odd, then with an increasing probability, H c, and Y have to touch each other in, c as c. The next proposition establishes this basic fact. Proposition 4.1 Let ɛ > 0 and consider m 1, m, and m 0 as specified above according to whether is even or odd. Then, there exists c 0 > 0 such that the probability that H c, and Y have at least one point of touch is greater than 1 ɛ for c > c 0 ; i.e., P Y τ = H c, τ for some τ [, c] 1, as c. Proof. We start with even. If H c, and Y do not touch each other at any point in, c, it follows that H c, is a polynomial of degree 1 in which case H c, is fully determined by H i i c, ±c = Y ±c, for i = 0,, / H i c, ±c =! i! c i, for i = /,, /. If we write the polynomial H c, as H c, t = α 1! t + α! t + + α 1 t + α 0, then α = 0 since H c, = H c, c. Because of the same symmetry, α 3 = α 5 = = α +1 = 0. Furthermore, it is easy to establish after some algebra that the coefficients α, α 4,, α are given by and for j =,, /. α j =! j! cj α =!! c, α For α,, α 0, we have different expressions: α = Y j! cj + + α j+ c! c Y c, 9

30 α = Y + Y c α! c + + α! c which can be viewed as the starting values for α j and α j given by α j = Y j and α j = Y j for j = 1,, /. c Y j α c c + Y j α j + 1! cj + + α j+1 c, 3! + j! c+j + + α j c! Let V denote the 1-fold integral of two-sided Brownian motion; i.e., Y t = V t +!! t, t R. We also introduce a j, for j = 1,, defined by and with a j = α j V j a j = α j, for j = 1,, / V j c, for j = + /,,. 4. The coefficients a j, for j =,, are given by the following recursive formula a j =! a j! cj j! cj + + a j+ c,! a =!! c. Now, using the expressions in 4.1 and 4., we can write the value of H c, at the point 0, H c, 0, as a function of the derivatives of V at the boundary points and c and the a j s: H c, 0 = α 0 = Y c + Y α a! + + a!! c + + α 30! c

31 V c + V V + a! c + V = V c + V V V c + V c + V c + a! c c! c! +! a! c! c + a 4 4! c a! = V c + V V c + V c! V c + V c! + a 0. By going bac to the definition of a j for j = 0,,, we can see that a j is proportional to c j. Hence, there exists λ such that a 0 = λ c. One can verify numerically that λ is negative. The following table shows a few values of λ and log λ. Table 1: Table of λ and log λ for some values of even integers. λ log λ Now, denote S c = V c + V V V c + V c + V c! c!. 31

32 However, we have Indeed, for 0 j, S c = O p c / as c. V j c d = 0 c t j dw t. 1 j! By using the change of variable u = ct and W cu d = cw u, we have V j c d = c j 0 d = c j/ 0 1 u j dw cu 1 j! 1 u j dw u. 1 j! Therefore, V j c = O p c j/ as c. Similarly, V j = O p c j/ and therefore S c = O p c /. But since λ < 0, it follows that P H c, 0 Y 0 = P S c + λ c 0 = P S c λ c 0 as c, that is, with probability converging to 1, H c, and Y have at least one point of touch as c. Now, suppose that is odd. The proof is similar but involves a different starting polynomial. Let us assume again that H c, and Y do not have any point of touch in, c. Then, H c, would be a polynomial of degree 1 which can be fully determined by the boundary conditions H i c, ±c =! i! c i, for i = /,, + 1/, 4.3 H c, c = c, 4.4 and H c, = Y, 4.5 H i i c, ±c = Y ±c, for i = 3/,, There exist coefficients α, α,, α 1, α 0 such that H c, t = α 1! t + α! t + + α 1 t + α 0, t [, c]. 3

33 The boundary conditions in 4.3 imply that α = α 3 = = α + = 0. Also, using the same conditions we obtain that and for j 1/ α j =! j! cj α =!! c α j! + + α j+ c.! The one-sided conditions 4.4 and 4.5 imply that for j = 1,, 1/ α = c α! c + + α ! c3 + α +1 c and α = Y α 1! c + + α +1 c α c! respectively. Finally, using the boundary conditions in 4.6 we obtain that and α j = Y j α j = Y j c Y j α c + Y j c j + 1! cj + + α j+ c 3 3! α + j 1! c+j + + α j+1 c! for j = 1,, 1/. Let V continue to denote the 1-fold integral of two-sided Brownian motion and consider a, a 4,, a +1, a, a,, a 0 given by a = a j = α j, for j = 1,, 1/ a = c a! + + a +3 c 3 + a +1 c 3!! a + 1! c+1 1! c + + α +1 c a c! and! a a j = + j + 1! c+j+1 + j 1! c+j + + a j+1 c! 33,

34 for j = 1,, 1/. It follows that H c, 0 = α 0 = Y + Y c = V + V c = S c + a 0 α! c + α 4 4! c α V + V c c! V + V c c! + a 0! c where a 0 =! a! c! c + + a! c. It is easy to see that the coefficients a, a 4,, a 0 are proportional to c, c 4,, c respectively. Therefore, there exists λ such that a 0 = λ c. We can verify numerically that λ > 0 the following table gives some values of λ and log λ for odd. But since it follows that S c = O p c /, P H c, 0 Y 0 = P S c + λ c 0 = P S c λ c = P S c λ c 0 as c, which completes the proof. Table : Table of λ and logλ for some values of odd integers. λ logλ

35 Corollary 4.1 Fix ɛ > 0 and let t, c. There exists c 0 > 0 such that the probability that the process H c, touches Y at two points of touch τ and τ + before and after the point t is larger than 1 ɛ for c > c 0. Proof. We focus on even as the arguments are very similar for odd. Consider first t = 0. We now by Proposition 4.1 that, with very large probability, there exists at least one point of touch before or after 0 as c. By symmetry of two-sided Brownian motion originating at 0 and hence by that of the process Y, there exist two points of touch before and after 0 with very large probability as c. Now, fix t 0 0 and consider the problem of minimizing Φ c,t0 f = 1 = 1 +t0 +t 0 f tdt +t0 +t 0 f tdt over the class of -convex functions satisfying +t0 +t 0 ftdx t +t0 +t 0 ftt dt + dw t f + t 0 =!! + t 0, f 4 + t 0 =! 4! + t 0 4,, f + t 0 = + t 0 and f c + t 0 =!! c + t 0, f 4 c + t 0 =! 4! c + t 0 4,, fc + t 0 = c + t 0. Since adding any constant to and c is irrelevant to the original minimization problem, all the above results hold and in particular that of existence of two points of touch τ and τ + before and after 0 with increasing probability as c. But using the change of variable u = t t 0, Φ c,t0 can be rewritten as Φ c,t0 f = 1 = 1 d = 1 +t0 f u + t 0 du f u + t 0 du g udu +t 0 ftt dt + dw t fu + t 0 u + t 0 dt + dw u + t 0 guu + t 0 dt + dw u 4.7 where in 4.7, we used stationarity of the increments of W and gu = fu + t 0 is -convex satisfying the above boundary conditions at and c. From the latter form of Φ c,t0, we can see that the true -convex is now t+t 0 defined on [, c]. However, the estimation problem is basically the same expect and hence there exist two points of touch before and after t 0 with increasing probability as c. 35

36 4. Tightness One very important element in proving the existence of the process H is tightness of the process H c, and its 1 derivatives when c. The process H can be defined as the limit of H c, as c the same way Groeneboom, Jongbloed, and Wellner 001a did for the special case =. In the latter case, tightness of the process H c, and its derivatives H c,, H c,, and H3 c, was implied by tightness of the distance between the points of touch of H c, with respect to Y. The authors could prove using martingale arguments, that for a fixed ɛ > 0, there exists M > 0 independent of t such that for any fixed t, c, lim sup P [t τ > M] [τ + t > M] ɛ 4.8 c where τ and τ + are respectively the last point of touch before t and the first point of touch after t. Before giving any further details about the difficulties of proving such a property when >, we explain the difference between the result proven in 4.8 and the one stated in Lemma 4.4 and Corollary 4.. By the first result, we only now that not both points of touch τ and τ + are out of control whereas our result implies that they both stay within a bounded distance from the point t with very large probability as c. Therefore, we are claiming a stronger result than the one proved by Groeneboom, Jongbloed, and Wellner 001a. Intuitively, tightness has to be a common property of both the points of touch and this can be seen by using symmetry of the process Y. Indeed, since the latter has the same law whether the Brownian motion W runs from to c or vice versa, it is not hard to be convinced that tightness of one point of touch implies tightness of the other. It should be mentioned here that for proving the existence of two points of touch before and after any fixed point t, the authors claimed that this follows from arguments that are similar to the ones used to show existence of at least one point of touch. We tried to reproduce such arguments but we found the situation somehow different. In fact, we found that the arguments used in the proof of Lemma.1 in Groeneboom, Jongbloed, and Wellner 001a cannot be used similarly to prove the existence of two points of touch unless one of these points of touch is under control. More formally, we need to mae sure that the existing point of touch is tight; i.e., there exists some M > 0 independent of t such that the distance between t and this point of touch is bounded by M with a large probability as c. We find that it is simpler to use a symmetry argument as in Corollary 4.1 to mae the conclusion. As mentioned before, proving tightness was the most crucial point that led in the end to showing the existence of the process H. Groeneboom, Jongbloed, and Wellner 001a were able to prove it by using martingale arguments but more importantly the fact that the process H c,, which is a cubic spline, can be explicitly determined on the excursion interval [τ, τ + ]. Indeed, in the special case of =, the four conditions H c, τ = Y τ, H c, τ + = Y τ + and H c, τ = Y τ, 36

37 H c, τ + = Y τ +, implied by the fact that H,c Y, yield a unique solution. The same conditions hold true for > but are obviously not enough to determine the -th spline H c,. To do so, it seems inevitable to consider the whole set of points of touch along with the boundary conditions at and c, which is rather infeasible since, in principle, the locations of the other points of touch are unnown. However, we shall see that we only need points to be able to determine the spline H c, completely. For >, it seems that the Gaussian problem becomes less local as we need more than one excursion interval in order to study the properties of H c, and its derivatives at a fixed point. Although the special case = gives a lot of insight into the general problem, the arguments by Groeneboom, Jongbloed, and Wellner 001a cannot be readapted directly for the general case of >. In the proof of Lemma 4.4, we sip many technical details as the tightness problem is very similar to the gap problem for the LSE and MLE studied in great detail in Balabdaoui and Wellner 004c. We will also restrict ourselves to even as the case odd can be handled similarly. In order to mae use of the techniques developed in Balabdaoui and Wellner 004c for solving the gap problem, it is very helpful to first change the minimization problem from its current version to a rescaled version. Now consider minimizing g tdt 1 +1 gtt dt + dw t 4.9 over the class of -convex functions on [ 1/+1, c 1/+1 ] satisfying gc 1 +1 = c +1, g c 1 +1 =!! c +1,, g c 1 +1 =!! c +1. Now using the change of variable t = c 1/+1 u, we can write d = c d = c d = c d = c d = c g tdt gtdx t g c 1 +1 udu g c 1 +1 udu g c 1 +1 udu g c 1 +1 udu g c 1 +1 udu gc 1 +1 uc u du + dw c 1 +1 u gc 1 +1 u c +1 gc 1 +1 u c u 1 du + c +1 dw u +1 u du + c +1 c gc 1 +1 u c u du + c +1 dw u +1 c gc 1 +1 uc +1 u dw u du +. c 37 1 dw u c

38 If we set gc 1 +1 u = c +1 hu, then the problem is equivalent to minimizing 1 c +1 h udu c +1 hu u dw u du + c or simply minimizing 1 h udu hu u dw u du +, 4.10 c over the class of -convex function on [, 1] satisfying h±1 = 1, h ±1 =!!,, h ±1 =!! With this new criterion function, the situation is very similar to the finite sample problem treated in Balabdaoui and Wellner 004c. Indeed, as the Gaussian noise vanishes at a rate of 1/ c as c, one can view t dt + dw t/ c as a continuous analogue to dg n t where G n is the empirical distribution of X 1,..., X n i.i.d. with monotone density g 0, and where the true -monotone density is replaced by the -convex function t. Existence and characterization of the minimizer of the criterion function in 4.10 with the boundary conditions 4.11 follows from arguments the same arguments used in the original problem. Furthermore, if h c denotes the minimizer, we claim that the number of jump points of h c that are in the neighborhood of a fixed point t increases to infinity, and the distance between two successive jump points is of the order c /+1 as c. To establish this result, we need the following definition and lemma: Definition 4.1 Let f be a sufficiently differentiable function on a finite interval [a, b], and t 1 t m be m points in [a, b]. The Lagrange interpolating polynomial is the unique polynomial P of degree m 1 which passes through t 1, ft 1,, t m, ft m. Furthermore, P is given by its Newton form or Lagrange form P t = m m ft j j=1 =1 j t t t j t P t = ft 1 + t t 1 [t 1, t ]f + + t t 1 t t m [t 1,, t m ]f where [x 1,, x p ]g denotes the divided difference of g of order p; see, e.g., de Boor 1978, page, Nürnberger 1989, page 4, or DeVore and Lorentz 1993, page

39 Lemma 4.1 Let g be an m-convex function on a finite interval [a, b]; i.e., g m exists and is convex on a, b, and let l m g, x, x 1,, x m be the Lagrange polynomial of degree m 1 interpolating g at the points x i, 1 i m, where a < x 1 x x m < b. Then m+i gx l m g, x, x 1,, x m 0, x [x i, x i+1 ], i = 1,, m 1. Proof. See, e.g., Ubhaya 1989, a, page 35 or Kopotun and Shadrin 003, Lemma 8.3, page 918. The following lemma gives consistency of the derivatives of the LS solution. It is very crucial for proving tightness of the distance between successive points of touch of H c, and Y. Lemma 4. For j {0,, 1} and t R, we have h j! c t j! t j 0, almost surely as c. Proof. We will prove the result for t = 0 as the arguments are similar in the general case. Let us denote ψ c h = 1 h tdt htdh c t where dh c t = t dt + dw t c. Since h c is the minimizer of ψ c, then implying that ψ h c + ɛ h c ψ h c lim = 0 ɛ 0 ɛ h ctdt = h c tdh c t. 4.1 Also, for any -convex function g defined on, 1 that satisfies the boundary conditions in 4.11, we have ψ1 ɛ h c + ɛg ψ h c lim 0 ɛ 0 ɛ 39