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We formulate and prove a generalization of Hardy's inequality (Hardy,1925) in terms of random variables and show that it contains the usual (or familiar) continuous and discrete forms of Hardy's inequality. Next we improve the recent version by Li and Mao of Hardy's inequality with weights for general Borel measures and mixed norms so that it implies the discrete version of Liao and the Hardy inequality with weights of Muckenhoupt as well as the mixed norm versions due to Hardy and Littlewood, Bliss, and Bradley. An equivalent formulation in terms of random variables is given as well. We also formulate a reverse version of Hardy's inequality, the closely related Copson inequality, a reverse Copson inequality and a Carleman-P\'olya-Knopp inequality via random variables. Finally we connect our Copson inequality with counting process martingales and survival analysis, and briefly discuss other applications., Comment: 47 pages

We introduce new shape-constrained classes of distribution functions on R, the bi-$s^*$-concave classes. In parallel to results of D\"umbgen, Kolesnyk, and Wilke (2017) for what they called the class of bi-log-concave distribution functions, we show that every $s$-concave density $f$ has a bi-$s^*$-concave distribution function $F$ for $s^*\leq s/(s+1)$. Confidence bands building on existing nonparametric bands, but accounting for the shape constraint of bi-$s^*$-concavity, are also considered. The new bands extend those developed by D\"umbgen et al. (2017) for the constraint of bi-log-concavity. We also make connections between bi-$s^*$-concavity and finiteness of the Cs\"org\H{o} - R\'ev\'esz constant of $F$ which plays an important role in the theory of quantile processes., Comment: 68 pages, 24 figures; replaces and extends arXiv:2006.03989 by Laha, Miao, and Wellner

Let $b(x)$ be the probability that a sum of independent Bernoulli random variables with parameters $p_1, p_2, p_3, \ldots \in [0,1)$ equals $x$, where $\lambda := p_1 + p_2 + p_3 + \cdots$ is finite. We prove two inequalities for the maximal ratio $b(x)/\pi_\lambda(x)$, where $\pi_\lambda$ is the weight function of the Poisson distribution with parameter $\lambda$.

In this paper, we develop a general approach to proving global and local uniform limit theorems for the Horvitz-Thompson empirical process arising from complex sampling designs. Global theorems such as Glivenko-Cantelli and Donsker theorems, and local theorems such as local asymptotic modulus and related ratio-type limit theorems are proved for both the Horvitz-Thompson empirical process, and its calibrated version. Limit theorems of other variants and their conditional versions are also established. Our approach reveals an interesting feature: the problem of deriving uniform limit theorems for the Horvitz-Thompson empirical process is essentially no harder than the problem of establishing the corresponding finite-dimensional limit theorems. These global and local uniform limit theorems are then applied to important statistical problems including (i) $M$-estimation (ii) $Z$-estimation (iii) frequentist theory of Bayes procedures, all with weighted likelihood, to illustrate their wide applicability., Comment: 46 pages

We present some new and explicit error bounds for the approximation of distributions. The approximation error is quantified by the maximal density ratio of the distribution $Q$ to be approximated and its proxy $P$. This non-symmetric measure is more informative than and implies bounds for the total variation distance. Explicit approximation problems include, among others, hypergeometric by binomial distributions, binomial by Poisson distributions, and beta by gamma distributions. In many cases we provide both upper and (matching) lower bounds., Comment: In Version 6 just one typo was corrected

Classical least squares estimators are well-known to be robust with respect to moment assumptions concerning the error distribution in a wide variety of finite-dimensional statistical problems; generally only a second moment assumption is required for least squares estimators to maintain the same rate of convergence that they would satisfy if the errors were assumed to be Gaussian. In this paper, we give a geometric characterization of the robustness of shape-restricted least squares estimators (LSEs) to error distributions with an $L_{2,1}$ moment, in terms of the `localized envelopes' of the model. This envelope perspective gives a systematic approach to proving oracle inequalities for the LSEs in shape-restricted regression problems in the random design setting, under a minimal $L_{2,1}$ moment assumption on the errors. The canonical isotonic and convex regression models, and a more challenging additive regression model with shape constraints are studied in detail. Strikingly enough, in the additive model both the adaptation and robustness properties of the LSE can be preserved, up to error distributions with an $L_{2,1}$ moment, for estimating the shape-constrained proxy of the marginal $L_2$ projection of the true regression function. This holds essentially regardless of whether or not the additive model structure is correctly specified. The new envelope perspective goes beyond shape constrained models. Indeed, at a general level, the localized envelopes give a sharp characterization of the convergence rate of the $L_2$ loss of the LSE between the worst-case rate as suggested by the recent work of the authors [25], and the best possible parametric rate., Comment: 44 pages, 1 figure

Firstly, we derive in dimension one a new covariance inequality of $L_{1}-L_{\infty}$ type that characterizes the isoperimetric constant as the best constant achieving the inequality. Secondly, we generalize our result to $L_{p}-L_{q}$ bounds for the covariance. Consequently, we recover Cheeger's inequality without using the co-area formula. We also prove a generalized weighted Hardy type inequality that is needed to derive our covariance inequalities and that is of independent interest. Finally, we explore some consequences of our covariance inequalities for $L_{p}$-Poincar\'{e} inequalities and moment bounds. In particular, we obtain optimal constants in general $L_{p}$-Poincar\'{e} inequalities for measures with finite isoperimetric constant, thus generalizing in dimension one Cheeger's inequality, which is a $L_{p}$-Poincar\'{e} inequality for $p=2$, to any real $p\geq 1$.

First we prove some kernel representations for the covariance of two functions taken on the same random variable and deduce kernel representations for some functionals of a continuous one-dimensional measure. Then we apply these formulas to extend Efron's monotonicity property, given in Efron [1965] and valid for independent log-concave measures, to the case of general measures on $\mathbb{R}^2$. The new formulas are also used to derive some further quantitative estimates in Efron's monotonicity property.

Yang (1978) considered an empirical estimate of the mean residual life function on a fixed finite interval. She proved it to be strongly uniformly consistent and (when appropriately standardized) weakly convergent to a Gaussian process. These results are extended to the whole half line, and the variance of the the limiting process is studied. Also, nonparametric simultaneous confidence bands for the mean residual life function are obtained by transforming the limiting process to Brownian motion., Comment: 18 pages

We study the performance of the Least Squares Estimator (LSE) in a general nonparametric regression model, when the errors are independent of the covariates but may only have a $p$-th moment ($p\geq 1$). In such a heavy-tailed regression setting, we show that if the model satisfies a standard `entropy condition' with exponent $\alpha \in (0,2)$, then the $L_2$ loss of the LSE converges at a rate \begin{align*} \mathcal{O}_{\mathbf{P}}\big(n^{-\frac{1}{2+\alpha}} \vee n^{-\frac{1}{2}+\frac{1}{2p}}\big). \end{align*} Such a rate cannot be improved under the entropy condition alone. This rate quantifies both some positive and negative aspects of the LSE in a heavy-tailed regression setting. On the positive side, as long as the errors have $p\geq 1+2/\alpha$ moments, the $L_2$ loss of the LSE converges at the same rate as if the errors are Gaussian. On the negative side, if $p<1+2/\alpha$, there are (many) hard models at any entropy level $\alpha$ for which the $L_2$ loss of the LSE converges at a strictly slower rate than other robust estimators. The validity of the above rate relies crucially on the independence of the covariates and the errors. In fact, the $L_2$ loss of the LSE can converge arbitrarily slowly when the independence fails. The key technical ingredient is a new multiplier inequality that gives sharp bounds for the `multiplier empirical process' associated with the LSE. We further give an application to the sparse linear regression model with heavy-tailed covariates and errors to demonstrate the scope of this new inequality., Comment: 50 pages, 1 figure

We introduce a new shape-constrained class of distribution functions on R, the bi-$s^*$-concave class. In parallel to results of D\"umbgen, Kolesnyk, and Wilke (2017) for what they called the class of bi-log-concave distribution functions, we show that every s-concave density f has a bi-$s^*$-concave distribution function $F$ and that every bi-$s^*$-concave distribution function satisfies $\gamma (F) \le 1/(1+s)$ where finiteness of $$ \gamma (F) \equiv \sup_{x} F(x) (1-F(x)) \frac{| f' (x)|}{f^2 (x)}, $$ the Cs\"org\H{o} - R\'ev\'esz constant of F, plays an important role in the theory of quantile processes on $R$., Comment: 30 pages, 11 figures

Lederer and van de Geer (2013) introduced a new Orlicz norm, the Bernstein-Orlicz norm, which is connected to Bernstein type inequalities. Here we introduce another Orlicz norm, the Bennett-Orlicz norm, which is connected to Bennett type inequalities. The new Bennett-Orlicz norm yields inequalities for expectations of maxima which are potentially somewhat tighter than those resulting from the Bernstein-Orlicz norm when they are both applicable. We discuss cross connections between these norms, exponential inequalities of the Bernstein, Bennett, and Prokhorov types, and make comparisons with results of Talagrand (1989, 1994), and Boucheron, Lugosi, and Massart (2013)., Comment: 24 pages, 2 figures

We study nonparametric maximum likelihood estimation of a log-concave density function $f_0$ which is known to satisfy further constraints, where either (a) the mode $m$ of $f_0$ is known, or (b) $f_0$ is known to be symmetric about a fixed point $m$. We develop asymptotic theory for both constrained log-concave maximum likelihood estimators (MLE's), including consistency, global rates of convergence, and local limit distribution theory. In both cases, we find the MLE's pointwise limit distribution at $m$ (either the known mode or the known center of symmetry) and at a point $x_0 \ne m$. Software to compute the constrained estimators is available in the R package \verb+logcondens.mode+. The symmetry-constrained MLE is particularly useful in contexts of location estimation. The mode-constrained MLE is useful for mode-regression. The mode-constrained MLE can also be used to form a likelihood ratio test for the location of the mode of $f_0$. These problems are studied in separate papers. In particular, in a separate paper we show that, under a curvature assumption, the likelihood ratio statistic for the location of the mode can be used for hypothesis tests or confidence intervals that do not depend on either tuning parameters or nuisance parameters., Comment: 71 pages, 3 figures. This replacement is a revision of the previous submission with a slightly modified title and some reorganizational changes and clarifications made to the paper

We study a likelihood ratio test for the location of the mode of a log-concave density. Our test is based on comparison of the log-likelihoods corresponding to the unconstrained maximum likelihood estimator of a log-concave density and the constrained maximum likelihood estimator where the constraint is that the mode of the density is fixed, say at $m$. The constrained estimation problem is studied in detail in Doss and Wellner [2018]. Here the results of that paper are used to show that, under the null hypothesis (and strict curvature of $-\log f$ at the mode), the likelihood ratio statistic is asymptotically pivotal: that is, it converges in distribution to a limiting distribution which is free of nuisance parameters, thus playing the role of the $\chi_1^2$ distribution in classical parametric statistical problems. By inverting this family of tests we obtain new (likelihood ratio based) confidence intervals for the mode of a log-concave density $f$. These new intervals do not depend on any smoothing parameters. We study the new confidence intervals via Monte Carlo methods and illustrate them with two real data sets. The new intervals seem to have several advantages over existing procedures. Software implementing the test and confidence intervals is available in the R package \verb+logcondens.mode+., Comment: 61 pages, 4 figures

We study the problem of estimating a multivariate convex function defined on a convex body in a regression setting with random design. We are interested in optimal rates of convergence under a squared global continuous $l_2$ loss in the multivariate setting $(d\geq 2)$. One crucial fact is that the minimax risks depend heavily on the shape of the support of the regression function. It is shown that the global minimax risk is on the order of $n^{-2/(d+1)}$ when the support is sufficiently smooth, but that the rate $n^{-4/(d+4)}$ is when the support is a polytope. Such differences in rates are due to difficulties in estimating the regression function near the boundary of smooth regions. We then study the natural bounded least squares estimators (BLSE): we show that the BLSE nearly attains the optimal rates of convergence in low dimensions, while suffering rate-inefficiency in high dimensions. We show that the BLSE adapts nearly parametrically to polyhedral functions when the support is polyhedral in low dimensions by a local entropy method. We also show that the boundedness constraint cannot be dropped when risk is assessed via continuous $l_2$ loss. Given rate sub-optimality of the BLSE in higher dimensions, we further study rate-efficient adaptive estimation procedures. Two general model selection methods are developed to provide sieved adaptive estimators (SAE) that achieve nearly optimal rates of convergence for particular "regular" classes of convex functions, while maintaining nearly parametric rate-adaptivity to polyhedral functions in arbitrary dimensions. Interestingly, the uniform boundedness constraint is unnecessary when risks are measured in discrete $l_2$ norms., Comment: 75 pages

We establish exponential bounds for the hypergeometric distribution which include a finite sampling correction factor, but are otherwise analogous to bounds for the binomial distribution due to Le\'on and Perron (2003) and Talagrand (1994). We also establish a convex ordering for sampling without replacement from populations of real numbers between zero and one: a population of all zeros or ones (and hence yielding a hypergeometric distribution in the upper bound) gives the extreme case., Comment: 38 pages, 5 figures

In this paper, we study the approximation and estimation of $s$-concave densities via R\'enyi divergence. We first show that the approximation of a probability measure $Q$ by an $s$-concave densities exists and is unique via the procedure of minimizing a divergence functional proposed by Koenker and Mizera (2010) if and only if $Q$ admits full-dimensional support and a first moment. We also show continuity of the divergence functional in $Q$: if $Q_n \to Q$ in the Wasserstein metric, then the projected densities converge in weighted $L_1$ metrics and uniformly on closed subsets of the continuity set of the limit. Moreover, directional derivatives of the projected densities also enjoy local uniform convergence. This contains both on-the-model and off-the-model situations, and entails strong consistency of the divergence estimator of an $s$-concave density under mild conditions. One interesting and important feature for the R\'enyi divergence estimator of an $s$-concave density is that the estimator is intrinsically related with the estimation of log-concave densities via maximum likelihood methods. In fact, we show that for $d=1$ at least, the R\'enyi divergence estimators for $s$-concave densities converge to the maximum likelihood estimator of a log-concave density as $s \nearrow 0$. The R\'enyi divergence estimator shares similar characterizations as the MLE for log-concave distributions, which allows us to develop pointwise asymptotic distribution theory assuming that the underlying density is $s$-concave., Comment: 65 pages

Let $\Omega$ be a bounded closed convex set in ${\mathbb R}^d$ with non-empty interior, and let ${\cal C}_r(\Omega)$ be the class of convex functions on $\Omega$ with $L^r$-norm bounded by $1$. We obtain sharp estimates of the $\epsilon$-entropy of ${\cal C}_r(\Omega)$ under $L^p(\Omega)$ metrics, $1\le p\frac{dr}{d+(d-1)r}$ is attained by the closed unit ball. While a general convex body can be approximated by inscribed polytopes, the entropy rate does not carry over to the limiting body. Our results have applications to questions concerning rates of convergence of nonparametric estimators of high-dimensional shape-constrained functions., Comment: 22 pages

In this note we prove the following law of the iterated logarithm for the Grenander estimator of a monotone decreasing density: If $f(t_0) > 0$, $f'(t_0) < 0$, and $f'$ is continuous in a neighborhood of $t_0$, then \begin{eqnarray*} \limsup_{n\rightarrow \infty} \left ( \frac{n}{2\log \log n} \right )^{1/3} ( \widehat{f}_n (t_0 ) - f(t_0) ) = \left| f(t_0) f'(t_0)/2 \right|^{1/3} 2M \end{eqnarray*} almost surely where $ M \equiv \sup_{g \in {\cal G}} T_g = (3/4)^{1/3}$ and $ T_g \equiv \mbox{argmax}_u \{ g(u) - u^2 \} $; here ${\cal G}$ is the two-sided Strassen limit set on $R$. The proof relies on laws of the iterated logarithm for local empirical processes, Groeneboom's switching relation, and properties of Strassen's limit set analogous to distributional properties of Brownian motion., Comment: 11 pages, 3 figures

We review a finite-sampling exponential bound due to Serfling and discuss related exponential bounds for the hypergeometric distribution. We then discuss how such bounds motivate some new results for two-sample empirical processes. Our development complements recent results by Wei and Dudley (2011) concerning exponential bounds for two-sided Kolmogorov - Smirnov statistics by giving corresponding results for one-sided statistics with emphasis on "adjusted" inequalities of the type proved originally by Dvoretzky, Kiefer, and Wolfowitz (1956) and by Massart (1990) for one-sample versions of these statistics., Comment: 16 pages

We prove that the convex least squares estimator (LSE) attains a $n^{-1/2}$ pointwise rate of convergence in any region where the truth is linear. In addition, the asymptotic distribution can be characterized by a modified invelope process. Analogous results hold when one uses the derivative of the convex LSE to perform derivative estimation. These asymptotic results facilitate a new consistent testing procedure on the linearity against a convex alternative. Moreover, we show that the convex LSE adapts to the optimal rate at the boundary points of the region where the truth is linear, up to a log-log factor. These conclusions are valid in the context of both density estimation and regression function estimation., Comment: 35 pages, 5 figures

We review and formulate results concerning log-concavity and strong-log-concavity in both discrete and continuous settings. We show how preservation of log-concavity and strongly log-concavity on $\mathbb{R}$ under convolution follows from a fundamental monotonicity result of Efron (1969). We provide a new proof of Efron's theorem using the recent asymmetric Brascamp-Lieb inequality due to Otto and Menz (2013). Along the way we review connections between log-concavity and other areas of mathematics and statistics, including concentration of measure, log-Sobolev inequalities, convex geometry, MCMC algorithms, Laplace approximations, and machine learning., Comment: 67 pages, 1 figure

We introduce new goodness-of-fit tests and corresponding confidence bands for distribution functions. They are inspired by multi-scale methods of testing and based on refined laws of the iterated logarithm for the normalized uniform empirical process $\mathbb{U}_n (t)/\sqrt{t(1-t)}$ and its natural limiting process, the normalized Brownian bridge process $\mathbb{U}(t)/\sqrt{t(1-t)}$. The new tests and confidence bands refine the procedures of Berk and Jones (1979) and Owen (1995). Roughly speaking, the high power and accuracy of the latter methods in the tail regions of distributions are essentially preserved while gaining considerably in the central region. The goodness-of-fit tests perform well in signal detection problems involving sparsity, as in Ingster (1997), Donoho and Jin (2004) and Jager and Wellner (2007), but also under contiguous alternatives. Our analysis of the confidence bands sheds new light on the influence of the underlying $\phi$-divergences.

We establish global rates of convergence for the Maximum Likelihood Estimators (MLEs) of log-concave and $s$-concave densities on $\mathbb{R}$. The main finding is that the rate of convergence of the MLE in the Hellinger metric is no worse than $n^{-2/5}$ when $-1 < s < \infty$ where $s=0$ corresponds to the log-concave case. We also show that the MLE does not exist for the classes of $s$-concave densities with $s < - 1$., Comment: 38 pages, 1 figure

The k-monotone classes of densities defined on (0, \infty) have been known in the mathematical literature but were for the first time considered from a statistical point of view by Balabdaoui and Wellner (2007, 2010). In these works, the authors generalized the results established for monotone (k=1) and convex (k=2) densities by giving a characterization of the Maximum Likelihood and Least Square estimators (MLE and LSE) and deriving minimax bounds for rates of convergence. For k strictly larger than 2, the pointwise asymptotic behavior of the MLE and LSE studied by Balabdaoui and Wellner (2007) would show that the MLE and LSE attain the minimax lower bounds in a local pointwise sense. However, the theory assumes that a certain conjecture about the approximation error of a Hermite spline holds true. The main goal of the present note is to show why such a conjecture cannot be true. We also suggest how to bypass the conjecture and rebuild the key proofs in the limit theory of the estimators.

Traditional statistical inference considers relatively small data sets and the corresponding theoretical analysis focuses on the asymptotic behavior of a statistical estimator when the number of samples approaches infinity. However, many data sets encountered in modern applications have dimensionality significantly larger than the number of training data available, and for such problems the classical statistical tools become inadequate. In order to analyze high-dimensional data, new statistical methodology and the corresponding theory have to be developed., Comment: Published in at http://dx.doi.org/10.1214/12-STS409 the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org)

We establish global rates of convergence of the Maximum Likelihood Estimator (MLE) of a multivariate distribution function in the case of (one type of) "interval censored" data. The main finding is that the rate of convergence of the MLE in the Hellinger metric is no worse than $n^{-1/3} (\log n)^{\gamma}$ for $\gamma = (5d - 4)/6$.

Case-cohort design, an outcome-dependent sampling design for censored survival data, is increasingly used in biomedical research. The development of asymptotic theory for a case-cohort design in the current literature primarily relies on counting process stochastic integrals. Such an approach, however, is rather limited and lacks theoretical justification for outcome-dependent weighted methods due to non-predictability. Instead of stochastic integrals, we derive asymptotic properties for case-cohort studies based on a general Z-estimation theory for semiparametric models with bundled parameters using modern empirical processes. Both the Cox model and the additive hazards model with time-dependent covariates are considered., Comment: 25 pages

We show that the density of $Z=\mathop {\operatorname {argmax}}\{W(t)-t^2\}$, sometimes known as Chernoff's density, is log-concave. We conjecture that Chernoff's density is strongly log-concave or "super-Gaussian", and provide evidence in support of the conjecture., Comment: Published in at http://dx.doi.org/10.3150/12-BEJ483 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

We develop asymptotic theory for weighted likelihood estimators (WLE) under two-phase stratified sampling without replacement. We also consider several variants of WLEs involving estimated weights and calibration. A set of empirical process tools are developed including a Glivenko-Cantelli theorem, a theorem for rates of convergence of M-estimators, and a Donsker theorem for the inverse probability weighted empirical processes under two-phase sampling and sampling without replacement at the second phase. Using these general results, we derive asymptotic distributions of the WLE of a finite-dimensional parameter in a general semiparametric model where an estimator of a nuisance parameter is estimable either at regular or nonregular rates. We illustrate these results and methods in the Cox model with right censoring and interval censoring. We compare the methods via their asymptotic variances under both sampling without replacement and the more usual (and easier to analyze) assumption of Bernoulli sampling at the second phase., Comment: Published in at http://dx.doi.org/10.1214/12-AOS1073 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Often of primary interest in the analysis of multivariate data are the copula parameters describing the dependence among the variables, rather than the univariate marginal distributions. Since the ranks of a multivariate dataset are invariant to changes in the univariate marginal distributions, rank-based estimators are natural candidates for semiparametric copula estimation. Asymptotic information bounds for such estimators can be obtained from an asymptotic analysis of the rank likelihood, that is, the probability of the multivariate ranks. In this article, we obtain limiting normal distributions of the rank likelihood for Gaussian copula models. Our results cover models with structured correlation matrices, such as exchangeable or circular correlation models, as well as unstructured correlation matrices. For all Gaussian copula models, the limiting distribution of the rank likelihood ratio is shown to be equal to that of a parametric likelihood ratio for an appropriately chosen multivariate normal model. This implies that the semiparametric information bounds for rank-based estimators are the same as the information bounds for estimators based on the full data, and that the multivariate normal distributions are least favorable., Comment: Published in at http://dx.doi.org/10.3150/12-BEJ499 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

Do there exist circular and spherical copulas in $R^d$? That is, do there exist circularly symmetric distributions on the unit disk in $R^2$ and spherically symmetric distributions on the unit ball in $R^d$, $d\ge3$, whose one-dimensional marginal distributions are uniform? The answer is yes for $d=2$ and 3, where the circular and spherical copulas are unique and can be determined explicitly, but no for $d\ge4$. A one-parameter family of elliptical bivariate copulas is obtained from the unique circular copula in $R^2$ by oblique coordinate transformations. Copulas obtained by a non-linear transformation of a uniform distribution on the unit ball in $R^d$ are also described, and determined explicitly for $d=2$., Comment: 32 pages; 15 figures submitted to: Symmetry

We derive an upper bound for the mean of the supremum of the empirical process indexed by a class of functions that are known to have variance bounded by a small constant $\delta$. The bound is expressed in the uniform entropy integral of the class at $\delta$. The bound yields a rate of convergence of minimum contrast estimators when applied to the modulus of continuity of the contrast functions., Comment: 11 pages; submitted to: Electronic Journal of Statistics

Suppose that $\m{U} = (U_1, \ldots , U_d) $ has a Uniform$([0,1]^d)$ distribution, that $\m{Y} = (Y_1 , \ldots , Y_d) $ has the distribution $G$ on $\RR_+^d$, and let $\m{X} = (X_1 , \ldots , X_d) = (U_1 Y_1 , \ldots , U_d Y_d )$. The resulting class of distributions of $\m{X}$ (as $G$ varies over all distributions on $\RR_+^d$) is called the {\sl Scale Mixture of Uniforms} class of distributions, and the corresponding class of densities on $\RR_+^d$ is denoted by $\{\cal F}_{SMU}(d)$. We study maximum likelihood estimation in the family ${\cal F}_{SMU}(d)$. We prove existence of the MLE, establish Fenchel characterizations, and prove strong consistency of the almost surely unique maximum likelihood estimator (MLE) in ${\cal F}_{SMU}(d)$. We also provide an asymptotic minimax lower bound for estimating the functional $f \mapsto f(\m{x})$ under reasonable differentiability assumptions on $f\in{\cal F}_{SMU} (d)$ in a neighborhood of $\m{x}$. We conclude the paper with discussion, conjectures and open problems pertaining to global and local rates of convergence of the MLE., Comment: 39 pages, 4 figures

A bracketing metric entropy bound for the class of Laplace transforms of probability measures on [0,\infty) is obtained through its connection with the small deviation probability of a smooth Gaussian process. Our results for the particular smooth Gaussian process seem to be of independent interest., Comment: 15 pages

We study estimation of multivariate densities $p$ of the form $p(x)=h(g(x))$ for $x\in \mathbb {R}^d$ and for a fixed monotone function $h$ and an unknown convex function $g$. The canonical example is $h(y)=e^{-y}$ for $y\in \mathbb {R}$; in this case, the resulting class of densities [\mathcal {P}(e^{-y})={p=\exp(-g):g is convex}] is well known as the class of log-concave densities. Other functions $h$ allow for classes of densities with heavier tails than the log-concave class. We first investigate when the maximum likelihood estimator $\hat{p}$ exists for the class $\mathcal {P}(h)$ for various choices of monotone transformations $h$, including decreasing and increasing functions $h$. The resulting models for increasing transformations $h$ extend the classes of log-convex densities studied previously in the econometrics literature, corresponding to $h(y)=\exp(y)$. We then establish consistency of the maximum likelihood estimator for fairly general functions $h$, including the log-concave class $\mathcal {P}(e^{-y})$ and many others. In a final section, we provide asymptotic minimax lower bounds for the estimation of $p$ and its vector of derivatives at a fixed point $x_0$ under natural smoothness hypotheses on $h$ and $g$. The proofs rely heavily on results from convex analysis., Comment: Published in at http://dx.doi.org/10.1214/10-AOS840 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

We study and compare three estimators of a discrete monotone distribution: (a) the (raw) empirical estimator; (b) the "method of rearrangements" estimator; and (c) the maximum likelihood estimator. We show that the maximum likelihood estimator strictly dominates both the rearrangement and empirical estimators in cases when the distribution has intervals of constancy. For example, when the distribution is uniform on $\{0, ..., y \}$, the asymptotic risk of the method of rearrangements estimator (in squared $\ell_2$ norm) is $y/(y+1)$, while the asymptotic risk of the MLE is of order $(\log y)/(y+1)$. For strictly decreasing distributions, the estimators are asymptotically equivalent., Comment: 39 pages. See also http://www.stat.washington.edu/www/research/reports/2009/ http://www.stat.washington.edu/jaw/RESEARCH/PAPERS/available.html

Functionals of Brownian bridge arise as limiting distributions in nonparametric statistics. In this paper we will give a derivation of distributions of extrema of the Brownian bridge based on excursion theory for Brownian motion. Only the Poisson character of the excursion process will be used. Particular cases of calculations include the distributions of the Kolmogorov-Smirnov statistic, the Kuiper statistic, and the ratio of the maximum positive ordinate to the minumum negative ordinate., Comment: 20 pages

We establish limit theory for the Grenander estimator of a monotone density near zero. In particular we consider the situation when the true density $f_0$ is unbounded at zero, with different rates of growth to infinity. In the course of our study we develop new switching relations by use of tools from convex analysis. The theory is applied to a problem involving mixtures., Comment: 28 pages. See also: http://www.stat.washington.edu/www/research/reports/2009/tr554r1 http://www.stat.washington.edu/jaw/RESEARCH/PAPERS/available.html

An important tool for statistical research are moment inequalities for sums of independent random vectors. Nemirovski and coworkers (1983, 2000) derived one particular type of such inequalities: For certain Banach spaces $(\B,\|\cdot\|)$ there exists a constant $K = K(\B,\|\cdot\|)$ such that for arbitrary independent and centered random vectors $X_1, X_2, ..., X_n \in \B$, their sum $S_n$ satisfies the inequality $ E \|S_n \|^2 \le K \sum_{i=1}^n E \|X_i\|^2$. We present and compare three different approaches to obtain such inequalities: Nemirovski's results are based on deterministic inequalities for norms. Another possible vehicle are type and cotype inequalities, a tool from probability theory on Banach spaces. Finally, we use a truncation argument plus Bernstein's inequality to obtain another version of the moment inequality above. Interestingly, all three approaches have their own merits., Comment: 23 pages, 1 figure. Revision for American Mathematical Monthly, February 2009. Mark Veraar added as co-author

This paper has been withdrawn by the authors due to a crucial error in a bound on page 19 and some other errors earlier in the paper., Comment: This paper has been withdrawn

In this paper, we study the nonparametric maximum likelihood estimator (MLE) of a convex hazard function. We show that the MLE is consistent and converges at a local rate of $n^{2/5}$ at points $x_0$ where the true hazard function is positive and strictly convex. Moreover, we establish the pointwise asymptotic distribution theory of our estimator under these same assumptions. One notable feature of the nonparametric MLE studied here is that no arbitrary choice of tuning parameter (or complicated data-adaptive selection of the tuning parameter) is required., Comment: Published in at http://dx.doi.org/10.3150/09-BEJ202 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)