Log-concave Density Research Articles

Optimization of relative arbitrage

In stochastic portfolio theory, a relative arbitrage is an equity portfolio which is guaranteed to outperform a benchmark portfolio over a finite horizon. When the market is diverse and sufficiently volatile, and the benchmark is the market or a buy-and-hold portfolio, functionally generated portfolios introduced by Fernholz provide a systematic way of constructing relative arbitrages. In this paper we show that if the market portfolio is replaced by the equal or entropy weighted portfolio among many others, no relative arbitrages can be constructed under the same conditions using functionally generated portfolios. We also introduce and study a shaped-constrained optimization problem for functionally generated portfolios in the spirit of maximum likelihood estimation of a log-concave density.

Stein’s method, logarithmic Sobolev and transport inequalities

We develop connections between Stein’s approximation method, logarithmic Sobolev and transport inequalities by introducing a new class of functional inequalities involving the relative entropy, the Stein kernel, the relative Fisher information and the Wasserstein distance with respect to a given reference distribution on \({\mathbb{R}^{d}}\). For the Gaussian model, the results improve upon the classical logarithmic Sobolev inequality and the Talagrand quadratic transportation cost inequality. Further examples of illustrations include multidimensional gamma distributions, beta distributions, as well as families of log-concave densities. As a by-product, the new inequalities are shown to be relevant towards convergence to equilibrium, concentration inequalities and entropic convergence expressed in terms of the Stein kernel. The tools rely on semigroup interpolation and bounds, in particular by means of the iterated gradients of the Markov generator with invariant measure the distribution under consideration. In a second part, motivated by the recent investigation by Nourdin, Peccati and Swan on Wiener chaoses, we address the issue of entropic bounds on multidimensional functionals F with the Stein kernel via a set of data on F and its gradients rather than on the Fisher information of the density. A natural framework for this investigation is given by the Markov Triple structure (E, μ, Γ) in which abstract Malliavin-type arguments may be developed and extend the Wiener chaos setting.

Log-concavity of a mixture of beta distributions

We provide a sufficient condition for a mixture of Beta distributions to have log-concave density. This is true whenever the Beta distributions to be mixed together have constant parameter sum and the mixing weights are themselves log-concave. The result has applications in economics and statistics, which we present in the last section.

Light tails: all summands are large when the empirical mean is large

It is well known that for a fixed number of independent identically distributed summands with light tail, large values of the sample mean are obtained only when all the summands take large values. This paper explores this property as the number of summands tends to infinity. It provides the order of magnitude of the sample mean for which all summands are in some interval containing this value and it also explores the width of this interval with respect to the distribution of the summands in their upper tail. These results are proved for summands with log-concave or nearly log concave densities. Making use of some extension of the Erdös-Rényi law of large numbers it also explores the forming of aggregates in a sequence of i.i.d. random variables. As a by product the connection is established between large exceedances of the local slope of a random walk on growing bins and the theory of extreme order statistics.

Computing confidence intervals for log-concave densities

In Balabdaoui, Rufibach, and Wellner (2009), pointwise asymptotic theory was developed for the nonparametric maximum likelihood estimator of a log-concave density. Here, the practical aspects of their results are explored. Namely, the theory is used to develop pointwise confidence intervals for the true log-concave density. To do this, the quantiles of the limiting process are estimated and various ways of estimating the nuisance parameter appearing in the limit are studied. The finite sample size behavior of these estimated confidence intervals is then studied via a simulation study of the empirical coverage probabilities.

Maximum-likelihood estimation of a log-concave density based on censored data

We consider nonparametric maximum-likelihood estimation of a log-concave density in case of interval-censored, right-censored and binned data. We allow for the possibility of a subprobability density with an additional mass at $+\infty$, which is estimated simultaneously. The existence of the estimator is proved under mild conditions and various theoretical aspects are given, such as certain shape and consistency properties. An EM algorithm is proposed for the approximate computation of the estimator and its performance is illustrated in two examples.

Estimation and application for a new extended Weibull distribution

In this paper, a new extended Weibull distribution with one scale parameter and two shape parameters is introduced. It is investigated that the new model has increasing and upside-down bathtub shaped hazard rate functions. The properties of skewness and Weibull type probability plot are discussed that serve as a tool for model identification. Given progressively type-II censored data, the maximum likelihood estimators of the parameters and corresponding confidence intervals are discussed; then the Gibbs sampling technique is derived to construct the Bayesian estimators of the parameters and the credible intervals based on the assumption that the prior of the scale parameter is gamma density function and priors of the shape parameters are the log-concave density function under the square error loss function. Two real data sets are analyzed for illustrating the application of the new model and the proposed Bayesian approach.

Some optimality properties of FDR controlling rules under sparsity

False Discovery Rate (FDR) and the Bayes risk are two different statistical measures, which can be used to evaluate and compare multiple testing procedures. Recent results show that under sparsity FDR controlling procedures, like the popular Benjamini-Hochberg (BH) procedure, perform also very well in terms of the Bayes risk. In particular asymptotic Bayes optimality under sparsity (ABOS) of BH was shown previously for location and scale models based on log-concave densities. This article extends previous work to a substantially larger set of distributions of effect sizes under the alternative, where the alternative distribution of true signals does not change with the number of tests $m$, while the sample size $n$ slowly increases. ABOS of BH and the corresponding step-down procedure based on FDR levels proportional to $n^{-1/2}$ are proved. A simulation study shows that these asymptotic results are relevant already for relatively small values of $m$ and $n$. Apart from showing asymptotic optimality of BH, our results on the optimal FDR level provide a natural extension of the well known results on the significance levels of Bayesian tests.

Reliable Single Chip Genotyping with Semi-Parametric Log-Concave Mixtures

The common approach to SNP genotyping is to use (model-based) clustering per individual SNP, on a set of arrays. Genotyping all SNPs on a single array is much more attractive, in terms of flexibility, stability and applicability, when developing new chips. A new semi-parametric method, named SCALA, is proposed. It is based on a mixture model using semi-parametric log-concave densities. Instead of using the raw data, the mixture is fitted on a two-dimensional histogram, thereby making computation time almost independent of the number of SNPs. Furthermore, the algorithm is effective in low-MAF situations.Comparisons between SCALA and CRLMM on HapMap genotypes show very reliable calling of single arrays. Some heterozygous genotypes from HapMap are called homozygous by SCALA and to lesser extent by CRLMM too. Furthermore, HapMap's NoCalls (NN) could be genotyped by SCALA, mostly with high probability. The software is available as R scripts from the website www.math.leidenuniv.nl/~rrippe.

Optimal insurance with adverse selection

We solve the principal–agent problem of a monopolist insurer selling to an agent whose riskiness (loss chance) is private information, a problem introduced in Stiglitz's (1977) seminal paper. For an arbitrary type distribution, we prove several properties of optimal menus, such as efficiency at the top and downward distortions elsewhere. We show that these results extend beyond the insurance problem we emphasize. We also prove that the principal always prefers an agent facing a larger loss and prefers a poorer one if the agent's risk aversion decreases with wealth. For the standard case of a continuum of types and a smooth density, we show that, under the mild assumptions of a log-concave density and decreasing absolute risk aversion, the optimal premium is backward-S-shaped in the amount of coverage—first concave, then convex. This curvature result implies that quantity discounts are consistent with adverse selection in insurance, contrary to the conventional wisdom from competitive models.

Optimal product variety in a Hotelling model

Consider the Hotelling linear spatial duopoly with firm uncertainty over the consumer mean. As uncertainty about the mean grows relative to the dispersion of consumers, competitive locations become socially optimal. A limit result for discontinuous, log-concave densities is also established.

A note on generating random variables with log-concave densities

We present a black-box style rejection method that is valid for generating random variables with any log-concave density, provided that one knows a mode of the density. When the density is only known up to a constant factor, that method is no longer applicable.

A Smooth ROC Curve Estimator Based on Log-Concave Density Estimates

We introduce a new smooth estimator of the ROC curve based on log-concave density estimates of the constituent distributions. We show that our estimate is asymptotically equivalent to the empirical ROC curve if the underlying densities are in fact log-concave. In addition, we empirically show that our proposed estimator exhibits an efficiency gain for finite sample sizes with respect to the standard empirical estimate in various scenarios and that it is only slightly less efficient, if at all, compared to the fully parametric binormal estimate in case the underlying distributions are normal. The estimator is also quite robust against modest deviations from the logconcavity assumption. We show that bootstrap confidence intervals for the value of the ROC curve at a fixed false positive fraction based on the new estimate are on average shorter compared to the approach by Zhou and Qin (2005), while maintaining coverage probability. Computation of our proposed estimate uses the R package logcondens that implements univariate log-concave density estimation and can be done very efficiently using only one line of code. These obtained results lead us to advocate our estimate for a wide range of scenarios.

Explicit error bounds for Markov chain Monte Carlo

We prove explicit, i.e. non-asymptotic, error bounds for Markov chain Monte Carlo methods. The problem is to compute the expectation of a function f with respect to a measure {\pi}. Different convergence properties of Markov chains imply different error bounds. For uniformly ergodic and reversible Markov chains we prove a lower and an upper error bound with respect to the L2 -norm of f . If there exists an L2 -spectral gap, which is a weaker convergence property than uniform ergodicity, then we show an upper error bound with respect to the Lp -norm of f for p > 2. Usually a burn-in period is an efficient way to tune the algorithm. We provide and justify a recipe how to choose the burn-in period. The error bounds are applied to the problem of the integration with respect to a possibly unnormalized density. More precise, we consider the integration with respect to log-concave densities and the integration over convex bodies. By the use of the Metropolis algorithm based on a ball walk and the hit-and-run algorithm it is shown that both problems are polynomial tractable.

Interpolating Thin-Shell and Sharp Large-Deviation Estimates for Lsotropic Log-Concave Measures

Given an isotropic random vector X with log-concave density in Euclidean space \({\mathbb{R}^n}\) , we study the concentration properties of |X| on all scales, both above and below its expectation. We show in particular that $$\begin{array}{l} \mathbb{P}\left ( \left | |X| - \sqrt{n} \right | \geq t\sqrt{n} \right ) \leq C \, {\rm exp} \left ( -cn^{1/2} {\rm min}(t^{3}, t) \right) \; \forall t \geq 0, \end{array}$$ for some universal constants c, C > 0. This improves the best known deviation results on the thin-shell and mesoscopic scales due to Fleury and Klartag, respectively, and recovers the sharp large-deviation estimate of Paouris. Another new feature of our estimate is that it improves when X is \({\psi_{\alpha}\, (\alpha \in(1, 2])}\) , in precise agreement with Paouris’ estimates. The upper bound on the thin-shell width \({\sqrt{\mathbb{V}{\rm ar}(|X|)}}\) we obtain is of the order of n1/3, and improves down to n1/4 when X is \({\psi_{2}}\) . Our estimates thus continuously interpolate between a new best known thin-shell estimate and the sharp large-deviation estimate of Paouris. As a consequence, a new best known bound on the Cheeger isoperimetric constant appearing in a conjecture of Kannan.Lovasz-Simonovits is deduced.

Multivariate log-concave distributions as a nearly parametric model

Abstract In this paper we show that the family P d (lc) of probability distributions on ℝ d with log-concave densities satisfies a strong continuity condition. In particular, it turns out that weak convergence within this family entails (i) convergence in total variation distance, (ii) convergence of arbitrary moments, and (iii) pointwise convergence of Laplace transforms. In this and several other respects the nonparametric model P d (lc) behaves like a parametric model such as, for instance, the family of all d-variate Gaussian distributions. As a consequence of the continuity result, we prove the existence of nontrivial confidence sets for the moments of an unknown distribution in P d (lc). Our results are based on various new inequalities for log-concave distributions which are of independent interest.

Some stochastic inequalities for weighted sums

We compare weighted sums of i.i.d. positive random variables according to the usual stochastic order. The main inequalities are derived using majorization techniques under certain log-concavity assumptions. Specifically, let $Y_i$ be i.i.d. random variables on $\mathbf{R}_+$. Assuming that $\log Y_i$ has a log-concave density, we show that $\sum a_iY_i$ is stochastically smaller than $\sum b_iY_i$, if $(\log a_1,...,\log a_n)$ is majorized by $(\log b_1,...,\log b_n)$. On the other hand, assuming that $Y_i^p$ has a log-concave density for some $p>1$, we show that $\sum a_iY_i$ is stochastically larger than $\sum b_iY_i$, if $(a_1^q,...,a_n^q)$ is majorized by $(b_1^q,...,b_n^q)$, where $p^{-1}+q^{-1}=1$. These unify several stochastic ordering results for specific distributions. In particular, a conjecture of Hitczenko [Sankhy\={a} A 60 (1998) 171--175] on Weibull variables is proved. Potential applications in reliability and wireless communications are mentioned.

Concentration of the information in data with log-concave distributions

A concentration property of the functional ${-}\log f(X)$ is demonstrated, when a random vector X has a log-concave density f on $\mathbb{R}^n$. This concentration property implies in particular an extension of the Shannon-McMillan-Breiman strong ergodic theorem to the class of discrete-time stochastic processes with log-concave marginals.

Optimal Product Variety in a Hotelling Model

In Hotelling style duopoly location games the product variety (or firm locations) is typically not socially optimal. This occurs because the competitive outcome is driven by the density of consumers at the margin while the socially optimal outcome depends on the whole distribution of consumer locations/tastes. We consider a natural extension of the standard model in which firms are imperfectly informed about the distribution of consumers, in particular firms are uncertain about the consumer mean. In the uniform case, as the aggregate uncertainty about the mean becomes large relative to the dispersion of consumers about the mean, competitive locations become socially optimal. A limit result on prices for discontinuous, log-concave densities shows the result will hold in a range of cases.

DUOPOLY MODELS WITH VERTICAL PRODUCT DIFFERENTIATION

This paper deals with duopoly models with vertical product differentiation. These are two-stage games. In the first stage, the quality game, the two firms choose their quality. In the second stage, the price game, they choose their prices. When the market is covered, the properties of this game are well known, even when the consumers are not uniformly distributed. When the market is uncovered the literature is not always clear about these models. In the present paper it is shown that the price game can be solved, when the consumers are distributed according to a distribution function with a log-concave density function.