Rates of convergence in the CLT for linear random fields

In this paper, we study sums of linear random fields defined on the lattice Z ² with values in a Hilbert space. The rate of convergence of distributions of such sums to the Gaussian law is discussed, and mild sufficient conditions to obtain an approximation of order n ^−p are presented. This can be considered as a complement of a recent result of [A.N. Nazarova, Logarithmic velocity of convergence in CLT for stochastic linear processes and fields in a Hilbert space, Fundam. Prikl. Mat., 8:1091–1098, 2002 (in Russian)], where the logarithmic rate of convergence was stated, and as a generalization of the result of [D. Bosq, Erratum and complements to Berry–Esseen inequality for linear processes in Hilbert spaces, Stat. Probab. Lett., 70:171–174, 2004] for linear processes.     

The Stratified Sampling Bootstrap for Measuring the Uncertainty in Mortality Forecasts

In this paper, we propose a procedure for reducing the uncertainty in mortality projections, on the basis of a log bilinear Poisson Lee Carter model (Renshaw and Haberman Appl Stat 52:119–137, 2003a). In the literature, because the non-linear nature of the quantities under consideration has prevented analytical solutions, simulation techniques have been used in order to provide prediction intervals for forecasted quantities (for example, Brouhns et al. Scand Actuar J 3:212–224, 2005, Renshaw and Haberman Insur Math Econ 42:797–816, 2008). In this respect, we adopt the bootstrap simulation approach in order to measure the uncertainty affecting mortality projections. In particular, we propose making the bootstrap procedure more efficient by using a specific variance reducing technique, the so-called Stratified Sampling technique. To this end, we propose a two stage simulation bootstrap procedure where variance reducing techniques are combined with the simple bootstrap of the Poisson Lee Carter version. Numerical applications are shown using the results for some datasets.     

The Dirichlet Distribution and Process through Neutralities

A new characterization of the Dirichlet distribution, based on the notion of complete neutrality and a regression version of neutrality, is derived. It unifies earlier characterizations by James and Mosimann (Ann. Stat. 8, 183–189, 1980) and by Seshadri and Wesołowski (Sankhyā, A 65, 248–291, 2003). Also new results on identification of the Dirichlet process in the class of neutral-to-the-right processes are obtained. The proof of the main result makes an extensive use of the method of moments.     

Classification of Refinable Splines in ℝ d

A refinable spline in ℝ ^d is a compactly supported refinable function whose support can be decomposed into simplices such that the function is a polynomial on each simplex. The best-known refinable splines in ℝ ^d are the box splines. Refinable splines play a key role in many applications, such as numerical computation, approximation theory and computer-aided geometric design. Such functions have been classified in one dimension in Dai et al. (Appl. Comput. Harmon. Anal. 22(3), 374–381, 2007), Lawton et al. (Comput. Math. 3, 137–145, 1995). In higher dimensions Sun (J. Approx. Theory 86, 240–252, 1996) characterized those splines when the dilation matrices are of the form A=mI, where m∈ℤ and I is the identity matrix. For more general dilation matrices the problem becomes more complex. In this paper we give a complete classification of refinable splines in ℝ ^d for arbitrary dilation matrices A∈M _d (ℤ).     

A short note on parameter approximation for von Mises-Fisher distributions: and a fast implementation of I s (x)

In high-dimensional directional statistics one of the most basic probability distributions is the von Mises-Fisher (vMF) distribution. Maximum likelihood estimation for the vMF distribution turns out to be surprisingly hard because of a difficult transcendental equation that needs to be solved for computing the concentration parameter κ. This paper is a followup to the recent paper of Tanabe et al. (Comput Stat 22(1):145–157, 2007), who exploited inequalities about Bessel function ratios to obtain an interval in which the parameter estimate for κ should lie; their observation lends theoretical validity to the heuristic approximation of Banerjee et al. (JMLR 6:1345–1382, 2005). Tanabe et al. (Comput Stat 22(1):145–157, 2007) also presented a fixed-point algorithm for computing improved approximations for κ. However, their approximations require (potentially significant) additional computation, and in this short paper we show that given the same amount of computation as their method, one can achieve more accurate approximations using a truncated Newton method. A more interesting contribution of this paper is a simple algorithm for computing I _s (x): the modified Bessel function of the first kind. Surprisingly, our na?ve implementation turns out to be several orders of magnitude faster for large arguments common to high-dimensional data, than the standard implementations in well-established software such as Mathematica ^?, Maple ^?, and Gp/Pari.     

FCLT and MDP for Stochastic Lotka–Volterra Model

In this paper, we continue an asymptotic analysis of a stochastic version of the Lotka–Volterra model for predator–prey interactions. While the fluid approximation and large deviations were shown in Klebaner and Liptser (Ann. Appl. Probab. 11, 1263–1291, 2001) here we establish the diffusion approximation and moderate deviations.     

On the regular variation of ratios of jointly Fréchet random variables

We provide a necessary and sufficient condition for the ratio of two jointly α-Fréchet random variables to be regularly varying. This condition is based on the spectral representation of the joint distribution and is easy to check in practice. Our result motivates the notion of the ratio tail index, which quantifies dependence features that are not characterized by the tail dependence index. As an application, we derive the asymptotic behavior of the quotient correlation coefficient proposed in Zhang (Ann Stat 36(2):1007–1030, 2008) in the dependent case. Our result also serves as an example of a new type of regular variation of products, different from the ones investigated by Maulik et al (J Appl Probab 39(4):671–699, 2002).     

On the Computational Complexity of the Minimum Committee Problem

Two special cases of the Minimum Committee Problem are studied, the Minimum Committee Problem of Finite Sets (MCFS) and the Minimum Committee Problem of a System of Linear Inequalities(MCLE). It is known that the first of these problems is NP-hard (see (Mazurov et al., Proc. Steklov Inst. Math., 1:67–101, 2002)). In this paper we show the NP-hardness of two integer optimization problems connected with it. In addition, we analyze the hardness of approximation to the MCFS problem. In particular, we show that, unless NP⊂TIME(n ^O(loglogn )), for every ε>0 there are no approximation algorithms for this problem with approximation ratio (1–ε)ln (m–1), where m is the number of inclusions in the MCFS problem. To prove this bound we use the SET COVER problem, for which a similar result is known (Feige, J. ACM, 45:634–652, 1998). We also show that the Minimum Committee of Linear Inequalities System (MCLE) problem is NP-hard as well and consider an approximation algorithm for this problem.      

Convergence of the Law of the Environment Seen by the Particle for Directed Polymers in Random Media in the <Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript> Region

We consider the model of directed polymers in an i.i.d. Gaussian or bounded environment (Imbrie and Spencer in J. Stat. Phys. 52(3/4), 609–626, 1988; Carmona and Hu in Probab. Theory Relat. Fields 124(3), 431–457, 2002; Comets et al. in Adv. Stud. Pure Math. 39, 115–142, 2004) in the L ² region. We prove the convergence of the law of the environment seen by the particle.     

Parametric approximation of isotropic and anisotropic elastic flow for closed and open curves

Deckelnick and Dziuk (Math. Comput. 78(266):645–671, 2009) proved a stability bound for a continuous-in-time semidiscrete parametric finite element approximation of the elastic flow of closed curves in \mathbbR^d, d 3 2{\mathbb{R}^d, d\geq2} . We extend these ideas in considering an alternative finite element approximation of the same flow that retains some of the features of the formulations in Barrett et al. (J Comput Phys 222(1): 441–462, 2007; SIAM J Sci Comput 31(1):225–253, 2008; IMA J Numer Anal 30(1):4–60, 2010), in particular an equidistribution mesh property. For this new approximation, we obtain also a stability bound for a continuous-in-time semidiscrete scheme. Apart from the isotropic situation, we also consider the case of an anisotropic elastic energy. In addition to the evolution of closed curves, we also consider the isotropic and anisotropic elastic flow of a single open curve in the plane and in higher codimension that satisfies various boundary conditions.     

Orders on Multisets and Discrete Cones

We study additive representability of orders on multisets (of size k drawn from a set of size n) which satisfy the condition of independence of equal submultisets (IES) introduced by Sertel and Slinko (Ranking committees, words or multisets. Nota di Laboro 50.2002. Center of Operation Research and Economics. The Fundazione Eni Enrico Mattei, Milan, 2002, Econ. Theory 30(2):265–287, 2007). Here we take a geometric view of those orders, and relate them to certain combinatorial objects which we call discrete cones. Following Fishburn (J. Math. Psychol., 40:64–77, 1996) and Conder and Slinko (J. Math. Psychol., 48(6):425–431, 2004), we define functions f(n,k) and g(n,k) which measure the maximal possible deviation of an arbitrary order satisfying the IES and an arbitrary almost representable order satisfying the IES, respectively, from a representable order. We prove that g(n,k) = n − 1 whenever n ≥ 3 and (n, k) ≠ (5, 2). In the exceptional case, g(5,2) = 3. We also prove that g(n,k) ≤ f(n,k) ≤ n and establish that for small n and k the functions g(n,k) and f(n,k) coincide.      

On Quasi-invariance of Infinite Product Measures

Quasi-invariance of infinite product measures is studied when a locally compact second countable group acts on a standard Borel space. A characterization of l ²-quasi-invariant infinite product measures is given. The group that leaves the measure class invariant is also studied. In the case where the group acts on itself by translations, our result extends previous ones obtained by Shepp (Ann. Math. Stat. 36:1107–1112, 1965) and by Hora (Math. Z. 206:169–192, 1991; J. Theor. Probab. 5:71–100, 1992) to all connected Lie groups.      

On Convergence Results for Weak Efficiency in Vector Optimization Problems with Equilibrium Constraints

In this note, we prove that the convergence results for vector optimization problems with equilibrium constraints presented in Wu and Cheng (J. Optim. Theory Appl. 125, 453–472, 2005) are not correct. Actually, we show that results of this type cannot be established at all. This is due to the possible lack, even under nice assumptions, of lower convergence of the solution map for equilibrium problems, already deeply investigated in Loridan and Morgan (Optimization 20, 819–836, 1989) and Lignola and Morgan (J. Optim. Theory Appl. 93, 575–596, 1997).     

Polynomial Birth–Death Distribution Approximation in the Wasserstein Distance

The polynomial birth–death distribution (abbreviated, PBD) on ℐ={0,1,2,…} or ℐ={0,1,2,…,m} for some finite m introduced in Brown and Xia (Ann. Probab. 29:1373–1403, 2001) is the equilibrium distribution of the birth–death process with birth rates {α _i } and death rates {β _i }, where α _i ≥0 and β _i ≥0 are polynomial functions of i∈ℐ. The family includes Poisson, negative binomial, binomial, and hypergeometric distributions. In this paper, we give probabilistic proofs of various Stein’s factors for the PBD approximation with α _i =a and β _i =i+bi(i−1) in terms of the Wasserstein distance. The paper complements the work of Brown and Xia (Ann. Probab. 29:1373–1403, 2001) and generalizes the work of Barbour and Xia (Bernoulli 12:943–954, 2006) where Poisson approximation (b=0) in the Wasserstein distance is investigated. As an application, we establish an upper bound for the Wasserstein distance between the PBD and Poisson binomial distribution and show that the PBD approximation to the Poisson binomial distribution is much more precise than the approximation by the Poisson or shifted Poisson distributions.      

An exact bootstrap for variance of finite-population <Emphasis Type="BoldItalic">L</Emphasis>-statistic

We give an exact formula of a finite-population bootstrap variance estimator for a general class of L-statistic. It is aimed to reduce the computational burden and to eliminate the approximation error, typically present in resampling approximations based on simulation. In the case of the classical nonparametric Efron bootstrap, a similar formula was shown by Hutson and Ernst [A.D. Hutson and M.D. Ernst, The exact bootstrap mean and variance of an L-estimator, J. R. Stat. Soc., Ser. B, 62:89–94, 2000].     

Functional BKR Inequalities,and their Duals,with Applications

The inequality conjectured by van den Berg and Kesten (J. Appl. Probab. 22, 556–569, 1985), and proved by Reimer (Comb. Probab. Comput. 9, 27–32, 2000), states that for A and B events on S, a finite product of finite sets, and P any product measure on S,

Comparison of ‘Tail Method’ Exact Confidence Limits for the Difference of Binomial Probabilities

Suppose that Y ₁ and Y ₂ are independent and have Binomial(n ₁,p ₁) and Binomial (n ₂,p ₂) distributions respectively. Also suppose that θ=p ₁−p ₂ is the parameter of interest. We consider the problem of finding an exact confidence limit (either upper or lower) for θ. The solution to this problem is very important for statistical practice in the health and life sciences. The ‘tail method’ provides a solution to this problem. This method finds the exact confidence limit by exact inversion of a hypothesis test based on a specified test statistic. Buehler (J. Am. Stat. Assoc. 52, 482–493, 1957) described, for the first time, a finite-sample optimality property of this confidence limit. Consequently, this confidence limit is sometimes called a Buehler confidence limit. An early tail method confidence limit for θ was described by Santner and Snell (J. Am. Stat. Assoc. 75, 386–394, 1980) who used the maximum likelihood estimator of θ as the test statistic. This confidence limit is known to be very inefficient (see e.g. Cytel Software, StatXact, version 6, vol. 2, 2004). The efficiency of the confidence limit resulting from the tail method depends greatly on the test statistic on which it is based. We use the results of Kabaila (Stat. Probab. Lett. 52, 145–154, 2001) and Kabaila and Lloyd (Aust. New Zealand J. Stat. 46, 463–469, 2004, J. Stat. Plan. Inference 136, 3145–3155, 2006) to provide a detailed explanation for the dependence of this efficiency on the test statistic. We consider test statistics that are estimators, Z-statistics and approximate upper confidence limits. This explanation is used to find the situations in which the tail method exact confidence limits based on test statistics that are estimators or Z-statistics are least efficient.     

<Emphasis Type="Italic">π</Emphasis>-formulas implied by Dougall’s summation theorem for <Subscript>5</Subscript><Emphasis Type="Italic">F</Emphasis><Subscript>4</Subscript>-series

The general summation theorem for well-poised ₅ F ₄-series discovered by Dougall (Proc. Edinb. Math. Soc. 25:114–132, 1907) is shown to imply several infinite series of Ramanujan-type for 1/π and 1/π ², including those due to Bauer (J. Reine Angew. Math. 56:101–121, 1859) and Glaisher (Q. J. Math. 37:173–198, 1905) as well as some recent ones by Levrie (Ramanujan J. 22:221–230, 2010).     

The Blackwell and Dubins Theorem and Rényi’s Amount of Information Measure: Some Applications

This survey paper provides first for an overview of how quantum-like concepts could be used in macroscopic environments like economics. The paper then argues for the use of the concept of a quantum mechanical wave function as an ‘information wave function’. A rationale is provided on why such interpretation is reasonable. After having defined the ‘information wave function’, Ψ(q), we argue how | Ψ(q)| ² can be interpreted as a Radon-Nikodym derivative. We consider how we can connect, using the | Ψ(q)| ², the Blackwell and Dubins (Ann. Math. Stat. 33:882–886, 1961) Theorem with Rényi’s (Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, 1961) measure of quantity of information. We also define ‘ambiguity of information’ and ‘multi-sourced information’.     

Stability and Instance Optimality for Gaussian Measurements in Compressed Sensing

In compressed sensing, we seek to gain information about a vector x∈ℝ ^N from d ≪ N nonadaptive linear measurements. Candes, Donoho, Tao et al. (see, e.g., Candes, Proc. Intl. Congress Math., Madrid, 2006; Candes et al., Commun. Pure Appl. Math. 59:1207–1223, 2006; Donoho, IEEE Trans. Inf. Theory 52:1289–1306, 2006) proposed to seek a good approximation to x via ℓ ₁ minimization. In this paper, we show that in the case of Gaussian measurements, ℓ ₁ minimization recovers the signal well from inaccurate measurements, thus improving the result from Candes et al. (Commun. Pure Appl. Math. 59:1207–1223, 2006). We also show that this numerically friendly algorithm (see Candes et al., Commun. Pure Appl. Math. 59:1207–1223, 2006) with overwhelming probability recovers the signal with accuracy, comparable to the accuracy of the best k-term approximation in the Euclidean norm when k∼d/ln N.