Central Limit Theorems for the Non‐Parametric Estimation of Time‐Changed Lévy Models

Abstract. Let {Z_t}_{t 0} be a Lévy process with Lévy measure ν and let be a random clock, where g is a non‐negative function and is an ergodic diffusion independent of Z. Time‐changed Lévy models of the form are known to incorporate several important stylized features of asset prices, such as leptokurtic distributions and volatility clustering. In this article, we prove central limit theorems for a type of estimators of the integral parameter β(?):=∫?(x)ν(dx), valid when both the sampling frequency and the observation time‐horizon of the process get larger. Our results combine the long‐run ergodic properties of the diffusion process with the short‐term ergodic properties of the Lévy process Z via central limit theorems for martingale differences. The performance of the estimators are illustrated numerically for Normal Inverse Gaussian process Z and a Cox–Ingersoll–Ross process .     

ESTIMATING A PARAMETER WHEN IT IS KNOWN THAT THE PARAMETER EXCEEDS A GIVEN VALUE

In some statistical problems a degree of explicit, prior information is available about the value taken by the parameter of interest, θ say, although the information is much less than would be needed to place a prior density on the parameter's distribution. Often the prior information takes the form of a simple bound, ‘θ > θ₁ ’ or ‘θ < θ₁ ’, where θ₁ is determined by physical considerations or mathematical theory, such as positivity of a variance. A conventional approach to accommodating the requirement that θ > θ₁ is to replace an estimator, , of θ by the maximum of and θ₁. However, this technique is generally inadequate. For one thing, it does not respect the strictness of the inequality θ > θ₁ , which can be critical in interpreting results. For another, it produces an estimator that does not respond in a natural way to perturbations of the data. In this paper we suggest an alternative approach, in which bootstrap aggregation, or bagging, is used to overcome these difficulties. Bagging gives estimators that, when subjected to the constraint θ > θ₁ , strictly exceed θ₁ except in extreme settings in which the empirical evidence strongly contradicts the constraint. Bagging also reduces estimator variability in the important case for which is close to θ₁, and more generally produces estimators that respect the constraint in a smooth, realistic fashion.     

On Projection‐type Estimators of Multivariate Isotonic Functions

Abstract. Let M be an isotonic real‐valued function on a compact subset of and let be an unconstrained estimator of M. A feasible monotonizing technique is to take the largest (smallest) monotone function that lies below (above) the estimator or any convex combination of these two envelope estimators. When the process is asymptotically equicontinuous for some sequence r_n→∞, we show that these projection‐type estimators are r_n‐equivalent in probability to the original unrestricted estimator. Our first motivating application involves a monotone estimator of the conditional distribution function that has the distributional properties of the local linear regression estimator. Applications also include the estimation of econometric (probability‐weighted moment, quantile) and biometric (mean remaining lifetime) functions.     

Negative Dependence in Sampling

Abstract. The strong Rayleigh property is a new and robust negative dependence property that implies negative association; in fact it implies conditional negative association closed under external fields (CNA+). Suppose that and are two families of 0‐1 random variables that satisfy the strong Rayleigh property and let . We show that {Z_i} conditioned on is also strongly Rayleigh; this turns out to be an easy consequence of the results on preservation of stability of polynomials of Borcea & Brändén (Invent. Math., 177, 2009, 521–569). This entails that a number of important π ps sampling algorithms, including Sampford sampling and Pareto sampling, are CNA+. As a consequence, statistics based on such samples automatically satisfy a version of the Central Limit Theorem for triangular arrays.     

BETWEEN OLSE AND BLUE

Several estimators of X β under the general Gauss–Markov model are considered. Particular attention is paid to those estimators whose efficiency lies between that of the ordinary least squares estimator and that of the best linear unbiased estimator.     

On the Validity of the Bootstrap in Non‐Parametric Functional Regression

Abstract. We consider the functional non‐parametric regression model Y= r( χ )+?, where the response Y is univariate, χ is a functional covariate (i.e. valued in some infinite‐dimensional space), and the error ? satisfies E(? | χ ) = 0. For this model, the pointwise asymptotic normality of a kernel estimator of r (·) has been proved in the literature. To use this result for building pointwise confidence intervals for r (·), the asymptotic variance and bias of need to be estimated. However, the functional covariate setting makes this task very hard. To circumvent the estimation of these quantities, we propose to use a bootstrap procedure to approximate the distribution of . Both a naive and a wild bootstrap procedure are studied, and their asymptotic validity is proved. The obtained consistency results are discussed from a practical point of view via a simulation study. Finally, the wild bootstrap procedure is applied to a food industry quality problem to compute pointwise confidence intervals.     

A REGRESSION APPROACH TO THE ESTIMATION OF THE FINITE POPULATION MEAN IN THE PRESENCE OF NON‐RESPONSE

This paper presents various estimators for estimating the population mean of the study variable y using information on the auxiliary variable x in the presence of non‐response. Properties of the suggested estimators are studied and compared with those of existing estimators. It is shown that the estimators suggested in this paper are among the best of all the estimators considered. An empirical study is carried out to demonstrate the performance of the suggested estimators and of others, and it is found that the empirical results support the theoretical study.     

RELATIONS BETWEEN ERGODICITY AND MEAN DRIFT FOR MARKOV CHAINS1

If {X_n} is an irreducible aperiodic Markov chain on a state apace denote the mean one step change of position, or “drift”, of {X_n} at j. The main result of this note is to show that, when |µ(j)| is bounded, {X_n} admits a stationary distribution {π_j}if and only if for some N > 0 and some state i, lim inf ∑when this holds, the limit infimum is in fact . Many of the known sufficient or necessary criteria for the existence of a stationary distribution can then be derived easily from this and related results.     

CERTAIN BOUNDS CONCERNING CHARACTERISTIC FUNCTIONS1

The largest value of the constant c for which holds over the class of random variables X with non-zero mean and finite second moment, is c=π. Let the random variable (r.v.) X with distribution function F(·) have non-zero mean and finite second moment. In studying a certain random walk problem (Daley, 1976) we sought a bound on the characteristic function of the form for some positive constant c. Of course the inequality is non-trivial only provided that . This note establishes that the best possible constant c =π. The wider relevance of the result is we believe that it underlines the use of trigonometric inequalities in bounding the (modulus of a) c.f. (see e.g. the truncation inequalities in §12.4 of Loève (1963)). In the present case the bound thus obtained is the best possible bound, and is better than the bound (2) |1-?(θ)| ≥ |θEX|-θ²EX²\2 obtained by applying the triangular inequality to the relation which follows from a two-fold integration by parts in the defining equation (*). The treatment of the counter-example furnished below may also be of interest. To prove (1) with c=π, recall that sin u > u(1-u/π) (all real u), so Since |E sinθX|-|E sin(-θX)|, the modulus sign required in (1) can be inserted into (4). Observe that since sin u > u for u < 0, it is possible to strengthen (4) to (denoting max(0,x) by x₊) To show that c=π is the best possible constant in (1), assume without loss of generality that EX > 0, and take θ > 0. Then (1) is equivalent to (6) c < θEX²/{EX-|1-?(θ)|/θ} for all θ > 0 and all r.v.s. X with EX > 0 and EX². Consider the r.v. where 0 < x < 1 and 0 < γ < ∞. Then EX=1, EX²=1+γx², From (4) it follows that |1-?(θ)| > 0 for 0 < |θ| <π|EX|/EX² but in fact this positivity holds for 0 < |θ| < 2π|EX|/EX² because by trigonometry and the Cauchy-Schwartz inequality, |1-?(θ)| > |Re(1-?(θ))| = |E(1-cosθX)| = 2|E sin²θX/2| (10) >2(E sinθX/2)² (11) >(|θEX|-θ²EX²/2π)²/2 > 0, the inequality at (11) holding provided that |θEX|-θ²EX²/2π > 0, i.e., that 0 < |θ| < 2π|EX|/EX². The random variable X at (7) with x= 1 shows that the range of positivity of |1-?(θ)| cannot in general be extended. If X is a non-negative r.v. with finite positive mean, then the identity shows that (1-?(θ))/iθEX is the c.f. of a non-negative random variable, and hence (13) |1-?(θ)| < |θEX| (all θ). This argument fans if pr{X < 0}pr{X> 0} > 0, but as a sharper alternative to (14) |1-?(θ)| < |θE|X||, we note (cf. (2) and (3)) first that (15) |1-?(θ)| < |θEX| +θ²EX²/2. For a bound that is more precise for |θ| close to 0, |1-?(θ)|²= (Re(1-?(θ)))²+ (Im?(θ))² <(θ²EX²/2)²+(|θEX| +θ²EX²_-/π)², so (16) |1-?(θ)| <(|θEX| +θ²EX²_-/π) + |θ|³(EX²)²/8|EX|.     

A Class of Multivariate Distributions

In this paper we introduce a class of multivariate distributions, known as the generalized Liouville distribution and defined by the functional form (0 ≤x_i < ∞, αi > 0, β_i > 0, q_i > 0). It is shown that such distributions can be used to derive both the Dirichlet distribution and the beta distribution of the second kind.     

Estimation bayesienne des effets dans le modele a effets aleatoires de classification double avec interaction

The problem of estimating the effects in a balanced two-way classification with interaction \documentclass{article}\pagestyle{empty}\begin{document}$i = 1, \ldots ,I;j = 1, \ldots ,J;k = 1, \ldots ,K$\end{document} using a random effect model is considered from a Bayesian view point. Posterior distributions of r_i, c_j and t_ij are obtained under the assumptions that r_i, c_j, t_ij and e_ijk are all independently drawn from normal distributions with zero meansand variances \documentclass{article}\pagestyle{empty}\begin{document}$\sigma _r^2 ,\sigma _c^2 ,\sigma _t^2 ,\sigma _e^2$\end{document} respectively. A non informative reference prior is adopted for \documentclass{article}\pagestyle{empty}\begin{document}$\mu ,\sigma _r^2 ,\sigma _c^2 ,\sigma _t^2 ,\sigma _e^2$\end{document}. Various features of thisposterior distribution are obtained. The same features of the psoterior distribution for a fixed effect model are also obtained. A numerical example is given.     

A Class of Normalized Random Measures with an Exact Predictive Sampling Scheme

Abstract. In this article, we define and investigate a novel class of non‐parametric prior distributions, termed the class . Such class of priors is dense with respect to the homogeneous normalized random measures with independent increments and it is characterized by a richer predictive structure than those arising from other widely used priors. Our interest in the class is mainly motivated by Bayesian non‐parametric analysis of some species sampling problems concerning the evaluation of the species relative abundances in a population. We study both the probability distribution of the number of species present in a sample and the probability of discovering a new species conditionally on an observed sample. Finally, by using the coupling from the past method, we provide an exact sampling scheme for the system of predictive distributions characterizing the class .     

How Many Iterations are Sufficient for Efficient Semiparametric Estimation?

Abstract. A common practice in obtaining an efficient semiparametric estimate is through iteratively maximizing the (penalized) full log‐likelihood w.r.t. its Euclidean parameter and functional nuisance parameter. A rigorous theoretical study of this semiparametric iterative estimation approach is the main purpose of this study. We first show that the grid search algorithm produces an initial estimate with the proper convergence rate. Our second contribution is to provide a formula in calculating the minimal number of iterations k ^* needed to produce an efficient estimate . We discover that (i) k ^* depends on the convergence rates of the initial estimate and the nuisance functional estimate, and (ii) k ^* iterations are also sufficient for recovering the estimation sparsity in high dimensional data. The last contribution is the novel construction of which does not require knowing the explicit expression of the efficient score function. The above general conclusions apply to semiparametric models estimated under various regularizations, for example, kernel or penalized estimation. As far as we are aware, this study provides a first general theoretical justification for the ‘one‐/two‐step iteration’ phenomena observed in the semiparametric literature.     

PLUG-IN ESTIMATION OF GENERAL LEVEL SETS

Given an unknown function (e.g. a probability density, a regression function, …) f and a constant c, the problem of estimating the level set L(c) ={f≥c} is considered. This problem is tackled in a very general framework, which allows f to be defined on a metric space different from . Such a degree of generality is motivated by practical considerations and, in fact, an example with astronomical data is analyzed where the domain of f is the unit sphere. A plug‐in approach is followed; that is, L(c) is estimated by L_n(c) ={f_n≥c} , where f_n is an estimator of f. Two results are obtained concerning consistency and convergence rates, with respect to the Hausdorff metric, of the boundaries ?L_n(c) towards ?L(c) . Also, the consistency of L_n(c) to L(c) is shown, under mild conditions, with respect to the L₁ distance. Special attention is paid to the particular case of spherical data.     

The inverse gaussian distribution: Some properties and characterizations

This article presents some structural properties of the inverse Gaussian distribution, together with several new characterizations based on constancy of regression of suitable functions on the sum of n independent identically distributed random variables. A decomposition of the statistic λσ (X^?1_i?X^?1) into n - 1 independent chi-squared random variables, each with one degree of freedom, is given when n is of the form 2^r.     

Measuring the Discrepancy of a Parametric Model via Local Polynomial Smoothing

Abstract. In the context of multivariate mean regression, we propose a new method to measure and estimate the inadequacy of a given parametric model. The measure is basically the missed fraction of variation after adjusting the best possible parametric model from a given family. The proposed approach is based on the minimum L ² ‐distance between the true but unknown regression curve and a given model. The estimation method is based on local polynomial averaging of residuals with a polynomial degree that increases with the dimension d of the covariate. For any d ≥ 1 and under some weak assumptions we give a Bahadur‐type representation of the estimator from which ‐consistency and asymptotic normality are derived for strongly mixing variables. We report the outcomes of a simulation study that aims at checking the finite sample properties of these techniques. We present the analysis of a dataset on ultrasonic calibration for illustration.     

TESTING THE EQUALITY OF PARAMETERS IN THE DISTRIBUTIONS WHEN THE RANGE DEPENDS UPON THE PARAMETER

The authors derive the null and non-null distributions of the test statistic v=y_min/y_max (where y_min= min x_ij, y_max= max x_ij, J=1,2, …, k) connected with testing the equality of scale parameters θ₁, θ₂, …θ_k in certain, class of density functions given by      

Estimators of location based on Kolmogorov-Smirnov-type statistics

Two classes of estimators of a location parameter ø₀ are proposed, based on a nonnegative functional H₁^* of the pair (D₁^ø_N, G^ø_N), where and where F_N is the sample distribution function. The estimators of the first class are defined as a value of ø minimizing H₁^*; the estimators of the second class are linearized versions of those of the first. The asymptotic distribution of the estimators is derived, and it is shown that the Kolmogorov-Smirnov statistic, the signed linear rank statistics, and the Cramérvon Mises statistics are special cases of such functionals H₁^*;. These estimators are closely related to the estimators of a shift in the two-sample case, proposed and studied by Boulanger in B2 (pp. 271–284).     

Geometrically Corrected Second Order Analysis of Events on a Linear Network,with Applications to Ecology and Criminology

Abstract. We study point patterns of events that occur on a network of lines, such as road accidents recorded on a road network. Okabe and Yamada developed a ‘network K function’, analogous to Ripley's K function, for analysis of such data. However, values of the network K‐function depend on the network geometry, making interpretation difficult. In this study we propose a correction of the network K‐function that intrinsically compensates for the network geometry. This geometrical correction restores many natural and desirable properties of K, including its direct relationship to the pair correlation function. For a completely random point pattern, on any network, the corrected network K‐function is the identity. The corrected estimator is intrinsically corrected for edge effects and has approximately constant variance. We obtain exact and asymptotic expressions for the bias and variance of under complete randomness. We extend these results to an ‘inhomogeneous’ network K‐function which compensates for a spatially varying intensity of points. We demonstrate applications to ecology (webs of the urban wall spider Oecobius navus) and criminology (street crime in Chicago).     

Coupling methods in approximations

Let X and Y be two arbitrary k-dimensional discrete random vectors, for k ≥ 1. We prove that there exists a coupling method which minimizes P( X ≠ Y ). This result is used to find the least upper bound for the metric d( X, Y ) = sup_A|P( X ∈ A ) ? P( Y ∈ A )| and to derive the inequality d(Σ X _i, Σ Y _i) ≤ Σd( X _i, Y _i). We thus obtain a unified method to measure the disparity between the distributions of sums of independent random vectors. Several examples are given.