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 .     

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 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 .     

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.     

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.     

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.     

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).     

Central Limit Theorems for Reduced U‐Statistics Under Dependence and Their Usefulness

A reduced ‐statistic is a ‐statistic with its summands drawn from a restricted but balanced set of pairs. In this article, central limit theorems are derived for reduced ‐statistics under ‐mixing, which significantly extends the work of Brown & Kildea in various aspects. It will be shown and illustrated that reduced ‐statistics are quite useful in deriving test statistics in various nonparametric testing problems.     

Confidence intervals following Box-Cox transformation

What is the interpretation of a confidence interval following estimation of a Box-Cox transformation parameter λ? Several authors have argued that confidence intervals for linear model parameters ψ can be constructed as if λ. were known in advance, rather than estimated, provided the estimand is interpreted conditionally given $\hat \lambda$. If the estimand is defined as $\psi \left( {\hat \lambda } \right)$, a function of the estimated transformation, can the nominal confidence level be regarded as a conditional coverage probability given $\hat \lambda$, where the interval is random and the estimand is fixed? Or should it be regarded as an unconditional probability, where both the interval and the estimand are random? This article investigates these questions via large-n approximations, small- σ approximations, and simulations. It is shown that, when model assumptions are satisfied and n is large, the nominal confidence level closely approximates the conditional coverage probability. When n is small, this conditional approximation is still good for regression models with small error variance. The conditional approximation can be poor for regression models with moderate error variance and single-factor ANOVA models with small to moderate error variance. In these situations the nominal confidence level still provides a good approximation for the unconditional coverage probability. This suggests that, while the estimand may be interpreted conditionally, the confidence level should sometimes be interpreted unconditionally.     

CONFIDENCE INTERVALS UTILIZING PRIOR INFORMATION IN THE BEHRENS–FISHER PROBLEM

Consider two independent random samples of size f + 1 , one from an N (μ₁, σ²₁) distribution and the other from an N (μ₂, σ²₂) distribution, where σ²₁/σ²₂∈ (0, ∞) . The Welch ‘approximate degrees of freedom’ (‘approximate t‐solution’) confidence interval for μ₁?μ₂ is commonly used when it cannot be guaranteed that σ²₁/σ²₂= 1 . Kabaila (2005, Comm. Statist. Theory and Methods 34 , 291–302) multiplied the half‐width of this interval by a positive constant so that the resulting interval, denoted by J₀, has minimum coverage probability 1 ?α. Now suppose that we have uncertain prior information that σ²₁/σ²₂= 1. We consider a broad class of confidence intervals for μ₁?μ₂ with minimum coverage probability 1 ?α. This class includes the interval J₀, which we use as the standard against which other members of will be judged. A confidence interval utilizes the prior information substantially better than J₀ if (expected length of J)/(expected length of J₀) is (a) substantially less than 1 (less than 0.96, say) for σ²₁/σ²₂= 1 , and (b) not too much larger than 1 for all other values of σ²₁/σ²₂ . For a given f, does there exist a confidence interval that satisfies these conditions? We focus on the question of whether condition (a) can be satisfied. For each given f, we compute a lower bound to the minimum over of (expected length of J)/(expected length of J₀) when σ²₁/σ²₂= 1 . For 1 ?α= 0.95 , this lower bound is not substantially less than 1. Thus, there does not exist any confidence interval belonging to that utilizes the prior information substantially better than J₀.     

Further Properties of Frequentist Confidence Intervals in Regression that Utilize Uncertain Prior Information

Consider a linear regression model with n‐dimensional response vector, regression parameter and independent and identically distributed errors. Suppose that the parameter of interest is where a is a specified vector. Define the parameter where c and t are specified. Also suppose that we have uncertain prior information that . Part of our evaluation of a frequentist confidence interval for is the ratio (expected length of this confidence interval)/(expected length of standard confidence interval), which we call the scaled expected length of this interval. We say that a confidence interval for utilizes this uncertain prior information if: (i) the scaled expected length of this interval is substantially less than 1 when ; (ii) the maximum value of the scaled expected length is not too much larger than 1; and (iii) this confidence interval reverts to the standard confidence interval when the data happen to strongly contradict the prior information. Kabaila and Giri (2009) present a new method for finding such a confidence interval. Let denote the least squares estimator of . Also let and . Using computations and new theoretical results, we show that the performance of this confidence interval improves as increases and decreases.     

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|.     

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.     

A generalized Fleming and Harrington's class of tests for interval‐censored data

The class $G^{\rho,\lambda }$ of weighted log‐rank tests proposed by Fleming & Harrington [Fleming & Harrington (1991) Counting Processes and Survival Analysis, Wiley, New York] has been widely used in survival analysis and is nowadays, unquestionably, the established method to compare, nonparametrically, k different survival functions based on right‐censored survival data. This paper extends the $G^{\rho,\lambda }$ class to interval‐censored data. First we introduce a new general class of rank based tests, then we show the analogy to the above proposal of Fleming & Harrington. The asymptotic behaviour of the proposed tests is derived using an observed Fisher information approach and a permutation approach. Aiming to make this family of tests interpretable and useful for practitioners, we explain how to interpret different choices of weights and we apply it to data from a cohort of intravenous drug users at risk for HIV infection. The Canadian Journal of Statistics 40: 501–516; 2012 © 2012 Statistical Society of Canada     

Multivariate Kendall's tau for change‐point detection in copulas

Statistical procedures for the detection of a change in the dependence structure of a series of multivariate observations are studied in this work. The test statistics that are proposed are $L_1$ , $L_2$ , and $L_{\infty }$ distances computed from vectors of differences of Kendall's tau; two multivariate extensions of Kendall's measure of association are used. Since the distributions of these statistics under the null hypothesis of no change depend on the unknown underlying copula of the vectors, a procedure based on the multiplier central limit theorem is used for the computation of p‐values; the method is shown to be valid both asymptotically and for moderate sample sizes. Alternative versions of the tests that take into account possible breakpoints in the marginal distributions are also investigated. Monte Carlo simulations show that the tests are powerful under many scenarios of change‐point. In addition, two estimators of the time of change are proposed and their efficiency is carefully studied. The methodologies are illustrated on simulated series from the Canadian Regional Climate Model. The Canadian Journal of Statistics 41: 65–82; 2013 © 2012 Statistical Society of Canada     

Asymptotics for the Hirsch Index

Abstract. The last decade methods for quantifying the research output of individual researchers have become quite popular in academic policy making. The h‐index (or Hirsch index) constitutes an interesting combined bibliometric volume/impact indicator that has attracted a lot of attention recently. It is now a common indicator, available for instance on the Web of Science. In this article, we establish the asymptotic normality of the empirical h‐index. The rate of convergence is non‐standard: , where f is the density of the citation distribution and n is the number of publications of a researcher. In case that the citations follow a Pareto‐type respectively a Weibull‐type distribution as defined in extreme value theory, our general result specializes well to results that are useful for practical purposes such as the construction of confidence intervals and pairwise comparisons for the h‐index. A simulation study for the Pareto‐type case shows that the asymptotic theory works well for moderate sample sizes already.     

Second‐order bias‐corrected AIC in multivariate normal linear models under non‐normality

This paper deals with a bias correction of Akaike's information criterion (AIC) for selecting variables in multivariate normal linear regression models when the true distribution of observation is an unknown non‐normal distribution. It is well known that the bias of AIC is $O(1)$ , and there are a number of the first‐order bias‐corrected AICs which improve the bias to $O(n^{-1})$ , where $n$ is the sample size. A new information criterion is proposed by slightly adjusting the first‐order bias‐corrected AIC. Although the adjustment is achieved by merely using constant coefficients, the bias of the new criterion is reduced to $O(n^{-2})$ . Then, a variance of the new criterion is also improved. Through numerical experiments, we verify that our criterion is superior to others. The Canadian Journal of Statistics 39: 126–146; 2011 © 2011 Statistical Society of Canada     

Wavelet Thresholding Estimation in a Poissonian Interactions Model with Application to Genomic Data

This paper deals with the study of dependencies between two given events modelled by point processes. In particular, we focus on the context of DNA to detect favoured or avoided distances between two given motifs along a genome suggesting possible interactions at a molecular level. For this, we naturally introduce a so‐called reproduction function h that allows to quantify the favoured positions of the motifs and that is considered as the intensity of a Poisson process. Our first interest is the estimation of this function h assumed to be well localized. The estimator based on random thresholds achieves an oracle inequality. Then, minimax properties of on Besov balls are established. Some simulations are provided, proving the good practical behaviour of our procedure. Finally, our method is applied to the analysis of the dependence between promoter sites and genes along the genome of the Escherichia coli bacterium.     

Goodness‐of‐Fit Test for Monotone Functions

Abstract. In this article, we develop a test for the null hypothesis that a real‐valued function belongs to a given parametric set against the non‐parametric alternative that it is monotone, say decreasing. The method is described in a general model that covers the monotone density model, the monotone regression and the right‐censoring model with monotone hazard rate. The criterion for testing is an ‐distance between a Grenander‐type non‐parametric estimator and a parametric estimator computed under the null hypothesis. A normalized version of this distance is shown to have an asymptotic normal distribution under the null, whence a test can be developed. Moreover, a bootstrap procedure is shown to be consistent to calibrate the test.