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     

Asymptotic results for random coefficient bifurcating autoregressive processes

The purpose of this paper is to study the asymptotic behaviour of the weighted least-squares estimators of the unknown parameters of random coefficient bifurcating autoregressive processes. Under suitable assumptions on the immigration and the inheritance, we establish the almost sure convergence of our estimators, as well as a quadratic strong law and central limit theorems. Our study mostly relies on limit theorems for vector-valued martingales.     

Social Indeterminacy

An extension of Condorcet's paradox by McGarvey (1953) asserts that for every asymmetric relation R on a finite set of candidates there is a strict‐preferences voter profile that has the relation R as its strict simple majority relation. We prove that McGarvey's theorem can be extended to arbitrary neutral monotone social welfare functions that can be described by a strong simple game G if the voting power of each individual, measured by the Shapley–Shubik power index, is sufficiently small. Our proof is based on an extension to another classic result concerning the majority rule. Condorcet studied an election between two candidates in which the voters' choices are random and independent and the probability of a voter choosing the first candidate is p>1/2. Condorcet's jury theorem asserts that if the number of voters tends to infinity then the probability that the first candidate will be elected tends to one. We prove that this assertion extends to a sequence of arbitrary monotone strong simple games if and only if the maximum voting power for all individuals tends to zero.     

Linear Aggregation of SSB Utility Functionals

A necessary and sufficient condition for linear aggregation of SSB utility functionals is presented. Harsanyi's social aggregation theorem for von Neumann–Morgenstern utility functions is shown to be a corollary to this result. Two generalizations of Fishburn and Gehrlein's conditional linear aggregation theorem for SSB utility functionals are also established.     

二阶常微分方程边值问题

本文通过对非线性常微分方程两边边值问题：ｘ〃＝ｆ（ｔ,ｘ,ｘ’）,ｘ（０）＝Ａ,ｘ（１）＝Ｂ的解的先验估计,并利用Ｌｅｒａｙ－Ｓｃｈａｕｄｅｒ度理论讨论了解的存在性及唯一性。     

Likelihood Factorizations for Mixed Discrete and Continuous Variables

Some general remarks are made about likelihood factorizations, distinguishing parameter-based factorizations and concentration-graph factorizations. Two parametric families of distributions for mixed discrete and continuous variables are discussed. Conditions on graphs are given for the circumstances under which their joint analysis can be split into separate analyses, each involving a reduced set of component variables and parameters. The result shows marked differences between the two families although both involve the same necessary condition on prime graphs. This condition is both necessary and sufficient for simplified estimation in Gaussian and for discrete log linear models.     

Asymptotically optimal Berry-Esseen-type bounds for distributions with an absolutely continuous part

Recursive and closed form upper bounds are offered for the Kolmogorov and the total variation distance between the standard normal distribution and the distribution of a standardized sum of n independent and identically distributed random variables. The method employed is a modification of the method of compositions along with Zolotarev's ideal metric. The approximation error in the CLT obtained vanishes at a rate O(n^−k/2+1), provided that the common distribution of the summands possesses an absolutely continuous part, and shares the same k−1 (k?3) first moments with the standard normal distribution. Moreover, for the first time, these new uniform Berry-Esseen-type bounds are asymptotically optimal, that is, the ratio of the true distance to the respective bound converges to unity for a large class of distributions of the summands. Thus, apart from the correct rate, the proposed error estimates incorporate an optimal asymptotic constant (factor). Finally, three illustrative examples are presented along with numerical comparisons revealing that the new bounds are sharp enough even to be used in practical statistical applications.     

Combining Multivariate Bioassays: Accurate Inference Using Small Sample Asymptotics

For several independent multivariate bioassays performed at different laboratories or locations, the problem of testing the homogeneity of the relative potencies is addressed, assuming the usual slope‐ratio or parallel line assay model. When the homogeneity hypothesis holds, interval estimation of the common relative potency is also addressed. These problems have been investigated in the literature using likelihood‐based methods, under the assumption of a common covariance matrix across the different studies. This assumption is relaxed in this investigation. Numerical results show that the usual likelihood‐based procedures are inaccurate for both of the above problems, in terms of providing inflated type I error probabilities for the homogeneity test, and providing coverage probabilities below the nominal level for the interval estimation of the common relative potency, unless the sample sizes are large, as expected. Correction based on small sample asymptotics is investigated in this article, and this provides significantly more accurate results in the small sample scenario. The results are also illustrated with examples.     

On the Mean Value Theorem for Estimating Functions

Abstract

Feng et al. revealed that the usual mean value theorem (MVT) should not be applied directly to a vector-valued function (e.g., the score function or a general estimating function under a multiparametric model). This note shows that the application of the Cramer–Wold’s device to a corrected version of the MVT is sufficient to obtain standard asymptotics for the estimators attained from vector-valued estimating functions.