An applied statistics course for systematics and ecology PhD students

Statistics education is under review at all educational levels. Statistical concepts, as well as the use of statistical methods and techniques, can be taught in at least two contrasting ways. Specifically, (1) teaching can be theoretically and mathematically oriented, or (2) it can be less mathematically oriented being focused, instead, on application and the use of data to solve real-world problems. The second approach is growing in practice and new goals have recently emerged. At present, statistics courses stress probability concepts, data analysis, and the interpretation and communication of results. Understanding the process of statistical investigation is established as a way of improving mastery of statistical reasoning. In this context, a project-based approach allows the design and implementation of participating learning scenarios in order to understand the statistical methodology and, as a consequence, improve research. This approach points out that statistics is a rational methodology used to solve practical problems. The purpose of this paper is to present the design and results of an applied statistics course for PhD students in ecology and systematics using a project-based approach. Examples involving character coding, species classification, and the interpretation of geographical variation, which are the principal systematic analyses requiring statistical techniques, are presented using the results from student projects. In addition, an example from conservation ecology is presented. Results indicate that the students understood the concepts and applied the systematic and statistical techniques accurately using a data oriented approach.     

Teaching of statistics at the secondary level An interdisciplinary approach

This paper examines the educational desirability of introducing the teaching of statistics at the secondary level. A syllabus is proposed for a course in statistics at the secondary level and brief comments are given on each topic, indicating how they may be presented to students.     

A decomposition for someU-type statistics

Define , $S_{k,n} = \Sigma _{1 \leqslant i_1< \cdot \cdot \cdot< l_k \leqslant n} X_{i_1 } \cdot \cdot \cdot X_{i_k } ,n \geqslant k \geqslant {\text{1}}$ where {X, X _n ,n≥1} are i.i.d. random variables withEX=0,EX ²=1 and letH _k (·) denote the Hermite polynomial of degreek. By establishing an LIL for products of correlated sums of i.i.d. random variables, the a.s. decomposition $$\begin{gathered} k!S_{k,n} = n^{k/2} H_k (S_{1n} /n^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} ) - \left( {\begin{array}{*{20}c} k \\ 2 \\ \end{array} } \right)S_{1.n}^{k - 2} \sum\limits_{i = 1}^n {(X_i^2 - 1)} \hfill \\ + O(n^{(k - 1)/2} (\log \log n)^{(k - 3/2} ) \hfill \\ \end{gathered} $$ valid whenEX ⁴<∞, elicits an LIL forη _k,n =k!S _k,n ?n ^k/2 H _k (S _1n /n ^1/2) under a reduced normalization. Moreover, whenE|X| ^p <∞ for somep in [2, 4], a Marcinkiewicz-Zygmund type strong law forη _k,n is obtained, likewise under a reduced normalization.     

Features of mathematical activities in interdisciplinary, project-based learning

Project-based learning in mathematics education leads to mathematical activities that are uncommon in regular lessons at school. Among these activities, the following are identified and examined more closely:

⊙ the elaboration and formulation of relevant mathematical problems, including necessary definitions.

⊙ the search for the mathematically feasible, and

⊙ the recognition of opportunities to apply mathematical methods.

Also, implications for the design of project-based learning environments are developed.     

A note on natural risk statistics

Recently Heyde, Kou and Peng [C.C. Heyde, S.G. Kou, X.H. Peng, What is a good external risk measure: Bridging the gaps between robustness, subadditivity, and insurance risk measures, 2007, preprint.] proposed the notion of a natural risk statistic associated with a finite sample that relaxes the subadditivity assumption in the classical coherent risk statistics. In this note we use convex analysis to provide alternate proofs of the representation results regarding natural risk statistics.     

A Berry-Esseen bound for symmetric statistics

Summary The rate of convergence of the distribution function of a symmetric function of N independent and identically distributed random variables to its normal limit is investigated. Under appropriate moment conditions the rate is shown to be (N^–1/2). This theorem generalizes many known results for special cases and two examples are given. Possible further extensions are indicated.Research supported by the U.S. Office of Naval Research, Contract N 00014-80-C-0163     

A martingale approach to scan statistics

Scan statistics are commonly used in biology, medicine, engineering and other fields where interest is in the probability of observing clusters of events in a window at an unknown location. Due to the dependent nature of the number of events in a large number of overlapping window locations, even approximate solutions for the simplest scan statistics may require elaborate calculations. We propose a new martingale method which allows one to approximate the distribution for a wide variety of scan statistics, including some for which analytical results are computationally infeasible.     

A general class of entropy statistics

To study the asymptotic properties of entropy estimates, we use a unified expression, called the -entropy. Asymptotic distributions for these statistics are given in several cases when maximum likelihood estimators are considered, so they can be used to construct confidence intervals and to test statistical hypotheses based on one or more samples. These results can also be applied to multinomial populations.     

The use of advance demand information in a project-based supply chain

In a project environment, a manufacturer is confronted with two types of demand: regular demand from many small orders and very irregular, lumpy demand from infrequent, large orders. Manufacturers who build to stock must carry large safety stocks to meet the lumpy demand. As part of the project engineering process, however, project engineers and implementers (e.g. installers) typically have developed information about material requirements well in advance of placement of orders. We analyze the inventory reduction that could be achieved if the installer were to communicate advance demand information (ADI) to the manufacturer. We look at it in particular when the bid is placed. We focus on the following characteristics of available ADI in project environments: First, ADI information is uncertain, because decisions on installer and manufacturer selection have not yet been finalized. Second, information is detailed, available at the item level. We show that ADI is particularly valuable when potential demand for large projects is irregular and when proposals for potential projects have a high probability of leading to orders.     

A generalization of Bernoulli's model occurring in order statistics II

The frequency of occurrences of events in a modified Bernoulli model under the assumption F_x(u) F_y(u) is studied. A homogeneity test for two samples based on properties of this frequency and its properties are investigated.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 6, pp. 779–786, June, 1991.     

A basic problem in the correlations between statistics and thermodynamics

By using a functional of probability distributions, several different statistical physics including extensive and nonextensive statistics are unified in a general formalism. It’s shown that the equivalence between the Maxent approach and the relation dU = TdS can be seen from the virtual work principle of mechanics. From dU = TdS, any entropic form which can exists for thermodynamical equilibrium system can be maximized in order to derive the probability distribution functions.     

Cooperation and competition in multidisciplinary optimization

This article aims to contribute to numerical strategies for PDE-constrained multiobjective optimization, with a particular emphasis on CPU-demanding computational applications in which the different criteria to be minimized (or reduced) originate from different physical disciplines that share the same set of design variables. Merits and shortcuts of the most-commonly used algorithms to identify, or approximate, the Pareto set are reviewed, prior to focusing on the approach by Nash games. A strategy is proposed for the treatment of two-discipline optimization problems in which one discipline, the primary discipline, is preponderant, or fragile. Then, it is recommended to identify, in a first step, the optimum of this discipline alone using the whole set of design variables. Then, an orthogonal basis is constructed based on the evaluation at convergence of the Hessian matrix of the primary criterion and constraint gradients. This basis is used to split the working design space into two supplementary subspaces to be assigned, in a second step, to two virtual players in competition in an adapted Nash game, devised to reduce a secondary criterion while causing the least degradation to the first. The formulation is proved to potentially provide a set of Nash equilibrium solutions originating from the original single-discipline optimum point by smooth continuation, thus introducing competition gradually. This approach is demonstrated over a testcase of aero-structural aircraft wing shape optimization, in which the eigen-split-based optimization reveals clearly superior. Thereafter, a result of convex analysis is established for a general unconstrained multiobjective problem in which all the gradients are assumed to be known. This results provides a descent direction common to all criteria, and adapting the classical steepest-descent algorithm by using this direction, a new algorithm is defined referred to as the multiple-gradient descent algorithm (MGDA). The MGDA realizes a phase of cooperative optimization yielding to a point on the Pareto set, at which a competitive optimization phase can possibly be launched on the basis of the local eigenstructure of the different Hessian matrices.     

A survey of techniques in applied computational complexity

An attempt is made to introduce the non-expert reader to the many aspects of a relatively new and varied field which seems to be at the same time analysis, algebra and computer science. Computational complexity can be roughly described as the theory of optimizing finite and infinite algorithms for use on digital computers. Even for “simple” problems like the finding of a zero of a real function or even the evaluation of a polynomial, surprisingly deep techniques are necessary. A representative sample of the presently existing bibliography on the subject is included at the end.     

Informative barycentres in statistics

Barycentres of a discrete probability measure on a dually flat statistical manifold are introduced. They are shown to be unique and to behave as barycentres in Euclidean space. The estimation of these barycentres is studied. Potential applicative usefulness of informative barycentres include the problem of interpolating a statistical manifold valued map and the problem of model merging, which consists in merging several statistical models into a unique one. The results are illustrated on the exponential family, for which a projection theorem is proved.     

Antieigenvalue techniques in statistics

In a number of his recent papers Karl Gustafson has outlined the similarities between the Antieigenvalue Theory he founded and several finite dimensional matrix optimization theorems for positive matrices arising in statistics. In this paper, we will show how the techniques that the author and Karl Gustafson have used for computation of Antieigenvalues can also be applied to prove and generalize these matrix optimization theorems in statistics. We will primarily focus on two techniques which we have used in Antieigenvalue computations in recent years. These two techniques are a two nonzero component property for certain class of functionals, and converting the matrix optimization problems in statistics to a convex programing problem. Indeed, these two techniques allow us to generalize some of the matrix optimization problems arising in statistics to strongly accretive operators on finite or infinite dimensional Hilbert spaces.     

意料之外,“数理”之中!

本文首先介绍“ParadeMagazine”杂志 1990年第 9期上刊登的一则“有奖竞猜”游戏 ,然后用图形的方式给出此游戏的直观解释     

A note on limiting distributions for spacings statistics

Summary Hájek's projection method is used to prove asymptotic normality for a class of spacings statistics.The author thanks Dr. John Petkau for useful discussions.