Considerations of the place of assumptions in correlational analysis.

The arguments of Nefzger and Drasgow (see 33: 2530) that it was not necessary to assume normality to compute Pearson's r "are not well supported at many points and inaccurate at others." Weaknesses of the Nefzger and Drasgow presentation are considered. Major topics are: The Role and Uses of Mathematical Models, Correlational Models, The Model of Nefzger and Drasgow, The Literature on Correlation. "The model for correlation presented by Nefzger and Drasgow is neither developed nor presented in a precise way… . some… statistical statements and arguments dealing with the model are in error. The writers have misinterpreted some of the literature on correlation." (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

A Neural Network for Solving Nonlinear Programming Problems

A neural network for solving convex nonlinear programming problems is proposed in this paper. The distinguishing features of the proposed network are that the primal and dual problems can be solved simultaneously, all necessary and sufficient optimality conditions are incorporated, and no penalty parameter is involved. Based on Lyapunov, LaSalle and set stability theories, we prove strictly an important theoretical result that, for an arbitrary initial point, the trajectory of the proposed network does converge to the set of its equilibrium points, regardless of whether a convex nonlinear programming problem has unique or infinitely many optimal solutions. Numerical simulation results also show that the proposed network is feasible and efficient. In addition, a general method for transforming non-linear programming problems into unconstrained problems is also proposed. ID="A1" Correspondence and offprint requests to: Dr Z Chen, Department of Electronic Engineering, Brunel University, Uxbridge, Middle-sex, UK     

Shapiro-Wilk-type test of normality under nuisance regression and scale

An extension of the Shapiro-Wilk test to verify the hypothesis of normality in the presence of nuisance regression and scale has been previously considered. Such a test is typically based on the pair of the maximum likelihood and BLUE estimators of the standard deviation in the linear regression model. It has been shown that the asymptotic null distribution of the test criterion, extended to the regression model, is equivalent to that of the original Shapiro-Wilk test for the location-scale model. A simulation study is performed in order to show that both criteria are close under the normality hypothesis for moderate as well for large data sets. The power of the test against various alternative distributions of the model errors is illustrated. Furthermore, it is shown that the probabilities of errors of both the first and second kinds do not depend on the design matrix or on the parameters of the linear model.     

Comment on "The Needless Assumption of Normality in Pearson's r".

Comment on article by M. D. Nefzger and J. Drasgow (see record 1959-02530-001), in which they emphasized the fact that the computation of r does not require an assumption of normality in the marginal distributions. They have, however, failed to take the equally desirable next step of pointing out that linearity also need not be assumed in the computation of r. The next term to be denned is "assumption." The implication apparently is that if you use a linear prediction equation you assume it to be the true relation in the sense just defined. I think this implication is unnecessary and that considerable confusion can be avoided by not making it. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

On Bartlett's Formula for Non-linear Processes

Bartlett's formula is widely used in time series analysis to provide estimates of the asymptotic covariance between sample autocovariances. However, it is derived under precise assumptions (namely linearity of the underlying process and vanishing of its fourth-order cumulants) and effectiv e computations show that the value given by this formula can deviate markedly from the true asymptotic covariance when the requirements on the underlying process are not satisfied. This is the case for a large class of models, for instance bilinear and autoregressive conditionally heteroscedastic processes. For these reasons we investigate the behaviour of smoothed empirical estimates of the covariance between two sample autocovariance s. We prove L ² and strong consistency for strongly mixing stationary processes and define for the covariance matrix of a vector of sample autocovariances a consistent estimate which is a non-negative definite matrix. The choice of the parameters is discussed, applications are given and comparisons are made through a simulation study     

GREEN'S FUNCTION FORMALISM IN GENERALIZED TAYLOR DISPERSION THEORY†

Green's function formalism is applied to obtain the asymptotically long-time solution of the multidimensional, phase-space, convective-diffusion equation governing transport and temporal evolution of the conditional probability density P of a Brownian tracer particle. In the long-time limit, P (as well as its infinite sequence of polyadic moments) is shown to be derivable from a pair of time-independent scalar fields characterizing the local-space transport process. These two functions are employed to obtain the global phenomenological coefficients occurring in the physical-space (global) equation governing the asymptotically long-time, locally-averaged, conditional probability density distribution     

Predicting the number of accidents at a road junction

We consider a model, within a Bayesian framework, which can be used to predict the number of accidents occurring at a road junction in a given period of time. The predictions are based on measurements of the traffic flows as well as on covariates which describe important features of the junctions. Various approximate and estimative methods, which use Gibbs sampling, posterior normality and Laplace approximations, are considered and compared. Procedures to assess the importance of the different covariates through the use of the Kullback-Leibler measure of divergence are also developed.     

Ergodic and adaptive control of nearest-neighbor motions

The self-tuning approach to adaptive control is applied to a class of Markov chains called nearest-neighbor motions. These have a countable state space and move from any state to at most finitely many neighboring states. For compact parameter and control spaces, the almost-sure optimality of the self-tuner for an ergodic cost criterion is established under two sets of assumptions.     

Selecting the median and two quartiles in a set of numbers

The median, the 0.25-percentile and the 0.75-percentile are three of the most relevant ranks in data analysis. MED2Q is a new in situ algorithm to solve this problem. It asymptotically performs an average of 2 5/8 n + o(n) comparisons when n numbers are given as input thus becoming the asymptotically fastest algorithm for this problem reported to date. The performance of MED2Q is compared with those of FIND¹ and ALGORITHM 489², two well-known selection algorithms adapted to this specific problem. From this performance comparison, MED2Q is best when input sets consist of more than 50,000 numbers.     

The needless assumption of normality in Pearson's r.

With regard to proper application of the Pearson technique, statistical textbook authors fall into 2 groups: those who indicate that it is necessary for each of the variables to be normally distributed, and those who point out that normality of distribution is required in neither of the correlated variables. Perhaps the erroneous assumption of normally distributed variables stems "from Pearson's first presentation of the method and has been perpetuated through 11 Editions of Fisher's classic text. An objection to Pearson's choice of a poor example and his failure to generalize beyond the restrictive limits of normal data were pointed out at the time by Yule." Psychologists and statisticians "are still harassed in many quarters by the needless assumption of normality for the proper computation of Pearson's r." (PsycINFO Database Record (c) 2010 APA, all rights reserved)