Tests based on equivariant estimators

Let ?⁽¹⁾ and ?⁽²⁾ be location-equivariant estimators of an unknown location parameter μ. It is shown that the test for H₀: μ ≤ μ₀ versus H_A : μ > μ₀ that rejects H₀ if ?⁽¹⁾ is large is uniformly more powerful than the one that rejects H₀ if ?⁽²⁾ is large if and only if ?⁽²⁾ is “more dispersed” than ?⁽¹⁾. A similar result is obtained for tests on scale using the star-shaped ordering. Examples are given.     

Why a time effect often has a limited impact on capture‐recapture estimates in closed populations

The author is concerned with log‐linear estimators of the size N of a population in a capture‐recapture experiment featuring heterogeneity in the individual capture probabilities and a time effect. He also considers models where the first capture influences the probability of subsequent captures. He derives several results from a new inequality associated with a dispersive ordering for discrete random variables. He shows that in a log‐linear model with inter‐individual heterogeneity, the estimator N is an increasing function of the heterogeneity parameter. He also shows that the inclusion of a time effect in the capture probabilities decreases N in models without heterogeneity. He further argues that a model featuring heterogeneity can accommodate a time effect through a small change in the heterogeneity parameter. He demonstrates these results using an inequality for the estimators of the heterogeneity parameters and illustrates them in a Monte Carlo experiment     

De l'unicité des estimateurs robustes en régression lorsque le paramètre d'échelle et le paramètre de la régression sont estimés simultanément

Several methods have been suggested to calculate robust M- and G-M -estimators of the regression parameter β and of the error scale parameter σ in a linear model. This paper shows that, for some data sets well known in robust statistics, the nonlinear systems of equations for the simultaneous estimation of β, with an M-estimate with a redescending ψ-function, and σ, with the residual median absolute deviation (MAD), have many solutions. This multiplicity is not caused by the possible lack of uniqueness, for redescending ψ-functions, of the solutions of the system defining β with known σ; rather, the simultaneous estimation of β and σ together creates the problem. A way to avoid these multiple solutions is to proceed in two steps. First take σ as the median absolute deviation of the residuals for a uniquely defined robust M-estimate such as Huber's Proposal 2 or the L₁-estimate. Then solve the nonlinear system for the M-estimate with σ equal to the value obtained at the first step to get the estimate of β. Analytical conditions for the uniqueness of M and G-M-estimates are also given.     

Some linear model techniques for analyzing small-circle spherical data

The author investigates least squares as a method for fitting small-circle models to a sample of unit vectors in R³. He highlights a local linear model underlying the estimation of the parameters of a circle. This model is used to construct an estimation algorithm and regression-type inference procedures for the parameters of a circle. It makes it possible to compare the fit of a small circle with that of a spherical ellipse. The limitations of the least-squares approach are emphasized: when the errors are bounded away from 0, the least-squares estimators are not consistent as the sample size goes to infinity. Two examples, concerned with the migration of elephant seals and with the classification of geological folds, are analyzed using the linear model techniques proposed in this work.     

Using orientation statistics to investigate variations in human kinematics

This paper applies orientation statistics to investigate variations in upper limb posture of human subjects drilling at six different locations on a vertical panel. Some of the drilling locations are kinematically equivalent in that the same posture could be used for these locations. Upper limb posture is measured by recording the co-ordinates of four markers attached to the subjects hand, forearm, arm and torso. A 3×3 rotation characterizes the relative orientation of one body segment with respect to another. Replicates are available since each subject drilled at the same location five times. Upper limb postures for the six drilling locations are compared by one-way analysis-of-variance tests for rotations. These tests rely on tangent space approximations at the estimated modal rotation of the sample. A parameterization of rotations in terms of unit quaternions simplifies the computations. The analysis detects significant differences in posture between all pairs of drilling locations. The smallest changes, less than 10° at all joints, are obtained for the kinematically equivalent pairs of locations. A short discussion of the biomechanical interpretation of these findings is presented.     

Products of random variables and star-shaped ordering

It is shown that if V₁ and V₂ are two positive random variables such that V₁ is “star-shaped” with respect to V₂, then for any random variable X with a distribution F(x) such that F(ax) has the monotone likelihood-ratio property, XV₁ is star-shaped with respect to XV₂. This result is then used to prove that the stable laws are star-shaped ordered.     

Propriétés statistiques des copules de valeurs extrêmes bidimensionnelles

Let (X, Y) be a bivariate random vector whose distribution function H(x, y) belongs to the class of bivariate extreme-value distributions. If F₁ and F₂ are the marginals of X and Y, then H(x, y) = C{F₁(x),F₂(y)}, where C is a bivariate extreme-value dependence function. This paper gives the joint distribution of the random variables Z = {log F₁(X)}/{log F₁(X)F₂(Y)} and W = C{F₁{(X),F₂(Y)}. Using this distribution, an algorithm to generate random variables having bivariate extreme-value distribution is présentés. Furthermore, it is shown that for any bivariate extreme-value dependence function C, the distribution of the random variable W = C{F₁(X),F₂(Y)} belongs to a monoparametric family of distributions. This property is used to derive goodness-of-fit statistics to determine whether a copula belongs to an extreme-value family.     

Improved log‐linear model estimators of abundance in capture‐recapture experiments

The authors review log‐linear models for estimating the size of a closed population and propose a new log‐linear estimator for experiments having between animal heterogeneity and a behavioral response. They give a general formula for evaluating the asymptotic biases of estimators of abundance derived from log‐linear models. They propose simple frequency modifications for reducing these asymptotic biases and investigate the modifications in a Monte Carlo experiment which reveals that they reduce both the bias and the mean squared error of abundance estimators.