Joint confidence regions in the multivariate calibration problem

Consider a vector valued response variable related to a vector valued explanatory variable through a normal multivariate linear model. The multivariate calibration problem deals with statistical inference on unknown values of the explanatory variable. The problem addressed is the construction of joint confidence regions for several unknown values of the explanatory variable. The problem is investigated when the variance covariance matrix is a scalar multiple of the identity matrix and also when it is a completely unknown positive definite matrix. The problem is solved in only two cases: (i) the response and explanatory variables have the same dimensions, and (ii) the explanatory variable is a scalar. In the former case, exact joint confidence regions are derived based on a natural pivot statistic. In the latter case, the joint confidence regions are only conservative. Computational aspects and the practical implementation of the confidence regions are discussed and illustrated using an example.     

Multivariate Quality Control Chart for Autocorrelated Processes

Traditional multivariate statistical process control (SPC) techniques are based on the assumption that the successive observation vectors are independent. In recent years, due to automation of measurement and data collection systems, a process can be sampled at higher rates, which ultimately leads to autocorrelation. Consequently, when the autocorrelation is present in the data, it can have a serious impact on the performance of classical control charts. This paper considers the problem of monitoring the mean vector of a process in which observations can be modelled as a first-order vector autoregressive VAR (1) process. We propose a control chart called Z-chart which is based on the single step finite intersection test (Timm, 1996). An important feature of the proposed method is that it not only detects an out of control status but also helps in identifying variable(s) responsible for the out of control situation. The proposed method is illustrated with the help of suitable illustrations.     

An estimating equation for parametric shared frailty models with marginal additive hazards

Summary.  Multivariate failure time data arise when data consist of clusters in which the failure times may be dependent. A popular approach to such data is the marginal proportional hazards model with estimation under the working independence assumption. In some contexts, however, it may be more reasonable to use the marginal additive hazards model. We derive asymptotic properties of the Lin and Ying estimators for the marginal additive hazards model for multivariate failure time data. Furthermore we suggest estimating equations for the regression parameters and association parameters in parametric shared frailty models with marginal additive hazards by using the Lin and Ying estimators. We give the large sample properties of the estimators arising from these estimating equations and investigate their small sample properties by Monte Carlo simulation. A real example is provided for illustration.     

Forecasting a class of doubly stochastic Poisson processes

Received: July 21, 2000; revised version: April 24, 2001     

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.     

A One-Sided Rank Test for Multivariate Censored Data

An asymptotically distribution-free test is proposed for testing the equality of two multivariate failure distributions against a particular one-sided alternative based on censored observations. This test may be interpreted as a multivariate one-sided Gehan test. The consistency of the test is established. An illustrative example is given.     

Pseudo confidence regions for the solution of a multivariate linear functional relationship

The problem of finding confidence regions (CR) for a q-variate vector γ given as the solution of a linear functional relationship (LFR) Λγ = μ is investigated. Here an m-variate vector μ and an m × q matrix Λ = (Λ₁, Λ₂,…, Λ_q) are unknown population means of an m(q+1)-variate normal distribution $\text{N}{}_{\mathrm{m(q+1)}}\text{(ζΩ?Σ)}$, where ζ′ = (μ′, Λ₁′, Λ₂′,…, Λ_q′Σ is an unknown, symmetric and positive definite m × m matrix and Ω is a known, symmetric and positive definite (q+1) × (q+1) matrix and ? denotes the Kronecker product. This problem is a generalization of the univariate special case for the ratio of normal means.A CR for γ with level of confidence 1 ? α, is given by a quadratic inequality, which yields the so-called ‘pseudo’ confidence regions (PCR) valid conditionally in subsets of the parameter space. Our discussion is focused on the ‘bounded pseudo’ confidence region (BPCR) given by the interior of a hyperellipsoid. The two conditions necessary for a BPCR to exist are shown to be the consistency conditions concerning the multivariate LFR. The probability that these conditions hold approaches one under ‘reasonable circumstances’ in many practical situations. Hence, we may have a BPCR with confidence approximately 1 ? α. Some simulation results are presented.     

Multivariate Tests of Mean-Variance Efficiency and Spanning With a Large Number of Assets and Time-Varying Covariances

We develop a finite-sample procedure to test the mean-variance efficiency and spanning hypotheses, without imposing any parametric assumptions on the distribution of model disturbances. In so doing, we provide an exact distribution-free method to test uniform linear restrictions in multivariate linear regression models. The framework allows for unknown forms of nonnormalities as well as time-varying conditional variances and covariances among the model disturbances. We derive exact bounds on the null distribution of joint F statistics to deal with the presence of nuisance parameters, and we show how to implement the resulting generalized nonparametric bounds tests with Monte Carlo resampling techniques. In sharp contrast to the usual tests that are not even computable when the number of test assets is too large, the power of the proposed test procedure potentially increases along both the time and cross-sectional dimensions.     

The Teacher's Corner: Interpolation in F-Tables

A lemma is given which simplifies the derivation of maximum likelihood estimates of the covariance matrix in the multivariate normal case.     

On influence diagnostics in elliptical multivariate regression models with equicorrelated random errors

In this paper we discuss estimation and diagnostic procedures in elliptical multivariate regression models with equicorrelated random errors. Two procedures are proposed for the parameter estimation and the local influence curvatures are derived under some usual perturbation schemes to assess the sensitivity of the maximum likelihood estimates (MLEs). Two motivating examples preliminarily analyzed under normal errors are reanalyzed considering appropriate elliptical distributions. The local influence approach is used to compare the sensitivity of the model estimates.