Some necessary conditions on the parameters of partial difference sets

In this paper, we study some necessary conditions on the parameters of nontrivial regular (υ, κ, λ, μ)-partial difference sets in abelian groups. In particular, we settle some undecided cases in Ma's table [Designs, Codes Cryptography, 4 (1994)]. Also, the case when λ ⩽ 1 is studied. Nonexistence results are obtained when λ = 0 and a complete characterization is given when λ = 1. Finally, parameters of partial difference sets with an odd μ are determined.     

Classification of Higher-order Data with Separable Covariance and Structured Multiplicative or Additive Mean Models

Although devised in 1936 by Fisher, discriminant analysis is still rapidly evolving, as the complexity of contemporary data sets grows exponentially. Our classification rules explore these complexities by modeling various correlations in higher-order data. Moreover, our classification rules are suitable to data sets where the number of response variables is comparable or larger than the number of observations. We assume that the higher-order observations have a separable variance-covariance matrix and two different Kronecker product structures on the mean vector. In this article, we develop quadratic classification rules among g different populations where each individual has κth order (κ ≥2) measurements. We also provide the computational algorithms to compute the maximum likelihood estimates for the model parameters and eventually the sample classification rules.     

Inference for Log-Gamma Distribution Based on Progressively Type-II Censored Data

We discuss the maximum likelihood estimates (MLEs) of the parameters of the log-gamma distribution based on progressively Type-II censored samples. We use the profile likelihood approach to tackle the problem of the estimation of the shape parameter κ. We derive approximate maximum likelihood estimators of the parameters μ and σ and use them as initial values in the determination of the MLEs through the Newton–Raphson method. Next, we discuss the EM algorithm and propose a modified EM algorithm for the determination of the MLEs. A simulation study is conducted to evaluate the bias and mean square error of these estimators and examine their behavior as the progressive censoring scheme and the shape parameter vary. We also discuss the interval estimation of the parameters μ and σ and show that the intervals based on the asymptotic normality of MLEs have very poor probability coverages for small values of m. Finally, we present two examples to illustrate all the methods of inference discussed in this paper.     

Adaptive autoregressive priors for area and time structured mortality data

Mortality counts by age and area are relevant to obtaining small area life tables and summary statistics such as life expectancy. A Bayesian approach to small area life tables is proposed here based on the principle of smoothing (or “pooling strength”) over adjacent ages or areas. Several schemes have been suggested to reflect dependence between age categories x or areas i  , such as conditional autoregressive priors based on the principle of local smoothing, determined by adjacency of age groups or spatial proximity. It is argued here that a more flexible approach is to allow a mix of local and global smoothing over age groups and areas, as determined by the data and additional parameters κ∈[0,1]$\kappa \in [0,1]$ and λ∈[0,1]$\lambda \in [0,1]$ for age and area, respectively. An extension is also proposed to reflect the fact that the appropriate mix between local and global smoothing may not be constant across age bands or across the region being studied. For example, local spatial smoothing will not be appropriate if an area is disparate from its neighbours (e.g. in terms of social distance), and so area specific mixing parameters _λi${\lambda }_i$ are introduced. The _λi${\lambda }_i$ may be modelled by logit regression on observed sources of disparity between neighbouring areas. The application considers small area life tables for males over 625 small areas (electoral wards) in London over 2003–2005.     

Lois limites pour les statistiques d'ordre dans le cas non identiquement distribue

Let X₁, X₂,… be a sequence of independent random variables with distribution functions F₁, where 1 ≤ i ≤ n, and for each n ≥ 1 let X_1,n ≤… ≤ X_n,n denote the order statistics of the first n random variables. Under suitable hypotheses about the F₁, we characterize the limit distribution functions H(x) for which P(X_k,n ? a_nx + b_n) → H(x), where a_n > 0 and b_n are real constants. We consider the cases where κ = κ(n) satisfies √n {κ(n)/n — λ} → 0 and √n {κ(n)/n — λ} → ∞ separately.     

Temporal pattern in number of staff on sick leave: the effect of downsizing

Summary.  The pattern of absenteeism in the downsizing process of companies is a topic in focus in economics and social science. A general question is whether employees who are frequently absent are more likely to be selected to be laid off or in contrast whether employees to be dismissed are more likely to be absent for the remaining time of their working contract. We pursue an empirical and microeconomic investigation of these theses. We analyse longitudinal data that were collected in a German company over several years. We fit a semiparametric transition model based on a mixture Poisson distribution for the days of absenteeism per month. Prediction intervals are considered and the primary focus is on the period of downsizing. The data reveal clear evidence for the hypothesis that employees who are to be laid off are more frequently absent before leaving the company. Interestingly, though, no clear evidence is seen that employees being selected to leave the company are those with a bad absenteeism profile.     

Graphical comparison of means with a standard in the one-way layout

Some simple test procedures are considered for comparing several group means with a standard value when the data are in a one-way layout. The underlying distributions are assumed to be normal with possibly unequal variances. The tests are based on a union-intersection formulation and can be applied in a form similar to a Shewhart control chart. Both two-sided and one-sided alternatives are considered. The power of the tests can be obtained from tables of a non-central t distribution. Implementation of the tests is illustrated with a numerical example. The tests help identify any group means different from the standard and might lead to a decision about rejecting the null hypothesis before all the group means are observed. The resulting savings in time and resources might be valuable in applications where the number of groups is large and the cost of acquiring data is high. For situations where the normality assumption is untenable, a non-parametric procedure, based on one-sample sign tests is considered.     

Sur l'estimation du risque encouru en reconstruisant une forme aléatoire

The problem of estimating the risk corresponding to the reconstruction of a random pattern is reviewed. It is shown that for a particular but important model, the problem is reduced to the estimation of two parameters closely related to those appearing in a two-state Markov chain, which is of independent interest. The estimation of the Markov chain's parameters is studied from the decision-theoretic point of view. Estimators which are better than others previously considered are obtained and adapted to the estimation of the corresponding risk. Examples are are analyzed; even if a very empirical method is used to give values to the parameters of an a priorilaw, some good estimators of the risk are obtained.     

Least squares parameter estimation in multiplicative noise models

A simple multiplicative noise model with a constant signal has become a basic mathematical model in processing synthetic aperture radar images. The purpose of this paper is to examine a general multiplicative noise model with linear signals represented by a number of unknown parameters. The ordinary least squares (LS) and weighted LS methods are used to estimate the model parameters. The biases of the weighted LS estimates of the parameters are derived. The biases are then corrected to obtain a second-order unbiased estimator, which is shown to be exactly equivalent to the maximum log quasi-likelihood estimation, though the quasi-likelihood function is founded on a completely different theoretical consideration and is known, at the present time, to be a uniquely acceptable theory for multiplicative noise models. Synthetic simulations are carried out to confirm theoretical results and to illustrate problems in processing data contaminated by multiplicative noises. The sensitivity of the LS and weighted LS methods to extremely noisy data is analysed through the simulated examples.     

Covariates and Random Effects in a Gamma Process Model with Application to Degradation and Failure

The gamma process is a natural model for degradation processes in which deterioration is supposed to take place gradually over time in a sequence of tiny increments. When units or individuals are observed over time it is often apparent that they degrade at different rates, even though no differences in treatment or environment are present. Thus, in applying gamma-process models to such data, it is necessary to allow for such unexplained differences. In the present paper this is accomplished by constructing a tractable gamma-process model incorporating a random effect. The model is fitted to some data on crack growth and corresponding goodness-of-fit tests are carried out. Prediction calculations for failure times defined in terms of degradation level passages are developed and illustrated.     

The Slippage Problem for the Circular Normal Distribution

The slippage problem occurs when an unspecified observation in a given random sample is from a distribution other than that for all the remaining observations. This paper studies the problem in terms of the 'slip' in the mean direction of a circular normal distribution. The slippage problem is first treated as a multiple decision problem with a prior which is invariant under the permutations of the hypotheses. The probabilities of accepting the various hypotheses for the Bayes rule with respect to this prior are explicitly obtained. The likelihood ratio tests for this slippage problem, for the cases when the mean directions are both known and unknown, are shown to be easily computable. The tests are illustrated through two well-known datasets. The performances of a range of tests are compared using extensive simulation.     

Estimating consumer acceptance limits

A methodology is developed for estimating consumer acceptance limits on a sensory attribute of a manufactured product. In concept these limits are analogous to engineering tolerances. The method is based on a generalization of Stevens' Power Law. This generalized law is expressed as a nonlinear statistical model. Instead of restricting the analysis to this particular case, a strategy is discussed for evaluating nonlinear models in general since scientific models are frequently of nonlinear form. The strategy focuses on understanding the geometrical contrasts between linear and nonlinear model estimation and assessing the bias in estimation and the departures from a Gaussian sampling distribution. Computer simulation is employed to examine the behavior of nonlinear least squares estimation. In addition to the usual Gaussian assumption, a bootstrap sample reuse procedure and a general triangular distribution are introduced for evaluating the effects of a non-Gaussian or asymmetrical error structure. Recommendations are given for further model analysis based on the simulation results. In the case of a model for which estimation bias is not a serious issue, estimating functions of the model are considered. Application of these functions to the generalization of Stevens’ Power Law leads to a means for defining and estimating consumer acceptance limits, The statistical form of the law and the model evaluation strategy are applied to consumer research data. Estimation of consumer acceptance limits is illustrated and discussed.     

A bayesian approach for testing periodicity of dichotomous observations

A Bayesian approach is utilized to test for periodicity in a dichotomous time series. Dichotomous data arise in a variety of circumstances when a variable takes on only two possible values. Conjugate and noninformative priors are considered as well as a hierarchical Bayes approach; the latter is considered the superior Bayes methodology. The situation of stochastic period lengths is also discussed. The generalization to the multinomial model is investigated to allow for the case that a variable takes on more than two possible values. In all cases decisions are made based on a Bayes factor. The proposed procedures are demonstrated on earthquake data in the central Virginia seismic zone     

Mini-max-risk and mini-mean-risk inferences for a partially piecewise regression

We consider a partially piecewise regression in which the main regression coefficients are constant in all subdomains, but the extraessential regression function is variable in different pieces and is difficult to be estimated. Under this situation, two new regression methodologies are proposed under the criteria of mini-max-risk and mini-mean-risk. The resulting models can describe the regression relations in maximum-risk and mean-risk environments, respectively. A two-stage estimation procedure, together with a composite method, is introduced. The asymptotic normality of the estimators is established, the standard convergence rate and efficiency are achieved. Some unusual features of the new estimators and predictions, and the related variable selection are discussed for a comprehensive comparison. Simulation studies and a real-financial example are given to illustrate the new methodologies.     

Tests for Comparing Mark-Specific Hazards and Cumulative Incidence Functions

It is of interest in some applications to determine whether there is a relationship between a hazard rate function (or a cumulative incidence function) and a mark variable which is only observed at uncensored failure times. We develop nonparametric tests for this problem when the mark variable is continuous. Tests are developed for the null hypothesis that the mark-specific hazard rate is independent of the mark versus ordered and two-sided alternatives expressed in terms of mark-specific hazard functions and mark-specific cumulative incidence functions. The test statistics are based on functionals of a bivariate test process equal to a weighted average of differences between a Nelson-Aalen-type estimator of the mark-specific cumulative hazard function and a nonparametric estimator of this function under the null hypothesis. The weight function in the test process can be chosen so that the test statistics are asymptotically distribution-free. Asymptotically correct critical values are obtained through a simple simulation procedure. The testing procedures are shown to perform well in numerical studies, and are illustrated with an AIDS clinical trial example. Specifically, the tests are used to assess if the instantaneous or absolute risk of treatment failure depends on the amount of accumulation of drug resistance mutations in a subject's HIV virus. This assessment helps guide development of anti-HIV therapies that surmount the problem of drug resistance.     

Estimates of regression coefficients based on the sign covariance matrix

Summary. A new estimator of the regression parameters is introduced in a multivariate multiple-regression model in which both the vector of explanatory variables and the vector of response variables are assumed to be random. The affine equivariant estimate matrix is constructed using the sign covariance matrix (SCM) where the sign concept is based on Oja's criterion function. The influence function and asymptotic theory are developed to consider robustness and limiting efficiencies of the SCM regression estimate. The estimate is shown to be consistent with a limiting multinormal distribution. The influence function, as a function of the length of the contamination vector, is shown to be linear in elliptic cases; for the least squares (LS) estimate it is quadratic. The asymptotic relative efficiencies with respect to the LS estimate are given in the multivariate normal as well as the t -distribution cases. The SCM regression estimate is highly efficient in the multivariate normal case and, for heavy-tailed distributions, it performs better than the LS estimate. Simulations are used to consider finite sample efficiencies with similar results. The theory is illustrated with an example.     

Statistical methods for regular monitoring data

Summary.  Meteorological and environmental data that are collected at regular time intervals on a fixed monitoring network can be usefully studied combining ideas from multiple time series and spatial statistics, particularly when there are little or no missing data. This work investigates methods for modelling such data and ways of approximating the associated likelihood functions. Models for processes on the sphere crossed with time are emphasized, especially models that are not fully symmetric in space–time. Two approaches to obtaining such models are described. The first is to consider a rotated version of fully symmetric models for which we have explicit expressions for the covariance function. The second is based on a representation of space–time covariance functions that is spectral in just the time domain and is shown to lead to natural partially nonparametric asymmetric models on the sphere crossed with time. Various models are applied to a data set of daily winds at 11 sites in Ireland over 18 years. Spectral and space–time domain diagnostic procedures are used to assess the quality of the fits. The spectral-in-time modelling approach is shown to yield a good fit to many properties of the data and can be applied in a routine fashion relative to finding elaborate parametric models that describe the space–time dependences of the data about as well.     

Permutational tests for correlation matrices

Permutational tests are proposed for the hypotheses that two population correlation matrices have common eigenvectors, and that two population correlation matrices are equal. The only assumption made in these tests is that the distributional form is the same in the two populations; they should be useful as a prelude either to tests of mean differences in grouped standardised data or to principal component investigation of such data.The performance of the permutational tests is subjected to Monte Carlo investigation, and a comparison is made with the performance of the likelihood-ratio test for equality of covariance matrices applied to standardised data. Bootstrapping is considered as an alternative to permutation, but no particular advantages are found for it. The various tests are applied to several data sets.