Local Conditional Influence

Through an investigation of normal curvature functions for influence graphs of a family of perturbed models, we develop the concept of local conditional influence. This concept can be used to study masking and boosting effects in local influence. We identify the situation under which the influence graph of the unperturbed model contains all the information on these effects. The linear regression model is used for illustration and it is shown that the concept developed is consistent with Lawrance's (1995) approach of conditional influence in Cook's distance.     

Normalized Quadratic Systems of Consumer Demand Functions

Empirically estimated demand systems frequently fail to satisfy the appropriate theoretical curvature conditions. We propose and estimate two demand systems for which these conditions can be imposed globally; the first is derived from a normalized quadratic reciprocal indirect utility function and the second is derived from a normalized quadratic expenditure function. The former is flexible if there are no restrictions on its free parameters, but loses flexibility if the curvature conditions need to be imposed. The latter is flexible, in the class of functions satisfying local money metric scaling, even if the curvature conditions need to be imposed.     

On Spiring's normal loss function

Loss functions express the loss to society, incurred through the use of a product, in monetary units. Underlying this concept is the notion that any deviation from target of any product characteristic implies a degradation in the product performance and hence a loss. Spiring (1993), in response to criticisms of the quadratic loss function, developed the reflected normal loss function, which is based on the normal density function. We give some modifications of these loss functions to simplify their application and provide a framework for the reflected normal loss function that accomodates a broader class of symmetric loss situations. These modifications also facilitate the unification of both of these loss functions and their comparison through expected loss. Finally, we give a simple method for determing the parameters of the modified reflected normal loss function based on loss information for multiple values of the product characteristic, and an example to illustrate the flexibility of the proposed model and the determination of its parameters.     

A semi-parametric approach to dual modeling when no replication exists

In many applications, it is of interest to simultaneously model the mean and variance of a response when no replication exists. Modeling the mean and variance simultaneously is commonly referred to as dual modeling. Parametric approaches to dual modeling are popular when the underlying mean and variance functions can be expressed explicitly. Quite often, however, nonparametric approaches are more appropriate due to the presence of unusual curvature in the underlying functions. In sparse data situations, nonparametric methods often fit the data too closely while parametric estimates exhibit problems with bias. We propose a semi-parametric dual modeling approach [Dual Model Robust Regression (DMRR)] for non-replicated data. DMRR combines parametric and nonparametric fits resulting in improved mean and variance estimation. The methodology is illustrated with a data set from the literature as well as via a simulation study.     

Conformal normal curvature and assessment of local influence

In 1986, R. D. Cook proposed differential geometry to assess local influence of minor perturbations of statistical models. We construct a conformally invariant curvature, the conformal normal curvature, for the same purpose. This curvature provides a measure of local influence ranging from 0 to 1, with objective bench-marks to judge largeness. We study various approaches to using the conformal normal curvature and the relationships between these approaches.     

A semi-parametric approach to robust parameter design

Parameter design or robust parameter design (RPD) is an engineering methodology intended as a cost-effective approach for improving the quality of products and processes. The goal of parameter design is to choose the levels of the control variables that optimize a defined quality characteristic. An essential component of RPD involves the assumption of well estimated models for the process mean and variance. Traditionally, the modeling of the mean and variance has been done parametrically. It is often the case, particularly when modeling the variance, that nonparametric techniques are more appropriate due to the nature of the curvature in the underlying function. Most response surface experiments involve sparse data. In sparse data situations with unusual curvature in the underlying function, nonparametric techniques often result in estimates with problematic variation whereas their parametric counterparts may result in estimates with problematic bias. We propose the use of semi-parametric modeling within the robust design setting, combining parametric and nonparametric functions to improve the quality of both mean and variance model estimation. The proposed method will be illustrated with an example and simulations.     

A Monte Carlo study of traditional and rank-based picked-points analyses

In analysis of covariance with heteroscedastic slopes a picked-points analysis is often performed. Least-squares based picked-points analyses often lose efficiency (at times substantial) for nonnormal error distributions. Robust rank-based picked-points analyses are developed which are optimizable for heavy-tailed and/or skewed error distributions. The results of a Monte Carlo investigation of these analyses are presented. The situations include the normal model and models which violate it in one or several ways. Empirically the rank-based analyses appear to be valid over all these situations and more powerful than the least squares analysis for all the nonnormal models, while losing little efficiency at the normal model.     

A Monte Carlo study of REML and robust rank-based analyses for the random intercept mixed model

Restricted maximum likelihood (REML) methods are traditionally used for analyzing mixed models. Based on a multivariate normal likelihood, these analyses are sensitive to outliers. Recently developed robust rank-based procedures offer a complete analysis of mixed model: estimation of fixed effects, standard errors, and estimation of variance components. The results of a large Monte Carlo study are presented, comparing these two analyses for many situations over multivariate normal and contaminated normal distributions. The rank-based analyses are much more powerful and efficient than the REML analyses over all non-normal situations, while losing little power for normal errors.     

Imposing and Testing Curvature Conditions on a Box–Cox Cost Function

We present a new method for imposing and testing concavity of cost functions using asymptotic least squares, which can be easily implemented even for nonlinear cost functions. We provide an illustration for a (generalized) Box–Cox cost function with six inputs: capital, labor disaggregated in three skill levels, energy, and intermediate materials. We present a parametric concavity test and compare price elasticities when curvature conditions are imposed versus when they are not. Although concavity is statistically rejected, estimates are not very sensitive to its imposition. We find stronger substitution between the different type of labor than between any other two inputs.     

Relative Curvature Measure for Heteroscedastic or Non Normal Nonlinear Regression

The Bates–Watts relative curvature measure can assess the validity of the linearized approximation in nonlinear regression models. However, it is developed based on an ordinary nonlinear regression in which the observation is assumed to be homoscedastically and normally distributed. In this article, we extend the original Bates–Watts relative curvature measure to one that can be applicable to nonlinear regression with heteroscedastic or non normal data, based on the transformation-both-sides (TBS) approach. In pharmacokinetic models, a diagnostic use of their measures is illustrated. By means of a simulation experiment, the performance of the relative curvature measure for the TBS approach is evaluated.     

Computation of the mean value and variance functions in geometric process

Geometric process (GP) is widely used as a non-stationary stochastic model in reliability analysis. In many of applications related with GP its mean value and variance functions are needed. Since there are no analytical forms of these functions in a lot of situations their computations are of importance. In this study, a numerical approximation and Monte Carlo estimation method based on the convolutions of distribution functions have been proposed for both the mean value and variance functions.     

Estimating smooth monotone functions

Many situations call for a smooth strictly monotone function f of arbitrary flexibility. The family of functions defined by the differential equation D  ² f  = w Df , where w is an unconstrained coefficient function comprises the strictly monotone twice differentiable functions. The solution to this equation is f = C ₀ + C ₁  D ⁻¹{exp( D ⁻¹ w )}, where C ₀ and C ₁ are arbitrary constants and D ⁻¹ is the partial integration operator. A basis for expanding w is suggested that permits explicit integration in the expression of f . In fitting data, it is also useful to regularize f by penalizing the integral of w ² since this is a measure of the relative curvature in f . Applications are discussed to monotone nonparametric regression, to the transformation of the dependent variable in non-linear regression and to density estimation.     

THE EPIDEMIOLOGY OF A DISEASE USING HAZARD FUNCTIONS1

A prime objective of this paper is to mathematically quantify certain epidemiological concepts and show how some of these can be applied to certain registry type data. Usually epidemiological quantities are expressed in terms of density functions. However it is often more natural to estimate hazard functions in practical situations. Hence the emphasis will be on expressing various measures in terms of hazard functions. This contrasts with the useful paper of Albert, Gertman & Louis (1978) where density functions are used throughout. The definitions of some key concepts such as cohort effect are also different which gives rise to an alternative analysis and some new results. Initially, incidence, mortality and prevalence are discussed and this is followed by a treatment of initiation which is a less commonly used measure. The results are applied to the lung cancer data in the South Australian Central Cancer Registry.     

Local Influence Under Parameter Constraints

Calculations of local influence curvatures and leverage have been well developed when the parameters are unrestricted. In this article, we discuss the assessment of local influence and leverage under linear equality parameter constraints with extensions to inequality constraints. Using a penalized quadratic function we express the normal curvature of local influence for arbitrary perturbation schemes and the generalized leverage matrix in interpretable forms, which depend on restricted and unrestricted components. The results are quite general and can be applied in various statistical models. In particular, we derive the normal curvature under three useful perturbation schemes for generalized linear models. Four illustrative examples are analyzed by the methodology developed in the article.     

Simple detection of outlying short time series

Sets of relatively short time series arise in many situations. One aspect of their analysis may be the detection of outlying series. We examine the performance of standard normal outlier tests applied to the means, or to simple functions of the means, of AR(1) series, not necessarily of equal lengths. Although unequal lengths of series implies that the means have unequal variances, that are only known approximately, it is shown that nominal significance levels hold good under most circumstances. Thus a standard outlier test can usefully be applied, avoiding the complication of estimating the time series' parameters. The test's power is affected by unequal lengths, being higher when the slippage occurs in one of the longer series     

Estimating conditional tail expectation with actuarial applications in view

We develop statistical inferential tools for estimating and comparing conditional tail expectation (CTE) functions, which are of considerable interest in actuarial science. In particular, we construct estimators for the CTE functions, develop the necessary asymptotic theory for the estimators, and then use the theory for constructing confidence intervals and bands for the functions. Both parametric and non-parametric approaches are explored. Simulation studies illustrate the performance of estimators in various situations. Results are obtained under minimal assumptions, and the general Vervaat process plays a crucial role in achieving these goals.     

Testing Whether One Regression Function is Larger Than Another

The problem of testing whether one regression function is larger than another on a specified set R is considered The regression functions must be linear functions of the parameters but need not be linear functions of the independent variables The proposed test has an exactly specified size in typical situations The test's critical value is a standard t percentile. The power function of the test is investigated.     

A new two-sample test for choosing between log-rank and Wilcoxon tests with right-censored data

When differences of survival functions are located in early time, a Wilcoxon test is the best test, but when differences of survival functions are located in late time, using a log-rank test is better. Therefore, a researcher needs a stable test in these situations. In this paper, a new two-sample test is proposed and considered. This test is distribution-free. This test is useful for choosing between log-rank and Wilcoxon tests. Its power is roughly the maximal power of the log-rank test and Wilcoxon test.     

Second-order inference for generalized least squares

Confidence regions for generalized least squares are commonly derived from a measure of departure calculated on the tangent plane at the MLE or on the tangent plane at the true value; the first gives approximate confidence regions, the second exact. For surfaces with curvature, indeed with varying curvature, the exact regions typically are not likelihood regions and can include parameter values of highest and of lowest likelihood. This paper develops an alternative approach to deriving exact confidence regions and uses both surface curvature and distance from the surface as supporting ingredients. For this, conditionality is invoked in two ways beyond that supported by the usual conditionality principle. For the case of normal error the ordinary chi-squared departure is replaced by a Von Mises-type angular (or cosine) departure which is assessed using curvature properties in the data direction and radial distance of the data from the regression surface. For the usual linear model (constant curvature equal to zero) the method coincides with the ordinary tests and confidence regions; for the case of constant nonzero curvature, the method generalizes to spheres and sphere-cylinders the Fisher (Statistical Methods and Scientific Inference, 1956) analysis of a rotationally symmetric normal on ?² with mean constrained to a circle. The effects of conditioning are indicated by a computer plot for obtaining 95% confidence.     

Notes on estimation of the intraclass correlation under the AB/BA crossover trial

Under the AB/BA crossover trial, we focus our attention on estimation of the intraclass correlation in normal data. We develop both point and interval estimators in closed form for the intraclass correlation. We employ Monte Carlo simulation to study the performance of these estimators in a variety of situations. We note that the estimators developed here for the intraclass correlation remain valid even when there are possibly unexpected carry-over effects.