Delta Method

We present a method for estimating the mean vector from a multivariate skew dis-tribution that includes some unobserved data below the detection limits. The method uses a Box-Cox transformation, of which the parameters are found by maximizing the likelihood function ...

Measurement error occurs when one or more regression model covariates are measured with error, and is a common problem in occupational health and other fields. If the effects of measurement error are not corrected, estimates of the regression coefficients and their variances are biased. We consider the situation when an external validation study provides information about the measurement error process. Two related approaches for correcting measurement error, both called regression calibration, have been proposed. Although the estimators for the corrected regression coefficients can be shown to be the same under the two approaches, the asymptotic variances are derived using two different methods and have not been compared. In this paper, we show that these two methods give algebraically identical estimators for the corrected regression coefficients and their asymptotic variances, in the general case of multiple covariates measured with error, for a generalized linear model of exponen...

The main objective of this work is to calculate and compare different measures of multivariate skewness for the skew-normal family of distributions. For this purpose, we consider the Mardia (1970) [10], Malkovich and Afifi (1973) [9], Isogai (1982) [17], Srivastava (1984) [15], Song (2001) [14], Móri et al. (1993) [11], Balakrishnan et al. (2007) [3] and Kollo (2008) [7] measures of skewness. The exact expressions of all measures of skewness, except for Song’s, are derived for the family of skew-normal distributions, while Song’s measure of shape is approximated by the use of delta method. The behavior of these measures, their similarities and differences, possible interpretations, and their practical use in testing for multivariate normal are studied by evaluating their power in the case of some specific members of the multivariate skew-normal family of distributions.► We calculate different measures of multivariate skewness for skew-normal distributions. ► Exact expressions of 7 measures of skewness are derived for multivariate skew-normal distributions. ► Behavior, differences, interpretations and tests for normality against skew-normal on all these measures are studied and their power properties are assessed.

In this paper, we consider the estimation of the stress–strength parameter R=P(Y<X) when X and Y are independent and both are modified Weibull distributions with the common two shape parameters but different scale parameters. The Markov Chain Monte Carlo sampling method is used for posterior inference of the reliability of the stress–strength model. The maximum-likelihood estimator of R and its asymptotic distribution are obtained. Based on the asymptotic distribution, the confidence interval of R can be obtained using the delta method. We also propose a bootstrap confidence interval of R. The Bayesian estimators with balanced loss function, using informative and non-informative priors, are derived. Different methods and the corresponding confidence intervals are compared using Monte Carlo simulations.

An index of repeatability is constructed to evaluate the relative magnitude of measurement error. This index is constructed as a ratio of two variance components. Estimation of the index is derived under the one-way random effects model. We compare the well-known maximum likelihood estimator to the Bayesian estimation procedure using non-informative prior. Large sample variance of the of the maximum likelihood estimator are obtained using the inverse of Fisher's information matrix and the delta method. Inference procedure using the. We also construct a test statistic on the equality of two repeatability indices using the Monte Carlo integration and sampling Importance re-sampling method. We illustrate the methodologies on the estimation of the index of repeatability of Gamma-glutamyl-transferase, an enzyme found in many organs all over the human body, with the highest concentrations found in the liver. This enzyme's level is raised in the blood in most diseases that cause damage to the liver or bile ducts and is considered an essential serum marker for alcohol-related liver disease.

This study centred on the analysis of ten pottery sherds from the collections of the Islamic Museum in Berlin (this analysis was made possible courtesy of Gisela Helmecke and Dr Volkmar Enderlein). Two of the analysed fragments were undoubtedly wasters arising from a fault in the technological process used in creating them- supporting pins survive stuck to the sherds. Based on macroscopic criteria and on the provenance of these fragments, it is possible to infer that they were produced at workshops in Fustat. Any certainty as to the provenance of the material in question can, however, only be attained by comparing the results of laboratory analysis for these fragments with the analysis results for kiln pottery recovered from occupation levels at the Fustat site. At this stage of research the attribution of the ten analyzed fragments is actually based mainly on comparisons with kiln pottery from other production centres (e.g. Raqqa) and on a comparison of the geochemical parameters o...

The goal of this paper is to propose a nonlinear dynamic model based on experimental data and NBR-6123-87 to accomplish a nonlinear dynamic analysis of slender structures subjected to wind loading. At first we compute the static answer given by the mean wind speed. In this part of the problem we consider the concept of effective stiffness to represent the physical nonlinearity of material and aP-Delta method to represent the geometrical nonlinearity. Considering the final stiffness obtained in thatP-Delta method, we compute the dynamic answer given by the floating wind speed, according to the discrete dynamic model given by NBR-6123-87. A 40 m RC telecommunication tower was analyzed, and the results obtained were compared with those given by linear static and dynamic models.

We study the problem of building confidence sets for ratios of parameters, from an identifi- cation robust perspective. In particular, we address the simultaneous confidence set estimation of a finite number of ratios. Results apply to a wide class of models suitable for estimation by consistent asymptotically normal procedures. Conventional methods (e.g. the delta method) derived by excluding the

In this paper we use a boundary integral method with single layer potentials to solve a class of Helmholtz transmission problems in the plane. We propose and analyze a novel and very simple quadrature method to solve numerically the equivalent system of integral equations which provides an approximation of the solution of the original problem with linear convergence (quadratic in some special cases). Furthermore, we also investigate a modified quadrature approximation based on the ideas of qualocation methods. This new scheme is again extremely simple to implement and has order three in weak norms.

An experimental station, based on the model of West Africa and adapted to the behaviour of the target species, was built in southern Vietnam. The study design was adapted from the WHO phase 2 guidelines. The study arms included a conventionally treated polyester net (CTN) ...

Many epidemiologic studies report the odds ratio as a measure of association for cross-sectional studies with common outcomes. In such cases, the prevalence ratios may not be inferred from the estimated odds ratios. This paper overviews the most commonly used procedures to ...

This paper is concerned with the numerical solution of boundary integral equations on smooth curves of the plane with some numerical methods having in common the use of sets of equally spaced periodic Dirac delta distributions as trial functions. In a functional frame of classical periodic pseudodifferential equations of nonpositive order, delta-spline and delta–delta methods are introduced and analysed with the overall aim of obtaining asymptotic expansions of the error in weak and strong norms. As a byproduct we obtain the convergence of the coefficients associated to the discrete delta approximation to pointwise values of the unknown, as well as superconvergent choices of positions of the delta distributions in relation with the discretization grid. Two numerical examples are explored to show nodal errors and the applicability of Richardson extrapolation.