Analysis of the Covariance Structure in Manufactured Parts

We discuss the use of variograms for covariance modeling under the Kriging model to assess tolerances on manufactured parts. The variogram is very informative about the spatial dependence and it is favored by researchers in the choice of a correlation function. It may give evidence of anisotropy and of nugget effect too. In this paper, various variogram estimators from literature are used for comparing and evaluating their properties in assessing the surface error of three workpieces manufactured with different machining precisions. Contrary to the common engineering belief, the variograms give evidence of both technological signatures and systematic errors of the measurement machines.     

A ROLE FOR VARIOGRAMS

In time series texts and journals, variograms are mentioned seldom, if at all. The autocovariance function is preferred. However there are situations where the variogram can be estimated with moderate precision but the autocovariance function cannot, because the variance of the process is not well known. If the problem to be solved does not require the process variance for its solution then it is generally more straightforward to use the variogram rather than the autocovariance function in solving this problem.     

On nonparametric variogram estimation

A critical step for geostatistical prediction is estimation of variogram from the data. One of the popular methods estimating variogram is a smoothed version of classical nonparametric variogram estimator. In this paper we investigate its theoretical and empirical properties to provide useful information for using it. The main results are based on asymptotic theories (i.e., risk and central limit theorem) under nearly infill domain sampling. Simulation is also employed to make our points.     

A regression approach for estimating the parameters of the covariance function of a stationary spatial random process

We consider the problem of estimating the parameters of the covariance function of a stationary spatial random process. In spatial statistics, there are widely used parametric forms for the covariance functions, and various methods for estimating the parameters have been proposed in the literature. We develop a method for estimating the parameters of the covariance function that is based on a regression approach. Our method utilizes pairs of observations whose distances are closest to a value h>0$h>0$ which is chosen in a way that the estimated correlation at distance h is a predetermined value. We demonstrate the effectiveness of our procedure by simulation studies and an application to a water pH data set. Simulation studies show that our method outperforms all well-known least squares-based approaches to the variogram estimation and is comparable to the maximum likelihood estimation of the parameters of the covariance function. We also show that under a mixing condition on the random field, the proposed estimator is consistent for standard one parameter models for stationary correlation functions.     

Diagnostic Checking for GARCH-Type Models

The asymptotic distributions of squared and absolute residual autocorrelations for GARCH model estimated by M-estimators are derived. Two diagnostic tests are developed which can be used to check the adequacy of GARCH model fitted by using M-estimators. Simulation results show that the empirical sizes of both tests are close to the nominal size in most of the cases. The power of test based on absolute residual autocorrelation is found better than test based on squared residual autocorrelations. Our results reveal that there are estimators that can fit GARCH-type models better than the commonly used quasi-maximum likelihood estimator under non normal errors. An application to real data set is also presented.     

Estimation of causal continuous‐time autoregressive moving average random fields

We estimate model parameters of Lévy‐driven causal continuous‐time autoregressive moving average random fields by fitting the empirical variogram to the theoretical counterpart using a weighted least squares (WLS) approach. Subsequent to deriving asymptotic results for the variogram estimator, we show strong consistency and asymptotic normality of the parameter estimator. Furthermore, we conduct a simulation study to assess the quality of the WLS estimator for finite samples. For the simulation, we utilize numerical approximation schemes based on truncation and discretization of stochastic integrals and we analyze the associated simulation errors in detail. Finally, we apply our results to real data of the cosmic microwave background.     

Least squares variogram fitting by spatial subsampling

Summary. Least squares methods are popular for fitting valid variogram models to spatial data. The paper proposes a new least squares method based on spatial subsampling for variogram model fitting. We show that the method proposed is statistically efficient among a class of least squares methods, including the generalized least squares method. Further, it is computationally much simpler than the generalized least squares method. The method produces valid variogram estimators under very mild regularity conditions on the underlying random field and may be applied with different choices of the generic variogram estimator without analytical calculation. An extension of the method proposed to a class of spatial regression models is illustrated with a real data example. Results from a simulation study on finite sample properties of the method are also reported.     

An empirical likelihood goodness-of-fit test for time series

Summary. Standard goodness-of-fit tests for a parametric regression model against a series of nonparametric alternatives are based on residuals arising from a fitted model. When a parametric regression model is compared with a nonparametric model, goodness-of-fit testing can be naturally approached by evaluating the likelihood of the parametric model within a nonparametric framework. We employ the empirical likelihood for an α -mixing process to formulate a test statistic that measures the goodness of fit of a parametric regression model. The technique is based on a comparison with kernel smoothing estimators. The empirical likelihood formulation of the test has two attractive features. One is its automatic consideration of the variation that is associated with the nonparametric fit due to empirical likelihood's ability to Studentize internally. The other is that the asymptotic distribution of the test statistic is free of unknown parameters, avoiding plug-in estimation. We apply the test to a discretized diffusion model which has recently been considered in financial market analysis.     

New compactly supported spatiotemporal covariance functions from SPDEs

Potential theory and Dirichlet’s priciple constitute the basic elements of the well-known classical theory of Markov processes and Dirichlet forms. This paper presents new classes of fractional spatiotemporal covariance models, based on the theory of non-local Dirichlet forms, characterizing the fundamental solution, Green kernel, of Dirichlet boundary value problems for fractional pseudodifferential operators. The elements of the associated Gaussian random field family have compactly supported non-separable spatiotemporal covariance kernels admitting a parametric representation. Indeed, such covariance kernels are not self-similar but can display local self-similarity, interpolating regular and fractal local behavior in space and time. The associated local fractional exponents are estimated from the empirical log-wavelet variogram. Numerical examples are simulated for illustrating the properties of the space–time covariance model class introduced.     

ON A TEST OF FIT IN TIME SERIES MODELS

It is shown that a test for adequacy of a fitted Arma (p, 4) model based on forecast errors, has the same asymptotic expected value whether the null hypothesis is true or false. A sampling study supports the conclusion in small to moderately large samples and indicates that for sample sizes commonly used by practitioners, the power of the test is very low.     

Analysis of a jump-diffusion option pricing model with serially correlated jump sizes

This paper extends the classical jump-diffusion option pricing model to incorporate serially correlated jump sizes which have been documented in recent empirical studies. We model the series of jump sizes by an autoregressive process and provide an analysis on the underlying stock return process. Based on this analysis, the European option price and the hedging parameters under the extended model are derived analytically. Through numerical examples, we investigate how the autocorrelation of jump sizes influences stock returns, option prices and hedging parameters, and demonstrate its effects on hedging portfolios and implied volatility smiles. A calibration example based on real market data is provided to show the advantage of incorporating the autocorrelation of jump sizes.     

Dynamic Graphics for Exploring Spatial Data with Application to Locating Global and Local Anomalies

We explore the application of dynamic graphics to the exploratory analysis of spatial data. We introduce a number of new tools and illustrate their use with prototype software, developed at Trinity College, Dublin. These tools are used to examine local variability—anomalies—through plots of the data that display its marginal and multivariate distributions, through interactive smoothers, and through plots motivated by the spatial auto-covariance ideas implicit in the variogram. We regard these as alternative and linked views of the data. We conclude that the most important single view of the data is the Map View: All other views must be cross-referred to this, and the software must encourage this. The view can be enriched by overlaying on other pertinent spatial information. We draw attention to the possibilities of one-many linking, and to the use of line-objects to link pairs of data points. We draw attention to the parallels with work on Geographical Information Systems.     

Analysis of the human sex ratio by using overdispersion models

For study of the human sex ratio, one of the most important data sets was collected in Saxony in the 19th century by Geissler. The data contain the sizes of families, with the sex of all children, at the time of registration of the birth of a child. These data are reanalysed to determine how the probability for each sex changes with family size. Three models for overdispersion are fitted: the beta–binomial model of Skellam, the 'multiplicative' binomial model of Altham and the double-binomial model of Efron. For each distribution, both the probability and the dispersion parameters are allowed to vary simultaneously with family size according to two separate regression equations. A finite mixture model is also fitted. The models are fitted using non-linear Poisson regression. They are compared using direct likelihood methods based on the Akaike information criterion. The multiplicative and beta–binomial models provide similar fits, substantially better than that of the double-binomial model. All models show that both the probability that the child is a boy and the dispersion are greater in larger families. There is also some indication that a point probability mass is needed for families containing children uniquely of one sex.     

Assessing Lund's critical values for testing for outliers in linear regression models

This note assesses Lund's critical bounds as approximations to the critical values of a size-a test for detecting single outlier in a multiple linear regression model. A Monte Carlo experiment shows that the empirical sizes are very close to the nominal sizes, and the regressors have little effect on the empirical sizes. Lund's tables are revised to higher accuracy.     

Inference on local average treatment effects for misclassified treatment

We develop point-identification for the local average treatment effect when the binary treatment contains a measurement error. The standard instrumental variable estimator is inconsistent for the parameter since the measurement error is nonclassical by construction. We correct the problem by identifying the distribution of the measurement error based on the use of an exogenous variable that can even be a binary covariate. The moment conditions derived from the identification lead to generalized method of moments estimation with asymptotically valid inferences. Monte Carlo simulations and an empirical illustration demonstrate the usefulness of the proposed procedure.     

A resampling approach to estimate variance components of multilevel models

In a multilevel model for complex survey data, the weight‐inflated estimators of variance components can be biased. We propose a resampling method to correct this bias. The performance of the bias corrected estimators is studied through simulations using populations generated from a simple random effects model. The simulations show that, without lowering the precision, the proposed procedure can reduce the bias of the estimators, especially for designs that are both informative and have small cluster sizes. Application of these resampling procedures to data from an artificial workplace survey provides further evidence for the empirical value of this method. The Canadian Journal of Statistics 40: 150–171; 2012 © 2012 Statistical Society of Canada     

Estimation of the Moment Generating Function

A number of statistical problems use the moment generating function (mgf) for purposes other than determining the moments of a distribution. If the distribution is not completely specified, then the mgf must be estimated from available data. The empirical mgf makes no assumptions concerning the underlying distribution except for the existence of the mgf. In contrast to the nonparametric approach provided by the empirical mgf, alternative estimators can be formed based on an assumed parametric model. Comparison of these approaches is considered for two parametric models; the normal and a one parameter gamma. Comparison criteria are efficiency and empirical confidence interval coverage. In general the parametric estimators outperform the empirical mgf when the model is correct. The comparisons are extended to underlying models which are two component mixtures from the distributional family assumed by the parametric estimators. Under the mixture models the superiority of the parametric estimator depends upon the model, value of the argument of the mgf, and the comparison criterion. The empirical mgf is the better estimator in some cases.     

AN EMPIRICAL LIKELIHOOD APPROACH FOR NON‐GAUSSIAN VECTOR STATIONARY PROCESSES AND ITS APPLICATION TO MINIMUM CONTRAST ESTIMATION

We develop the empirical likelihood approach for a class of vector‐valued, not necessarily Gaussian, stationary processes with unknown parameters. In time series analysis, it is known that the Whittle likelihood is one of the most fundamental tools with which to obtain a good estimator of unknown parameters, and that the score functions are asymptotically normal. Motivated by the Whittle likelihood, we apply the empirical likelihood approach to its derivative with respect to unknown parameters. We also consider the empirical likelihood approach to minimum contrast estimation based on a spectral disparity measure, and apply the approach to the derivative of the spectral disparity. This paper provides rigorous proofs on the convergence of our two empirical likelihood ratio statistics to sums of gamma distributions. Because the fitted spectral model may be different from the true spectral structure, the results enable us to construct confidence regions for various important time series parameters without assuming specified spectral structures and the Gaussianity of the process.     

A note on the estimation methods of crop production at the block level

This paper presents a method of estimation of crop-production statistics at smaller geographical levels like a community development block (generally referred to as a block) to make area-specific plans for agricultural development programmes in India. Using available district-level data on crop yield from crop-cutting experiments and data on auxiliary variables from various administrative sources, a suitable regression model is fitted. The fitted model is then used to predict the crop production at the block level. Some scaled estimators are also developed using predicted estimates. An empirical study is also carried out to judge the merits of the proposed estimators.