Robust Estimation in Analysis of Variance

This paper addresses two central problems in simulation methodology: the generation of confidence intervals for the steady state means of the output sequences and the sequential use of these confidence intervals to control the run length. The variance of the sample mean of a covariance stationary process is given approximately by p(0)/N, where p(f) is the spectral density at frequency f and N is the sample size. In an earlier paper we developed a method of confidence interval generation based on the estimation of p(0) through the least squares fit of a quadratic to the logarithm of the periodogram. This method was applied in a run length control procedure to a sequence of batched means. As the run length increased the batch means were rebatched into larger batch sizes so as to limit storage requirements. In this rebatching the shape of the spectral density changes, gradually becoming flat as N increases. Quadratics were chosen as a compromise between small sample bias and large sample stability. In this paper we consider smoothing techniques which adapt to the changing spectral shape in an attempt to improve both the small and large sample behavior of the method. The techniques considered are polynomial smoothing with the degree selected sequentially using standard regression statistics, polynomial smoothing with the degree selected by cross validation, and smoothing splines with the amount of smoothing determined by cross validation. These techniques were empirically evaluated both for fixed sample sizes and when incorporated into the sequential run length control procedure. For fixed sample sizes they did not improve the small sample behavior and only marginally improved the large sample behavior when compared with the quadratic method. Their performance in the sequential procedure was unsatisfactory. Hence, the straightforward quadratic technique recommended in the earlier paper is still recommended as an effective, practical technique for simulation confidence interval generation and run length control.

Performance measures of phylogenetic estimation methods such as accuracy, consistency, and power are an attempt at summarizing an ensemble of a given estimator's behavior. These summaries characterize an ensemble behavior with a single number, leading to a variety of definitions. In particular, the relationships between different performance measures such as accuracy and consistency or accuracy and error depend on the exact definition of these measures. In addition, it is relatively common to use large-sample behavior to infer similar behavior for small samples. In fact, large-sample results such as the claimed asymptotic efficiency of the maximum-likelihood estimator are often uninformative for small samples. Conversely, small-sample behavior using simulations is sometimes used to imply large-sample behavior such as consistency. However, such extrapolation is often difficult. How the performance of a phylogenetic estimator scales with the addition of taxa must be qualified with respect to whether the whole tree is being estimated or a fixed subset of taxa is being estimated. It must also be qualified with respect to how tree models are sampled. Over the ensemble of all possible trees of a given size, the performance of the estimators for the whole tree estimate suffers when the tree size becomes larger. However, under certain models of cladogenesis, the estimate can improve with the addition of taxa. In fact, at all numbers of taxa there are subsets of tree models that are easier to estimate than others. This suggests that with judicious addition or subtraction of taxa we can move from tree models that are more difficult to estimate at one number of taxa to those that are easier to estimate at another number of taxa.

Finding a new drug which is both safe and efficient is an expensive and time-consuming endeavour. In particular, establishing the ‘dose-effect relationship’ – how beneficial a drug is at different dosages – can be challenging. Predicting this curve requires gathering experimental data by exposing and recording how cells respond to various levels of the drug. However, extreme values are often observed at low and high dosages, potentially introducing errors that are hard to correct in the prediction process. Yet, these extreme observations are sometimes genuine so researchers cannot just ignore them. To improve dose-effect estimation, Zhou, Liu, Fang et al. developed a new general-purpose approach. It uses advanced statistical modelling to account for extremes in lab data. This strategy outperformed other methods when dealing with these observations while also providing higher efficiency in data analysis with more uniform data in experiments. To facilitate implementation, Zhou, Liu, Fang et al. set up a user-friendly tool baptized ‘REAP’; this free online resource allows scientists without advanced statistical experience to harness the new approach and to perform dose-effect analysis more easily and accurately. This could boost research across many different disciplines that examine the effects of chemicals on cells.

Bootstrap is a widely applicable computational statistical method. The focus of this paper lies in developing a bootstrap method for linear mixed models under homoscedasticity violation (heteroscedasticity) in variance of errors. We assume that the form of heteroscedasticity is unknown. To generate bootstrap response data as close as possible to the actual response data, we transform the marginal residuals to ensure that the variance of the modified marginal residuals in bootstrap samples is an unbiased estimator for the variance of the response variable in the linear mixed models. The consistency is proved, implying that the parameter estimators by the proposed method are asymptotically unbiased. The simulations are conducted with varying error terms and sample sizes to show improvement of the proposed method in parameter estimation. The simulation results are compared to demonstrate that the proposed method outperforms the other considered approaches including the wild method in the heteroscedastic linear mixed models under small samples and performs competitively with the parametric method and both methods outperform the other considered methods in homoscedastic linear mixed models. The proposed bootstrap method is applied to squid data for illustration of its effectiveness.

Aims: This study considered the optimal maintenance cost of water boreholes for a multiple pumping scenario in South-East Nigeria. The objective of the study was to determine the optimal maintenance cost of water boreholes for the Southeastern region of Nigeria.
 Place and Duration of Study: The data used in this study is secondary data obtained from selected community and commercial boreholes in the Abia State, Anambra State, Ebonyi State, Enugu State and Imo State.
 Methodology: The linear model and the quadratic model were used to determine the cost parameters for the optimization of the maintenance cost of the water boreholes.
 Results: The result of the study showed that the maintenance cost obtained for Anambra State is more adequate than the other states since it recorded a higher coefficient of determination value using the linear and quadratic model. Also, it was found the quadratic model was better than the linear model for the estimation of cost parameters used for determining the optimum maintenance cost of water boreholes in South-East Nigeria. The findings showed that the quadratic model recorded the best net savings for Abia, Anambra, and Enugu with the corresponding cost of ₦4,079,246.817, ₦5,198,716.141, and ₦4,767,680.392 respectively. It was found that the linear model recorded the best net savings for Ebonyi and Imo with the corresponding cost of ₦1,593,013.72 and ₦8,692,452.91 respectively. Hence, the quadratic models was identified as the best model for the estimation of cost parameters and enhance the performance of the optimal total operating cost of the water borehole model in South-East Nigeria.
 Conclusion: The study concludes by recommending the quadratic model for the estimation of cost parameters for the optimization of maintenance cost of water boreholes until further studies prove otherwise.

Pharmacokinetic-Pharmacodynamic Modeling and Simulation

The linear regression model is widely used in empirical work in Economics, Statistics, and many other disciplines. Researchers often include many covariates in their linear model specification in an attempt to control for confounders. We give inference methods that allow for many covariates and heteroskedasticity. Our results are obtained using high-dimensional approximations, where the number of included covariates are allowed to grow as fast as the sample size. We find that all of the usual versions of Eicker-White heteroskedasticity consistent standard error estimators for linear models are inconsistent under this asymptotics. We then propose a new heteroskedasticity consistent standard error formula that is fully automatic and robust to both (conditional) heteroskedasticity of unknown form and the inclusion of possibly many covariates. We apply our findings to three settings: parametric linear models with many covariates, linear panel models with many fixed effects, and semiparametric semi-linear models with many technical regressors. Simulation evidence consistent with our theoretical results is also provided. The proposed methods are also illustrated with an empirical application.

ABSTRACTThe linear regression model is widely used in empirical work in economics, statistics, and many other disciplines. Researchers often include many covariates in their linear model specification in an attempt to control for confounders. We give inference methods that allow for many covariates and heteroscedasticity. Our results are obtained using high-dimensional approximations, where the number of included covariates is allowed to grow as fast as the sample size. We find that all of the usual versions of Eicker–White heteroscedasticity consistent standard error estimators for linear models are inconsistent under this asymptotics. We then propose a new heteroscedasticity consistent standard error formula that is fully automatic and robust to both (conditional) heteroscedasticity of unknown form and the inclusion of possibly many covariates. We apply our findings to three settings: parametric linear models with many covariates, linear panel models with many fixed effects, and semiparametric semi-linear models with many technical regressors. Simulation evidence consistent with our theoretical results is provided, and the proposed methods are also illustrated with an empirical application. Supplementary materials for this article are available online.

We prove theorems about the Gaussian asymptotics of an empirical bridge built from linear model regressors with multiple regressor ordering. We study the testing of the hypothesis of a linear model for the components of a random vector: one of the components is a linear combination of the others up to an error that does not depend on the other components of the random vector. The results of observations of independent copies of a random vector are sequentially ordered in ascending order of several of its components. The result is a sequence of vectors of higher dimension, consisting of induced order statistics (concomitants) corresponding to different orderings. For this sequence of vectors, without the assumption of a linear model for the components, we prove a lemma of weak convergence of the distributions of an appropriately centered and normalized process to a centered Gaussian process with almost surely continuous trajectories. Assuming a linear relationship of the components, standard least squares estimates are used to compute regression residuals -- the differences between response values and the predicted ones by the linear model. We prove a theorem of weak convergence of the process of regression residuals under the necessary normalization to a centered Gaussian process.

This study examines the endogeneity effect on autoregressive linear models of AR (1) in small samples, making use of the Ordinary Least Square (OLS) estimator, Two-Stage Least Squares (2SLS) estimator, and Generalized Method of Moment (GMM) estimator, based on the sensitivity analysis of sample size and specification errors in estimator determination in linear regression model through the use of Monte Carlo simulation and application to real-life data. The simulation indicates that 2SLS and GMM estimators show the smallest biases when the sample size is varied from n = 10, 25, 50 to 100. The estimator that performs best when sample size n = 10 across autocorrelation (ρ) and significant correlation (α) at all levels of replication of 10,000 is GMM. In the real-life data, OLS and 2SLS exhibit higher endogeneity characteristics from the dataset used. The empirical analysis base on MSE criteria GMM is the best estimator for dealing with external shock factors to inflations embedded with endogeneity in the linear model. When endogeneity and autocorrelation are bedeviled in a linear AR (1) model, in small samples, using the GMM estimator will provide the best results in small samples than using 2SLS and OLS.

The linear-fractional model (LFM) is a fraction function whose numerator and denominator are linear in parameters. The LFM is a group of models nonlinear in parameters. The estimation methods for nonlinear models can be applied to the FLM. However, the parameters in an LFM can naturally be divided into two groups: those in the numerator and those in the denominator. When the parameters in the denominator are known, the standard least squares algorithm for the linear model can be used to estimate the parameters in the numerator. On the other hand, when parameters in the numerator are known, by a reciprocal transformation, the standard least squares algorithm for the linear model can again be used to estimate the parameters in the denominator. From this observation, we develop a recursive least-squares algorithm for estimation of parameters in the LFM when both groups has unknown parameters. The basic idea is to estimate the parameters in the numerators for a given initial parameters in the denominator using the standard least squares algorithm for the linear model, and then to estimate the parameters in the denominator with the previous estimates of parameters in the denominator using the standard least squares algorithm for the linear model when new data is available. The simulation results validated the convergence of the proposed algorithm and also showed the superior performance of the algorithm proposed over some existing algorithm.

The main goal of this research article is to discuss the mathematical and statistical aspects of linear models and their characteristic properties. Linear model is the most common modeling used in science. Actually linear models have many different meanings depend on the context. Linear model is often preferred than other model such as quadratic model because of its ability to interpret easily. In the other hand most of the real life cases have linear relationship .Modeling the cases using linear model will able us to computethe relative influence of one or more independent variables to the dependent variable. In the present talk an attempt has been made to propose the specific forms of simple and multiple linear regression models. In this conversation mathematical aspects of linear statistical models have been extensively depicted. Different types of mathematical models are depictedhere and the methods of fitting transformed models are proposed.Furthermore specific form of linear statistical model is presented and the crucial assumptions of general linear model are extensively discussed.At the last stage of this article the procedure of OLS estimation of parameters of a linear model has been proposed

Non-parametric estimation of a survival function from left truncated data subject to right censoring has been extensively studied in the literature. It is commonly assumed in such studies that the lifetime variables are a sample of independent and identically distributed random variables from the target population. This assumption is often prone to failure in practical studies. For instance, when recruited subjects are all from the same institute or the same geographical region. To the best of our knowledge, there is no study in the past literature addressing such situations. In this article, we study large and small sample behavior of Tsai-Jewell-Wang estimator under positive and negative association.

Modeling and Identification of a Nuclear Reactor

Prediction of spatial soil property information from ancillary sensor data using ordinary linear regression: Model derivations, residual assumptions and model validation tests