The L<sub>1</sub> regression is a robust alternative to the least squares regression whenever there are outliers in the values of the response variable, or the errors follow a long-tailed distribution. To ...The L<sub>1</sub> regression is a robust alternative to the least squares regression whenever there are outliers in the values of the response variable, or the errors follow a long-tailed distribution. To calculate the standard errors of the L<sub>1</sub> estimators, construct confidence intervals and test hypotheses about the parameters of the model, or to calculate a robust coefficient of determination, it is necessary to have an estimate of a scale parameterτ. This parameter is such that τ<sup>2</sup>/n is the variance of the median of a sample of size n from the errors distribution. [1] proposed the use of , a consistent, and so, an asymptotically unbiased estimator of τ. However, this estimator is not stable in small samples, in the sense that it can increase with the introduction of new independent variables in the model. When the errors follow the Laplace distribution, the maximum likelihood estimator of τ, say , is the mean absolute error, that is, the mean of the absolute residuals. This estimator always decreases when new independent variables are added to the model. Our objective is to develop asymptotic properties of under several errors distributions analytically. We also performed a simulation study to compare the distributions of both estimators in small samples with the objective to establish conditions in which is a good alternative to for such situations.展开更多
In this article, we develop numerical method by constructing ninth degree spline function using extended cubic spline Bickley’s method to find the approximate solution of seventh order linear boundary value problems ...In this article, we develop numerical method by constructing ninth degree spline function using extended cubic spline Bickley’s method to find the approximate solution of seventh order linear boundary value problems at different step lengths. The approximate solution is compared with the solution obtained by eighth degree splines and exact solution. It has been observed that the approximate solution is an excellent agreement with exact solution. Low absolute error indicates that our numerical method is effective for solving high order linear boundary value problems.展开更多
The efficiency of an optimization method for acoustic emission/microseismic（AE/MS） source location is determined by the compatibility of its error definition with the errors contained in the input data.This compatib...The efficiency of an optimization method for acoustic emission/microseismic（AE/MS） source location is determined by the compatibility of its error definition with the errors contained in the input data.This compatibility can be examined in terms of the distribution of station residuals.For an ideal distribution,the input error is held at the station where it takes place as the station residual and the error is not permitted to spread to other stations.A comparison study of two optimization methods,namely the least squares method and the absolute value method,shows that the distribution with this character constrains the input errors and minimizes their impact,which explains the much more robust performance by the absolute value method in dealing with large and isolated input errors.When the errors in the input data are systematic and/or extreme in that the basic data structure is altered by these errors,none of the optimization methods are able to function.The only means to resolve this problem is the early detection and correction of these errors through a data screening process.An efficient data screening process is of primary importance for AE/MS source location.In addition to its critical role in dealing with those systematic and extreme errors,data screening creates a favorable environment for applying optimization methods.展开更多
文摘The L<sub>1</sub> regression is a robust alternative to the least squares regression whenever there are outliers in the values of the response variable, or the errors follow a long-tailed distribution. To calculate the standard errors of the L<sub>1</sub> estimators, construct confidence intervals and test hypotheses about the parameters of the model, or to calculate a robust coefficient of determination, it is necessary to have an estimate of a scale parameterτ. This parameter is such that τ<sup>2</sup>/n is the variance of the median of a sample of size n from the errors distribution. [1] proposed the use of , a consistent, and so, an asymptotically unbiased estimator of τ. However, this estimator is not stable in small samples, in the sense that it can increase with the introduction of new independent variables in the model. When the errors follow the Laplace distribution, the maximum likelihood estimator of τ, say , is the mean absolute error, that is, the mean of the absolute residuals. This estimator always decreases when new independent variables are added to the model. Our objective is to develop asymptotic properties of under several errors distributions analytically. We also performed a simulation study to compare the distributions of both estimators in small samples with the objective to establish conditions in which is a good alternative to for such situations.
文摘In this article, we develop numerical method by constructing ninth degree spline function using extended cubic spline Bickley’s method to find the approximate solution of seventh order linear boundary value problems at different step lengths. The approximate solution is compared with the solution obtained by eighth degree splines and exact solution. It has been observed that the approximate solution is an excellent agreement with exact solution. Low absolute error indicates that our numerical method is effective for solving high order linear boundary value problems.
文摘The efficiency of an optimization method for acoustic emission/microseismic（AE/MS） source location is determined by the compatibility of its error definition with the errors contained in the input data.This compatibility can be examined in terms of the distribution of station residuals.For an ideal distribution,the input error is held at the station where it takes place as the station residual and the error is not permitted to spread to other stations.A comparison study of two optimization methods,namely the least squares method and the absolute value method,shows that the distribution with this character constrains the input errors and minimizes their impact,which explains the much more robust performance by the absolute value method in dealing with large and isolated input errors.When the errors in the input data are systematic and/or extreme in that the basic data structure is altered by these errors,none of the optimization methods are able to function.The only means to resolve this problem is the early detection and correction of these errors through a data screening process.An efficient data screening process is of primary importance for AE/MS source location.In addition to its critical role in dealing with those systematic and extreme errors,data screening creates a favorable environment for applying optimization methods.
