Fitting regression models using the LINEX loss function: Properties and applications



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Researchers in statistical shape analysis often analyze outlines of objects. Even though these contours are infinite-dimensional in theory, they must be discretized in practice. When discretizing, it is important to reduce the number of sampling points considerably to reduce computational costs, but to not use too few points so as to result in too much approximation error. Unfortunately, determining the minimum number of points needed to achieve sufficiently approximate the contours is computationally expensive. In this paper, we fit regression models to predict these lower bounds using characteristics of the contours that are computationally cheap as predictor variables.

However, least squares regression is inadequate for this task because it treats overestimation and underestimation equally, but underestimation of lower bounds is far more serious. Instead, to fit the models, we use the LINEX loss function, which allows us to penalize underestimation at an exponential rate while penalizing overestimation only linearly. We present a novel approach to select the shape parameter of the loss function and tools for analyzing how well the model fits the data. Through validation methods, we show that the LINEX models work well for reducing the underestimation for the lower bounds.

The LINEX regression model works well in reducing the underestimation with an appropriate shape parameter value. Our study does not focus on model selection in regression analysis under the LINEX loss function. The main goal is to develop the regression model using the Ordinary Least Squares (OLS) regression model as the basic model, which will enhance the prediction performance by reducing the underestimation or overestimation by using the LINEX loss function. Since there is no closed-form solution for LINEX estimators, from the simulation study, we show that regardless of the error distribution, it provides a bias estimator. We also show that increasing the shape parameter value in a large amount will not reduce the underestimation a lot after some point. Finally, we illustrate the usage of the LINEX loss function to reduce the underestimation in data fitting for dynamical models rather than using the squared error loss function.

Embargo status: Restricted until 09/2027. To request the author grant access, click on the PDF link to the left.



Linear Regression, LINEX Loss Function, Planar Contours