Abstract
This paper extends the simple linear regression model with wrapped Cauchy error to the functional case when both variables are subjected to wrapped Cauchy errors. Assuming the ratio between the two error variances is known and the slope parameter equals one the maximum likelihood estimates are obtained. The closed-form expression for the maximum likelihood estimators are not available and the estimates are obtained iteratively by choosing a suitable initial values. The quality of estimates and the accuracy of the model are illustrated via simulations and the results revealed an acceptable performance of the estimators where they are unbiased, consistent and robust. The sampling variances of the model parameters are obtained via bootstrapping methods and consequently the confidence intervals were constructed. The proposed model is illustrated with an application on the analysis of wind directions data at two cities in the Gaza strip, Palestine.
Highlights
The functional relationship model is a part of the general class of Error-In-Variables Models (EIVM) which known as the measurement error or random regression models
In the ordinary regression we supposed that the explanatory variables measured without error and all the errors are in the response variable, but the EIVM assumed that the errors are in both response and explanatory variables
Robustness of an estimator is a useful property which gives a fair assurance that the existence of any possible outlier or violation of model assumptions will not have much effect on the parameters estimates
Summary
The functional relationship model is a part of the general class of Error-In-Variables Models (EIVM) which known as the measurement error or random regression models. The un-replicated linear functional relationship for two circular variables was first introduced by (Hussin 1997) assuming that errors of both variables are independently distributed with von Mises distribution with equal variances. Later, (Hussin 2003) improved the model by estimating the concentration parameters of error for any ratio using the asymptotic properties of the Bessel function. (Caires and Wyatt 2003) proposed the simple linear circular functional relationship model in which the value of the slope parameter is fixed to be one, and the errors are assumed to follow von Mises distribution for any ratio . The heavily tailed property of the wrapped Cauchy distribution motivated (Abuzaid and Allahham 2015) to propose the simple linear regression for circular variables in the following form: yi = + xi + i (mod 2 ), where i circular random error having a wrapped Cauchy distribution with circular mean 0 an concentration parameter.
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