On parameter estimation, confidence intervals and outlier detection for some circular models

This study focuses on the parameter estimation, confidence interval estimation and outlier detection for several types of the circular model. The models consider in this study
are Down and Mardia Circular Regression Model, Circular Functional Relationship Model (CFRM) and a new model known as the Simultaneous Circular Functional Relationship Model (CSFRM). Firstly, for the Down and Mardia (DM) Circular
Regression model, we propose the roots of a polynomial function (polyroot) and minimizing the negative value of the log-likelihood function (ms) to obtain the parameter
estimates. The confidence interval for this model are obtained based on the normal distribution where the parameter are estimated using polyroot or minimum sum (ms) and estimation using the bootstrap technique. Secondly, we consider the parameters estimation and confidence intervals for CFRM. We used the polyroot and ms to estimate the
concentration, angular and slope parameters. The confidence intervals for the parameters are obtained based on the normal distribution where the parameters are estimated using
polyroot or ms and estimation using the bootstrap technique. Thirdly, we consider the outlier problem in CFRM using two statistics namely the Functional Difference Mean Circular Error Using Cosine (FDMCEc) statistic and Functional Difference Mean Circular Error Using Sine (FDMCEs) statististic. This methods also used the row deletion method
to calculate FDMCEc and FDMCEs values for the full and reduced data set. In our CFRM, we determine the cut-off points for several values of sample size and concentration
parameter designed using simulation study. Lastly, a new simultaneous circular functional relationship model (CSFRM) which is an extended version of a CFRM is developed. We
want to study the relationship for more than two circular variables. We derived the maximum likelihood function of the model and the estimate the variance-covariance of parameters based on Fisher information matrix. Then, the efficiency of the model is assessed using the biasness for the circular variables using circular mean, circular distance and mean resultant length and mean, absolute estimated bias and estimated root mean square error for continuous variables respectively. Model verification and assessment of all methods and model proposed in this study are examined using the simulation study. The wind and wave direction data set used for illustration and application to real data set