Abstract
Department of Actuarial ScienceInstituto Tecnolo´gico Auto´nomo de M´exicoR´io Hondo No. 1, Col. Progreso Tizap´an, C.P. 01080, M´exico D.F., M´exicoe-mail: mercedes@itam.mxAbstract: In Fern´andez-Duran [4], a new family of circular distributionsbased on nonnegative trigonometric sums (NNTS models) is developed.Because the parameter space of this family is the surface of the hypersphere,an efficient Newton-like algorithm on manifolds is generated in order toobtain the maximum likelihood estimates of the parameters.AMS 2000 subject classifications: Primary 49M15, 62G07; secondary49Q99.Keywords and phrases: Differential geometry, maximum likelihood es-timation, Newton algorithm, nonnegative Fourier series, smooth Riemannmanifold.Received November 2010.
Highlights
IntroductionThe θ parameter space corresponds to the surface of a 2M + 1 dimensional hypersphere
The probability density function, f (x; θ), of a circular random variable X ∈ (0, 2π] must be nonnegative and periodic (f (x + 2kπ; θ) = f (x; θ)) for any integer k where θ is the vector of parameters
The circular density function based on nonnegative trigonometric sums (NNTS density) is expressed as
Summary
The θ parameter space corresponds to the surface of a 2M + 1 dimensional hypersphere This family of circular distributions has the advantage of being able to fit datasets that present multimodality and/or skewness because the density function can be expressed as a mixture of multimodal circular distributions. The second section presents a convenient, alternative way to express likelihood functions for continuous and grouped data in the univariate case. Given these convenient expressions for likelihood functions, in the third section an efficient Newton-like algorithm is developed for maximizing the log-likelihood function on the surface of the hypersphere. The conclusions of the present work are presented in the fifth section
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