QUANTILES FROM THE MAXIMUM STUDENTIZED RANGE DISTRIBUTION

Daniel Furtado FERREIRA[1]

Clarice Garcia Borges DEMÉTRIO[2]

Bryan Frederick John MANLY[3]

Amauri de Almeida MACHADO[4]

§   ABSTRACT: The Copenhaver and Holland (1988) algorithm for calculating quantiles and the distribution function of the maximum of c statistics having studentized range distributions of r sample means obtained from random a sample of size n from normal homocedastic distributions was adapted and implemented in Pascal. The algorithm uses Gauss-Legendre quadrature to obtain the distribution function and uses the secant method to calculate the 100p% quantiles from that distribution. The main advantage of the implemented algorithm is that it is not limited to the range of quantiles from 90% to 99%, and it is more accurate than the iterative procedure proposed by Lund and Lund (1983). When c = 1 the algorithm supplies the cumulative distribution function and the quantiles of the studentized range distribution, except for the particular case where the number of degrees of freedom is v=1. In general, the accuracy of the algorithm decreases as r increases, as v decreases and as p approaches 1. Also r and v have a larger effect on accuracy. Differences among the results from Tukey and maximum studentized range quantiles were shown in a real example.  These differences were due to the guarantee of the overall protection for the global confidence coefficient only in the maximum studentized range distribution.

§   KEYWORDS: Studentized range; quantiles



[1]Departamento de Ciências Exatas, Universidade Federal de Lavras – UFLA, CEP 37200‑000, Lavras, MG, Brazil. E‑mail: danielff@ufla.br

[2]Departamento de Ciências Exatas, Escola Superior de Agricultura “Luiz de Queiroz” da Universidade de São Paulo – ESALQ/USP, CEP 13418-900, Piracicaba, SP, Beasil. E-mail: clarice@carpa.ciagri.usp.br

[3] Western EcoSystems Technology Inc. of Wyoming, USA – CAPES visiting Professor in Departamento de Ciências Exatas, Escola Superior de Agricultura “Luiz de Queiroz” da Universidade de São Paulo – ESALQ/USP, E-mail: bmanly@compuserve.com

[4]Departamento de Matemática Estatística e Informática, Instituto de Física e Matemática, Universidade Federal de Pelotas – UFPEL, Caixa Postal 354, CEP 96010-900, Pelotas, RS, Brasil. E-mail: amachado@ufpel.edu.br