Confidence region on the location of the
 stationary point in response surfaces,
a Bayesian bootstrap approach

David José MIQUELLUTI [1]

Silvio Sandoval ZOCCHI [2]

Clarice Garcia Borges DEMÉTRIO 2

§    ABSTRACT: The procedures for the construction of confidence regions for the coordinates of the stationary point were studied in different situations, considering the shape of the surfaces and the distribution of the variance errors. The methodology of Box and Hunter (1954) (BH), bootstrap and Bayesian bootstrap with Mahalanobis distance among the coordinates of the stationary point of the observed sample and those obtained using bootstrap estimates(BCM and BBM) and bootstrap and Bayesian bootstrap with non-parametric methods for density estimation (BCNP and BBNP) were compared. The methodology evaluation was realized by means of simulation and applied to a peanuts data set. The BH methodology presented a good performance in the analyzed situations, having concordance among the nominal and real confidence regions. This behavior was also observed for the BCM and BBM methods. The BCNP and BBNP methods did not presented a satisfactory performance, resulting in a real significance level lower than the nominal for the lower eigenvalue. The inverse was observed using higher eigenvalue. In the analysis of the peanuts data set the BH, BCM and BCNP methods presented confidence regions larger than the BBM and BBNP methods. The Bayesian bootstrap estimate values were closer of the minimum square estimates and presented less dispersion.

§    KEYWORDS: Respose surface; stationary point; confidence regions; Bayesian bootstrap; non parametric density.

[1] Departamento de Solos e Recursos Naturais, Centro de Ciências Agroveterinárias, Universidade do Estado de Santa Catarina -- UDESC, CEP: 88520-000, Lages, SC, Brasil. E-mail:

[2] Departamento de Ciências Exatas, Escola Superior de Agricultura “Luiz de Queiroz” – ESALQ, Universidade de São Paulo – USP, CEP: 13418-900, Piracicaba, São Paulo, Brasil. E-mail: /