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**1 - 5**of**5**### Pré-Publicações do Departamento de Matemática Universidade de Coimbra Preprint Number 06–61 ON THE FINITE SAMPLE BEHAVIOUR OF FIXED BANDWIDTH BICKEL-ROSENBLATT TEST FOR UNIVARIATE AND MULTIVARIATE UNIFORMITY

"... Abstract: The Bickel-Rosenblatt (BR) goodness-of-fit test with fixed bandwidth was introduced by Fan in 1998 [Econometric Theory 14, 604–621, 1998]. Although its asymptotic properties have being studied by several authors, little is known about its finite sample performance. Restricting our attentio ..."

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Abstract: The Bickel-Rosenblatt (BR) goodness-of-fit test with fixed bandwidth was introduced by Fan in 1998 [Econometric Theory 14, 604–621, 1998]. Although its asymptotic properties have being studied by several authors, little is known about its finite sample performance. Restricting our attention to the test of uniformity in the d-unit cube for d ≥ 1, we present in this paper a description of the finite sample behaviour of the BR test as a function of the bandwidth h. For d = 1 our analysis is based not only on empirical power results but also on the Bahadur’s concept of efficiency. The numerical evaluation of the Bahadur local slopes of the BR test statistic for different values of h for a set of Legendre and trigonometric alternatives give us some additional insight about the role played by the smoothing parameter in the detection of departures from the null hypothesis. For d> 1 we develop a Monte-Carlo study based on a set of meta-type uniforme alternative distributions and a rule-of-thumb for the practical choice of the bandwidth is proposed. For both univariate and multivariate cases, comparisons with existing uniformity tests are presented. The BR test reveals an overall good comparative performance, being clearly superior to the considered competitors tests for bivariate data.

### WHITE NOISE ASSUMPTIONS REVISITED: REGRESSION METAMODELS AND EXPERIMENTAL DESIGNS IN PRACTICE

"... Classic linear regression metamodels and their concomitant experimental designs assume a univariate (not multivariate) simulation response and white noise. By definition, white noise is normally (Gaussian), independently (implying no common random numbers), and identically (constant variance) distri ..."

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Classic linear regression metamodels and their concomitant experimental designs assume a univariate (not multivariate) simulation response and white noise. By definition, white noise is normally (Gaussian), independently (implying no common random numbers), and identically (constant variance) distributed with zero mean (valid metamodel). This advanced tutorial tries to answer the following questions: (i) How realistic are these classic assumptions in simulation practice? (ii) How can these assumptions be tested? (iii) If assumptions are violated, can the simulation’s I/O data be transformed such that the analysis becomes correct? (iv) If such transformations cannot be applied, which alternative statistical methods (for example, generalized least squares, bootstrapping, jackknifing) can then be applied? 1

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"... the choice of the smoothing parameter for the BHEP goodness-of-fit test∗ ..."

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