Tenreiro, Carlos On the finite sample behavior of fixed bandwidth Bickel-Rosenblatt test for univariate and multivariate uniformity. (English) Zbl 1126.62036 Commun. Stat., Simulation Comput. 36, No. 4, 827-846 (2007). Summary: The Bickel-Rosenblatt (BR) goodness-of-fit test with fixed bandwidth was introduced by Y. Fan [Goodness-of-fit tests based on kernel density estimators with fixed smoothing parameters. Econometric Theory 14, 604–621 (1998)]. Although its asymptotic properties have been 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 \(d \geq\) 1, we present a description of the finite sample behavior 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 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 gives 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 uniform 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 competiting tests for bivariate data. Cited in 7 Documents MSC: 62G10 Nonparametric hypothesis testing 62G20 Asymptotic properties of nonparametric inference 65C60 Computational problems in statistics (MSC2010) 65C05 Monte Carlo methods Keywords:Bahadur efficiency; goodness-of-fit test; kernel density estimator; uniformity test; tables Software:LAPACK PDF BibTeX XML Cite \textit{C. Tenreiro}, Commun. Stat., Simulation Comput. 36, No. 4, 827--846 (2007; Zbl 1126.62036) Full Text: DOI References: [1] Anderson E., LAPACK User’s Guide., 3. ed. (1999) [2] Anderson N.H., J. Multivariate Anal. 50 pp 41– (1994) · Zbl 0798.62055 · doi:10.1006/jmva.1994.1033 [3] DOI: 10.2307/2281537 · Zbl 0059.13302 · doi:10.2307/2281537 [4] DOI: 10.1214/aos/1176342558 · Zbl 0275.62033 · doi:10.1214/aos/1176342558 [5] Eubank R. L., Ann. Statist. 20 pp 2071– (1992) · Zbl 0769.62033 · doi:10.1214/aos/1176348903 [6] Fan Y., Econometric Theor. 10 pp 316– (1994) · Zbl 04520945 · doi:10.1017/S0266466600008434 [7] Fan Y., Econometric Theor. 14 pp 604– (1998) [8] DOI: 10.1080/03067310290024030 · doi:10.1080/03067310290024030 [9] Gouriéroux C., J. Multivariate Anal. 78 pp 161– (2001) · Zbl 1081.62529 · doi:10.1006/jmva.2000.1950 [10] DOI: 10.1214/aos/1176343744 · Zbl 0371.62033 · doi:10.1214/aos/1176343744 [11] DOI: 10.1016/S0167-7152(97)00015-1 · Zbl 0955.62060 · doi:10.1016/S0167-7152(97)00015-1 [12] DOI: 10.1006/jmva.1997.1684 · Zbl 0874.62043 · doi:10.1006/jmva.1997.1684 [13] DOI: 10.1080/03610929008830400 · Zbl 0738.62068 · doi:10.1080/03610929008830400 [14] Johnson M. E., Multivariate Statistical Simulation (1987) · Zbl 0604.62056 [15] Kim J.-T., J. Amer. Statist. Assoc. 95 pp 829– (2000) · Zbl 0996.62049 · doi:10.2307/2669466 [16] Koroljuk V. S., Theory of U-Statistics (1989) [17] Liang J.-J., Math. Comp. 70 pp 337– (2001) · Zbl 0958.65016 · doi:10.1090/S0025-5718-00-01203-5 [18] Madras N., Lectures on Monte Carlo Methods (2002) · Zbl 0987.65003 [19] DOI: 10.1080/02331880500178562 · Zbl 1084.62041 · doi:10.1080/02331880500178562 [20] Miller F. L.jun., Commun. Statist. Simul. Comput. B 8 pp 271– (1979) · Zbl 0467.62019 · doi:10.1080/03610917908812119 [21] DOI: 10.1017/CBO9780511530081 · doi:10.1017/CBO9780511530081 [22] DOI: 10.1214/aoms/1177704472 · Zbl 0116.11302 · doi:10.1214/aoms/1177704472 [23] DOI: 10.1214/aoms/1177729394 · Zbl 0047.13104 · doi:10.1214/aoms/1177729394 [24] DOI: 10.1214/aoms/1177728190 · Zbl 0073.14602 · doi:10.1214/aoms/1177728190 [25] Shorack G. R., Empirical Processes with Applications to Statistics (1986) · Zbl 1170.62365 [26] DOI: 10.2307/2286009 · doi:10.2307/2286009 [27] Stephens M. A., Goodness-of-Fit Techniques pp 331– (1986) [28] Tenreiro C., SORT 29 pp 201– (2005) [29] Tenreiro C., J. Statist. Plann. Inf. 137 pp 103– (2007) · Zbl 1098.62055 · doi:10.1016/j.jspi.2005.11.006 [30] Watson G. S., Biometrika 48 pp 109– (1961) · Zbl 0212.21905 · doi:10.1093/biomet/48.1-2.109 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.