×

Limit laws for disparities of spacings. (English) Zbl 1024.62020

Summary: Disparities of spacings mean the \(\varphi\)-disparities \(D_\varphi (\overline q_n,p_n)\) of discrete hypothetical and empirical distributions \(\overline q_n\) and \(p_n\) defined by \(m\)-spacings on i.i.d. samples of size \(n\) where \(\varphi: (0,\infty)\mapsto R\) is twice continuously differentiable in a neighborhood of 1 and strictly convex at 1. It is shown that a slight modification of the disparity statistics introduced for testing the goodness-of-fit by P. Hall [J. Multivariate Anal. 19, 201-224 (1986; Zbl 0605.62038)] are the \(\varphi\)-disparity statistics \(D_n(\varphi)=n D_\varphi (\overline q_n,p_n)\). These modified statistics can be ordered for \(1 \leq m\leq n\) as to their sensitivity to alternatives. The limit laws governing for \(n\to \infty\) the distributions of the statistics under local alternatives are shown to be unchanged by the modification, which allows to construct the asymptotically \(\alpha\)-level goodness-of-fit tests based on \(D_n(\varphi)\).
In spite of that the limit laws depend only on the local properties of \(\varphi\) in a neighborhood of 1, we show by a simulation that for small and medium sample sizes \(n\) the true test sizes and powers significantly depend on \(\varphi\) and also on the alternatives, so that an adaptation of \(\varphi\) to concrete situations can improve performance of the \(\varphi\)-disparity test. Relations of \(D_n(\varphi)\) to some other \(m\)-spacing statistics known from the literature are discussed as well.

MSC:

62G20 Asymptotic properties of nonparametric inference
62G30 Order statistics; empirical distribution functions
62B10 Statistical aspects of information-theoretic topics
62B99 Sufficiency and information
62F99 Parametric inference

Citations:

Zbl 0605.62038
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Ali S. M., Journal of Royal Statistical Society, Series B 286 pp 131– (1966)
[2] Csiszár I., Publications of the Mathematical Institute of Hungarian Academy of Sciences, Ser A 8 pp 85– (1963)
[3] DOI: 10.1016/S0378-3758(00)00202-0 · Zbl 0996.62052 · doi:10.1016/S0378-3758(00)00202-0
[4] DOI: 10.2307/2287597 · Zbl 0484.62035 · doi:10.2307/2287597
[5] DOI: 10.1080/02331889908802689 · Zbl 0949.60047 · doi:10.1080/02331889908802689
[6] DOI: 10.1007/BF02614112 · Zbl 0702.62041 · doi:10.1007/BF02614112
[7] DOI: 10.2307/3315481 · Zbl 0695.62125 · doi:10.2307/3315481
[8] DOI: 10.1016/S0167-7152(01)00172-9 · Zbl 0994.62009 · doi:10.1016/S0167-7152(01)00172-9
[9] DOI: 10.1016/0047-259X(86)90027-8 · Zbl 0605.62038 · doi:10.1016/0047-259X(86)90027-8
[10] DOI: 10.1137/1135111 · Zbl 0780.62043 · doi:10.1137/1135111
[11] Liese F., Convex Statistical Distances (1987) · Zbl 0656.62004
[12] DOI: 10.1214/aos/1176325512 · Zbl 0807.62030 · doi:10.1214/aos/1176325512
[13] DOI: 10.1016/S0167-7152(96)00169-1 · Zbl 0899.62031 · doi:10.1016/S0167-7152(96)00169-1
[14] DOI: 10.1080/03610929808832117 · Zbl 1126.62300 · doi:10.1080/03610929808832117
[15] Morales D., IEEE Transactions on System, Man and Cybernetics 26 pp 1– (1996)
[16] DOI: 10.1214/aos/1176343006 · Zbl 0305.62013 · doi:10.1214/aos/1176343006
[17] DOI: 10.1214/aos/1176349550 · Zbl 0576.62061 · doi:10.1214/aos/1176349550
[18] Read R. C., Goodness-of-fit Statistics for Discrete Multivariate Data (1988) · Zbl 0663.62065 · doi:10.1007/978-1-4612-4578-0
[19] DOI: 10.1016/0167-7152(94)00156-3 · Zbl 0828.60021 · doi:10.1016/0167-7152(94)00156-3
[20] Vajda I., Theory of Statistical Inference Information (1989) · Zbl 0711.62002
[21] Vajda I., Tatra Mountains Mathematical Journal (2003)
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.