caa a22 4500 477077080 CHVBK 20180405111441.0 cr unu---uuuuu 170330e19960201xx s 000 0 eng 10.1007/BF00047166 doi (NATIONALLICENCE)springer-10.1007/BF00047166 Consistency of a rank test against general alternatives of change points (surfaces) and continuous trend [Elektronische Daten] [A. Katsevich, A. Ramm] Considered are modifications of a rank test of randomness for the one- and multi-dimensional regular design cases as well as for the one- and multi-dimensional random design cases. The null hypothesis is that all observations are independent and identically distributed. The main result is the proof of consistency of the test in each of the above cases against two general alternatives.Alternative 1: there exists a pairwise disjoint partion U i =1 m D i =D, where D ⊂ ℝd≥1, is a bounded domain inside which one makes observations, such that (1) if an observation point falls insideD i , then the corresponding observed value is the realization of a random variable ξi i = l,...,m; (2) there exists an ordering % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaamXvP5wqonvsaeHbfv3ySLgzaGqbaiab-Tha7jabe67a4Hqbdiab% +LgaPnaaBaaaleaacaWGRbaabeaakiab-1ha9naaDaaaleaacaWGRb% Gaeyypa0JaaGymaaqaaiaad2gaaaaaaa!4C2D!\[\{ \xi i_k \} _{k = 1}^m \], where % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiabe67a4nXvP5wqonvsaeHbfv3ySLgzaGqbdiab-LgaPnaaBaaa% leaacaWGRbaabeaaaaa!454D!\[\xi i_k \] is stochastically smaller than % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiabe67a4nXvP5wqonvsaeHbfv3ySLgzaGqbdiab-LgaPnaaBaaa% leaacaWGRbaabeaakmaaBaaaleaacqGHRaWkcaaIXaaabeaakiaacY% cacaWGRbGaeyypa0JaaGymaiaacYcacaGGUaGaaiOlaiaac6cacaGG% SaGaamyBaiabgkHiTiaaigdaaaa!509B!\[\xi i_k _{ + 1} ,k = 1,...,m - 1\], (3) the partition is independent of the number of observation points. Note thatm, this ordering, and the sets D i are not known a priori: one tests only for the existence of such a partition. Note also that in the one-dimensional case the initial sequence need not be stochastically monotone under the alternative.Alternative 2: there exists an arbitrary ‘asymptotically continuous' trend in location. ‘Asymptotically continuous' means that the trend converges to some continuous, not identically constant function as the number of data points goes to infinity. This function need not be monotone. A numerical example illustrating the use of the obtained results for image analysis (edge detection) is presented. Kluwer Academic Publishers, 1996 62G10 nationallicence rank test nationallicence test of randomness nationallicence change points nationallicence trend nationallicence consistency nationallicence Katsevich A. Los Alamos National Laboratory, MS B265, CIC-3 Division, 87545, Los Alamos, NM, U.S.A. aut Ramm A. Mathematics Department, Kansas State University, 66506-2602, Manhattan, KS, U.S.A. aut Acta Applicandae Mathematica Kluwer Academic Publishers; gopher://gopher.wkap.nl 42/2(1996-02-01), 105-137 0167-8019 42:2<105 1996 42 10440 https://doi.org/10.1007/BF00047166 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/BF00047166 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Katsevich A. Los Alamos National Laboratory, MS B265, CIC-3 Division, 87545, Los Alamos, NM, U.S.A aut NATIONALLICENCE 700 1- Ramm A. Mathematics Department, Kansas State University, 66506-2602, Manhattan, KS, U.S.A aut NATIONALLICENCE 773 0- Acta Applicandae Mathematica Kluwer Academic Publishers; gopher://gopher.wkap.nl/ 42/2(1996-02-01), 105-137 0167-8019 42:2<105 1996 42 10440 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer