![]() 2, 4th edn, Charles Griffin, London, 1979. and Stuart, A.: The Advanced Theory of Statistics, Vol. G.: Mathematical results in signal and image processing, Dokl. G.: Nonparametric estimation of the singularities of a signal from noisy measurements, Proc. G.: Consistency of rank tests against some general alternatives, Math. ![]() G.: Multi-dimensional algorithm for finding discontinuities of signals from noisy discrete data, Math. Quality Control and Reliability, North-Holland, Amsterdam, 1988, pp. K., Sen (eds), Handbook of Statistics, Vol. Q.: Review about estimation of change points, in P. K.: Time Series, 3rd edn, Edward Arnold, London, 1990. I.: Kernel estimation of the singularities of a signal, Panam. C.: The contiguity ratio and statistical mappings, The Incorporated Statistician 5 (1954), 115–145. S.: Testing for serial correlation in least squares regression I, Biometrika 37 (1950), 409–428. K.: Spatial Processes: Models and Applications, Pion, London, 1981.ĭurbin, J. and Horváth, L.: Nonparametric methods for changepoint problems, in P. Nonparametric Methods, North-Holland, Amsterdam, 1984, pp. ![]() K.: Tests of randomness against trend or serial correlation, in P. L., and Albers, W.: Asymptotic relative efficiencies of rank tests for trend alternative, J. J.: Asymptotic efficiency of rank tests of randomness against autocorrelation, Ann. This function need not be monotone.Ī numerical example illustrating the use of the obtained results for image analysis (edge detection) is presented.Īiyar, R. ‘Asymptotically continuous’ means that the trend converges to some continuous, not identically constant function as the number of data points goes to infinity. Alternative 2: there exists an arbitrary ‘asymptotically continuous’ trend in location. Note also that in the one-dimensional case the initial sequence need not be stochastically monotone under the alternative. Note that m, this ordering, and the sets D i are not known a priori: one tests only for the existence of such a partition. Alternative 1: there exists a pairwise disjoint partion U i =1 m D i= D, where D ⊂ ℝ d≥1, is a bounded domain inside which one makes observations, such that (1) if an observation point falls inside D i, then the corresponding observed value is the realization of a random variable ξ i 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!\, (3) the partition is independent of the number of observation points. The main result is the proof of consistency of the test in each of the above cases against two general alternatives. The null hypothesis is that all observations are independent and identically distributed. 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.
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