By G. Shilov

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Records tests for trend in location. , Trenkler, G. ) Mathematical Statistics with Applications in Biometry, Festschrift in Honour of Prof. Dr. Siegfried Schach, pp. 163–178. : Distribution-free tests in time-series based on the breaking of records. J. Roy. Stat. Soc. : Techniques of trend analysis for monthly water quality data. Water Resour. Res. , Slack. : A nonparametric trend test for seasonal data with serial dependence. Water Resour. Res. : A new measure of rank correlation. : Rank correlation methods.

For a more detailed description see Collett [3] and Leissen [9]. 2 The modified ordered logrank test Liu et al. r) j ), r = 1, . . r) j = d1 j + . . r) j = e1 j + . . + er j , so that L(r) corresponds to the logrank statistic Li according to (3) for the combined sample 1, . . , r. The variance in the nominator of the test statistic is given by Liu et al. [10] for the special case of no ties in the data. A general derivation can be found in Leissen [9]. It can be shown that the statistic LIT is a special case of the statistic LRT(w) given in (6).

Leissen et al. in the marginal distributions. For each group consider the sum of the observed and expected number of events over the single event times m Li = ∑ (di j − ei j ), i = 1, . . , k, (3) j=1 and the weighted sum of the Li over the groups LT(w) = k k i=1 i=1 ∑ wi Li = ∑ wi (di. − ei. ) . (4) If H0 is true, it holds that E(LT(w) ) = 0 and that d. j (n. j − d. j ) ∑ (n. j − 1) j=1 m Var(LT(w) ) = k ∑ w2i i=1 ni j − n. j k ∑ wi i=1 ni j n. j 2 (5) (cf. Leissen [9]), so that the statistic LRT(w) = LT(w) Var(LT(w) ) (6) follows a standard normal distribution asymptotically.