By Ramis E., Deschamps C., Odoux J.
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R. The variance in the nominator of the test statistic is given by Liu et al.  for the special case of no ties in the data. A general derivation can be found in Leissen . It can be shown that the statistic LIT is a special case of the statistic LRT(w) given in (6). For the proof it is needed that m e.. = k m k m i=1 j=1 d. j ∑ ∑ ei j = ∑ n. j ∑ ni j = ∑ d. j = d.. j=1 i=1 j=1 (9) Now consider that the numerator of LIT can be rewritten as k−1 m m r=1 j=1 j=1 ∑ L(r) = ∑ (d1 j − e1 j ) + .
Xn . Additionally, many time series show seasonal effects and so X1 , . . , Xn are not identically distributed, even if there is no monotonic trend. We modify the hypothesis of randomness for seasonal data to handle at least the second problem: Firstly, if there is a cycle of k periods, the random sample X = (X1 , . . , Xn ) is splitted into k parts X = (X1 , X2 , . . , Xk ) with X j = (X1, j , X2, j , . . , Xn j , j ) and Xi, j = Xk(i−1)+ j (1) On Nonparametric Tests for Trend Detection 21 for j = 1, .
Hettmansperger and Norton  look at the k-sample test problem with location shifts in the alternative HHN 0 : θ1 = . . = θk vs. HHN 1 : θi = θ0 + θ ci (θ > 0, θ0 ∈ R) with arbitrary but fixed ci . Amongst others they propose the test statistic HNT = HN Var(HN) (gi − gi ) Uii , ni ni i=1 i =i+1 k−1 with HN = k ∑ ∑ (18) where Uii is defined as in (12) and gi = λi (ci − c¯w ) with λi = nni , n = ∑ki=1 ni , and c¯w = ∑ki=1 λi ci . , ci = 1, . . , ci = k in the case of an increasing alternative.
Cours de mathematiques speciales: algebre by Ramis E., Deschamps C., Odoux J.