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Journal ArticleDOI

Nonparametric tests against trend

01 Jul 1945-Econometrica (ECONOMETRICA)-Vol. 13, Iss: 3, pp 245-259

About: This article is published in Econometrica.The article was published on 1945-07-01. It has received 8002 citation(s) till now. The article focuses on the topic(s): Nonparametric statistics.
Topics: Nonparametric statistics (51%)
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Book ChapterDOI
Wassily Hoeffding1Institutions (1)
Abstract: Let X 1 …, X n be n independent random vectors, X v = , and Φ(x 1 …, x m ) a function of m(≤n) vectors . A statistic of the form , where the sum ∑″ is extended over all permutations (α1 …, α m ) of different integers, 1 α≤ (αi≤ n, is called a U-statistic. If X 1, …, X n have the same (cumulative) distribution function (d.f.) F(x), U is an unbiased estimate of the population characteristic θ(F) = f … f Φ(x 1,…, x m ) dF(x 1) … dF(x m ). θ(F) is called a regular functional of the d.f. F(x). Certain optimal properties of U-statistics as unbiased estimates of regular functionals have been established by Halmos [9] (cf. Section 4)

2,291 citations

Cites background from "Nonparametric tests against trend"

  • ...Mann [17] has suggested a test of H1 against H2 based on the number T of inequalities Ya < Yp, where a <p3....


Journal ArticleDOI
Abstract: Some of the characteristics that complicate the analysis of water quality time series are non-normal distributions, seasonality, flow relatedness, missing values, values below the limit of detection, and serial correlation. Presented here are techniques that are suitable in the face of the complications listed above for the exploratory analysis of monthly water quality data for monotonie trends. The first procedure described is a nonparametric test for trend applicable to data sets with seasonality, missing values, or values reported as ‘less than’: the seasonal Kendall test. Under realistic stochastic processes (exhibiting seasonality, skewness, and serial correlation), it is robust in comparison to parametric alternatives, although neither the seasonal Kendall test nor the alternatives can be considered an exact test in the presence of serial correlation. The second procedure, the seasonal Kendall slope estimator, is an estimator of trend magnitude. It is an unbiased estimator of the slope of a linear trend and has considerably higher precision than a regression estimator where data are highly skewed but somewhat lower precision where the data are normal. The third procedure provides a means for testing for change over time in the relationship between constituent concentration and flow, thus avoiding the problem of identifying trends in water quality that are artifacts of the particular sequence of discharges observed (e.g., drought effects). In this method a flow-adjusted concentration is defined as the residual (actual minus conditional expectation) based on a regression of concentration on some function of discharge. These flow-adjusted concentrations, which may also be seasonal and non-normal, can then be tested for trend by using the seasonal Kendall test.

2,259 citations

Journal ArticleDOI
Khaled H. Hamed1, A. Ramachandra Rao1Institutions (1)
Abstract: One of the commonly used tools for detecting changes in climatic and hydrologic time series is trend analysis. A number of statistical tests exist to assess the significance of trends in time series. One of the commonly used non-parametric trend tests is the Mann-Kendall trend test. The null hypothesis in the Mann-Kendall test is that the data are independent and randomly ordered. However, the existence of positive autocorrelation in the data increases the probability of detecting trends when actually none exist, and vice versa. Although this is a well-known fact, few studies have addressed this issue, and autocorrelation in the data is often ignored. In this study, the effect of autocorrelation on the variance of the Mann-Kendall trend test statistic is discussed. A theoretical relationship is derived to calculate the variance of the Mann-Kendall test statistic for autocorrelated data. The special cases of AR(1) and MA(1) dependence are discussed as examples. An approximation to the theoretical relationship is also presented in order to reduce computation time for long time series. Based on the modified value of the variance of the Mann-Kendall trend test statistic, a modified non-parametric trend test which is suitable for autocorrelated data is proposed. The accuracy of the modified test in terms of its empirical significance level was found to be superior to that of the original Mann-Kendall trend test without any loss of power. The modified test is applied to rainfall as well as streamflow data to demonstrate its performance as compared to the original Mann-Kendall trend test.

1,662 citations

Journal ArticleDOI
Sheng Yue1, Paul Pilon1, George CavadiasInstitutions (1)
Abstract: In many hydrological studies, two non-parametric rank-based statistical tests, namely the Mann–Kendall test and Spearman's rho test are used for detecting monotonic trends in time series data. However, the power of these tests has not been well documented. This study investigates the power of the tests by Monte Carlo simulation. Simulation results indicate that their power depends on the pre-assigned significance level, magnitude of trend, sample size, and the amount of variation within a time series. That is, the bigger the absolute magnitude of trend, the more powerful are the tests; as the sample size increases, the tests become more powerful; and as the amount of variation increases within a time series, the power of the tests decrease. When a trend is present, the power is also dependent on the distribution type and skewness of the time series. The simulation results also demonstrate that these two tests have similar power in detecting a trend, to the point of being indistinguishable in practice. The two tests are implemented to assess the significance of trends in annual maximum daily streamflow data of 20 pristine basins in Ontario, Canada. Results indicate that the P -values computed by these different tests are almost identical. By the binomial distribution, the field significant downward trend was assessed at the significance level of 0.05. Results indicate that a higher number of sites show evidence of decreasing trends than one might expect due to chance alone.

1,429 citations

Journal ArticleDOI
Abstract: This study investigated using Monte Carlo simulation the interaction between a linear trend and a lag-one autoregressive (AR(1)) process when both exist in a time series. Simulation experiments demonstrated that the existence of serial correlation alters the variance of the estimate of the Mann–Kendall (MK) statistic; and the presence of a trend alters the estimate of the magnitude of serial correlation. Furthermore, it was shown that removal of a positive serial correlation component from time series by pre-whitening resulted in a reduction in the magnitude of the existing trend; and the removal of a trend component from a time series as a first step prior to pre-whitening eliminates the influence of the trend on the serial correlation and does not seriously affect the estimate of the true AR(1). These results indicate that the commonly used pre-whitening procedure for eliminating the effect of serial correlation on the MK test leads to potentially inaccurate assessments of the significance of a trend; and certain procedures will be more appropriate for eliminating the impact of serial correlation on the MK test. In essence, it was advocated that a trend first be removed in a series prior to ascertaining the magnitude of serial correlation. This alternative approach and the previously existing approaches were employed to assess the significance of a trend in serially correlated annual mean and annual minimum streamflow data of some pristine river basins in Ontario, Canada. Results indicate that, with the previously existing procedures, researchers and practitioners may have incorrectly identified the possibility of significant trends. Copyright  2002 Environment Canada. Published by John Wiley & Sons, Ltd.

1,403 citations

Cites background or methods from "Nonparametric tests against trend"

  • ...…D n 1∑ iD1 n∑ jDiC1 sgn Xj Xi 1 where the Xj are the sequential data values, n is the length of the data set, and sgn D { 1 if > 0 0 if D 0 1 if 0 2 Mann (1945) and Kendall (1975) have documented that when n ½ 8, the statistic S is approximately normally distributed with the mean and the…...


  • ...…of green-house gases on the environment, researchers and practitioners have employed the nonparametric Mann–Kendall (MK) statistical test (Mann, 1945; Kendall, 1975) to identify whether monotonic trends exist in hydro-meteorological data such as temperature, precipitation, and…...


  • ...The MK test, also called Kendall’s tau test due to Mann (1945) and Kendall (1975), is the rank-based nonparametric test for assessing the significance of a trend, and has been widely used in hydrological trend detection studies....


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