Estimates of the Regression Coefficient Based on Kendall's Tau
01 Dec 1968-Journal of the American Statistical Association (Taylor & Francis Group)-Vol. 63, Iss: 324, pp 1379-1389
TL;DR: In this article, a simple and robust estimator of regression coefficient β based on Kendall's rank correlation tau is studied, where the point estimator is the median of the set of slopes (Yj - Yi )/(tj-ti ) joining pairs of points with ti ≠ ti.
Abstract: The least squares estimator of a regression coefficient β is vulnerable to gross errors and the associated confidence interval is, in addition, sensitive to non-normality of the parent distribution. In this paper, a simple and robust (point as well as interval) estimator of β based on Kendall's [6] rank correlation tau is studied. The point estimator is the median of the set of slopes (Yj - Yi )/(tj-ti ) joining pairs of points with ti ≠ ti , and is unbiased. The confidence interval is also determined by two order statistics of this set of slopes. Various properties of these estimators are studied and compared with those of the least squares and some other nonparametric estimators.
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Monash University, Clayton campus1, Bureau of Meteorology2, Met Office3, Meteorological Service of Canada4, National Oceanic and Atmospheric Administration5, Royal Netherlands Meteorological Institute6, University of East Anglia7, Indian Institute of Tropical Meteorology8, National Institute of Water and Atmospheric Research9, University of Reading10, University of the West Indies11, University of Oxford12, China Meteorological Administration13, Facultad de Ciencias Exactas y Naturales14, National Autonomous University of Mexico15
TL;DR: A suite of climate change indices derived from daily temperature and precipitation data, with a primary focus on extreme events, were computed and analyzed as discussed by the authors, and the results showed widespread significant changes in temperature extremes associated with warming.
Abstract: A suite of climate change indices derived from daily temperature and precipitation data, with a primary focus on extreme events, were computed and analyzed. By setting an exact formula for each index and using specially designed software, analyses done in different countries have been combined seamlessly. This has enabled the presentation of the most up-to-date and comprehensive global picture of trends in extreme temperature and precipitation indices using results from a number of workshops held in data-sparse regions and high-quality station data supplied by numerous scientists world wide. Seasonal and annual indices for the period 1951-2003 were gridded. Trends in the gridded fields were computed and tested for statistical significance. Results showed widespread significant changes in temperature extremes associated with warming, especially for those indices derived from daily minimum temperature. Over 70% of the global land area sampled showed a significant decrease in the annual occurrence of cold nights and a significant increase in the annual occurrence of warm nights. Some regions experienced a more than doubling of these indices. This implies a positive shift in the distribution of daily minimum temperature throughout the globe. Daily maximum temperature indices showed similar changes but with smaller magnitudes. Precipitation changes showed a widespread and significant increase, but the changes are much less spatially coherent compared with temperature change. Probability distributions of indices derived from approximately 200 temperature and 600 precipitation stations, with near-complete data for 1901-2003 and covering a very large region of the Northern Hemisphere midlatitudes (and parts of Australia for precipitation) were analyzed for the periods 1901-1950, 1951-1978 and 1979-2003. Results indicate a significant warming throughout the 20th century. Differences in temperature indices distributions are particularly pronounced between the most recent two periods and for those indices related to minimum temperature. An analysis of those indices for which seasonal time series are available shows that these changes occur for all seasons although they are generally least pronounced for September to November. Precipitation indices show a tendency toward wetter conditions throughout the 20th century.
3,722 citations
Cites methods from "Estimates of the Regression Coeffic..."
...Therefore, we used a non-parametric Kendall’s tau based slope estimator [Sen, 1968] to compute trends since this method does not assume a distribution for the residuals and is robust to the effect of outliers in the series....
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TL;DR: In this paper, the median of the squared residuals is used to resist the effect of nearly 50% of contamination in the data in the special case of simple least square regression, which corresponds to finding the narrowest strip covering half of the observations.
Abstract: Classical least squares regression consists of minimizing the sum of the squared residuals. Many authors have produced more robust versions of this estimator by replacing the square by something else, such as the absolute value. In this article a different approach is introduced in which the sum is replaced by the median of the squared residuals. The resulting estimator can resist the effect of nearly 50% of contamination in the data. In the special case of simple regression, it corresponds to finding the narrowest strip covering half of the observations. Generalizations are possible to multivariate location, orthogonal regression, and hypothesis testing in linear models.
3,713 citations
TL;DR: This paper presents a generic framework for permutation inference for complex general linear models (glms) when the errors are exchangeable and/or have a symmetric distribution, and shows that, even in the presence of nuisance effects, these permutation inferences are powerful while providing excellent control of false positives in a wide range of common and relevant imaging research scenarios.
2,756 citations
Cites methods from "Estimates of the Regression Coeffic..."
...Statistics that can be used include, for instance, those based on ranks of observations (Brunner and Munzel, 2000; Rorden et al., 2007), derived from regression methods other than least squares (Cade and Richards, 1996) or that are robust to outliers (Sen, 1968; Theil, 1950)....
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..., 2007), derived from regression methods other than least squares (Cade and Richards, 1996) or that are robust to outliers (Sen, 1968; Theil, 1950)....
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TL;DR: The seasonal Kendall test as discussed by the authors is a nonparametric test for trend applicable to data sets with seasonality, missing values, or values reported as "less than" or values below the limit of detection.
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.
2,482 citations
Cites background from "Estimates of the Regression Coeffic..."
...The problem of testing water quality monitoring data for trend in time has received considerable attention in the last decade (see, for example, Wolman [1971], Steele et al. [1974], Lettenmaier [1977], and Liebetrau [1979])....
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...Examples of their combined use are presented by Smith et al. [1982]. The three procedures are as follows....
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TL;DR: In this paper, the effect of autocorrelation on the variance of the Mann-Kendall trend test statistic is discussed, and a modified non-parametric trend test is proposed.
2,252 citations
References
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Book•
01 Jan 1948
TL;DR: The measurement of rank correlation was introduced in this paper, and rank correlation tied ranks tests of significance were applied to the problem of m ranking, and variate values were used to measure rank correlation.
Abstract: The measurement of rank correlation introduction to the general theory of rank correlation tied ranks tests of significance proof of the results of chapter 4 the problem of m ranking proof of the result of chapter 6 partial rank correlation ranks and variate values proof of the result of chapter 9 paired comparisons proof of the results of chapter 11 some further applications.
6,404 citations
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01 Jan 1963
TL;DR: In this article, a tabular summary of parametric families of distributions is presented, along with a parametric point estimation method and a nonparametric interval estimation method for point estimation.
Abstract: 1 probability 2 Random variables, distribution functions, and expectation 3 Special parametric families of univariate distributions 4 Joint and conditional distributions, stochastic independence, more expectation 5 Distributions of functions of random variables 6 Sampling and sampling distributions 7 Parametric point estimation 8 Parametric interval estimation 9 Tests of hypotheses 10 Linear models 11 Nonparametric method Appendix A Mathematical Addendum Appendix B tabular summary of parametric families of distributions Appendix C References and related reading Appendix D Tables
4,571 citations
TL;DR: In this article, a tabular summary of parametric families of distributions is presented, along with a parametric point estimation method and a nonparametric interval estimation method for point estimation.
Abstract: 1 probability 2 Random variables, distribution functions, and expectation 3 Special parametric families of univariate distributions 4 Joint and conditional distributions, stochastic independence, more expectation 5 Distributions of functions of random variables 6 Sampling and sampling distributions 7 Parametric point estimation 8 Parametric interval estimation 9 Tests of hypotheses 10 Linear models 11 Nonparametric method Appendix A Mathematical Addendum Appendix B tabular summary of parametric families of distributions Appendix C References and related reading Appendix D Tables
3,211 citations
TL;DR: In this article, the authors considered the problem of estimating a U-statistic of the population characteristic of a regular functional function, where the sum ∑″ is extended over all permutations (α 1, α m ) of different integers, 1 α≤ (αi≤ n, n).
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)
2,439 citations
TL;DR: Rank tests such as the two Wilcoxon tests or the Kruskal-Wallis H-test have been shown to be more robust against gross errors than that of the t-and F-tests, even in the rare case in which the suspicion of the possibility of gross errors is unfounded.
Abstract: A serious objection to many of the classical statistical methods based on linear models or normality assumptions is their vulnerability to gross errors. For certain testing problems this difficulty is suc-cessfully overcome by rank tests such as the two Wilcoxon tests or the Kruskal- Wallis H-test. Their power is more robust against gross errors than that of the t- and F-tests, and their efficiency loss is quite small even in the rare case in which the suspicion of the possibility of gross errors is unfounded.
1,086 citations