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Journal ArticleDOI

L-Moments: Analysis and Estimation of Distributions Using Linear Combinations of Order Statistics

Jonathan R. M. Hosking1
01 Sep 1990-Journal of the royal statistical society series b-methodological (John Wiley & Sons, Ltd)-Vol. 52, Iss: 1, pp 105-124
TL;DR: The authors define L-moments as the expectations of certain linear combinations of order statistics, which can be defined for any random variable whose mean exists and form the basis of a general theory which covers the summarization and description of theoretical probability distributions.
Abstract: L-moments are expectations of certain linear combinations of order statistics. They can be defined for any random variable whose mean exists and form the basis of a general theory which covers the summarization and description of theoretical probability distributions, the summarization and description of observed data samples, estimation of parameters and quantiles of probability distributions, and hypothesis tests for probability distributions. The theory involves such established procedures as the use of order statistics and Gini's mean difference statistic, and gives rise to some promising innovations such as the measures of skewness and kurtosis and new methods of parameter estimation
Citations
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Journal ArticleDOI
TL;DR: In this article, a new climatic drought index, the standardized precipitation evapotranspiration index (SPEI), is proposed, which combines multiscalar character with the capacity to include the effects of temperature variability on drought assessment.
Abstract: The authors propose a new climatic drought index: the standardized precipitation evapotranspiration index (SPEI). The SPEI is based on precipitation and temperature data, and it has the advantage of combining multiscalar character with the capacity to include the effects of temperature variability on drought assessment. The procedure to calculate the index is detailed and involves a climatic water balance, the accumulation of deficit/surplus at different time scales, and adjustment to a log-logistic probability distribution. Mathematically, the SPEI is similar to the standardized precipitation index (SPI), but it includes the role of temperature. Because the SPEI is based on a water balance, it can be compared to the self-calibrated Palmer drought severity index (sc-PDSI). Time series of the three indices were compared for a set of observatories with different climate characteristics, located in different parts of the world. Under global warming conditions, only the sc-PDSI and SPEI identified an...

5,088 citations


Cites methods from "L-Moments: Analysis and Estimation ..."

  • ...For this purpose, L-moment ratio diagrams were used because they allow for comparison of the empirical frequency distribution of D series computed at different time scales with a number of theoretical distributions (Hosking 1990)....

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  • ...The PWMs of order s are calculated as w s 5 1 N N i51 (1 F i )sD i , where Fi is a frequency estimator calculated following the approach of Hosking (1990): F i 5 i 0.35 N , where i is the range of observations arranged in increasing order and N is the number of data points....

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Journal ArticleDOI
TL;DR: In this paper, the authors presented a European land-only daily high-resolution gridded data set for precipitation and minimum, maximum, and mean surface temperature for the period 1950-2006.
Abstract: We present a European land-only daily high-resolution gridded data set for precipitation and minimum, maximum, and mean surface temperature for the period 1950-2006. This data set improves on previous products in its spatial resolution and extent, time period, number of contributing stations, and attention to finding the most appropriate method for spatial interpolation of daily climate observations. The gridded data are delivered on four spatial resolutions to match the grids used in previous products as well as many of the rotated pole Regional Climate Models (RCMs) currently in use. Each data set has been designed to provide the best estimate of grid box averages rather than point values to enable direct comparison with RCMs. We employ a three-step process of interpolation, by first interpolating the monthly precipitation totals and monthly mean temperature using three-dimensional thin-plate splines, then interpolating the daily anomalies using indicator and universal kriging for precipitation and kriging with an external drift for temperature, then combining the monthly and daily estimates. Interpolation uncertainty is quantified by the provision of daily standard errors for every grid square. The daily uncertainty averaged across the entire region is shown to be largely dependent on the season and number of contributing observations. We examine the effect that interpolation has on the magnitude of the extremes in the observations by calculating areal reduction factors for daily maximum temperature and precipitation events with return periods up to 10 years. Copyright 2008 by the American Geophysical Union.

2,359 citations

Journal ArticleDOI
TL;DR: In this article, statistical downscaling of hydrologic extremes is considered, and future challenges such as the development of more rigorous statistical methodology for regional analysis of extremes, as well as the extension of Bayesian methods to more fully quantify uncertainty in extremal estimation are reviewed.

1,458 citations


Cites background from "L-Moments: Analysis and Estimation ..."

  • ...Hosking [40] discussed more generally the advantages of PWM over ML....

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  • ..., linear combinations of ‘‘L-moments’’, the term that commonly appears in the hydrology literature) also has a well-established tradition in hydrology [40]....

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  • ...Probability weighted moments (PWM) (or L-moments) are more popular than ML in applications to hydrologic extremes, both because of their computational simplicity and because of their good performance for small samples [40,43]....

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Journal ArticleDOI
TL;DR: In this paper, the authors present a collection of 25 figs from the genus "Figs" and discuss the relationship between the two types of figs: figs and figs.
Abstract: 48 pags., 25 figs. Available online 21 December 2013. The definitive version is available at: http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)1097-0088

1,083 citations


Cites methods from "L-Moments: Analysis and Estimation ..."

  • ...Originally, Hosking (1990) proposed use of Probability Weighted Moments (PWMs) based on the plotting position method....

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References
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Book
01 Jan 1965
TL;DR: Algebra of Vectors and Matrices, Probability Theory, Tools and Techniques, and Continuous Probability Models.
Abstract: Algebra of Vectors and Matrices. Probability Theory, Tools and Techniques. Continuous Probability Models. The Theory of Least Squares and Analysis of Variance. Criteria and Methods of Estimation. Large Sample Theory and Methods. Theory of Statistical Inference. Multivariate Analysis. Publications of the Author. Author Index. Subject Index.

8,300 citations

Journal ArticleDOI
TL;DR: The theory of least squares and analysis of variance has been studied in the literature for a long time, see as mentioned in this paper for a review of some of the most relevant works. But the main focus of this paper is on the analysis of variance.
Abstract: Algebra of Vectors and Matrices. Probability Theory, Tools and Techniques. Continuous Probability Models. The Theory of Least Squares and Analysis of Variance. Criteria and Methods of Estimation. Large Sample Theory and Methods. Theory of Statistical Inference. Multivariate Analysis. Publications of the Author. Author Index. Subject Index.

5,182 citations

Book
08 Dec 1980
TL;DR: In this paper, the basic sample statistics are used for Parametric Inference, and the Asymptotic Theory in Parametric Induction (ATIP) is used to estimate the relative efficiency of given statistics.
Abstract: Preliminary Tools and Foundations. The Basic Sample Statistics. Transformations of Given Statistics. Asymptotic Theory in Parametric Inference. U--Statistics. Von Mises Differentiable Statistical Functions. M--Estimates. L--Estimates. R--Estimates. Asymptotic Relative Efficiency. Appendix. References. Author Index. Subject Index.

4,827 citations

Book ChapterDOI
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

Journal ArticleDOI

2,260 citations