James R. Wallis

Estimation of the Generalized Extreme-Value Distribution by the Method of Probability-Weighted Moments

We use the method of probability-weighted moments to derive estimators of the parameters and quantiles of the generalized extreme-value distribution. We investigate the properties of these estimators in large samples, via asymptotic theory, and in small and moderate samples, via computer simulation. Probability-weighted moment estimators have low variance and no severe bias, and they compare favorably with estimators obtained by the methods of maximum likelihood or sextiles. The method of probability-weighted moments also yields a convenient and powerful test of whether an extreme-value distribution is of Fisher-Tippett Type I, II, or III.

Parameter and quantile estimation for the generalized pareto distribution

The generalized Pareto distribution is a two-parameter distribution that contains uniform, exponential, and Pareto distributions as special cases. It has applications in a number of fields, including reliability studies and the analysis of environmental extreme events. Maximum likelihood estimation of the generalized Pareto distribution has previously been considered in the literature, but we show, using computer simulation, that, unless the sample size is 500 or more, estimators derived by the method of moments or the method of probability-weighted moments are more reliable. We also use computer simulation to assess the accuracy of confidence intervals for the parameters and quantiles of the generalized Pareto distribution.

Some statistics useful in regional frequency analysis

Regional frequency analysis uses data from a number of measuring sites. A “region” is a group of sites each of which is assumed to have data drawn from the same frequency distribution. The analysis involves the assignment of sites to regions, testing whether the proposed regions are indeed homogeneous, and choice of suitable distributions to fit to each regions data. This paper describes three statistics useful in regional frequency analysis: a discordancy measure, for identifying unusual sites in a region; a heterogeneity measure, for assessing whether a proposed region is homogeneous; and a goodness-of-fit measure, for assessing whether a candidate distribution provides an adequate fit to the data. Tests based on the statistics provide objective backing for the decisions involved in regional frequency analysis. The statistics are based on the L moments [Hosking, 1990] of the at-site data.

Hydro-Climatological Trends in the Continental United States, 1948-88

Abstract Spatial patterns in trends of four monthly variables: average temperature, precipitation, streamflow, and average of the daily temperature range were examined for the continental United States for the period 1948–88. The data used are a subset of the Historical Climatology Network (1036 stations) and a stream gage network of 1009 stations. Trend significance was determined using the nonparametric seasonal Kendalls test on a monthly and annual basis, and a robust slope estimator was used for determination of trend magnitudes. A bivariate test was used for evaluation of relative changes in the variables, specifically, streamflow relative to precipitation, streamflow relative to temperature, and precipitation relative to temperature. Strong trends were found in all of the variables at many more stations than would be expected due to chance. There is a strong spatial and seasonal structure in the trend results. For instance, although annual temperature increases were found at many stations, mostly i...

AN APPROACH TO STATISTICAL SPATIAL-TEMPORAL MODELING OF METEOROLOGICAL FIELDS

Abstract In this article we develop a random field model for the mean temperature over the region in the northern United States covering eastern Montana through the Dakotas and northern Nebraska up to the Canadian border. The readings are temperatures at the stations in the U.S. historical climatological network. The stochastic structure is modeled by a stationary spatial-temporal Gaussian random field. For this region, we find little evidence of temporal dependence while the spatial structure is temporally stable. The approach strives to incorporate the uncertainty in estimating the covariance structure into the predictive distributions and the final inference. As an application of the model, we derive posterior distributions of the areal mean over time. A posterior distribution for the static areal mean is presented as a basis for calibrating temperature shifts by the historical record. For this region and season, the distribution indicates that under the scenario of a gradual increase of 5°F over 50 ye...

An appraisal of the regional flood frequency procedure in the UK Flood Studies Report

ABSTRACT The algorithm for estimating the regional flood frequency hazard contained in the 1975 Natural Environment Research Council Flood Studies Report (FSR) can occasionally lead to upper quantile estimates that appear unrealistic when compared with engineering judgement. Tests with the FSR algorithm were made for several sets of observed flood sequences and a great variety of synthetic data in a Monte Carlo simulation study. Similar tests were conducted with many other regional and at-site flood frequency estimation procedures including a regional generalized extreme value distribution (GEV) procedure and a regional Wakeby distribution (WAK) procedure, both of which used biased probability weighted moments (PWM) in their formulation. For the Monte Carlo simulations, for which the true quantiles to be estimated were known, it was found that the FSR algorithm yielded quantile estimates that were always more variable, often by a factor of as much as 4 or 5, than those obtained by either the GEV/PWM or WA...

A daily hydroclimatological data set for the continental United States

Previous attempts to validate general circulation model simulations of land surface hydrology have often been limited by the absence of systematic historical data, especially for runoff, precipitation, and temperature. Because hydrological response times for unregulated watersheds in the United States vary from a few hours to a few days at most, climatological studies dealing with land surface hydrology require data at relatively short time intervals. We describe a set of 1009 U.S. Geological Survey streamflow stations, and 1036 National Oceanic and Atmospheric Administration climatological stations, for which long-term (1948–1988) observations have been assembled into a consistent daily data base with missing observations estimated using a simple closest-station prorating rule. Care was taken in selection of the streamflow stations to assure that the records were essentially free from regulation. The climatological stations are a subset of the historical climatology network for which monthly data are described by Quinlan et al. (1987). The data format is provided to facilitate development of alternative data retrieval algorithms, Estimated values for missing data, as well as suspicious observations, are flagged. The data are retrievable by station list, state, latitude-longitude range, and hydrologic unit code from compact digital read-only memory (CD ROM). CD-ROM copies are available from the second author.

Regional Precipitation Quantile Values for the Continental United States Computed from L-Moments

Abstract Precipitation quantile values have been computed for 9 probabilities, 8 durations, 12 starting months, and 1 1 1 regions across the United States. L-moment methodology has been used for the calculations. Discussed are the rationale for selecting the Pearson type III (gamma) and Wakeby distributions, and the confidence that can be placed in the quantile values. Results show that distribution functions become more asymmetrical as the duration decreases, indicating that the median may be a better measure of central tendency than the mean. Portraying the quantile values as a percentage of the median value leads to smooth spatial fields. Computation of quantile values was the first known large-scale application of L-moment methodology. In spite of the complexity of the techniques and the extensive use of personnel and computer resources, the results justify the procedures in terms of preparing easy to use probability statements that reflect underlying physical processes.

A Comparison of Unbiased and Plotting‐Position Estimators of L Moments

Plotting-position estimators of L moments and L moment ratios have several disadvantages compared with the “unbiased” estimators. For general use, the “unbiased” estimators should be preferred. Plotting-position estimators may still be useful for estimating extreme upper tail quantiles in regional frequency analysis.

Geographical Patterning of Interannual Rainfall Variability in the Tropics and Near Tropics: An L-Moments Approach

Abstract Interannual rainfall variability has important effects for the evolution of biotic and human communities. Historical records of monthly rainfall totals for 1492 stations within 30° of the equator were analyzed using the method of L-moments. The 0.1 quantile (QU10), or the proportion of mean annual rainfall expected in the driest year in 10, was selected as the measure of variability. A nonlinear regression was fit to the relationship between QU10 and mean annual rainfall, and regions were categorized into three classes on the basis of the residuals: the 25% with the most negative, the 25% with the most positive, and the middle 50%. Maps of the global and regional patterns of rainfall variability show marked geographical patterning of variability and identify areas where rainfall variability may be a particularly important environmental feature.