Probability estimates based on small normal-distribution samples

Beard, Leo R.

doi:10.1029/jz065i007p02143

Cited by 38 publications

(16 citation statements)

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“…The standard noninformative prior for Ix and 0-has them independently distributed with Ix and log (0-) each uniformly distributed [Box and Tiao, 1973;Zellner, 1971a Some caution must be exercised here concerning the meaning of the exceedence probability 1 -p. In the derivation of (14), x, 2, and s are all viewed as random variables, while /x and tr are fixed, though their values were not specified. Thus if one calculates many design values and observes the frequency with which future flows exceed those design values, it frequently will be 1 -p. This classical interpretation of (14) supports Beard's denoting 1 -p as an 'expected probability' [Beard, 1960, Appendix A, 1978Thomas, 1976]. Hardison and Jennings [1972] used the term 'average exceedance probability.'…”

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confidence: 57%

Design events with specified flood risk

Stedinger¹

1983

Water Resources Research

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A maximum likelihood or method of moments estimate of the 99 percentile of a flood flow distribution is the traditional choice for the 100-year design flood event in flood plain planning. Because such estimates ignore the small-sample properties of quantile estimators, the resulting design event values will be exceeded with a probability in excess of that intended. For a sample size of 16 the estimated 100-year flood is, in expectation, the 50-year flood. Bayesian and classical statistical approaches can be used to develop design flood values which, given available hydrologic information, will (on average) be exceeded with the specified 1% design probability. Bias in estimates of expected flood damages and procedures for using regional information at gauged and ungauged sites are discussed. INTRODUCTIONIn flood plain planning, flood plain mapping, and levee and flood control reservoir design, a single estimate of the 100year flood event (and perhaps the 10-, 50-, and 500-year events) often serves as the design event upon which a design and management policy are based. The traditional approach for selecting such design values has been to use a statistically efficient or minimum variance estimate of the 99 percentile of the flood flow distribution [Benson, 1968; Water Resources Council (Hydrology Committee), 1976, pp. 1-20; Kite, 1977]. Such estimates are most often variations of the method of moments or maximum likelihood parameterestimation methods. Other investigators, adopting a Bayesian point of view, have developed the posterior distribution for future flood flows; these distributions integrate the natural variability of flood flows and one's uncertainty concerning the parameters of that distribution [Shane and Gaver, 1970; Wood, 1978; Wood and Rodriguez-Iturbe, 1975a; Woad et al., 1974; Vicens et al., 1975; Benjamin and Cornell, 1970, p. 640]. Some work has even incorporated model uncertainty into the analysis [Wood and Rodriquez-Iturbe, 1975b; Bodo and Unny, 1976; Bogardi et al., 1977]. Classical analogues of the Bayesian approach have been presented by Moran [1957], Beard [1960, 1974 (Appendix A), 1978], Hardison and Jennings [1972], and Thomas [1948]. On the other hand, Wood [1978] and Kuczera [1982a, b] formulated a Bayesian analog of the classical approach. These four approaches for obtaining design flood flow values result from different interpretations of the hydrologic information requirements of flood plain planning studies and the use of different statistical philosophies for developing design flood values which meet those requirements; there is considerable controversy and confusion among practitioners concerning the appropriate procedure for estimating design floods [Jackson, 1981; Thomas, 1976]. It seems that what planners and legal statutes often ask for from statistical hydrologists is a design flood value xp ø which, given the available regional hydrologic and geomorphologic information, will be exceeded with an allowable or design probability of 1 -p. Then a flood control or flood management...

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confidence: 57%

Design events with specified flood risk

Stedinger¹

1983

Water Resources Research

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“…However, an unbiased estimator of the T -yr event will not, in general, be exceeded with an average probability of p = I/T. Beard (1960), Beard (1978), IACWD ["Guidelines" (1982), Appendix 11], Stedinger (1983), Cunnane (1991), andStedinger et al (1993) discuss this issue in greater detail.…”

Section: Expected Probability Adjustmentmentioning

confidence: 99%

Flood‐Flow Frequency Model Selection in Southwestern United States

Vogel

Thomas

McMahon

1993

J. Water Resour. Plann. Manage.

118

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Uniform flood frequency guidelines in the United States recommend the use of the log Pearson type 3 (LP3) distribution in flood frequency investigations. Many investigators have suggested alternate models such as the generalized extreme value (GEV) distribution as an improvement over the LP3 distribution. Using flood-flow data at 383 sites in the southwestern United States, we explore the suitability of various flood frequency models using L-moment diagrams. We also repeat the experiment performed in the original Water Resource Council report (Bulletin 17B, issued in 1982), which led to the LP3 mandate. All our evaluations consistently reveal that the LP3, GEV, and the two-and three-parameter lognormal models (LN2 and LN3) provide a good approximation to flood-flow data in this region. Other models such as the normal, Pearson, and Gumbel distributions are shown to perform poorly. Recent research indicates that regional index-flood procedures should be more accurate and more robust than the type of at-site procedures evaluated here. Nevertheless, this study reveals that index-flood procedures need not be restricted to the GEV distribution because the LN2, LN3, and LP3 distributions appear to be suitable alternatives, at least in the southwestern United States.

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“…By studying variations in random samples drawn from a known Gaussian normal distribution, variations of true probabilities from probabilities computed from sample data can be evaluated (Beard, 1960;Hardison and Jennings, 1972). Figure 2 illustrates these variations in general terms for a small sample size.…”

Section: Expected Probabilitymentioning

confidence: 99%

Practical flood frequency estimates

Beard

1977

Geophysical Surveys

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Abstract. The best information on which to base estimates of future flood frequencies is records of past flood events. Where there is a substantial record at the location for which estimates are desired the estimation process is generally straightforward, although a variety of methods is used and there is major uncertainty in the estimates. In general, the frequency of future events is assumed to be indicated by the observed frequency of past events under constant controlling watershed conditions.Techniques are available for using information on historical (pre-record) flood data to improve the reliability of flood frequency estimates. There are methods for detecting and managing extremely unusual actual events (outliers) and for improving the reliability of short-record estimates based on long-record data at related locations. Regional correlation analysis is usable for establishing flood frequency estimates for locations where records are not available.Detailed hydrologic analysis, usually involving rainfall-runoff studies, is required for establishing flood frequency relationships for modified conditions of the watershed or, in many cases, for establishing flood frequency estimates for newly formed drainage systems such as in urban areas and airports.The principal use of flood frequency functions is to compare expected changes in flood damages (due to a contemplated action) with the economic and social costs or benefits of the contemplated action.

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Probability estimates based on small normal-distribution samples

Cited by 38 publications

References 1 publication

Design events with specified flood risk

Design events with specified flood risk

Flood‐Flow Frequency Model Selection in Southwestern United States

Practical flood frequency estimates

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