On the informative value of the largest sample element of log-Gumbel distribution

Extremes of stream flow and precipitation are commonly modeled by heavytailed distributions. While scrutinizing annual flow maxima or the peaks over threshold, the largest sample elements are quite often suspected to be low quality data, outliers or values corresponding to much longer return periods than the observation period. Since the interest is primarily in the estimation of the right tail (in the case of floods or heavy rainfalls), sensitivity of upper quantiles to largest elements of a series constitutes a problem of special concern. This study investigated the sensitivity problem using the log-Gumbel distribution by generating samples of different sizes (n) and different values of the coefficient of variation by Monte Carlo experiments. Parameters of the log-Gumbel distribution were estimated by the probability weighted moments (PWMs) method, method of moments (MOMs) and maximum likelihood method (MLM), both for complete samples and the samples deprived of their largest elements. In the latter case, the distribution censored by the non-exceedance probability threshold, FT , was considered. Using FT instead of the censored threshold T creates possibility of controlling estimator property. The effect of the FT value on the performance of the quantile estimates was then examined. It is shown that right censoring of data need not reduce an accuracy of large quantile estimates if the method of PWMs or MOMs is employed. Moreover allowing bias of estimates one can get the gain in variance and in mean square error of large quantiles even if ML method is used

Flood frequency analysis supported by the largest historical flood

The use of non-systematic flood data for statistical purposes depends on the reliability of the assessment of both flood magnitudes and their return period. The earliest known extreme flood year is usually the beginning of the historical record. Even if one properly assesses the magnitudes of historic floods, the problem of their return periods remains unsolved. The matter at hand is that only the largest flood (XM) is known during whole historical period and its occurrence marks the beginning of the historical period and defines its length (<i>L</i>). It is common practice to use the earliest known flood year as the beginning of the record. It means that the <i>L</i> value selected is an empirical estimate of the lower bound on the effective historical length <i>M</i>. The estimation of the return period of XM based on its occurrence (<i>L</i>), i.e. <span style="position:relative; margin-left:-0.45m; top:-0.3em;">^</span><span style="position:relative; margin-left:-1.0em; top:0.3em;"><i>M</i></span> = <i>L</i>, gives a severe upward bias. The problem arises that to estimate the time period (<i>M</i>) representative of the largest observed flood XM. <br><br> From the discrete uniform distribution with support 1, 2, ... , <i>M</i> of the probability of the <i>L</i> position of XM, one gets <span style="position: relative; margin-left: -0.45m; top: -0.3em;">^</span><span style="position:relative; margin-left:-1.0em; top:0.3em;"><i>L</i></span> = <i>M</i>/2. Therefore <span style="position: relative; margin-left: -0.45m; top: -0.3em;">^</span><span style="position:relative; margin-left:-1.0em; top:0.3em;"><i>M</i></span> = 2<i>L</i> has been taken as the return period of XM and as the effective historical record length as well this time. As in the systematic period (<i>N</i>) all its elements are smaller than XM, one can get <span style="position: relative; margin-left: -0.45m; top: -0.3em;">^</span><i><span style="position:relative; margin-left:-1.0em; top:0.3em;">M</i></span> = 2<i>t</i>( <i>L</i>+<i>N</i>). <br><br> The efficiency of using the largest historical flood (XM) for large quantile estimation (i.e. one with return period <i>T</i> = 100 years) has been assessed using the maximum likelihood (ML) method with various length of systematic record (<i>N</i>) and various estimates of the historical period length <span style="position: relative; margin-left: -0.45m; top: -0.3em;">^</span><span style="position:relative; margin-left:-1.0em; top:0.3em;"><i>M</i> </span> comparing accuracy with the case when systematic records alone (<i>N</i>) are used only. The simulation procedure used for the purpose incorporates <i>N</i> systematic record and the largest historic flood (XM_<i>i</i>) in the period <i>M</i>, which appeared in the <i>L</i>_<i>i</i> year of the historical period. The simulation results for selected two-parameter distributions, values of their parameters, different <i>N</i> and <i>M</i> values are presented in terms of bias and root mean square error RMSEs of the quantile of interest are more widely discussed

On accuracy of upper quantiles estimation

Flood frequency analysis (FFA) entails the estimation of the upper tail of a probability density function (PDF) of annual peak flows obtained from either the annual maximum series or partial duration series. In hydrological practice, the properties of various methods of upper quantiles estimation are identified with the case of known population distribution function. In reality, the assumed hypothetical model differs from the true one and one cannot assess the magnitude of error caused by model misspecification in respect to any estimated statistics. The opinion about the accuracy of the methods of upper quantiles estimation formed from the case of known population distribution function is upheld. The above-mentioned issue is the subject of the paper. The accuracy of large quantile assessments obtained from the four estimation methods is compared to two-parameter log-normal and log-Gumbel distributions and their three-parameter counterparts, i.e., three-parameter log-normal and GEV distributions. The cases of true and false hypothetical models are considered. The accuracy of flood quantile estimates depends on the sample size, the distribution type (both true and hypothetical), and strongly depends on the estimation method. In particular, the maximum likelihood method loses its advantageous properties in case of model misspecification

Physics Of Flood Frequency Analysis. Part I. Linear Convective Diffusion Wave Model

It is hypothesized that the impulse response of a linearized convective diffusion wave (CD) model is a probability distribution suitable for flood frequency analysis. This flood frequency model has two parameters, which are derived using the methods of moments and maximum likelihood. Also derived are errors in quantiles for these methods of parameter estimation. The distribution shows an equivalency of the two estimation methods with respect to the mean value - an important property in the case of unknown true distribution function. As the coefficient of variation tends to zero (with the mean fixed), the distribution tends to a normal one, similar to the lognormal and gamma distributions

Model Error In Flood Frequency Estimation

Asymptotic bias in large quantiles and moments for three parameter estimation methods, including the maximum likelihood method (MLM), moments method (MOM) and linear moments method (LMM), is derived when a probability distribution function (PDF) is falsely assumed. It is illustrated using an alternative set of PDFs consisting of five two-parameter PDFs that are lower-bounded at zero, i.e., Log-Gumbel (LG), Log-logistic (LL), Log-normal (LN), Linear Diffusion (LD) and Gamma (Ga) distribution functions. The stress is put on applicability of LG and LL in the real conditions, where the hypothetical distribution (H) differs from the true one (T). Therefore, the following cases are considered: H=LG; T=LL, LN, LD and Ga, and H=LL, LN, LD and Ga, T=LG. It is shown that for every pair (H;T) and for every method, the relative bias (RB) of moments and quantiles corresponding to the upper tail is an increasing function of the true value of the coefficient of variation (cᵧ), except that RB of moments for MOM is zero. The value of RB is smallest for MOM and the largest for MLM. The bias of LMM occupies an intermediate position. Since MLM used as the approximation method is irreversible, the asymptotic bias of the MLM-estimate of any statistical characteristic is not asymmetric as is for the MOM and LMM. MLM turns out to be the worst method if the assumed LG or LL distribution is not the true one. It produces a huge bias of upper quantiles, which is at least one order higher than that of the other two methods. However, the reverse case, i.e., acceptance of LN, LD or Ga as a hypothetical distribution while LG or LL as the true one, gives the MLM-bias of reasonable magnitude in upper quantiles. Therefore, one should be highly reluctant in choosing the LG and LL in flood frequency analysis, especially if MLM is to be applied