Three-parameter Distributions Research Articles

Estimating rainfall-runoff parameters for ungauged catchments: a case study in Turkiye

ABSTRACT Determining water yield and flood discharges in catchments is a vital aspect of hydrology. This entails considering precipitation and runoff as key hydrological parameters. Constructing infrastructure like hydroelectric plants and regulators over streams necessitates continuous, accurate flow, and meteorological observations spanning at least 25 years. However, in developing nations, economic factors often impede such observations. This study proposes a method to estimate peak rainfall and flow values for ungauged basins with varying return periods by utilizing from gauged basins and spatial variables. Flood calculations were carried out for the ungauged Rabat River basin. In this study, regional flood frequency analysis was carried out using the flow values of the flow gauging stations neighboring the basin. In addition, maximum flow values were calculated using Moscus and the DSI synthetic method. Two- and three-parameter distributions were used to estimate 50-, 100-, 200- and 500-year flood return values at stations with observation periods ranging from 15 to 64 years. Kolmogorov–Smirnov and Probability Line Correlation coefficient (Chi-square) tests were applied to check the suitability of these distributions and the most appropriate distributions were found. This yielded an estimation for the flow values of the Rabat River, indicating the method's reliability for forecasting runoff-rainfall in the ungauged basin.

A New Inverse Extended Weibull Distribution for Modelling Insurance Loss Data

This paper proposes a new three-parameter inverse Weibull-type distribution via a similar way of constructing a skew normal distribution. The additional shape parameter increases the distribution’s modelling capability as other three-parameter inverse Weibull-type distributions. The mathematical properties of this inverse extended Weibull distribution are studied and proven. In the empirical study of four insurance datasets from Australia, Denmark and the USA, we show that the proposed distribution performs very well compared with other three-parameter competitors.

The odd log- logistic Power Inverse Lindley distribution: Model, Properties and Applications

In this article, we introduce a new three-parameter odd log-logistic power inverse Lindley distribution and discuss some of its properties. These include the shapes of the density and hazard rate functions, mixture representation, the moments, the quantile function, and order statistics. Maximum likelihood estimation of the parameters and their estimated asymptotic standard errors are derived. Three algorithms are proposed for generating random data from the proposed distribution. A simulation study is carried out to examine the bias and root mean square error of the maximum likelihood estimators of the parameters. An application of the model to three real data sets is presented finally and compared with the fit attained by some other well-known two and three-parameter distributions for illustrative purposes. It is observed that the proposed model has some advantages in analyzing lifetime data as compared to other popular models in the sense that it exhibits varying shapes and shows more flexibility than many currently available distributions.

Stochastic simulation of wind wave parameters for energy production

A combination of stochastic and deterministic models is applied for the study of ocean wind waves. Timeseries of significant wave height and mean zero up-crossing period, obtained from globally scattered floating buoys, are analyzed in order to construct a double periodic model, and select an optimal marginal distribution and dependence function for the description of the stochastic structure of wind waves. It is concluded that wind waves, in contrast to the atmospheric wind speed process, are mostly governed by the seasonal periodicity rather than the diurnal periodicity, which is often weak and can be neglected. Also, the Pareto-Burr-Feller distribution is found to be a fair selection among other common three-parameter marginal distributions. The dependence function is simulated through the Hurst-Kolmogorov (HK) dynamics using the climacogram (i.e., variance of the averaged process in the scale domain), a stochastic tool that can robustly estimate both the short-term fractal and long-range dependence behaviors both apparent at the wind wave process. To test the validity of the model, a stochastic synthesis of the wind wave process is performed through the Symmetric Moving Average scheme, focused on the explicit preservation of the probabilistic and the dependence structures. Finally, the stochastic model is applied for simulation to an offshore station southeast of Australia having one of the largest record lengths. The energy potential is also estimated through the significant wave height and the mean zero up-crossing period of both the synthetic and observed timeseries, and the effectiveness of the model is further discussed.

An Assessment of Uncertainties in Flood Frequency Estimation Using Bootstrapping and Monte Carlo Simulation

Reducing uncertainty in design flood estimates is an essential part of flood risk planning and management. This study presents results from flood frequency estimates and associated uncertainties for five commonly used probability distribution functions, extreme value type 1 (EV1), generalized extreme value (GEV), generalized pareto distribution (GPD), log normal (LN) and log Pearson type 3 (LP3). The study was conducted using Monte Carlo simulation (MCS) and bootstrapping (BS) methods for the 10 river catchments in eastern Australia. The parameters were estimated by applying the method of moments (for LP3, LN, and EV1) and L-moments (for GEV and GPD). Three-parameter distributions (e.g., LP3, GEV, and GPD) demonstrate a consistent estimation of confidence interval (CI), whereas two-parameter distributions show biased estimation. The results of this study also highlight the difficulty in flood frequency analysis, e.g., different probability distributions perform quite differently even in a smaller geographical area.

Use of the Halphen distribution family for mean wind speed estimation with application to Eastern Canada

We introduce the family of three-parameter heavy-tailed distributions, the Halphen distribution family (HDF), to model the mean wind speed for the purpose of wind energy estimation. The HDF has a number of properties favorable to model wind speed data, such as lower bound at zero (absence of location parameter), flexibility to cover a large range of shapes, and an explicit form of moment generating functions. We examined 126 stations in Eastern Canada (125 stations were heavy-tailed with positive excess kurtosis) and found that HDF provides fit superior to the most commonly used distribution for this purpose, the two-parameter Weibull distribution, in 100% of the stations according to the Akaike information criterion (AIC). HDF was compared against 4 two-parameter models (Gaussian, Weibull, Gamma, and inverse Gamma) and 3 three-parameter models (generalized extreme value, generalized Gamma and Burr). The most common best-fit distribution for the stations in the Eastern Canada case study is the HDF (46% according to AIC). The results of the case study show that wind observations cannot be fit by the Halphen inverse type B, but can be modelled by Halphen A and especially well by Halphen B (minimum of Kolmogorov-Smirnov statistic among candidate distributions). 87 stations exhibited class D tail behavior. No correlation between tail behavior class and best-fit distribution was observed. We encourage the use of Halphen A and Halphen B as candidate distributions for wind resource estimation studies that are based on mean wind speed estimation.

Signal power distributions for simulated outdoor sound propagation in varying refractive conditions.

Probability distributions of acoustic signals propagating through the near-ground atmosphere are simulated by the parabolic equation method. The simulations involve propagation at four angles relative to the mean wind, with frequencies of 100, 200, 400, and 800 Hz. The environmental representation includes realistic atmospheric refractive profiles, turbulence, and ground interactions; cases are considered with and without parametric uncertainties in the wind velocity and surface heat flux. The simulated signals are found to span a broad range of scintillation indices, from near zero to exceeding ten. In the absence of uncertainties, the signal power (or intensity) is fit well by a two-parameter gamma distribution, regardless of the frequency and refractive conditions. When the uncertainties are included, three-parameter distributions, namely, the compound gamma or generalized gamma, are needed for a good fit to the simulation data. The compound gamma distribution appears preferable because its parameters have a straight forward interpretation related to the saturation and modulation of the signal by uncertainties.

Comparison of Three-Parameter Distributions in Controlled Catchments for a Stationary and Non-Stationary Data Series

Flood Frequency Analysis (FFA) and the non-stationary FFA approaches are used in flood study, water resource planning, and the design of hydraulic structures. However, there is still a need to develop these methods and to find new procedures that can be used in estimating simple distributions in controlled catchments. The aim of the study is a comparison of three-parameter distributions in controlled catchments for stationary and non-stationary data series and further to develop the procedure of the estimation the simple distributions. Ten rivers from the Czech Republic and Poland were selected because of their existing or planned reservoirs as well as for flood protection reasons. The annual maximum method and the three-parameter Weibull, Log-Normal, Generalized extreme value, and Pearson Type III distributions were used in this study. The analyzed time series are stationary and non-stationary. The methodology used in this study, which makes use of the Maximum Likelihood Estimation, allows one to simplify the analysis whenever there is a series of data that is both stationary and non-stationary. The novelty in our research is the standardization and development of a new procedure for a stationary and non-stationary data series, taking into account to read a specific value of the maximum flow with a given exceedance probability from the lower or upper tail. It determines the optimal choice of the theoretical distribution that can be used, for example in the design of weirs in rural areas (lower quantiles) or in the design of hydrotechnical structures in areas at risk of flooding (upper quantiles).

A reduced distribution of the modified Weibull distribution and its applications to medical and engineering data.

In this work, we suggest a reduced distribution with two parameters of the modified Weibull distribution to avoid some estimation difficulties. The hazard rate function of the reduced distribution exhibits decreasing, increasing or bathtub shape. The suggested reduced distribution can be applied to many problems of modelling lifetime data. Some statistical properties of the proposed distribution have been discussed. The maximum likelihood is employed to estimate the model parameters. The Fisher information matrix is derived and then applied to construct confidence intervals for parameters. A simulation is conducted to illustrate the performance of maximum likelihood estimation. Four sets of real data are tested to prove the proposed distribution advantages. According to the statistical criteria, the proposed distribution fits the tested data better than some well-known two-and three-parameter distributions.

Multivariate and multi-temporal analysis of meteorological drought in the northeast of Thailand

This study evaluates drought characteristics in the Mun River Basin, located in the northeast of Thailand. The basin is important from an agriculture perspective, as about 75% of the total area is under cultivation (a majority of which is without irrigation facilities). It is, however, also one of the most drought-prone regions in the country, and a proper understanding of drought characteristics can aid in planning and implementing drought mitigation and management strategies and measures in the basin. Using the Standardized Precipitation Evapotranspiration Index (SPEI), calculated from high-resolution climatic data during 1979–2017, we find that major drought events recur approximately every 6 years in the basin. A new approach to the use of thresholds to remove the effects of minor droughts is introduced. It is found to have improved drought characterization for prolonged drought events with low intensity. Drought events are evaluated for severity, duration, and frequency (SDF) using univariate and multivariate approaches at nine timescales (1-week to 24-months). Generalized Extreme Value (GEV) and Log-logistic (LL) are the most suitable distributions for SPEI calculation in the study area, though all three-parameter distributions have good fit at longer timescales. SPEI calculated at weekly timescales, especially during the vegetative stage of crop growth, show a good correlation with yields; thus, they can be useful for planning agricultural activities. Drought vulnerable assessment using joint occurrence probabilities of drought duration and severity indicates that most of the area in the western part and some area in the eastern part are the highest drought-prone regions in the basin. It is interesting to note that the drought prone areas of the basin are independent of the annual rainfall amount but are influenced by its temporal variability.

A Probabilistic Model for Maximum Rainfall Frequency Analysis

As determining the probability of the exceedance of maximum precipitation over a specified duration is critical to hydrotechnical design, particularly in the context of climate change, a model was developed to perform a frequency analysis of maximum precipitation of a specified duration. The PMAXΤP model (Precipitation MAXimum Time (duration) Probability) harbors a pair of computational modules fulfilling different roles: (i) statistical analysis of precipitation series, and (ii) estimation of maximum precipitation for a specified duration and its probability of exceedance. The input data consist of homogeneous 30-element series of precipitation values for 16 different durations: 5, 10, 15, 30, 45, 60, 90, 120, 180, 360, 720, 1080, 1440, 2160, 2880, and 4320 min, obtained through Annual Maximum Precipitation (AMP) and Peaks-Over-Threshold (POT) approaches. The statistical analysis of the precipitation series includes: (i) detecting outliers using the Grubbs-Beck test; (ii) checking for the random variable’s independence using the Wald-Wolfowitz test and the Anderson serial correlation coefficient test; (iii) checking the random variable’s stationarity using nonparametric tests, e.g., the Kruskal-Wallis test and Spearman rank correlation coefficient test for trends of mean and variance; (iv) identifying the trend of the random variables using correlation and regression analysis, including an evaluation of the form of the trend function; and (v) checking for the internal correlation of the random variables using the Anderson autocorrelation coefficient test. To estimate maximum precipitations of a specified duration and with a specified probability of exceedance, three-parameter theoretical probability distributions were used: a shifted gamma distribution (Pearson type III), a log-normal distribution, a Weibull distribution (Fisher-Tippett type III), a log-gamma distribution, as well as a two-parameter Gumbel distribution. The best distribution was selected by: (i) maximum likelihood estimation of parameters; (ii) tests of the hypothesis of goodness of fit of the theoretical probability distribution function with the empirical distribution using Pearson’s χ2 test; (iii) selection of the best-fitting function within each type according to the criterion of minimum Kolmogorov distance; (iv) selection of the most credible probability distribution function from the set of various types of best-fitting functions according to the Akaike information criterion; and (v) verification of the most credible function using single-dimensional tests of goodness of fit: the Kolmogorov-Smirnov test, the Anderson-Darling test, the Liao-Shimokawa test, and Kuiper’s test. The PMAXTP model was tested on data from two meteorological stations in northern Poland (Chojnice and Bialystok) drawn from a digital database of high-resolution precipitation records for the period of 1986 to 2015, available for 100 stations in Poland (i.e., the Polish Atlas of Rainfall Intensities (PANDa)). Values of maximum precipitation with a specified probability of exceedance obtained from the PMAXTP model were compared with values obtained from the probabilistic Bogdanowicz-Stachý model. The comparative analysis was based on the standard error of fit, graphs of the density function for the probability of exceedance, and estimated quantile errors. The errors of fit were lower for the PMAXTP compared to the Bogdanowicz-Stachý model. For both stations, the smallest errors were obtained for the quantiles determined on the basis of maximum precipitation POT using PMAXTP.

Statistical Approach to the Analysis of the Corrosive Behaviour of NiTi Alloys under the Influence of Different Seawater Environments

Probabilistic models of corrosion rate estimation in the case of two NiTi alloys obtained by different technological processes are analysed in this paper. The depth of corrosion was measured by focused ion beam analysis on a metal surface that was not protected by an anti-corrosion coating. The samples were exposed to the influence of three different seawater environments, and empirical data were obtained in a systematic measurement procedure after 6, 12 and 18 months. Assuming that corrosion processes begin immediately after exposure of the samples to the influence of the seawater environment, and observing the corrosion rate as a random variable affected by various stochastic processes, the formed standard linear corrosion model was analysed by a statistical approach. The three best-fitted three-parameter distributions which can describe the changes in the corrosion rate for NiTi alloys exposed to the influence of the seawater environment adequately were obtained by fitting the continuous theoretical distributions. Adequate statistical tests showed similarities and differences in the behaviour of the two observed NiTi alloys from the point of view of corrosion processes caused by air, tide and sea.

Assessing the impact of climate on annual and seasonal discharges at the Sremska Mitrovica station on the Sava River, Serbia

Abstract Flood frequency analysis was performed on annual maxima series for 90 years (1928–2017) of discharge data recorded at the Sremska Mitrovica gauging station on the Sava River. The three-parameter distributions (PearsonIII, Log-PearsonIII) are more suitable for modelling annual maxima than distribution functions with only two parameters (Normal, Log-normal, Gumbel). The Mann–Kendall test statistic indicated that there is no statistically significant trend identified in annual maximum discharges or average annual discharges. A positive increasing trend was observed in annual temperature, while annual precipitation shows a decreasing trend which is non-significant. The seasonality analysis found a statistically non-significant weak negative trend in discharge in spring, summer and autumn and a statistically non-significant weak positive trend in winter. During winter, spring, and summer a non-significant negative trend in precipitation was observed, while autumn has experienced a statistically significant increasing trend. Temperatures show a positive trend in all seasons, but only temperatures during the warm period show a statistically significant increase. The results demonstrate that decreasing discharges of the Sava River at the Sremska Mitrovica gauging station are mainly the consequence of decreasing precipitation and increasing temperature (increasing evaporation), which is consistent with the results of other studies of the region.

Selection of the best fit probability distributions for temperature data and the use of L-moment ratio diagram method: a case study for NSW in Australia

This paper explores different goodness-of-fit (GOF) criteria’s used in the various fields of science to compare candidate probability density functions (pdfs) to annual maximum temperature records and discusses their usefulness and drawbacks. The L-moment ratio diagram method is also proposed as alternative approach for the GOF of the pdfs. The advantage this method allows for an easy comparison of the fit of many pdfs for several stations on a single diagram. To gain knowledge about higher order moments (i.e. shape, skewness and kurtosis) of the station data set, plotting the position of a given temperature data set in L-moment ratio diagram space is prompt and effective and can provide a useful addition to the GOF criterion. Both the L-moment ratio diagrams and many GOF criteria are used on real data to assess the fit of the pdfs for temperature data in the state of New South Wales, Australia. The analysis of the L-moment ratio diagrams reveals that the generalized extreme value and normal distributions generally fit best the annual maximum temperature series. The other two- and three-parameter distributions also showed viable fits in some instances. Results obtained from L-moment diagrams, temperature frequency histograms, cumulative density plots and the simulation study are compared with those obtained from GOF statistics, and a good agreement is generally observed between all these approaches. In conclusion the L-moment ratio diagram can represent a simple, effective and efficient approach to be used as a complementary method along with the traditional GOF criteria.

Bayesian and Classical Inference for the Generalized Log-Logistic Distribution with Applications to Survival Data.

The generalized log-logistic distribution is especially useful for modelling survival data with variable hazard rate shapes because it extends the log-logistic distribution by adding an extra parameter to the classical distribution, resulting in greater flexibility in analyzing and modelling various data types. We derive the fundamental mathematical and statistical properties of the proposed distribution in this paper. Many well-known lifetime special submodels are included in the proposed distribution, including the Weibull, log-logistic, exponential, and Burr XII distributions. The maximum likelihood method was used to estimate the unknown parameters of the proposed distribution, and a Monte Carlo simulation study was run to assess the estimators' performance. This distribution is significant because it can model both monotone and nonmonotone hazard rate functions, which are quite common in survival and reliability data analysis. Furthermore, the proposed distribution's flexibility and usefulness are demonstrated in a real-world data set and compared to its submodels, the Weibull, log-logistic, and Burr XII distributions, as well as other three-parameter parametric survival distributions, such as the exponentiated Weibull distribution, the three-parameter log-normal distribution, the three-parameter (or the shifted) log-logistic distribution, the three-parameter gamma distribution, and an exponentiated Weibull distribution. The proposed distribution is plausible, according to the goodness-of-fit, log-likelihood, and information criterion values. Finally, for the data set, Bayesian inference and Gibb's sampling performance are used to compute the approximate Bayes estimates as well as the highest posterior density credible intervals, and the convergence diagnostic techniques based on Markov chain Monte Carlo techniques were used.

Performance comparisons of distribution-free Shewhart-type Lepage and Cucconi schemes in monitoring complex process distributions

In recent years, researchers introduced several distribution-free schemes for simultaneously monitoring the location and scale parameters of distribution in the literature related to process monitoring and control. To this end, the Shewhart-Lepage (SL) and Shewhart-Cucconi (SC) schemes are two fundamental distribution-free schemes. These schemes are primarily designed to monitor the location-scale family of densities. In practice, apart from the location and scale parameters, we often encounter the presence of a shape (or skewness) parameter. In this article, we investigate the performance of the SL and SC schemes in monitoring such models. We consider some skewed distributions in the location-scale family with one or two additional parameters, some three-parameter time-to-event processes, such as three-parameter Weibull and Gamma, which are very common in various measurement and control literature. First, we present the in-control performance of the two competing schemes and then carry out a comprehensive out-of-control performance study by considering different combinations of shifts. Several recent investigations showed that the SC scheme performs just as well or better than the SL scheme in joint monitoring of the location and scale parameters for a large number of process distributions. The current study shows that in the presence of an additional parameter, especially when the shift in the shape parameter is substantial, the SL scheme is better; for a small change in shape, the SC scheme is more competitive. In general, the SL scheme performs better in monitoring the three-parameter distributions for time-to-event processes. Finally, a real application and some concluding remarks are presented.

A Probabilistic Method for Estimating the Percentage of Corrosion Depth on the Inner Bottom Plates of Aging Bulk Carriers

This paper presents an approach for the model estimating the probabilistic percent corrosion depth for inner bottom plates of fuel oil tanks located in the double bottom of aging bulk carriers. Assuming that corrosion begins after four years of exploitation, a statistical approach to investigations on the ratio of the corrosion rate and the average initial inner bottom plate’s thickness of considered bulk carriers is given. We consider this ratio to be a random variable since it is included in the usual linear corrosion model. By applying adequate statistical tests to the available empirical dataset, three best fitted three-parameter distributions for estimating the cumulative density function and the probability density function of the random variable were obtained. These three distributions were further used to estimate the studied percentage of corrosion depth. Lastly, we present the corresponding numerical and graphical results concerning the obtained statistical and empirical results and give concluding remarks.

The analysis of time series of river water mineralization in the Dnipro basin with the use of theoretical laws of random variables distribution

The analysis of current scientific work on the use of statistical methods in hydrochemical research has shown that this approach is sufficiently substantial, both in Ukraine and abroad. The purpose of this work is to determine the main statistical parameters and to research the possibility of applying theoretical laws of distribution to the time series of water mineralization.This research presents the results of the application of standard statistical methods of hydrometeorological information processing for data on water mineralization at 28 gauges of the Dnipro basin (within Ukraine) for the period from 1990 to 2015. The dynamics of the obtained statistical parameters (long-term annual average, coefficients of variation, asymmetry and autocorrelation) within the Dnipro basin in Ukraine has been analyzed. The average annual values of mineralization vary substantially within the studied part of the Dnipro basin - in the northern part the maximum value of the annual average mineralization is 447 mg/l, as it moves to the south, the mineralization increases and in the sub-basin of the Middle Dnipro it reaches a maximum of 971 mg/l; the highest values are observed in the south (sub-basin of the Lower Dnipro), where they can reach extremely high values for particular small rivers (the Solon River - Novopavlivka village, 3356 mg / l). The long-term variability of mineralization in the rivers of the studied area is insignificant, and the autocorrelation coefficients of the mineralization series are quite high, in most cases they are significant and tend to decrease from the sub-basin of the Prypyat’ river in the north to the sub-basin of the Lower Dnipro river in the south. Within the framework of the presented research, the possibility of using theoretical distribution curves known in hydrology to describe the series of river mineralization, using the example of the Dnipro basin, has also been analyzed. Using Pearson’s fitting criterion, the Pearson type III distributions and the three-parameter distributions by S.M.Krytsky and M.F.Menkel have been verified on their correspondence with the empirical series of mineralization. As a result, it was found that in 85% of cases the Pearson type III distribution can be used, and the three-parameter by S.M.Krytsky and M.F.Menkel can be used in 60% of cases.

A four-parameter negative binomial-Lindley distribution for modeling over and underdispersed count data with excess zeros

Count data often exhibits the property of dispersion and have large number of zeros. In order to take these properties into account, a new generalized negative binomial-Lindley distribution with four parameters is proposed, of which the two-parameter and three-parameter negative binomial-Lindley distributions are special cases. Several statistical properties of the proposed distribution are presented. The dispersion index for the proposed distribution is derived and based on the index, it is clear that the proposed distribution can adequately fit the data with properties of overdispersion or underdispersion depending on the choice of the parameters. The proposed distribution is fitted to three overdispersed datasets with large proportion of zeros. The best fitted model is selected based on the values of AIC, mean absolute error and root mean squared error. From the model fittings, it can be concluded that the proposed distribution outperforms Poisson and negative binomial distributions in fitting the count data with overdispersion and large number of zeros.

Modelling Insurance Losses with a New Family of Heavy-Tailed Distributions

The actuaries always look for heavy-tailed distributions to model data relevant to business and actuarial risk issues. In this article, we introduce a new class of heavy-tailed distributions useful for modeling data in financial sciences. A specific sub-model form of our suggested family, named as a new extended heavy-tailed Weibull distribution is examined in detail. Some basic characterizations, including quantile function and raw moments have been derived. The estimates of the unknown parameters of the new model are obtained via the maximum likelihood estimation method. To judge the performance of the maximum likelihood estimators, a simulation analysis is performed in detail. Furthermore, some important actuarial measures such as value at risk and tail value at risk are also computed. A simulation study based on these actuarial measures is conducted to exhibit empirically that the proposed model is heavy-tailed. The usefulness of the proposed family is illustrated by means of an application to a heavy-tailed insurance loss data set. The practical application shows that the proposed model is more flexible and efficient than the other six competing models including (i) the two-parameter models Weibull, Lomax and Burr-XII distributions (ii) the three-parameter distributions Marshall-Olkin Weibull and exponentiated Weibull distributions, and (iii) a well-known four-parameter Kumaraswamy Weibull distribution.