Type Of Probability Distribution Research Articles

Type of probability distribution reflects how close mixing dynamics in river chemistry are to thermodynamic equilibrium

The distribution of geochemical species are typically either (log)normally distributed or follow power laws. Here we link these types of distributions to the dynamics of the system that generates these distributions, showing that power laws can emerge in dissipative systems far from equilibrium while (log)normal distributions are found for species for which the concentrations are close to equilibrium. We use observations of the chemical composition of river water from the sampling space in central Italy as well as discharge data to test this interpretation. We estimate the dissipation rate that results when groundwater drains into the river and the dissolved chemical species mix with the river water. We show that calcium (Ca2+) and bicarbonate (HCO3−) concentrations are close to saturation along most of the downstream length of the Arno river, with decreasing dissipation rates and a lognormal distribution, while sodium (Na+) and chloride (Cl−) concentrations increase substantially downstream, show increased dissipation rates, and are power-law distributed. This supports our hypothesis that power law distributions appear to be indicative of dissipative systems far from thermodynamic equilibrium, while (log)normal distributions indicate weakly dissipative systems close to equilibrium. What this implies is that probability distributions are likely to be indicative of the thermodynamics of the system and the magnitude of disequilibrium constrains the range over which power-law scaling may be observed. This should help us to better identify the generalities and mechanisms that result in these common types of distributions and to better classify variability in systems according to how dissipative these are.

Uncertainty quantification and global sensitivity analysis of bearing capacity of push-out specimen considering randomness in bond-slip behaviour

As an artificial composite material, the uncertainty in mechanical property of concrete significantly impacts the bond-slip behaviour of the shaped steel concrete structure. To investigate the effect of the uncertainty of concrete mechanical properties on the bearing capacity of push-out specimens, based on the data of a small number of push-out test with identical design parameters, the improved Bootstrap method is utilized to execute sample augmentation for the bond stress-slip curves, and the stochastic bond slip constitutive model is established after considering five parameters (K1, …, K5) in the deterministic bond-slip model as random variables. Simultaneously, considering the randomness in elastic modulus of concrete (Ec), uncertainty quantification (UQ) of the bearing capacities of the push-out specimens is conducted using the arbitrary polynomial chaos (aPC) method. Subsequently, the global sensitivity analysis of the ultimate and residual bearing capacities is accomplished through the aPC surrogate model. The results indicate that the probability distribution types of the parameters in the stochastic bond-slip constitutive model can be arbitrary when considering the randomness of concrete material, and the aPC method with at least a third-order expansion is recommended to execute UQ for the push-out test due to the strong nonlinearty between the bearing capacities and the random parameters. The ultimate bearing capacity is primarily affected by the parameter K2, followed by K4, K3, and K5. Whereas the residual bearing capacity is predominantly influenced by the parameter K4, followed by K5, the remaining parameters exhibit negligible influence. In contrast, K1 and Ec have the greatest influence on equivalent plastic strain of the concrete near the loading end of the push-out specimen.

An empirical study on the stochastic long-term travel demands of a large-scale metro network

The widely-existed uncertainty of origin–destination (OD) demand in transportation networks has attracted extensive attention. Most characterizations or modeling of stochastic OD demands in networks assume a homogenous probability distribution, but it lacks empirical studies of large-scale networks to justify this assumption. Given that the long-term continuous automatic fare collection data of metro networks can provide complete OD passenger demand information, this study takes the Shanghai Metro network as an example to empirically examine the stochasticity characteristics of OD passenger demands of metro network. Based on the morning peak OD demand data for 250 weekdays, a local outlier factor method is used to identify and remove outliers in the data. A clustering method is used to cluster the OD pairs, study the fluctuation and distribution characteristics of the OD passenger demands, and select the optimal distribution type through goodness-of-fit indices. The results show that 1) the coefficients of variation of morning peak OD demands in the network are mainly distributed in the range of 0.2–0.6, and different OD pairs have different fluctuations, and the degree of demand fluctuation decreases as the mean increases; 2) the probability distribution types of OD demands based on statistical characteristics are heterogeneous; and 3) the optimal distribution type of OD demands is Poisson, Lognormal/Gamma, and Normal for OD pairs characterized by a small mean and right-skewness, a small mean and skewness close to 0, and a large mean, respectively. In contrast to the simple average-based data processing of OD passenger demand in metro networks, this paper presents a new perspective of mining long-term continuous data to understand the inherent stochasticity of OD passenger demands. The results can provide more realistic and practical inputs and assumptions for theoretical research on stochastic OD demands in metro networks.

Hedging Interest Rate Options with Reinforcement Learning: an investigation of a heavy-tailed distribution

Purpose: The study intends to model an interest rate index option using a heavy-tailed distribution. The goal is to calculate the interest rate path-dependent option prices that are consistent with market data and to develop a reinforcement learning strategy to discretely hedge the position considering transaction costs. Methodology: This paper presents a mathematical framework to calculate the price of interest rate path-dependent options. The research adapted a Fourier cosine series formula to employ the characteristic function of the present value of the forward index, which is modeled as a variance-gamma process and uses deep Q-learning to hedge such options. Findings: There is market evidence that the implied volatility curve is not flat. The study demonstrated that the variance-gamma process generates an increasing volatility smile, which is consistent with market observations. Additionally, hedging results show that the path-dependent options generated from the variance-gamma process can be efficiently hedged with advanced Q-learning techniques. Research limitations/implications: The study comprised only the variance-gamma process. Other probability distributions, such as the Normal Inverse Gaussian model, should be investigated. Practical implications: This study reveals which type of probability distribution should be present in a pricing engine to be consistent with implied volatilities. The approach provided here can assist managers in evaluating and comprehending market pricing behavior as well as achieving discrete hedging with costs. Originality: The paper addressed the merging of a fast pricing method for the interest rate options with a heavy-tailed distribution and the discrete interest rate derivatives hedging with reinforcement learning.

A digital twin-based analysis method to assess geometric variations for parts in batch production

Background In mass production, engineers are more concerned with the statistical distribution accuracy of parts in mass production rather than just the qualification of individual parts. However, currently, the statistical analysis methods designed for product accuracy are relatively scattered, and most of them focus on nominal part models. Therefore, there is a need to design a statistical analysis method for parts in mass production based on the Digital Twin model. >Methods This paper presents a novel method to analyse the geometric variations of parts in batch production in the production line, which is based on digital twins to model and evaluate deviations contributed by the geometrical condition, assembly condition and material condition. Firstly, the statistical descriptions of the parts, particularly the features of a digital twin for parts in batch production related to the geometry and position, are classified into various hierarchies. Secondly, a covariance method is employed to analyse the law of their shape from the descriptions. Thirdly, the parts' shape feature similarity for different terms is derived, including the linear features of pose constraint, rotation deviation, and geometric deviation and the curve features like a geometric deviation. Finally, the probability distribution of discrete points on the manufacturing error caused by different reasons is calculated. Results Two case studies of reducer and rail highlight the applicability of the proposed approach. The standard deviation of the points has similar trend with sample cases according to normal distribution. Conclusions This paper categorizes the deviations of batch parts into the linear features of pose constraint, rotation deviation, and geometric deviation. When batch parts exhibit any of these deviation types, the eigenvalues and eigenvectors of their covariance matrix show certain patterns, enabling the identification of the deviation type and calculation of the statistical deviation probability distribution for the corresponding features.

Experimental study on the corrosion characteristics of steel bars in concrete considering the effects of multiple factors

In this paper, twenty reinforced concrete specimens were designed for accelerated corrosion test, and a systematic study was conducted to investigate the effects of water-cement ratio, thickness of concrete cover, and spacing and quantity of steel bars on the corrosion characteristics of steel bars. The results showed that the mass loss of steel bars approximately increased with the water-cement ratio and decreased with an increase in the thickness of concrete cover. As the spacing between the steel bars in lower layer of the specimens increased, the mass loss of the steel bars in lower layer approximately showed an increasing trend. When there were more than two steel bars arranged in lower layer of the specimens, the mass loss of the middle steel bars was smaller than that of the steel bars located in the corner areas on both sides. The probability distribution type of the corrosion depth was mainly normal distribution, and the influence of water-cement ratio, thickness of concrete cover, and spacing and quantity of steel bars in lower layer of the specimens on the probability distribution was not obvious. The probability distribution type of the width of corrosion-induced cracks was mainly exponential distribution, and the influence of the spacing and quantity of steel bars in lower layer on the probability distribution was not obvious. When the water-cement ratio was large, the width of corrosion-induced cracks was less discrete from its mean value. With the increase of the quantity of steel bars in lower layer, the shape of the probability distribution curve of the width of corrosion-induced cracks tended to be flat. There was a good positive linear relationship between the mass loss of steel bars and the average width of corrosion-induced cracks, and the influence of water-cement ratio and the spacing between the steel bars in lower layer on this linear relationship was irregular. A good positive linear relationship between the maximum width of corrosion-induced cracks and the average corrosion depth of steel bars was established, which was irregularly affected by the water-cement ratio and the spacing between the steel bars in lower layer.

Simplified reliability assessment approach for tunnel structures considering the effects of adjacent excavation and soil uncertainty

Reliability assessment of tunnel structures is a fundamental concern in civil engineering, as tunnels are inevitably affected by adjacent excavation projects, resulting to additional deformations. The complexity of this subject is further amplified when considering the soil uncertainty. In this study, a direct Monte-Carlo simulation (DMCS) approach is proposed to evaluate the reliability of tunnels structures after disturbed by adjacent excavation. Random variables are employed to capture the uncertainty in soil mechanics parameters, while the tunnel is modeled as a Timoshenko beam placed on a Pasternak foundation, allowing for the consideration of generalized deflections and shearing deformation, as well as the bending deformation. A comparison between DMCS, Response Surface method (RSM) and First-order Second-moment (FOSM) method is performed through a typical engineering example. The results demonstrate that DMCS accurately captures the reliability of tunnel structures and requires less computational effort, as it eliminates the need for re-establishing response surface functions through numerical simulations. Further parametric analysis investigates the influence of coefficient of variation (COV), cross-correlation, and probability distribution types of soil parameters on reliability of tunnel, which indicated that the COV and probability distribution types had a significant influence, while the impact of cross-correlation was limited. The reliability of tunnel structures is more sensitive to the variability of elastic modulus. Additionally, the reliability results obtained when assuming soil parameters follow a normal distribution are lower than those obtained when considering them as log-normal distributions. The proposed method offers an accurate and practical tool for evaluating the reliability of operational tunnel structures in the presence of adjacent excavations.

Sensitivity analysis of factors affecting gravity dam anti-sliding stability along a foundation surface using Sobol method

The anti-sliding stability of a gravity dam along its foundation surface is a key problem in the design of gravity dams. In this study, a sensitivity analysis framework was proposed for investigating the factors affecting gravity dam anti-sliding stability along the foundation surface. According to the design specifications, the loads and factors affecting the stability of a gravity dam were comprehensively selected. Afterwards, the sensitivity of the factors was preliminarily analyzed using the Sobol method with Latin hypercube sampling. Then, the results of the sensitivity analysis were verified with those obtained using the Garson method. Finally, the effects of different sampling methods, probability distribution types of factor samples, and ranges of factor values on the analysis results were evaluated. A case study of a typical gravity dam in Yunnan Province of China showed that the dominant factors affecting the gravity dam anti-sliding stability were the anti-shear cohesion, upstream and downstream water levels, anti-shear friction coefficient, uplift pressure reduction coefficient, concrete density, and silt height. Choice of sampling methods showed no significant effect, but the probability distribution type and the range of factor values greatly affected the analysis results. Therefore, these two elements should be sufficiently considered to improve the reliability of the dam anti-sliding stability analysis.

Service life prediction and time-variant reliability of reinforced concrete structures in harsh marine environment considering multiple factors: A case study for Qingdao Bay Bridge

Chloride attack is the primary cause of performance deterioration and life attenuation of reinforced concrete (RC) structures in marine environments. In an attempt to accurately predict the reliability and service life of RC structures under the co-effects of multiple factors, a comprehensively modified chloride-transport-based model is proposed in this paper. Based on Fick's second law, a multifactor integrated deterministic chloride transport model is developed considering the time dependence, concrete mix proportion, chloride binding capacity, freeze–thaw cycles, sustained load, temperature, humidity and concrete deterioration. Based on extensive literature review combined with the results of multitudinous long-term field exposure tests used in the durability design of Hong Kong-Zhuhai-Macao Bridge and Shenzhen-Zhongshan Bridge, the probability distribution types and parameter values of the governing variables are determined. Taking Qingdao Bay Bridge as an example, the time-variant reliability index, failure probability and reliability-index-based corrosion initiation are determined by using the equivalent normalization method (JC method) and Monte Carlo simulation technique for different environmental zones including atmospheric/splash/tidal/submerged zones. The sensitivity of the reliability index to the control variables and the effects of age index, chloride binding capacity, numbers of freeze–thaw cycles, stress level and concrete deterioration on the reliability index are quantitatively investigated. The results show that the time-variant reliability of different environmental zones is sequenced as: atmospheric zone > submerged zone > tidal zone > splash zone; the sensitivity of reliability to the governing variables sequenced as: concrete cover thickness > surface chloride concentration > chloride diffusion coefficient > critical chloride concentration; the time dependence of the diffusion coefficient has a significant effect on the time to corrosion initiation and cannot be ignored in the predictive model for both the high-performance or ordinary concrete; the number of freeze–thaw cycles and the tensile stress level are relatively insensitive but substantially detrimental to the reliability of corrosion initiation.

Selecting the best probability distribution of infection with MERS-COV in Wasit Governorate

This research aims to study the probability distribution of cases infected with Coronavirus in Wasit Governorate for the period from 2020 to 2021 using three types of probability distributions: the logistic distribution, the two-parameter Weibull distribution, and the least-valued Campbell distribution. Maximum, where the parameters of these distributions were estimated using the maximum likelihood method. Applying several criteria to determine the optimal distribution of cases infected with Coronavirus in the governorate, as these criteria were consistent Akaike (CAIC), Bayesian Akaike (BIC), and Akaike (AIC), and the probability distribution with the lowest value for these criteria is considered the best to represent a good model for studying this data. Based on the applied part, the researchers concluded that the most appropriate probability distribution for coronavirus infection data in Wasit Governorate is the two-parameter Weibull distribution. The research is to provide a better understanding of the pattern of spread of the virus in the governorate, and to enable health officials to take more effective measures to limit the spread of infection in the future.

A digital twin-based analysis method to assess geometric variations for parts in batch production

Background: In mass production, engineers are more concerned with the statistical distribution accuracy of parts in mass production rather than just the qualification of individual parts. However, currently, the statistical analysis methods designed for product accuracy are relatively scattered, and most of them focus on nominal part models. Therefore, there is a need to design a statistical analysis method for parts in mass production based on the Digital Twin model. Methods: This paper presents a novel method to analyse the geometric variations of parts in batch production in the production line, which is based on digital twins to model and evaluate deviations contributed by the geometrical condition, assembly condition and material condition. Firstly, the statistical descriptions of the parts, particularly the features of a digital twin for parts in batch production related to the geometry and position, are classified into various hierarchies. Secondly, a covariance method is employed to analyse the law of their shape from the descriptions. Thirdly, the parts' shape feature similarity for different terms is derived, including the linear features of pose constraint, rotation deviation, and geometric deviation and the curve features like a geometric deviation. Finally, the probability distribution of discrete points on the manufacturing error caused by different reasons is calculated. Results: Two case studies of reducer and rail highlight the applicability of the proposed approach. The standard deviation of the points has similar trend with sample cases according to normal distribution. Conclusions: This paper categorizes the deviations of batch parts into the linear features of pose constraint, rotation deviation, and geometric deviation. When batch parts exhibit any of these deviation types, the eigenvalues and eigenvectors of their covariance matrix show certain patterns, enabling the identification of the deviation type and calculation of the statistical deviation probability distribution for the corresponding features.

Wind Power Interval Prediction Based on Robust Kernel Density Estimation

Wind power output has a high degree of randomness, so it is difficult to describe it accurately with some typical probability distribution. The extreme values influence the sample’s general non-parametric kernel density estimation method, and the estimated results are relatively conservative. A wind power interval prediction method based on robust kernel density estimation is proposed to improve the compactness and accuracy of interval prediction. In probability estimation, this method will assign a small weight to the extreme sample data to reduce its influence on the probability density estimation accuracy and has better robustness than the general kernel density estimation method. In addition, the bandwidth of robust kernel density estimation is a dynamic parameter, which can be adjusted with the sample data to improve the prediction accuracy and avoid over-conservative prediction.

The growth and size of orogenic gold systems: probability and dynamical behaviour

Every nonlinear system grows by increments, and the final probability distributions for components of that system emerge from an amalgamation of these increments. The resulting probability distribution depends on the constraints imposed on each increment by the physical and chemical processes that produce the system. Hence there is the potential that the observed probability distribution can reveal information on these processes. Complex systems that grow by competition between the supply and consumption of energy and mass have growth laws that are cumulative probability distributions for their component parts that reflect such competition. We show that the type of probability distribution is characteristic of the endowment of orogenic gold deposits with the sequence: Weibull → Fréchet → gamma → log normal representative of increasing endowment. Further, the differential entropy of the probability distribution is indicative of the quality of the deposit, with low-quality deposits represented by high entropy and high-quality deposits represented by low or negative entropy. The type of probability distribution gives an indication of the processes that operated to produce the deposit. These conclusions hold for mineralisation as well as for the associated alteration assemblages. We suggest that the probability distribution for the mineralisation or the alteration assemblage gives a good indication of the endowment and quality of a deposit from a single drill hole. KEY POINTS A single drill hole from a deposit can provide information on endowment and organisation. Weibull → Fréchet → gamma → log normal probability distributions are representative of increasing gold endowment. The differential entropies of these distributions characterise the organisation of the system.

Stability analysis of slopes based on cloud model-Monte Carlo coupling

Risk analysis of dam slopes is crucial for ensuring the safety and stability of hydraulic engineering. To improve the accuracy and reliability of risk analysis, we adopt the cloud theory approach and conduct a study on the distribution types of soil shear strength indicators based on indoor geotechnical tests. We propose a “cloud model-Monte Carlo” coupling model that uses the cloud model to describe the uncertainty of risk factors and determine the probability distribution types of shear strength parameters, while the Monte Carlo method is used to simulate random variables in the model. The effectiveness of the proposed model is validated through a risk analysis of a slope of an earth-rock dam, with results showing significantly greater accuracy and reliability compared to traditional methods. The calculation results show that the risk probability corresponding to the design flood level of the dam is 9.01×10-6, exceeding its allowable risk standard of 0.5×10-6, hence the need for reinforcement treatment. The proposed model can accurately evaluate the risk of dams and provide the scientific basis for decision-making in dam safety management.

Vulnerability Assessment of Equipment Excited by Transient Electromagnetic Disturbances Based on Distribution-Free QMU Method

Based on the framework of quantification of margins and uncertainties (QMU) method, a distribution-free QMU method is proposed for the vulnerability evaluation of electrical and electronic equipment excited by transient electromagnetic disturbances. The proposed method can provide a comprehensive evaluation of the equipment considering the stress and the strength with uncertainties. The stress is represented by the induced responses on the equipment conductors, the strength is quantified by the occurrence probability of failure obtained from the experiment. The quantile-based QMU metric can take account of the inherent probability characteristics of stress and strength, as well as the aleatory uncertainties caused by sampling variability. The quantiles and their tolerance bounds are estimated by a nonparametric estimation method based on order statistics, which is a data-driven method without being constrained by the type of probability distribution. A case study of an electronic equipment in the power plant is carried out to illustrate the evaluation procedures of the proposed method.

МЕТОД ВЕРОЯТНОСТНОГО АНАЛИЗА НАДЕЖНОСТИ УЗЛОВ СТАЛЬНЫХ ФЕРМ НА ОСНОВЕ Р-БЛОКОВ

The article presents a new approach to probabilistic analysis of reliability for steel trusses’ joints based on p-boxes. The p-box is an area formed by two boundary cumulative distribution functions (CDFs) inside which there is a real cumulative distribution function (which is unknown due to the epistemic uncertainty of statistical data). The research demonstrates an algorithm for arithmetic operations with p-boxes by discretizing them into the Dempster-Shafer structures. Information about the failure probability of trusses’ joints is necessary to consider the truss reliability as a mechanical serial system in terms of reliability theory. The use of p-boxes makes it possible to obtain a more cautious model of a random variable due to the absence of the need to adopt some statistical hypotheses about the type of probability distribution of various random variables in mathematical models.

Corrosion depth of steel bars in recycled aggregate concrete beams under static load

To study the distribution of corrosion depth along the longitudinal direction of tensile steel bars embedded in recycled aggregate concrete (RAC) beams under static loads, a total of 12 RAC beams were designed for accelerated corrosion testing by considering three types of recycled coarse aggregate (RCA) replacement ratios (i.e. 0%, 50% and 100%) and four kinds of static load levels (i.e. 0, 0.2, 0.4 and 0.6). The results demonstrated that the corrosion depth of tensile reinforcement showed an increasing trend with the increase of static load level. The maximum corrosion depth had a good quadratic polynomial relationship with the average corrosion depth of tensile reinforcement and increased with the average corrosion depth. For beams with the same RCA replacement ratio, there was a good linear relationship between the overall uneven corrosion coefficient and average mass loss of tensile reinforcement. The probability distribution type of corrosion depth along the longitudinal direction of tensile reinforcement was mainly a lognormal distribution. An increase in both RCA replacement ratio and static load level could make the probability distribution curve of corrosion depth move to the right, as well as increase the dispersion of corrosion depth around its mean value.

Optimal cap on claim settlements based on social benefit maximization

Caps on damages constitute one of the proposed reforms of the tort system, especially in the U.S. In this paper, we establish models to capture both the claimant's and the insurer's behaviors under a system whereby damages are capped. We also establish an objective function of maximizing the social benefit, expressed as the sum of claimant and insurer benefits. We examine the optimal upper limit that balances both claimant and insurer benefits. Our model implies that the optimal value of the upper limit should be the expected value of the random claim payment independent of the type of probability distribution the claim payment follows. Finally, we apply empirical data to our theory and arrive at an estimate of the optimal damage cap.

A Framework for Estimating the Probability Distribution of Event Runoff Coefficient in Ungauged Catchments

AbstractThe derived distribution method is a promising approach for flood estimation in ungauged catchments. In this approach, event runoff coefficient (ERC) is often adopted to behave as the runoff generation component since the behavior of ERC is closely related to flood generation mechanisms and its distribution reflects catchment climatological and landscape characteristics. However, there is a lack of understanding of how to transfer the information on ERC characteristics from gauged to ungauged catchments at catchment scale. Hence, here, we propose a generalized framework to estimate the probability distribution of ERC for ungauged catchments based on knowledge of the spatial and temporal controls on ERC from gauged catchments. Key components of the framework include cluster analysis based on ERC characteristics, linking clusters with catchment attributes, and constructing typical probability distribution of ERC conditioned on its temporal indicators. A total of 290,743 rainfall‐runoff events observed in 431 GB catchments during the period 1990–2014 have been employed to verify the framework. Good estimations are obtained with the median value of coefficient of determination (R2) reaching 0.85 across all test catchments. The results indicate that similar pre‐event catchment conditions may cause distinct runoff response in different catchments, thus predicting the correct spatial cluster is crucial to the estimation accuracy. This work sheds light on constructing a stochastic generator model of ERC according to its spatial pattern and temporal dynamic, facilitating a new alternative for flood estimation method in ungauged catchments.

Assessment of the Uncertainty Associated with Statistical Modeling of Precipitation Extremes for Hydrologic Engineering Applications in Amman, Jordan

Estimates of extreme precipitation are commonly associated with different sources of uncertainty. One of the primary sources of uncertainty in the statistical modeling of precipitation extremes comes from extreme data series (i.e., sampling uncertainty). Therefore, this research aimed to quantify the sampling uncertainty in terms of confidence intervals. In addition, this article examined how the data record length affects predicted extreme precipitation estimates and data set statistics. A nonparametric bootstrap resample was utilized to quantify the precipitation quantile sampling distribution at a particular non exceedance probability. This sampling distribution can provide a point estimation of the precipitation quantile and the confidence interval at a particular non exceedance probability. It has been shown that the different types of probability distributions fit the extreme precipitation data series of various weather stations. Therefore, the uncertainty analysis should be conducted using the best-fit probability distribution for extreme precipitation data series rather than a predefined single probability distribution for all stations based on modern extreme value theory. According to the 95% confidence intervals, precipitation quantiles are subject to significant uncertainty and the band of the uncertainty intervals increases with the return period. These uncertainty bounds need to be integrated into any frequency analysis from historical data. The average, standard deviation, skewness and kurtosis are highly affected by the data record length. Thus, a longer record length is desirable to decrease the sampling uncertainty and, therefore, decrease the error in the predicted quantile values. Moreover, the results suggest that a series of at least 40 years of data records is needed to obtain reasonably accurate estimates of the distribution parameters and the precipitation quantiles for 100 years return periods and higher. Using only 20 to 25 years of data to obtain estimates of the higher return period quantile is risky, since it created high sampling variability relative to the full data length.