Three-parameter Lognormal Distribution Research Articles

Analysis of Probability Distributions for Modelling Extreme Rainfall Events and Detecting Climate Change: Insights from Mathematical and Statistical Methods

Exploring the realm of extreme weather events is indispensable for various engineering disciplines and plays a pivotal role in understanding climate change phenomena. In this study, we examine the ability of 10 probability distribution functions—including exponential, normal, two- and three-parameter log-normal, gamma, Gumbel, log-Gumbel, Pearson type III, log-Pearson type III, and SQRT-ET max distributions—to assess annual maximum 24 h rainfall series obtained over a long period (1972–2022) from three nearby meteorological stations. Goodness-of-fit analyses including Kolmogorov–Smirnov and chi-square tests reveal the inadequacy of exponential and normal distributions in capturing the complexity of the data sets. Subsequent frequency analysis and multi-criteria assessment enable us to discern optimal functions for various scenarios, including hydraulic engineering and sediment yield estimation. Notably, the log-Gumbel and three-parameter log-normal distributions exhibit superior performance for high return periods, while the Gumbel and three-parameter log-normal distributions excel for lower return periods. However, caution is advised regarding the overuse of log-Gumbel, due to its high sensitivity. Moreover, as our study considers the application of mathematical and statistical methods for the detection of extreme events, it also provides insights into climate change indicators, highlighting trends in the probability distribution of annual maximum 24 h rainfall. As a novelty in the field of functional analysis, the log-Gumbel distribution with a finite sample size is utilised for the assessment of extreme events, for which no previous work seems to have been conducted. These findings offer critical perspectives on extreme rainfall modelling and the impacts of climate change, enabling informed decision making for sustainable development and resilience.

Brash ice macroporosity and piece size distribution in ship channels

Merchant vessels' performance on brash ice plays an important role in navigation in fast ice conditions in Northern Baltic ports. Among many parameters, the piece size distribution and macroporosity of the brash ice influence the accuracy of the model simulations of ship performance in brash ice. These properties also govern the brash ice accumulation and consolidation in ship channels. The current work presents analyses of macroporosity and piece size distribution from three full-scale brash ice channels investigated during winters 2020–21 and 2021–22. The results provide insights into brash ice macroporosity and piece size distribution. Smaller brash ice pieces exhibited higher porosity. On each measured cross-section, the average macroporosity ranged from 1.4% to 23%, with standard deviations of 3.8% and 16%, respectively. Porosity variations were observed in relation to equivalent brash ice thickness, the number of ship passages, and cumulative freezing air temperatures. Total porosity and the channel's brash ice porosity initially increased with breaking events and then stabilized at approximately 23% and 30%, respectively, after 9 passages. In contrast, side ridge porosity increased initially (up to 23%) and subsequently decreased. The initial total, brash ice and side ridge maximal porosities after the breaking event were estimated equal to 25%, 35% and 30%, respectively. The first two channels, which were navigated a total of 9 and 10 times, had an average degree of brash ice consolidation equal to 0.84 and 0.76, and an average degree of side ridge consolidation equal to 1.1 and 1.0, respectively. The third channel, characterized by frequent navigation, exhibited a consolidation degree of 0.82 for brash ice and 1.35 in the side ridges. The average vertical piece size across all cross-sections ranged from 0.28 m to 0.52 m with standard deviations of 0.16 m and 0.25 m. The average horizontal piece size ranged from 0.33 m to 0.4 m with standard deviations between 0.2 m and 0.34 m. The horizontal piece size distribution of brash ice was best described by the probability density function of a three-parameter lognormal distribution.

A novel statistical model for flood prediction in the Eel River watershed, New Brunswick, Canada

ABSTRACT A strong correlation between the effect of climate change and the increase in flooding frequency and magnitude has been reported in Canada. Consequently, there is a crucial need to examine the effects of future climate change scenarios on flooding conditions. The main objective of this research is to better understand the destructive effects of flood events under historical and future climate change conditions for a small watershed (Eel River watershed) in New Brunswick (NB), Eastern Canada. A practical model had been developed using the modified Artificial Neural Network (ANN) in MATLAB by the authors of this study. The architecture and data structure of ANN is characterized by a back propagation with the Levenberg–Marquardt method. The observed daily total precipitation, daily maximum and minimum air temperatures, daily discharge for the period 1967 to 1983, the simulated monthly maximum and minimum air temperatures, and monthly total precipitation for the period of 1996–2099 from the CanESM2, the second-generation Canadian Earth System Model (CGCM), were used as input of the model. The Representative Concentration Pathways (RCP 4.5 and 8.5), as suitable climate change scenarios, were selected based on the Intergovernmental Panel on Climate Change (IPCC) recommendations for flood studies. Daily values of temperatures, precipitations, and discharges were converted to monthly mean values for better prediction of the output results. In addition, two series of observed discharges were prepared using mean monthly (Qavg) and daily maximum discharges (Qd) as the Target of the model. For more accurate analysis, the time frames of 1996–2012 (for the historical) and 2022–2038, 2039–2055, 2056–2072, 2073–2089, and 2083–2099 (for the future) were considered with a duration of 16 years for each time frame. The output results of ANN were predicted daily maximum (Qd) and mean (Qavg) discharges under the impact of climate change scenarios. As a part of the developed model, Flood Frequency Analysis (FFA) was undertaken using the generalized extreme value (GEV) and the three-parameter lognormal (LN3) distributions based on the predicted and observed discharges. The performance of FFA and ANN were demonstrated using the Anderson–Darling (AD), the Chi-square (CS) tests and coefficient of correlation (R) and mean squared error (MSE), respectively. In conclusion, the three most critical time frames with the highest values of predicted discharges were 2022–2038, 2056–2072, and 2073–2089 for RCP4.5 and 2039–2055, 2073–2089, and 2083–2099 for RCP8.5. Also, based on the FFA, the magnitudes of flood recurrence for the future time period of 100 years will dramatically increase according to the most critical time frames of 2056–2072 and 2039–2055 for RCP 4.5 and 8.5, respectively. Findings indicated that the Eel River watershed will encounter severe floods, and about a 50% increase in mean discharge, especially for the critical time frames. Finally, flood occurrences show increasing trends due to climate change effects in the most critical time frames.

Capacity characteristics of long-term work zones on signalized intersection approaches

This study explored the capacity characteristics of five long-term WZ configurations at a major signalized intersection in Toronto. In total, 1,766 cycles and 22,057 inter-vehicle headway observations were recorded using over 140 h of real-time traffic videos. The saturation flow rate ranged from 1318.7 to 1767.3 pc/hr/ln across the study sites. Interestingly, the Highway Capacity Manual WZ model was found to significantly overestimate the traffic capacity at the study sites. The analysis assessed the hypothesis that the saturation headway is further compressed when long queues are present on site. The average saturation headway during oversaturated conditions was found to be shorter by 5 to 11% across the study sites as compared to undersaturated conditions. Other influential factors such as day versus night and late-merge were assessed. The saturation flow under daytime conditions was higher by 3 to 5% as compared to nighttimes. Vehicles merging late from the closed lanes had a shorter headway as compared to vehicles approaching from the open lane by around 8 to 10%. Probability distribution functions of the saturation headway were developed, the best-fit curve for all cases was found to be the three-parameter lognormal distribution. The study also developed multiple regression equations to estimate WZ capacity reduction induced by heavy vehicles for a wide range of heavy vehicle percentages. On average, oversaturated conditions yielded a 5.4% additional flow reduction as compared to undersaturated conditions for the same HV%. Moreover, the impact of different snow conditions on the saturation flow was estimated. The fully slushy, partly slushy, and bare-and-wet snow categories resulted in headway elongation of 35.1 to 42%, 21.2 to 23.4%, and 6.2 to 12.6%, respectively. The study explains how the findings can assist transportation agencies and practitioners in improving the policies, practices, and guidelines related to WZ activities and planning.

Estimation methods for the ratio of medians of three-parameter lognormal distributions containing zero values and their application to wind speed data from northern Thailand.

Wind speed has an important impact on the formation and dispersion of fine particulate matter (PM), which can cause several health problems. During the transition from the winter to the summer season in northern Thailand, the wind speed has been low for longer than usual, which has resulted in fine PM accumulating in the air. Motivated by this, we have identified a need to investigate wind speed due to its effect on PM formation and dispersion and to raise awareness among the general public. The hourly windspeed can be approximated by using confidence intervals for the ratio of the medians of three-parameter lognormal distributions containing zero values. Thus, we constructed them by using fiducial, normal approximation, and Bayesian methods. By way of comparison, the performance measures for all ofthe proposed methods (the coverage percentage, lower and upper error probabilities (LEP and UEP,respectively), and expected length) were assessed via Monte Carlo simulation. The results of Monte Carlo simulation studies show that the Bayesian method provided coverage percentages close to the nominal confidence level and shorter intervals than the other methods. Importantly, it maintained a good balance between LEP and UEP even for large variation and percentage of zero-valued observations. To illustrate the efficacy of our proposed methods, we applied them to hourly wind speed data from northern Thailand.

Efficient method for fully quantifying the uncertainty of failure probability

It has been recognized that the uncertainty of distribution parameters has a significant effect on the outcome of structural reliability analysis. In this study, an efficient and accurate method is proposed to fully quantify failure probability, considering the uncertainty of the distribution parameters. This method obtains predictive failure probability by integrating the probability space of the conditional reliability index instead of directly utilizing the point-estimate method on the conditional failure probability as in previous studies. In the proposed method, a three-parameter lognormal distribution is suggested to approach the distribution of the conditional reliability index. The first three central moments of the conditional reliability index are estimated utilizing the point-estimate method combined with the bivariate dimension-reduction method. Based on this, the analytical solutions for the quantiles and probability distribution of the conditional failure probability can be readily derived without redundant calculations. Four illustrative examples were investigated to verify the efficiency and accuracy of the proposed method. The results show that the proposed method produces sufficiently accurate results in an efficient and simple way. It also presents a complete picture of the structural reliability evaluation results considering distribution parameter uncertainty in a wide range of applications.

Comparative fire-following-earthquake performance of code-compliant low-rise base-isolated structures

Seismic base isolation is an effective way of protecting structures against seismic loads and is very effective in terms of both collapse mitigation of structures and protection of non-structural elements under severe ground shaking. In this study, the structural demand of base-isolated three- and four-story steel moment-resisting frames is determined in case of a fire event followed by an earthquake, and compared with the results of fixed-base frames, which have similar geometry, load, and seismic hazard for a location in California. Four compartments are selected as possible locations for fire events in each building, and beams and columns in those compartments are exposed to a representative temperature increase in time, which includes a cooling phase as well. Maximum, minimum, and residual axial force, and moment demands on elements of the fire compartments, and drift demand of structural frames on the first and second floor, where the fire is assumed to occur, are determined and compared. Results are given in terms of parameters of three-parameter log-normal distribution, hence fragility curves can be constructed for each response considered in the study. Seismic isolation is effective in reducing both maximum and residual drift demand of the frames, and axial force in the beam element for each compartment considered. Fixed-base frames have 20% more maximum axial load on beams. Beam and column elements in the four-story configuration are under relatively more moments in case of a fire, while the performance of three-story frames depends on the location of the assumed fire.

Phosphorus transport in the Three Gorges Reservoir over the past two decades

Phosphorus (P) is an essential nutrient in aquatic ecosystems and its dynamics in river systems are greatly affected by human activities. Based on the in situ measurements of water discharge (Q), suspended sediment concentrations (SSC), and concentrations (C) of total P (TP), particulate P (PP), and total dissolved P (TDP) from 1997 to 2017, P transport in the Three Gorges Reservoir (TGR) and the combined effects of cascade reservoirs operation and anthropogenic emissions have been explored. The whole period was divided into three sequential periods of January 1997 to May 2003 (period 1), June 2003 to September 2012 (period 2), and October 2012 to December 2017 (period 3), corresponding to the period prior to impoundment of the TGR, during the operation of the TGR, and after the impoundment of upstream cascade reservoirs, respectively. Results showed that the P concentrations were positively skewed and generally followed the three-parameter gamma and log-normal distributions. There was significant difference for SSC, TP, PP, and TDP between adjacent periods but not for Q. The SSC, TP, and PP presented downward trends during the periods affected by the impoundment of cascade reservoirs, while the TDP increased in period 2 especially after 2007 and decreased in period 3 due to the upstream reservoirs regulation and the pollution management in the tributaries. For the monthly variation, the SSC, TP, and PP became more evenly distributed throughout the year after the impoundment of cascade reservoirs, while the TDP tended to show a pattern of “higher in dry season and lower in flood season”. The C-Q relationships indicated that the TP in the TGR was positively correlated with Q (transport limited), while negatively with Q for TDP (source limited). Generally, the TP, PP, and TDP were positively correlated with both SSC and anthropogenic P emissions, thus, the P concentrations declined with sediment retention in the reservoir but increased with time due to the increasing pollutant emissions. Correspondingly, the P dynamics were better depicted after considering the effects of sediment and pollutant emissions than using a simple C-Q relationship.

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.

Scaling Relationships between van Genuchten Model Parameters and Hydraulic Conductivity

The soil-water characteristic curve (SWCC) is indispensable for predicting the behaviour of groundwater flow in vadose zones. It remains difficult to determine SWCC, particularly for fractured rocks, under field conditions when little or no test data is available. It is generally believed that the van Genuchten model (VG) parameters α and n are related to pore size distribution (PSD) of soil and can be estimated from PSD data. In this study, correlations are proposed for predicting the VG parameters α and n from hydraulic conductivity K. The VG parameters were obtained by curve-fitting the experimental data of 993 soil samples collected from UNSODA database and published papers. The hydraulic conductivity K of the soil samples varies in six orders of magnitude from 1.30 × 10 −9 m/s to 1.22 × 10 −3 m/s. The statistical distribution of the VG parameters α and n is, respectively, examined using hypothesis test and Anderson-Darling test, showing that α follows a normal distribution after a Box-Cox transformation, and n follows a three-parameter lognormal distribution. The mean and standard deviation of α can be best estimated from K by and σ α = 1.4K 0.13, and the parameter n can be similarly estimated by and σ n = 6.6K 0.18. These correlations are also validated against site-scale test data at Yucca Mountain and published data on fractured rocks and applied to estimate the VG parameters of fractured rocks in the left bank slope at the site of Changbo Hydropower Project. The compiled data and the proposed equations are useful for estimating a narrower range of VG parameters from the more easily-obtained K values at site scale or in case of unavailable test data, and these estimates can be used as input for inverse modelling and simulation of unsaturated flow in porous or fractured media.

A Driving Behavior Distribution Fitting Method Based on Two-Stage Hybrid User Classification

Determining the distribution fitting of traditional private vehicle user driving behavior is an effective way to understand the differences between different users and provides valuable information on user travel demands. The classification of users is significant to product improvement, precision marketing, and driving recommendations. This study proposed a method which includes four aspects: (1) data collection; (2) data preprocessing; (3) data analysis—a two-stage hybrid user classification, and (4) distribution fitting method. A two-stage hybrid user classification method is used to cluster traditional vehicle users. First, the first-stage classification of the classification method extracts the daily typical time–mileage-series travel patterns (TMTP) to obtain user driving time characteristics. This first-stage classification also extracts the mean and standard deviation of the daily vehicle mileage traveled (DVMT) to express user driving demands. Next, users are divided by K-means based on the driving time characteristics and driving demands from the first stage. Finally, a three-parameter log-normal distribution is used to fit the DVMT of different user types. Comparison with traditional clustering based on the mean and standard deviation and the proportion of each vehicle’s time series in the TMTP types, this study reveals that the new methods provide significant advantages in analyzing driving behavior and high reference value for enterprises making electric vehicle driving range recommendations, car market segmentation, and policy making decisions.

A structural demand model for seismic fragility analysis based on three-parameter lognormal distribution

Seismic fragility analysis, which quantitatively describes the dynamic relationship between earthquake intensity and structural failure probability, plays an important role in the overall seismic risk assessment. An appropriate structural demand model can significantly improve the efficiency of seismic fragility analysis, and currently the most commonly used model is the lognormal model. However, recent studies have shown that this model's rationality is still in doubt. In this study, a novel structural demand model for generating analytical fragility curves in the form of a three-parameter lognormal distribution is proposed and investigated based on a large amount of nonlinear dynamic analytical data using observed Japanese ground motions with a wide range of magnitudes and epicenter distances. It shows a better fitting effect than the existing lognormal model on structural demand distribution and is expected to be more applicable to seismic fragility analysis. Through numerical fragility analysis examples on two common structural types, i.e., a steel frame structure and a reinforced concrete frame structure, the accuracy and feasibility of the proposed model are proven.

An orthogonal normal transformation of correlated non-normal random variables for structural reliability

In this paper, an efficient and explicit technique is proposed for transforming correlated non-normal random variables into independent standard normal variables based on the three-parameter (3P) lognormal distribution. In contrast with the classic Nataf transformation, the derived equivalent correlation coefficient in non-orthogonal standard normal space of the proposed transformation is expressed as an explicit formula, thereby avoiding tedious iteration algorithm or multifarious empirical formulas. Meanwhile, the applicable range of the original correlation coefficient is determined based on fundamental properties of the proposed expression of correlation distortion and the definition of correlation coefficient. The proposed transformation requires only the first three moments (i.e., mean, standard deviation, and skewness) of basic random variables, as well as their correlation matrix. Therefore, the proposed transformation can also be applied even when the joint distribution or marginal distributions of the basic random variables are unknown. Several numerical examples are presented to demonstrate the user-friendliness, efficiency, and accuracy of the proposed transformation applied in structural reliability analysis involving correlated non-normal random variables.

Experimental study on the fatigue performance of aluminum foam sandwich with 304 stainless steel face-sheet

This work focused on aluminum foam sandwich (AFS) with different foam core densities and different face-sheet thicknesses subjected to constant amplitude three-point bending cyclic loading to study its fatigue performance. The experiments were conducted out by a high frequency fatigue test machine named GPS-100. The experimental results showed that the fatigue life of AFS decreased with the increasing loading level and the structure was sensitive to cyclic loading, especially when the loading level was under 20%. S-N curves of nine groups of AFS specimens were obtained and the fatigue life of AFS followed three-parameter lognormal distribution well. AFS under low cyclic loading showed pronounced cyclic hardening and the static strength after fatigue test increased. For the same loading level, effects of foam core density and face-sheet thickness on the fatigue life of AFS structure were trade-off and for the same loading value, the fatigue life of AFS increased with aluminum foam core density or face-sheet thickness monotonously. Core shear was the main failure mode in the present study.

Statistical analysis of monthly rainfall in Central West Brazil using probability distributions

In this study, eight probability distributions such as two-parameter Weibull, three-parameter Weibull, two-parameter Rayleigh, two-parameter Gamma, three-parameter Gamma, two-parameter Lognormal, three-parameter lognormal and maximum Gumbel were applied for modelling the monthly rainfall data of the pluviometric station of Campo Grande, MS, Brazil, during the period from 1975 to 2013. The rainfall data were obtained from National Water Agency (ANA) ( https://www.ana.gov.br ). Parameters of these distributions were estimated using the maximum likelihood estimation method. Six goodness of fit tools such as chi-squared test, Kolmogorov–Smirnov test, Akaike information criterion, Bayesian information criterion, root mean square error and coefficient of determination were used to identify the best fitted probability distribution. The goodness of fit tools indicated that although no distribution provides the best fit to the rainfall data for all months, the three-parameter lognormal distribution shows generally better fit than the other distributions. The two-parameter lognormal distribution has the worst fit among the distributions.

A Bayesian hierarchical model for estimating the statistical parameters in a three-parameter log-normal distribution for monthly average streamflows

We develop a Bayesian hierarchical model (BHM) for estimating the statistical parameters for monthly average streamflows. We assume monthly average streamflow can be characterized by a three-parameter log-normal distribution (LN3). The three underlying statistical parameters are shift, shape, and location. When estimating a parameter, such as the shape, of a given month the BHM utilizes historical observations not only from the month under consideration but also from all other months. This is different from traditional statistical parameter estimation methods that only use historical observations for the month under consideration. We apply the proposed BHM for parameter estimation to eight watersheds in the United States, where historical unimpaired streamflows have been collected. We also carry out parameter estimation using traditional methods, such as the maximum likelihood estimation, the method of moments, and the L-moment method. Using cross-validation with test data log-likelihood as the measure of performance, the results show that BHM outperforms traditional estimation methods. In addition, we show that as available observation data decreases, the more the proposed method improves relative to traditional methods. Since BHM utilizes information contained in the entire data set, it is especially suited for parameter estimation where historical observations are limited. Furthermore, we conduct a comparative analysis between BHM and an autoregressive model to demonstrate the advantage of BHM.

Limit States of Structures and Global Sensitivity Analysis Based on Cramér-von Mises Distance

This article presents a stochastic computational model for the analysis of the reliability of a drawn steel bar. The whole distribution of the limit state function is studied using global sensitivity analysis based on Cramér-von Mises distance. The algorithm for estimating the sensitivity indices is based on one loop of the Latin Hypercube Sampling method in combination with numerical integration. The algorithm is effective due to the approximation of resistance using a threeparameter lognormal distribution. Goodness-of-fit tests and other comparative studies demonstrate the significant accuracy and suitability of the three-parameter lognormal distribution, which provides better results and faster response than sampling-based methods. Global sensitivity analysis is evaluated for two load cases with proven dominant effect of the long-term variation load action, which is introduced using Gumbel probability density function. The Cramér-von Mises indices are discussed in the context of other types of probability-oriented sensitivity indices whose performance has been studied earlier.

Modeling Responses and Response Times in Tests With the Hierarchical Model and the Three-Parameter Lognormal Distribution.

The hierarchical model of van der Linden is the most popular model for responses and response times in tests. It is composed of two separate submodels-one for the responses and one for the response times-that are joined at a higher level. The submodel for the response times is based on the lognormal distribution. The lognormal distribution is a skew distribution with a support from zero to infinity. Such a support is unrealistic as the solution process demands a minimal processing time that sets a response time threshold. Ignoring this response time threshold misspecifies the model and threatens the validity of model-based inferences. In this article, the response time model of van der Linden is replaced by a model that is based on the three-parameter lognormal distribution. The three-parameter lognormal distribution extends the lognormal distribution by an additional location parameter that bounds the support away from zero. Two different approaches to model fitting are proposed and evaluated with regard to parameter recovery in a simulation study. The extended model is applied to two data sets. In both data sets, the extension improves the fit of the hierarchical model.

Fatigue behavior and damage mechanism of aluminum foam sandwich with carbon-fiber face-sheets

Aluminum foam sandwich (AFS) has been used in engineering field, where cyclic loading is in most of the applications. However, limited research has been done on the fatigue behavior of AFS. In the present study, fatigue performance of AFS made of aluminum foam core and carbon-fiber face-sheet was studied experimentally by a high frequency fatigue test machine and the damage mechanisms of the structure were studied by SEM. Results indicated that the fatigue life of AFS decreased with the increasing loading level. The S-N curve of AFS obeys three-parameter lognormal distribution in most of the cases. AFS was sensitive to cyclic loading level and the load capacity after cyclic loading improved significantly when the cyclic loading level was lower. The main life of AFS was its crack initiation stage. Three categories of damage mechanisms were observed. This study is of great significance to the design and application of AFS.

Formulation of Autoconversion and Drop Spectra Shape in Shallow Cumulus Clouds

Abstract Two-moment autoconversion parameterizations as compared to accretion parameterizations exhibit significant errors suggesting that additional moments are needed to increase their accuracy. We develop a three-moment autoconversion parameterization using output from an LES model with size-resolved microphysics. Adding the third moment decreases the errors of parameterization and improves precipitation prediction. However, the errors are still significantly larger than errors of accretion rate. An analysis of the cloud drop size distributions (DSDs) in the simulated tropical convective cloud system reveals that most DSDs have a significant fraction of cloud liquid water content qc in the midsize droplet range (radii from 20 to 40 μm). Our data indicate that more than 30% of DSDs have over half of qc contained in the midsize range and about 60% of spectra have, at least, one-third of qc in this range. Even when the rain/drizzle mode is small (radar reflectivity Z < −10 dBZ), there is a significant number of spectra in which fraction of qc in the midsize range is as large as 60%. These DSDs are more complex than the frequently used gamma or lognormal distributions, which exhibit a single mode and can be defined by three microphysical moments. The need to define DSDs by more than three moments explains the large errors in the three-moment autoconversion parameterization. The limitation of three-parameter gamma or lognormal distributions should be kept in mind when applying them in precipitating shallow cumulus clouds.