Three-parameter Lognormal Distribution Research Articles

Binary regression with continuous outcomes.

Clinical research often involves continuous outcome measures, such as blood cholesterol, that are amenable to statistical techniques of analysis based on the mean, such as the t-test or multiple linear regression. Clinical interest, however, frequently focuses on the proportion of subjects who fall below or above a clinically relevant cut-off value, as a measure of the risk of disease. The customary approach to analyse such data is to dichotomize the continuous outcome measure and use statistical techniques based on binary data and the binomial distribution. In this paper, we use a parametric approach and the framework of generalized linear models to fit various regression models, including the logistic, on the basis of the original continuous outcome. We consider the Gaussian and the three-parameter log-normal distributions for the continuous outcome, assessing both precision and bias under various conditions. In simulation analyses, we find that we are unable to fit some of the samples with the 'dichotomous' approach, but we can with the 'continuous' approach, and that the latter yields estimates between 25 and 85 per cent more efficient than the former. We illustrate the method, programmed using GLIM macros, with data from clinical studies of the risk of hypoxaemia during open thoracic surgery and the risk of nocturnal hypoglycaemia among diabetic children.

Reservoir storage capacity with gamma inflows

Reservoir storage capacity determined by the double-cycling sequent peak algorithm has been investigated in previous studies for normal and log-normal inflows. Since annual streamflows have also been found to follow the gamma distribution, it is necessary to consider this situation and to cater for other possible situations. This paper considers the distribution of the storage capacity of reservoirs with gamma inflows that are either independent or first-order autoregressive. Empirical formulae for the mean and standard deviation of the non-dimensional capacity are obtained from Monte Carlo experiments for both full and partial regulation and for the entire realistic range of the lag-one autocorrelation coefficient. It is also confirmed that the standardized capacity may be represented by a three-parameter log-normal distribution.

Estimation and discrimination of alternative air pollution models

This paper provides essential background to and illustration of the procedures available within the PROFIT computer package. The package aids the practitioner in selecting an appropriate probability distribution from a set of alternatives to respresent the frequency of air pollutant concentrations; it also estimates the parameters of the distribution, predicts any desired percentile values, and calculates the minimum errors associated with that prediction. As an illustration, two- and three-parameter gamma, Weibull and lognormal distributions are applied to annual sets of air pollutant concentrations recorded at several monitoring sites in Melbourne, Australia. Six types of pollutants, namely carbon dioxide, nitrogen monoxide, nitrogen dioxide, nitrogen oxides, sulphur dioxide and β-scattering, from up to five monitoring stations are analysed using a comprehensive model selection procedure. This procedure incorporates recently developed estimation and discrimination methods, together with analysis of the effects of misspecifying the distribution and of errors in estimation of observed upper percentile values. The emphasis is placed on how to select an appropriate distribution for a given pollutant over different years and, whenever possible, over different sites. An important issue addressed here is the need for compromise between using discrimination criteria and analysing relative root mean square values in fitting percentiles. The results are useful for general purposes, such as summarizing or smoothing data, particularly the upper percentiles, as well as providing statistical information to construct hybrid models when the data contain trends.

Comparison of various flood frequency distributions using annual flood peaks data of rivers in Anatolia

The two-parameter log-normal, three-parameter log-normal, smemax, two-step-power, log-Boughton, Gumbel, general extreme-value, Pearson-3, log-Pearson-3, log-logisticm and Wakeby distributions were applied to the annual flood peaks series ⩾30 observations of 45 unregulated streams in Anatolia. The parameters of most of these distributions were estimated by methods of: (1) moments; (2) maximum-likelihood; (3) probability-weighted moments. Methods of entropy, Bobee, and mixed-moments also were employed for the log-Pearson-3 distribution. Average values and standard deviations of the absolute relative differences of the 0.01, 0.001, and 0.0001 exceedance-probability peaks computed from 3333 and 1000 synthetically generated series of 30- and 100-element lengths, respectively, from the peaks by the base distribution used in generating the synthetic data revealed that the three-parameter and two-parameter log-normal, and the Gumbel distributions predicted these extreme right-tail events better than the other distributions.

Estimation, reference ranges and goodness of fit for the three-parameter log-normal distribution.

The three-parameter log-normal distribution (3PL) is an appropriate model for many of the continuous variables encountered in medicine. It is shown how to obtain different types of estimate and approximate (sometimes conservative) confidence intervals for the parameters of the 3PL and for certain functions of them, particularly in the calculation of reference ranges of clinical measurements. A simple non-iterative estimate of the shift parameter is described. The Shapiro-Wilk test of non-normality is modified to allow it to be used for testing for departure from the 3PL. Its power is compared with that of other well-known tests. The methods are illustrated using several data sets.

Generalized probability distribution of reservoir capacity

The estimate of reservoir capacity required to meet a prescribed demand for a given model of annual inflows is a random variable. The probability distribution and parameters of this ideal reservoir capacity are determined by simulation experiments using synthetic streamflow series generated by four different models. It is shown that in all cases the reservoir capacity standardized by subtracting the mean and dividing by the standard deviation follows the three-parameter lognormal distribution with a constant lower bound. The mean and standard deviation of the reservoir capacity are also given as functions of the non-dimensional net mean inflow and length of simulation period.

Existence of Maximum Likelihood Estimates for Interval-Censored Data from Some Three-Parameter Models with a Shifted Origin

SUMMARY Given data in the form of interval-censored observations from a three-parameter distribution, practical criteria are established for the existence of maximum likelihood estimates for three kinds of parameter space. These criteria can be derived by analysing the behaviour of the log-likelihood near a type of boundary of the parameter space, called the probability contents inner boundary. The effectiveness of the criteria is discussed. In the case where the underlying distribution is a three-parameter log-normal distribution, the efficiency of the criteria is evaluated by simulation.

Statistical treatment of outlier data in epithermal gold deposit reserve estimation

A common characteristic of gold deposits is highly skewed frequency distributions. Lognormal and three-parameter lognormal distributions have worked well for Witwatersrand-type deposits. Epithermal gold deposits show evidence of multiple pulses of mineralization, which make fitting simple distribution models difficult. A new approach is proposed which consists of the following steps: (1) ordering the data in descending order. (2) Finding the cumulative coefficient of variation for each datum. Look for the quantile where there is a sudden acceleration of the cumulative C.V. Typically, the quantile will be above 0.85. (3) Fitting a lognormal model to the data above that quantile. Establish the mean above the quantile, Z H * . This is done by fitting a single or double truncated lognormal model. (4) Use variograms to establish the spatial continuity of below-quantile data (ZL) and indicator variable (1 if below quantile, 0 if above). (5) Estimate grade of blocks by (1*) (Z L * )+(1 − 1*) (Z H * ), where 1* is the kriged estimate of the indicator, and Z L * is the kriged estimate of the below quantile portion of the distribution. The method is illustrated for caldera, Carlin-type, and hot springs-type deposits. For the latter two types, slight variants of the above steps are developed.

A probability model for acid rain data

A class of probability models is proposed for making inferences about the regional distribution of lake alkalinity. This class includes the three-parameter lognormal distribution which is frequently used as a regional model for lake chemistry. The adequacy of the lognormal distribution can be assessed by testing the significance of a single parameter in the proposed model. It is shown that the three-parameter lognormal does not fit the majority of data sets from Eastern Canada. Furthermore, the pH-alkalinity relationship developed by Small and Sutton is used to yield a class of probability models representing the derived distributions for pH. It is shown that there is an improved agreement between the pH-alkalinity relationship and the observations when the water colours are less than or equal to 30 Hazen units.

A comparative evaluation of the estimators of the three-parameter lognormal distribution by Monte Carlo simulation

The parameters and quantiles of the three-parameter lognormal distribution were estimated by the method of moments, modified method of moments, maximum likelihood estimation, modified maximum likelihood estimation and entropy for Monte Carlo generated samples. The performance of these estimators was statistically evaluated, with the objective of identifying the most robust estimator from amongst them.

Percentile estimation of the three-parameter gamma and lognormal distributions: Methods of moments versus maximum likelihood

Percentile estimation of the three-parameter gamma and lognormal distributions: Methods of moments versus maximum likelihood

Bayesian analysis of theT-year events for flood data fitted by a three-parameter lognormal distribution

Abstract The three-parameter lognormal distribution is widely used in hydrology to fit maximum annual flood events. The method most commonly used to estimate the parameters and 7”-year flood event is the method of maximum likelihood. In this paper, Bayes estimates of the parameters and of the 7”-yearflood and their posterior variances of estimates are obtained by using Lindley's Bayesian approximation procedure. These estimates are then compared to estimates obtained by the method of maximum likelihood. Twelve flood series from all over Canada that are shown to be well-fitted by the three-parameter lognormal distribution are used as numerical examples. In all cases the posterior variances of the Bayes estimates of the 7”-year flood events are less than the corresponding variances of their maximum likelihood counterparts.

Small sample properties of the maximum likelihood estimators for an alternative parameterization of the three-parameter lognormal distribution

The small sample properties of the score function approximation to the maximum likelihood estimator for the three-parameter lognormal distribution using an alternative parameterization are considered. The new set of parameters is a continuous function of the usual parameters. However, unlike with the usual parameterization, the score function technique for this parameterization is extremely insensitive to starting values. Further, it is shown that whenever the sample third moment is less than zero, a local maximum to the likelihood function exists at a boundary point. For the usual parameterization, this point is unattainable. However, the alternative parameter space can be expanded to include these boundary points. This procedure results in good estimates of the expected value, variance, extreme percentiles and other parameters of the distribution even in samples where, with the typical parameterization, the estimation procedure fails to converge.

Asymptotic Properties of a Family of Minimum Quantile Distance Estimators

Abstract A family of estimators based upon M specified quantiles is defined. These procedures take as the estimate the vector that minimizes a quadratic distance measure between M sample quantiles and a parametric family of quantile functions. Under regularity conditions these estimators are consistent, asymptotically normal, and robust. For a specific quadratic form the estimator is optimal among a class of asymptotically normal estimators, and it approaches full efficiency as M approaches infinity. The asymptotic relative efficiency is computed for various sets of quantiles and various parameter values of the three-parameter lognormal distribution. The small-sample properties and robustness of the optimal M-quantile estimator for the three-parameter lognormal distribution are investigated in Monte Carlo studies.

Asymptotic Properties of a Family of Minimum Quantile Distance Estimators

Abstract A family of estimators based upon M specified quantiles is defined. These procedures take as the estimate the vector that minimizes a quadratic distance measure between M sample quantiles and a parametric family of quantile functions. Under regularity conditions these estimators are consistent, asymptotically normal, and robust. For a specific quadratic form the estimator is optimal among a class of asymptotically normal estimators, and it approaches full efficiency as M approaches infinity. The asymptotic relative efficiency is computed for various sets of quantiles and various parameter values of the three-parameter lognormal distribution. The small-sample properties and robustness of the optimal M-quantile estimator for the three-parameter lognormal distribution are investigated in Monte Carlo studies.

Comparative study of aggregations under different dependency assumptions for assessment of undiscovered recoverable oil resources in the world

The U.S. Geological Survey assessed all significant sedimentary basins in the world for undiscovered conventionally recoverable crude-oil resources. Probabilistic methodology was applied to each basin assessment to produce estimates in the form of probability distributions. Basin probability distributions were computer aggregated to produce resource estimates for the entire world. Aggregation was approximated by a three-parameter lognormal distribution by combining the first three central moments of basin distributions. For purposes of experiment and study, world aggregation was conducted under four different sets of assumptions. The four cases are (1) dependent assessments of all basins, (2) dependent assessments within continental areas, but independent assessments among continental areas, (3) dependent assessments within countries, but independent assessments among countries, and (4) independent assessments of all basins. Mean estimate remained the same in all four cases, but the width of interval estimate formed using the 95th and 5th fractiles decreased with reduced dependency in going from first to fourth case.

Modified Moment Estimation for the Three-Parameter Weibull Distribution

A modification of moment estimators in which the third moment is replaced by a function of the first order statistic is presented for parameters of the three-parameter lognormal distribution. The modified estimators enjoy advantages over traditional mom..

Fitting three-parameter lognormal models by numerical global optimization — an improved algorithm

The theoretical maximum likelihood point estimates of the parameters of the three-parameter lognormal distribution (LD) for complete, ungrouped samples are infeasible values for which the likelihood function is positively infinite. This has prompted the search for usable point estimates which are called local-maximum likelihood estimates (LMLE's) and which correspond to the largest finite local maximum of the log-likelihood function (LLF). Previously published work has shown that an effective way to obtain LMLE's is to numerically maximize the conditional LLF of the observations using sequential interior penalty and barrier function methods. These methods are robust but necessitate the numerical solution of a sequence of maximization problems. This paper describes an algorithm for computing LMLE's which involves only a single numerical maximization of a suitably transformed conditional LLF. The results of computational tests indicate that the algorithm, when implemented using computationally efficient and reliable univariate global optimization methods, is a computationally robust method for computing LMLE's of LD parameters.

Estimation of log-normal quantiles: Monte Carlo results and first-order approximations

Monte Carlo estimates of the average bias and root mean square error achieved with different quantile estimation procedures for the three-parameter log-normal distribution allow a comparison of the methods' performance. The maximum likelihood method, method of moments, and two quantile-lower-bound estimators in combination with two moments in real or in log space were considered. The maximum likelihood method and Stedinger's quantile-lower-bound methods provide more accurate estimators of the 1 and 10 percentiles of the distribution. No method was best for estimating the 99 and 99.8 percentiles. First-order approximations provide reasonable estimates of the estimators' sampling variance in most cases. All quantile estimators of the 99 or 99.8 percentiles of a flood-flow distribution were shown to yield design flood values which will be exceeded with probabilities substantially in excess of 1% and 0.2%, respectively.

Probabilistic analysis of the uncertainty in the fatigue capacity of welded joints

This paper presents the results of a study of the uncertainty in the fatigue capacity (constant amplitude fatigue life) of welded steel joints, due to uncertainties related to geometrical and material parameters. An efficient method of probabilistic fracture mechanics analysis is described and applied. A linearelastic fracture mechanics model and the Paris-Erdogan law of crack propagation were adopted. Stressintensity factors were evaluated by employing an influence function method, which is very cost-effective. The main parameters were treated as stochastic variables. Data for weld and crack geometry of the non-load carrying fillet weld cruciform joint selected as the example joint in the study, were recorded from specimens. Other data were compiled from the literature. The uncertainties associated with the basic variables were transformed into a measure of uncertainty of the fatigue capacity by employing the Monte Carlo simulation technique. The relative contributions to the uncertainty in the fatigue capacity from the various factors were also compared. The S-N data established analytically compared fairly well with test data obtained with 42 specimens. The probabilistic fracture mechanics analysis provided a sufficient sample of data to allow a test of analytical probability distributions to the fatigue life. The fit of two- and three-parameter lognormal and Weibull distributions was examined. Only the three-parameter lognormal pdf passed the chi-square test on the 5% confidence level.