Four-parameter Distribution Research Articles

A new estimate of skewness with mean-squared error smaller than that of the sample skewness

Recently, a new procedure for distribution fitting, based on matching of the first two moments, partial and complete, was introduced (Shore, 1995). When the sampling skewness of the fitted distribution is compared to the sample skewness, and both are regarded as estimates of the skewness of the underlying distribution, the mean-squared-error of the former is appreciably lower than that of the latter. In this paper we present some simulation results to support this claim and demonstrate its magnitude. An alternative two-moment distributional fitting procedure, based on a new family of four-parameter distributions, is also introduced and studied. Since three-moment distribution fitting is very common practice in simulation studies, these results may have important implications for the current state-of-the-art of simulation

Fitting a distribution by the first two moments (partial and complete)

Given a sample of observations from an unknown population, a common practice to derive distributional representation for the given data is to fit a four-parameter distribution via matching of the first four moments. However, third and fourth sample moments are notorious for their large standard errors, which require sample sizes that in a typical industrial setting are rarely available. In this paper we propose an alternative approach that employs only the first two moments (partial and complete) to fit a certain four-parameter distribution to the given sample data. The fitted distribution is a mixture of two components, where each is a linear transformation of a symmetrically distributed standardized variable. Separate transformations are used for each half of the distribution. Estimation of the parameters is carried out by matching of the mean, the variance, and the first and second partial moments. This fitting procedure is shown to be approximately a least squares solution, that provides good-estimates for the fractiles of the approximated distribution. Moreover, the linear transformations may provide mathematically manageable solutions to stochastic optimization problems (like inventory problems) that would otherwise require complex solution procedures. Some numerical examples and a simulation study attest to the effectiveness of the new approach when sample data are scarce.

A generalization of the binomial distribution

This paper presents a new departure in the generalization of the binomial distribution by adopting the assumption that the underlying Bernoulli trials take on the values α or β where α < β, rather than the conventional values 0 or 1. The adoption of this more general assumption renders the binomial distribution a four-parameter distribution of the form B(n,p,α,β), and requires the generalization of Romanovsky's (1923) reduction formula for central moments. This paper assesses the usefulness of B(n,p,α,β), and its reduction formula, in the numerical analysis of two problems of interest to decision theorists.

The four-parameter kappa distribution

Many common probability distributions, including some that have attracted recent interest for flood-frequency analysis, may be regarded as special cases of a four-parameter distribution that generalizes the three-parameter kappa distribution of P.W. Mielke. This four-parameter kappa distribution can be fitted to experimental data or used as a source of artificial data in simulation studies. This paper describes some of the properties of the four-parameter kappa distribution, and gives an example in which it is applied to modeling the distribution of annual maximum precipitation data.

Optimum utilization of the channel capacity of a satellite link in the presence of amplitude scintillations and rain attenuation

By considering the global fading process on the link caused by rain attenuation and amplitude scintillations, particularly at K/sub a/ band, it is possible to derive a long-term statistical model of the satellite channel capacity. The four-parameter distribution, which combines amplitude scintillations and rain fade within an up/down link system, is presented. Also presented are the degradation (and improvement) of bit error rate (BER) in the presence of amplitude scintillations, thus complementing the flat fade effect due to rain only. By implementation of adaptive communication systems, a more efficient channel capacity utilization is possible. The concepts and the use of novel analytical expressions combining a log-normal model of rain fade with a Moulsley-Vilar distribution for scintillations are illustrated. These are then applied to a very-small-aperture terminal (VSAT) example of a 29/19-GHz digital communications link through the Olympus satellite using M-ary phase shift keying (PSK) modulation schemes. >

Germination of Aristida armata Under Constant and Alternating Temperatures and Its Analysis With the Cumulative Weibull Distribution as a Model

Germination of Aristida armata was compared at different temperatures on a thermogradient plate. Temperatures ranged from 10°C to 50°C with day/night differentials of 0, 5, 10 and 15°C. Alternating temperatures improved overall germination, particularly at the extremes of temperature. Average temperatures of 35°C and higher were fatal to many seeds. Day temperatures of 17.5°C and lower inhibited germination but did not prevent subsequent germination under warmer conditions. There was little variation in the rate of germination with incubation under constant temperatures. Under alternating temperatures, maximum germination occurred at lower temperatures than those under which germination rate was greatest. A four-parameter cumulative Weibull distribution was used to summarise cumulative germination. The distribution has the general form: Y = M(1-exp[- {k(t - I)}c]), where Y is the total germination at time t, M is the final total germination, k is germination rate, I is the interval between the start of incubation and the start of germination, and c is a shape parameter. In nearly all cases, the fitted function had a coefficient of determination greater than 0.98. The Weibull distribution allows reconstruction of the original germination data with minimal distortion and its use is recommended for both the analysis and modelling of germination responses.

A heuristic model of the Tromp (distribution) curve

A five-parameter mathematical model for the Tromp curve has been derived by using heuristic arguments. It can be transformed into a four-parameter self-similar distribution or partition function for the generalized Tromp curves in a reduced specific gravity. The generalized model can be linearized for graphical curve fitting of data. For more accurate analysis, a nonlinear optimization technique is required. However, the structure of the proposed model facilitates the parameter search in four- or five-dimensional space. The model exhibits close fit to a variety of Tromp curve shapes based on plant data for various coal-cleaning equipment and permits explicit expressions for error area and probable error in terms of its parameters.

The production rate of a two-stage system with stochastic processing times

SUMMARY This paper extends the earlier papers of Rao (1975,1976) and Knott (1976) on the output rate of a two-stage production system with stochastic processing times and no buffer storage between the stages. Justifications for using different probability distributions to represent stochastic task times are examined, and convenient procedures are presented for computing the system's expected output rate for a large variety of processing times distributions, including some very versatile four-parameter probability distributions.

A Class of Variate Transformations Causing Unbounded Likelihood

Abstract Hill (1963) proved the existence of paths in the parameter space of the three-parameter lognormal distribution along which the likelihood for any given sample tends to infinity. Lambert (1970) considered a similar phenomenon in the case of the four-parameter lognormal distribution. It is shown that this anomaly is caused by the nature of the variate transformations used and not by the normal distribution itself.

A Class of Variate Transformations Causing Unbounded Likelihood

Hill (1963) proved the existence of paths in the parameter space of the three-parameter lognormal distribution along which the likelihood for any given sample tends to infinity. Lambert (1970) considered a similar phenomenon in the case of the four-parameter lognormal distribution. It is shown that this anomaly is caused by the nature of the variate transformations used and not by the normal distribution itself.

The Use of Versatile Distribution Families in Some Stochastic Inventory Calculations

Previous works on stochastic inventory problems have often assumed that an item's lead time demand follows a distribution such as the normal, the γ or the Weibull. First, this paper argues that these convenient distributions may be overly restrictive and unrealistic, and points out the versatility and realism of using four-parameter distributions of the Pearson's and the Schmeiser-Deutsch's systems. Second, using these four-parameter distributions, this paper presents practical `manual methods for computing the stock-out probability, reorder level and expected lost sales of an inventory item and for solving the lot-size reorder-point model. Some of these methods are actually simpler than the ones developed previously for the more restrictive distributions.

Methods for modelling and generating probabilistic components in digital computer simulation when the standard distributions are not adequate

Methods of modelling probabilistic components which are not adequately represented by the standard continuous distributions (such as normal, gamma, and Weibull) are surveyed. The methods are categorized as systems of distributions, approximations to the cumulative distribution function, and four-parameter distributions. Emphasis is on generality, determination of appropriate parameter values, and random variate generation.

Applications of the four-parameter distribution to electrostatic charge and particle size distribution

A four-parameter model has been developed to describe the size and charge distribution of aerosols. In the limiting case the distribution reduces to a log-normal distribution for the particle size and a normal distribution for the particle charge. A procedure for systematically determining the distribution is presented. Finally, applications of the distribution to model the optical diameter of asbestos fibers, the bimodal charge distribution of NaCl aerosols, and atmospheric aerosol size distributions are presented.

The use of interior penalty functions to overcome lognormal distribution parameter estimation anomalies

The maximum likelihood statistical modelling problems for the three-and four-parameter lognormal distributions possess the unusual future that the theoretical maximum likelihood parameter estimates are inadmissible values for which the likelihood function is postively infinite. Accordingly, “local-maximum” likelihood parameter estimates corresponding to the primary relative maximum of the likelihood function must be derived. Although these estimates often possess many of the properties associated with ordinary maximum likelihood estimates, their numerical computation is not altogether straightforward since solutions corresponding to the global maximum of the likelihood function must be avoided. A procedure for overcoming these estimation anomalies is presented. The numerical examples also presented and discussed indicate that the procedure, which employs interior penalty function techniques, is robust. Extensive practical experience with the procedure supports this conclusion.

Estimation of Parameters in Truncated Pearson Frequency Distributions

A method based on higher moments is presented in this paper by which the type of a univariate Pearson frequency distribution (population) can be determined and its parameters estimated from truncated samples with known points of truncation and an unknown number of missing observations. Estimating equations applicable to the four-parameter distributions involve the first six moments of a doubly truncated sample or the first five moments of a singly truncated sample. When the number of parameters to be estimated is reduced, there is a corresponding reduction in the order of the sample moments required. A sample is described as singly or doubly truncated according to whether one or both "tails" are missing. Estimates obtained by the method of this paper enjoy the property of being consistent and they are relatively simple to calculate in practice. They should be satisfactory for (a) rough estimation, (b) graduation, and (c) first approximations on which to base iterations to maximum likelihood estimates. Previous investigations of truncated univariate distributions include studies of truncated normal distributions by Pearson and Lee [1], [2], Fisher [3], Stevens [4], Cochran [5], Ipsen [6], Hald [7], and this writer [8], [9]. In addition, the truncated binomial distribution has been studied by Finney [10], and the truncated Type III distribution by this writer [11].