Continuous Univariate Distributions Research Articles

A Robust Measure of Skewness

The asymmetry of a univariate continuous distribution is commonly measured by the classical skewness coefficient. Because this estimator is based on the first three moments of the dataset, it is strongly affected by the presence of one or more outliers. This article investigates the medcouple, a robust alternative to the classical skewness coefficient. We show that it has a 25% breakdown value and a bounded influence function. We present a fast algorithm for its computation, and investigate its finite-sample behavior through simulated and real datasets.

Robust measures of tail weight

The kurtosis coefficient is often regarded as a measure of the tail heaviness of a distribution relative to that of the normal distribution. However, it also measures the peakedness of a distribution, hence there is no agreement on what kurtosis really estimates. Another disadvantage of the kurtosis is that its interpretation and consequently its use is restricted to symmetric distributions. Moreover, the kurtosis coefficient is very sensitive to outliers in the data. To overcome these problems, several measures of left and right tail weight for univariate continuous distributions are proposed. They can be applied to symmetric as well as asymmetric distributions that do not need to have finite moments. Their interpretation is clear and they are robust against outlying values. The breakdown value and the influence functions of these measures and the resulting asymptotic variances are discussed and used to construct goodness-of-fit tests. Simulated as well as real data are employed for further comparison of the proposed measures.

Moments of truncated continuous univariate distributions

Truncated moment expressions (TMEs), defined as moment equations for truncated or incomplete distributions, are derived for several continuous univariate distributions commonly applied to hydrologic problems, including normal, lognormal, Pearson type III, log Pearson type III, and extreme value (Weibull and Gumbel) distributions. Solutions for gamma, tanks-in-series, and exponential distributions result as special cases. For most of the distributions considered here, closed form TMEs are presented for Nth order moments for the general case of double truncation (both upper and lower bounds). For the normal and Gumbel distributions, TMEs are presented only for moments of order N={0,1,2,3} and {0,1}, respectively. The derived TMEs are used to evaluate the effect of truncation on measured moments. The relative error between the first four truncated and complete moments is calculated as a function of both upper and lower truncation point.

Formulas for Rényi information and related measures for univariate distributions

In a recent paper Song [J. Stat. Plan. Infer. 93 (2001) 51] considered Rényi information of order λ and established its connection to the loglikelihood. From this relation an intrinsic distribution measure was proposed and analytic expressions of this measure and Rényi information were derived for some standard continuous distributions. In this paper, we derive analytical formulas for Rényi and Shannon entropies, as well as, for Song’s measure for 26 flexible families of univariate continuous distributions. We believe that the results presented here will serve as an important reference for scientists and engineers in many areas.

Short universal generators via generalized ratio-of-uniforms method

We use inequalities to design short universal algorithms that can be used to generate random variates from large classes of univariate continuous or discrete distributions (including all log-concave distributions). The expected time is uniformly bounded over all these distributions for a particular generator. The algorithms can be implemented in a few lines of high level language code.

Fixed Width Confidence Interval of P ( X < Y ) in Partial Sequential Sampling Scheme

The article is related to nonparametric fixed-width confidence interval estimation of the parameter , where F and G are two unknown univariate continuous distribution functions, by adopting a partial sequential sampling scheme. Different asymptotic results associated with the proposed procedures are formulated and examined.

Characterizations of univariate continuous distributions. II

This is a follow up to our previous work characterizing univariate continuous distributions based on a simple relationship between two truncated moments.

A note on determining the number of outliers in an exponential sample by least squares procedure

In this paper, we suggest a least squares procedure for the determination of the number of upper outliers in an exponential sample by minimizing sample mean squared error. Moreover, the method can reduce the masking or “swamping” effects. In addition, we have also found that the least squares procedure is easy and simple to compute than test procedure Tk, suggested by Zhang (1998) for determining the number of upper outliers, since Zhang (1998) need to use the complicated null distribution of Tk. Moreover, we give three practical examples and a simulated example to illustrate the procedures. Further, simulation studies are given to show the advantages of the proposed method. Finally, the proposed least squares procedure can also determine the number of upper outliers in other continuous univariate distributions (for example, Pareto, Gumbel, Weibull, etc.).

Consistency of Generalized Maximum Spacing Estimates

General methods for the estimation of distributions can be derived from approximations of certain information measures. For example, both the maximum likelihood (ML) method and the maximum spacing (MSP) method can be obtained from approximations of the Kullback–Leibler information. The ideas behind the MSP method, whereby an estimation method for continuous univariate distributions is obtained from an approximation based on spacings of an information measure, were used by Ranneby &amp; Ekstrom (1997) (using simple spacings) and Ekstrom (1997b) (using high order spacings) to obtain a class of methods, called generalized maximum spacing (GMSP) methods. In the present paper, GMSP methods will be shown to give consistent estimates under general conditions, comparable to those of Bahadur (1971) for the ML method, and those of Shao &amp; Hahn (1999) for the MSP method. In particular, it will be proved that GMSP methods give consistent estimates in any family of distributions with unimodal densities, without any further conditions on the distributions.

A simple universal generator for continuous and discrete univariate T-concave distributions

We use inequalities to design short universal algorithms that can be used to generate random variates from large classes of univariate continuous or discrete distributions (including all log-concave distributions). The expected time is uniformly bounded over all these distributions. The algorithms can be implemented in a few lines of high-level language code. In opposition to other black-box algorithms hardly any setup step is required, and thus it is superior in the changing-parameter case.

CHARACTERIZATIONS OF NIVARIATE CONTINUOUS DISTRIBUTIONS

Most of the well-known univariate continuous distributions are characterized based on a simple relationship between two truncated moments.

Maximum product of spacings method: A unified formulation with illustration of strong consistency

A simple unified formulation for generating an estimator of a univariate distribution (not necessarily continuous) is proposed. It is a variant of the maximum likelihood method and also a generalization of the maximum product of spacings (MPS) method for estimating continuous univariate distributions. The general formulation is then applied to monotone hazard rate families which arise frequently in applications. It is shown that any asymptotic MPS estimator for any family of distributions with monotone hazard rate is always strongly consistent (whether the setting is parametric or not). The MPS estimator of a distribution function with a monotone hazard rate can be derived explicitly and is asymptotically minimax for the Kolmogorov-Smirnov type loss functions.

Discrete Approximations to Continuous Univariate Distributions—an Alternative to Simulation

Summary A method to replace a continuous univariate distribution with a discrete distribution that takes MN different values is analysed. Both distributions share the same rth moments for r=0, . . ., 2N−1 and their corresonding distribution functions coincide at least at M+1 points. Several statistical and engineering examples are considered in which the discrete approximation may be used to avoid a simulation study that would be much more demanding computationally.

Sequential-type nonparametric test using Mann-Whitney statistics

The paper provides a nonparametric test for the identity of two continuous univariate distribution functions when observations are drawn in pairs from the populations, by adopting a sampling scheme which, using Mann-Whitney scores, generalizes the existing inverse binomial sampling technique. Some exact performance characteristics of the proposed test are formulated and compared numerically with existing competitors of the proposed test. The applicability of the proposed test is illustrated using real-life data.

On the consistency of the maximum spacing method

The main result of this paper is a consistency theorem for the maximum spacing method, a general method of estimating parameters in continuous univariate distributions, introduced by Cheng and Amin ( J. Roy. Statist. Soc. Ser. A 45 (1983) 394–403) and independently by Ranneby ( Scand. J. Statist. 11 (1984) 93–112). This main result generalizes a theorem of Ranneby ( Scand. J. Statist. 11 (1984) 93–112). Also, some examples are given, which shows that this estimation method works also in cases where the maximum likelihood method breaks down.

CLASSICAL PROBABILITY APPLICATION: A DISCRETE DYNAMICAL SYSTEM DEFINED BY A MARKOV CHAIN

ABSTRACT Each student at the United States Military Academy takes a four course sequence in core mathematics. As the final core mathematics capstone course, probability and statistics (P&S) not only includes the standard material of such a course, but also asks the students to reflect upon previous core mathematics topics. Since our probability course covers continuous univariate and multivariate distributions, calculus (core courses 2 and 3) already assumes a significant role. It was our desire that the students utilize their modeling skills gained in the first core course, Discrete Dynamical Systems (DDS). This led to a P&S project that requires DDS to find the steady-state probabilities of a Markov Chain. These steady-state probabilities are then used in a larger Bayes' Theorem and conditional probability scenario.

On Uniqueness and Symmetry of Self-Consistent Points of Univariate Continuous Distributions

On Uniqueness and Symmetry of Self-Consistent Points of Univariate Continuous Distributions

Book reviews

Journal Article Book reviews Get access Continuous univariate distributions (Volume 1, 2nd edition) Norman L. Johnson, Samuel Kotz and N. Balakrishnan John Wiley & Sons, New York ( 1994), 756 pp., ISBN 0-471-58495-9 (v.1), £66 GWYN JONES GWYN JONES London Guildhall University Search for other works by this author on: Oxford Academic Google Scholar IMA Journal of Management Mathematics, Volume 8, Issue 1, January 1997, Pages 96–97, https://doi.org/10.1093/imaman/8.1.96 Published: 01 January 1997

Three Information–Theoretical Methods to Estimate a Random Variable

Three information–theoretical methods to estimate a continuous univariate distribution are proposed for estimation when the distribution type is uncertain, when data are scarce, or when extremes are important. The first is a new version of Jaynes» MaxEnt Method. The second, minimizing Shannon's information measure, yields a minimally informative estimate. The third analysis produces a pair of distributions that together minimize the relative entropy (=cross-entropy) satisfying the principle of least information. Thus, one distribution (p) provides minimal information among all chosen candidates about these events, while the other (q) minimizes the information among all distributions that satisfies the sample rule, which is an essential constraint. Modelpis also identical to the ««maximum product of spacings»» estimate. Distributionqis determined frompby simple algebra. The principle of least information yields a unique solution (p,q) when other methods fail. The algorithm is the same for all types of distributions; the estimation process introduces a minimum of information, approaching objectivity, and the solution is invariant under monotonic variable transformations. All three methods are computationally simple but involve optimization. There are also several approximate information–theoretical methods that retain some of the advantages cited and are computationally simpler.

Discussion of the Paper by Lee and Nelder

Discussion of the Paper by Lee and Nelder