One-parameter Distribution Research Articles

One-Parameter Weibull-Type Distribution, Its Relative Entropy with Respect to Weibull and a Fractional Two-Parameter Exponential Distribution

A new one-parameter distribution is presented with similar mathematical characteristics to the two parameter conventional Weibull. It has an estimator that only depends on the sample mean. The relative entropy with respect to the Weibull distribution is derived in order to examine the level of similarity between them. The performance of the new distribution is compared to the Weibull and in some cases the Gamma distribution using real data. In addition, the Exponential distribution is modified to include an extra parameter via a simple transformation using fractional mathematics. It will be shown that the modified version also exhibits Weibull characteristics for particular values of the second parameter.

On Properties and Applications of a Two-Parameter Xgamma Distribution

An existing one-parameter probability distribution can be very well generalized by adding an extra parameter in it and, in turn, the two-parameter family of distributions, thus obtained, provides added flexibility in modeling real life data. In this article, we propose and study a two-parameter generalization of xgamma distribution [1] and utilize it in modeling time-to-event data sets. Along with the different structural and distributional properties of the proposed two-parameter xgamma distribution, we concentrate in studying useful survival and reliability properties, such as hazard rate, reversed hazard rate, stress-strength reliability etc. Two methods of estimation, viz. maximum likelihood and method of moments, are been suggested for estimating unknown parameters. Distributions of order statistics, stochastic order relationships are investigated for the proposed model. A Monte-Carlo simulation study is carried out to observe the trends in estimation process. Two real life time-to-event data sets are analyzed and the proposed model is compared with some other two-parameter lifetime models in the literature.

Simple approach to the chaos-order contributions and symmetry breaking in nuclear spectra

The simple one-parameter nearest neighbor-spacing distribution (NNSD) is suggested for statistical analysis of nuclear spectra. This distribution is derived within the Wigner-Dyson approach in the linear approximation for the level repulsion density of quantum states. The obtained NNSD gives the individual information on the Wigner and Poisson contributions in agreement with that of the statistical experimental distributions of collective states in deformed nuclei. Using this NNSD, one finds that the symmetry breaking due to the fixing of projections of the angular momentum of collective states enhances a chaos as a shift of the NNSD from the Poisson to Wigner distribution behavior.

Method of Assessing the Optical Resistance of an Irradiated Surface Under Laser Ablation

A methodology is proposed for the measurement of the optical resistance of the surface of a substance irradiated by powerful pulsed laser radiation. The measurement methodology is based on the use of the one-parameter Weibull-Gnedenko statistical distribution. An installation is created by which this method is implemented. The optical resistance of multilayered nanodimensional coverings is measured using this methodology.

How predictions of cosmological models depend on Hubble parameter data sets

We explore recent estimations of the Hubble parameter $H$ depending on redshift $z$, which include 31 $H(z)$ data points measured from differential ages of galaxies and 26 data points, obtained with other methods. We describe these data together with Union 2.1 observations of Type Ia supernovae and observed parameters of baryon acoustic oscillations with 2 cosmological models: the standard cold dark matter model with the $\Lambda$ term ($\Lambda$CDM) and the model with generalized Chaplygin gas (GCG). For these models with different sets of $H(z)$ data we calculate two-parameter and one-parameter distributions of $\chi^2$ functions for all observed effects, estimate optimal values of model parameters and their $1\sigma$ errors. For both considered models the results appeared to be strongly depending on a choice of Hubble parameter data sets if we use all 57 $H(z)$ data points or only 31 data points from differential ages. This strong dependence can be explained in connection with 4 $H(z)$ data points with high redshifts $z>2$.

On the Bell distribution and its associated regression model for count data

• We define the one-parameter discrete Bell distribution, which has a simple form for its probability mass function. • It is shown to be a particular solution of a multiple Poisson process. • It is useful for modeling count data with greater variability than the Poisson distribution allows. • We introduce a Bell regression model with statistical inference procedures, diagnostic tools, and computer implementation. • An application to real data suggests that the new model is useful in practice. In this paper we define and study the one-parameter discrete Bell distribution, which has a very simple form for its probability mass function. In particular, we show that this discrete distribution is a particular solution of a multiple Poisson process. By considering the distribution studied in this article, we introduce a new regression model where the response variable is a count. Further, residuals are also proposed for the new count regression model. An empirical application is considered to show the usefulness of the Bell regression model in practice.

The Topp–Leone odd log-logistic family of distributions

ABSTRACTWe introduce a new class of continuous distributions named the Topp–Leone odd log-logistic family, which extends the one-parameter distribution pioneered by Topp and Leone [A family of J-shaped frequency functions. J Amer Statist Assoc. 1955;50:209–219]. We study some of its mathematical properties and describe two special cases. Further, we propose a regression model based on the new Topp–Leone odd log-logistic Weibull distribution. The usefulness and flexibility of the proposed family are illustrated by means of three real data sets.

A Three-Parameter Weighted Lindley Distribution and its Applications to Model Survival Time

Abstract In this paper a three-parameter weighted Lindley distribution, including Lindley distribution introduced by Lindley (1958), a two-parameter gamma distribution, a two-parameter weighted Lindley distribution introduced by Ghitany et al. (2011) and exponential distribution as special cases, has been suggested for modelling lifetime data from engineering and biomedical sciences. The structural properties of the distribution including moments, coefficient of variation, skewness, kurtosis and index of dispersion have been derived and discussed. The reliability properties, including hazard rate function and mean residual life function, have been discussed. The estimation of its parameters has been discussed using the maximum likelihood method and the applications of the distribution have been explained through some survival time data of a group of patients suffering from head and neck cancer, and the fit has been compared with a one-parameter Lindley distribution and a two-parameter weighted Lindley distribution.

A THREE-PARAMETER WEIGHTED LINDLEY DISTRIBUTION AND ITS APPLICATIONS TO MODEL SURVIVAL TIME

In this paper a three-parameter weighted Lindley distribution, including Lindley distribution introduced by Lindley (1958), a two-parameter gamma distribution, a two-parameter weighted Lindley distribution introduced by Ghitany et al. (2011) and exponential distribution as special cases, has been suggested for modelling lifetime data from engineering and biomedical sciences. The structural properties of the distribution including moments, coefficient of variation, skewness, kurtosis and index of dispersion have been derived and discussed. The reliability properties, including hazard rate function and mean residual life function, have been discussed. The estimation of its parameters has been discussed using the maximum likelihood method and the applications of the distribution have been explained through some survival time data of a group of patients suffering from head and neck cancer, and the fit has been compared with a one-parameter Lindley distribution and a two-parameter weighted Lindley distribution.

The Behavior of Chi-Square Statistics Using Unbiased Estimators in the Case of One-Parameter Distribution from Exponential Family

We propose two new variants of chi-square-type statistics for testing the hypothesis on the shape of a distribution belonging to the one-parameter exponential family. Each variant assumes the use of unbiased estimators of the probabilities for the random variable to hit the atoms (intervals) of decomposition of its range. We establish the convergence of each statistics’ distribution to the chi-square distribution with (r − 1) degrees of freedom, where r is the number of decomposition atoms.

An absolutely continuous bivariate Topp–Leone distribution: A useful model on a bounded domain

ABSTRACTWe introduce an absolutely continuous bivariate generalization of the Topp–Leone distribution, which is a special member of the proportional reversed hazard family using a one-parameter bivariate exchangeable distribution. We show that a copula approach could also be used in defining the bivariate Topp–Leone distribution. The marginal distributions of the new bivariate distribution have also Topp–Leone distributions. We study its distributional and dependence properties. We estimate the parameters by maximum-likelihood procedure, perform a simulation study on the estimators, and apply them to a real data set. Furthermore, we give a way of generating bivariate distributions using the proposed distribution.

Sujatha Distribution and its Applications

AbstractIn this paper a new one-parameter lifetime distribution named “Sujatha Distribution” with an increasing hazard rate for modelling lifetime data has been suggested. Its first four moments about origin and moments about mean have been obtained and expressions for coefficient of variation, skewness, kurtosis and index of dispersion have been given. Various mathematical and statistical properties of the proposed distribution including its hazard rate function, mean residual life function, stochastic ordering, mean deviations, Bonferroni and Lorenz curves, and stress-strength reliability have been discussed. Estimation of its parameter has been discussed using the method of maximum likelihood and the method of moments. The applications and goodness of fit of the distribution have been discussed with three real lifetime data sets and the fit has been compared with one-parameter lifetime distributions including Akash, Shanker, Lindley and exponential distributions.

Sujatha distribution and its applications

In this paper a new one-parameter lifetime distribution named “Sujatha Distribution” with an increasing hazard rate for modelling lifetime data has been suggested. Its first four moments about origin and moments about mean have been obtained and expressions for coefficient of variation, skewness, kurtosis and index of dispersion have been given. Various mathematical and statistical properties of the proposed distribution including its hazard rate function, mean residual life function, stochastic ordering, mean deviations, Bonferroni and Lorenz curves, and stress-strength reliability have been discussed. Estimation of its parameter has been discussed using the method of maximum likelihood and the method of moments. The applications and goodness of fit of the distribution have been discussed with three real lifetime data sets and the fit has been compared with one-parameter lifetime distributions including Akash, Shanker, Lindley and exponential distributions.

Role of the pressure gradient in flows controlled by a near-wall body force

Steady two-dimensional flows of a viscous incompressible fluid over a plane boundary in the presence of a near-wall body force are studied. The calculations are performed for a very simple one-parameter force distribution simulating the action of a single plasma actuator. The important role played by the pressure gradient in the flows controlled by body forces is shown.

Topp–Leone generated family of distributions: Properties and applications

ABSTRACTWe introduce a new family of distributions based on a one-parameter distribution exhibiting bathtub-shaped hazard rates. We study the mathematical properties of the family and estimate its parameters by the method of maximum likelihood. Finally, the usefulness of the family is illustrated using a real dataset.

The probabilistic distribution of metal whisker lengths

Significant reliability concerns in multiple industries are related to metal whiskers, which are random high aspect ratio filaments growing on metal surfaces and causing shorts in electronic packages. We derive a closed form expression for the probabilistic distribution of metal whisker lengths. Our consideration is based on the electrostatic theory of metal whiskers, according to which whisker growth is interrupted when its tip enters a random local “dead region” of a weak electric field. Here, we use the approximation neglecting the possibility of thermally activated escapes from the “dead regions,” which is later justified. We predict a one-parameter distribution with a peak at a length that depends on the metal surface charge density and surface tension. In the intermediate range, it fits well the log-normal distribution used in the experimental studies, although it decays more rapidly in the range of very long whiskers. In addition, our theory quantitatively explains how the typical whisker concentration is much lower than that of surface grains. Finally, it predicts the stop-and-go phenomenon for some of the whiskers growth.

New Results for Computing Blaker’s Exact Confidence Interval for One Parameter Discrete Distributions

The authors state new general results for computing Blaker’s exact confidence interval limits for usual one-parameter discrete distributions. Specific results for implementing an accurate and fast algorithm are made explicit for the binomial, negative binomial, Poisson and hypergeometric model.

Analyzing bivariate ordinal data with CUB margins

Statistical modelling for ordinal data has received a considerable attention in the literature, and a consolidated theory relying on Generalized Linear Model approach has been developed. In this article, we present an innovative technique for modelling bivariate ordinal data. In particular, we consider the method introduced by Plackett for constructing a one-parameter bivariate distribution from given margins, and we apply it in order to represent correlated ordinal variables which individually follows a CUB model. This is a univariate mixture distribution defined by the convex Combination of a Uniform and a shifted Binomial distribution whose parameters may be related to rater's covariates. The article shows how the bivariate distribution can be defined and how its characterizing parameter, which describes the association between the component random variables, can be related to the subject's covariates. The proposed approach is applied to the study of two key drivers of extra virgin olive oil consumption in Italy. The technique allows a representation of the data whose meaning can be easily interpreted providing useful information for management support.

Analytical design of refractive optical elements generating one-parameter intensity distributions.

We calculate a refractive surface to generate a one-parameter intensity distribution described by a vector function of one argument from an incident beam with a plane wavefront. The calculation of the refractive surface is reduced to the solution of two explicit ordinary differential equations of the first order. An approach to solving the problem of the existence of solutions for various one-parameter intensity distributions is proposed. Optical elements that form a variety of intensity distributions (a line, a circular arc, and a double-arc) are designed. The results of numerical simulations of the designed optical elements demonstrate the high quality of the generated line-shaped patterns.

Stochastic frontiers with a Rayleigh distribution

We introduce a stochastic frontier model with a one-parameter distribution known as the Rayleigh distribution which has a non-zero mode and yet it is easy to estimate and use. We show how this model can be estimated using various estimation methods. The Rayleigh model with environmental variables and time-varying features is also considered. It is also tested against exponential and half-normal models using two real data sets.