Classes of Ordinary Differential Equations Obtained for the Probability Functions of 3-Parameter Weibull Distribution

In this paper, the differential calculus was used to obtain some classes of ordinary differential equations (ODE) for the probability density function, quantile function, survival function, inverse survival function, hazard function and reversed hazard function of the 3-parameter Weibull distribution. The stated necessary conditions required for the existence of the ODEs are consistent with the various parameters that defined the distribution. Solutions of these ODEs by using numerous available methods are a new ways of understanding the nature of the probability functions that characterize the distribution

Ordinary Differential Equations of the Probability Functions of the Weibull Distribution and their Application in Ecology

Weibull distribution has been applied to many areas in ecological studies and engineering. Application of the Weibull and other probability distributions in ecology are mainly in fitting ecological data which is very vital in revealing latent characteristics of the object of study. The use of the ordinary differential equations (ODE) in fitting has not been studied in ecological studies. Ordinary differential calculus was used to obtain the homogenous ODE of the probability density function (PDF), quantile function (QF), survival function (SF), inverse survival function (ISF), hazard function (HF) and reversed hazard function (RHF) whose solutions are their respective functions of the Weibull distribution. Different classes of ODEs were obtained. The novelty of this proposed method is applied to radiation data

Some Extended Classes of Distributions: Characterizations and Properties

Based on a simple relationship between two truncated moments and certain functions of the th order statistic, we characterize some extended classes of distributions recently proposed in the statistical literature, videlicet Beta-G, Gamma-G, Kumaraswamy-G and McDonald-G. Several properties of these extended classes and some special cases are discussed. We compare these classes in terms of goodness-of-fit criteria using some baseline distributions by means of two real data sets

ENHANCEMENT AND COMPARISON OF ANT COLONY OPTIMIZATION FOR SOFTWARE RELIABILITY MODELS

In Common parlance, the traditional software reliability estimation methods often rely on assumptions like statistical distributions that are often dubious and unrealistic. The ability to predict the number of faults during development phase and a proper testing process helps in specifying timely release of software and efficient management of project resources. In the Present Study Enhancement and Comparison of Ant Colony Optimization Methods for Software Reliability Models are studied and the estimation accuracy was calculated. The Enhanced method shows significant advantages in finding the goodness of fit for software reliability model such as finite and infinite failure Poisson model and binomial models

The hjorth's IDB generator of distributions: properties, characterizations, regression modeling and applications

We introduce a new flexible class of continuous distributions via the Hjorth’s IDB model. We provide some mathematical prop-erties of the new family. Characterizations based on two truncated moments, conditional expectation as well as in terms of thehazard function are presented. The maximum likelihood method is used for estimating the model parameters. We assess the per-formance of the maximum likelihood estimators in terms of biases and mean squared errors by means of the simulation study.A new regression model as well as residual analysis are presented. Finally, the usefulness of the family is illustrated by means offour real data sets. The new model provides consistently better fits than other competitive models for these data sets

On the Burr XII-Power Cauchy Distribution: Properties and Applications

We propose a new four-parameter lifetime model with flexible hazard rate called the Burr XII Power Cauchy (BXII-PC) distribution. We derive the BXII-PC distribution via (i) the T-X family technique and (ii) nexus between the exponential and gamma variables. The new proposed distribution is flexible as it has famous sub-models such as Burr XII-half Cauchy, Lomax-power Cauchy, Lomax-half Cauchy, Log-logistic-power Cauchy, log-logistic-half Cauchy. The failure rate function for the BXII-PC distribution is flexible as it can accommodate various shapes such as the modified bathtub, inverted bathtub, increasing, decreasing; increasing-decreasing and decreasing-increasing-decreasing. Its density function can take shapes such as exponential, J, reverse-J, left-skewed, right-skewed and symmetrical. To illustrate the importance of the BXII-PC distribution, we establish various mathematical properties such as random number generator, moments, inequality measures, reliability measures and characterization. Six estimation methods are used to estimate the unknown parameters of the proposed distribution. We perform a simulation study on the basis of the graphical results to demonstrate the performance of the maximum likelihood, maximum product spacings, least squares, weighted least squares, Cramer-von Mises and Anderson-Darling estimators of the parameters of the BXII-PC distribution. We consider an application to a real data set to prove empirically the potentiality of the proposed model

A New Parametric Lifetime Distribution with Modified Chi-square Type Test for Right Censored Validation, Characterizations and Different Estimation Methods

A new three-parameter extension of the generalized Nadarajah-Haghighi model is introduced and studied. Some of its statistical properties are derived. Characterization results are presented. The failure rate can be increasing , decreasing , bathtub , upside-down , upside-down-constant , increasing-constant or constant . Different non-Bayesian estimation methods under uncensored scheme are considered. Numerical simulations are performed for comparing the estimation methods using different sample sizes. The censored Barzilai-Borwein algorithm is employed via a simulation study. Using the approach of the Bagdonavicius-Nikulin chi-square goodness-of-fit test for validation under the right censored data, we propose a modified chi-square goodness-of-fit test for the new model. Based on the maximum likelihood estimators on initial data, the modified Bagdonavicius-Nikulin chi-square goodness-of-fit test recovers the loss in information. The modified Bagdonavicius-Nikulin test for validation under the right censored data is applied to four real and right censored data sets. The new model is compared with many other competitive models by means of a real data set

On the Generalized Log Burr III Distribution: Development, Properties, Characterizations and Applications

In this paper, we present a generalized log Burr III (GLBIII) distribution developed on the basis of a generalized log Pearson differential equation (GLPE). The density function of the GLBIII is exponential, arc, J, reverse-J, bimodal, left-skewed, right- skewed and symmetrical shaped. The hazard rate function of GLBIII distribution has various shapes such as constant, increasing, decreasing, increasing-decreasing, upside- down bathtub and modified bathtub. Descriptive measures such as quantile function, sub- models, ordinary moments, moments of order statistics, incomplete moments, reliability and uncertainty measures are theoretically established. The GLBIII distribution is characterized via different techniques. Parameters of the GLBIII distribution are estimated using maximum likelihood method. A simulation study is performed to illustrate the performance of the maximum likelihood estimates (MLEs). Goodness of fit of this distribution through different methods is studied. The potentiality and usefulness of the GLBIII distribution is demonstrated via its applications to two real data sets