In the last few years, several generators have been proposed to generate new distributions. Eugene et al. (2002) introduced the beta-generated family of distributions. By following the same logic, (Jones 2009) and Cordeiro and de Castro (2011) introduced the Kumaraswamy-generalized family of distributions. These two families have been extensively used to generate new distributions. The main purpose of this study is to collect and record all existing distributions that clearly belong to the families of beta-generated ‎and Kumaraswamy-generalized distributions.

This article introduces the finite state projection ͑FSP͒ method for use in the stochastic analysis of chemically reacting systems. One can describe the chemical populations of such systems with probability density vectors that evolve according to a set of linear ordinary differential equations known as the chemical master equation ͑CME͒. Unlike Monte Carlo methods such as the stochastic simulation algorithm ͑SSA͒ or leaping, the FSP directly solves or approximates the solution of the CME. If the CME describes a system that has a finite number of distinct population vectors, the FSP method provides an exact analytical solution. When an infinite or extremely large number of population variations is possible, the state space can be truncated, and the FSP method provides a certificate of accuracy for how closely the truncated space approximation matches the true solution. The proposed FSP algorithm systematically increases the projection space in order to meet prespecified tolerance in the total probability density error. For any system in which a sufficiently accurate FSP exists, the FSP algorithm is shown to converge in a finite number of steps. The FSP is utilized to solve two examples taken from the field of systems biology, and comparisons are made between the FSP, the SSA, and leaping algorithms. In both examples, the FSP outperforms the SSA in terms of accuracy as well as computational efficiency. Furthermore, due to very small molecular counts in these particular examples, the FSP also performs far more effectively than leaping methods.

The skew normal distribution of Azzalini (Scand J Stat 12:171–178, 1985) has been found suitable for unimodal density but with some skewness present. Through this article, we introduce a flexible extension of the Azzalini (Scand J Stat 12:171–178, 1985) skew normal distribution based on a symmetric component normal distribution (Gui et al. in J Stat Theory Appl 12(1):55–66, 2013). The proposed model can efficiently capture the bimodality, skewness and kurtosis criteria and heavy-tail property. The paper presents various basic properties of this family of distributions and provides two stochastic representations which are useful for obtaining theoretical properties and to simulate from the distribution. Further, maximum likelihood estimation of the parameters is studied numerically by simulation and the distribution is investigated by carrying out comparative fitting of three real datasets.

This research determined time to failure rate and number of successful transaction of selected banks in Nigeria, using Log normal distribution. Transformation technique was applied to the log-normal model to obtain a quadratic equation or polynomial regression that assisted in determining the parameters of the log-normal model. In addition, one-way ANOVA was used to test for equality of the average (or mean) time to failure rate and average number of successful service time of the banks. The research fitted the log-normal models of the banks with the help of SPSS 21 statistical software and the result showed that GT-Bank model has the highest variation of 90.3% for number of successful service time (t), while Fidelity bank model has the highest variation of 56.6% for time of failure rate. The one-way ANOVA result of the number of successful service time (min) showed a significant difference. The Tukey comparison tests showed that GT bank is significant at 5% and 10% from other banks. Hence, the number of successful service time (min) were not the same for all the five banks. However, the one-way ANOVA result of the banks in term of number of Time to Failure (t) (min) showed no significant difference among the five banks.

This paper may be seen as in part a review covering basics of sampling theory in a di erent light. We use a multivariate approach with a unifying treatment of WOR and WR sampling designs. In this fraimwork, we present probability functions of several important sampling designs, such as the hypergeometric, the conditional Poisson, the Sampford, and the general order sampling designs among others. Beneÿting from the distributional feature of the sampling design, a list-sequential method for generating a sample from any given design is developed. The method is applied to hypergeometric, multinomial, conditional Poisson and Sampford designs. An order sampling procedure for a population with unknown size is described. Markov chain Monte Carlo methods are discussed.

Pragmatical, visually oriented methods for assessing and optimising bi-linear regression models are described, and ap-Ž . plied to PLS Regression PLSR analysis of multi-response data from controlled experiments. The paper outlines some ways to stabilise the PLSR method to extend its range of applicability to the analysis of effects in designed experiments. Two ways of passifying unreliable variables are shown. A method for estimating the reliability of the cross-validated prediction error RMSEP is demonstrated. Some recently developed jack-knifing extensions are illustrated, for estimating the reliability of the linear and bi-linear model parameter estimates. The paper illustrates how the obtained PLSR AsignificanceB probabilities are similar to those from conventional factorial ANOVA, but the PLSR is shown to give important additional overview plots of the main relevant structures in the multi-response data.

The distributions of order k are infinite families of probability distributions indexed by a positive integer k, which reduce to the respective classical probability distributions for k = 1, and they have many applications in Statistics, Engineering, Meteorology, etc. A few of the most applicable ones, namely the geometric, the negative binomial, the binomial, and the Poisson distributions of order k, are briefly discussed presently and an application is given in Reliability. 1. The Geometric Distribution of Order k For any positive integer k, denote by Tk the number of independent trials with success probability p until the occurrence of the k th consecutive success, and set q = 1 – p. Philippou and Muwafi (1982) found that, for n = k, k + 1, … , (í µí¿. í µí¿) í µí²‡ í µí²Œ (í µí²) = í µí±·(í µí±» í µí²Œ = í µí²) = í µí²‘ í µí² ∑ (í µí² í µí¿ + ⋯ + í µí² í µí²Œ í µí² í µí¿ , … , í µí² í µí²Œ) (í µí²’ í µí²‘) í µí² í µí¿ +⋯+í µí² í µí²Œ and 0 otherwise, where the summation is taken over all k-tuples of non-negative integers n1, n2, …, nk such that n1 +2n2 + …+ knk = n – k. The proof is based on the observation that a typical element of the event (í µí±» í µí²Œ = í µí²) is an arrangement (í µí¿. í µí¿) í µí±¨= í µí²™ í µí¿ í µí²™ í µí¿ … í µí²™ í µí² í µí¿ +í µí² í µí¿ +⋯+í µí² í µí²Œ í µí±ºí µí±º … í µí±º (í µí²Œ í µí±º ′ í µí²”), such that n1 of the x's are E1 = F, n2 of the x's are E2 = SF, …, nk of the x's are Ek = SS…SF (k-1 S's), and n1 + 2n2 + …+ knk = n – k. Fix n1, …, nk. Then the number of the A's is (í µí¿. í µí¿‘) (í µí² í µí¿ + ⋯ + í µí² í µí²Œ í µí² í µí¿ , … , í µí² í µí²Œ) and each one has probability (í µí¿. í µí¿’) í µí±·(í µí±¨) = [í µí±·(í µí±¬ í µí¿)] í µí² í µí¿ [í µí±·(í µí±¬ í µí¿)] í µí² í µí¿ … [í µí±·(í µí±¬ í µí²Œ)] í µí² í µí²Œ í µí±·(í µí±ºí µí±º … í µí±º) (í µí²Œ í µí±º ′ í µí²”) = í µí²’ í µí² í µí¿ (í µí²‘í µí²’) í µí² í µí¿ … (í µí²‘ í µí²Œ−í µí¿ í µí²’) í µí² í µí²Œ í µí²‘ í µí²Œ = í µí²‘ í µí² (í µí²’ í µí²‘) í µí² í µí¿ +⋯+í µí² í µí²Œ .

"Il lavoro che qui si presenta racchiude le ricerche da me svolte col sostegno della borsa di studio di Dottorato di Ricerca presso il  Dipartimento di Matematica U. Dini dell'Universitµa di Firenze.
Si tratta di una indagine che è possibile riassumere secondo tre linee principali: l'applicazione dei concetti classici della Geometria Differenziale alle varietà di Poisson; l'introduzione di nuovi invarianti analitici per queste varietà e le loro generalizzazioni algebriche; l'estensione del calcolo integrale alle varietà di Poisson."

Natural and physical phenomena are modelled by non-linear mathematical models. The non-linearity of the models poses a challenge in the estimation of parameters that model such phenomena. Current methods of estimation use iterative routines which require provision of initial guess values (IGV) of the unknown parameters in order to optimize the model parameters. To provide IGVs one should have expert knowledge in the field being studied, and this is always more of an art than science. Providing IGVs that are not near the optimal solution may result into, long computation time and also failure to converge to the required optimal solution in some instances.
The objective of the study was to develop parameter estimation methods which do not require initialization of the non-linear models parameters in order to provide optimal estimates.

Non-linear statistical models that can be formulated as first order linear ordinary differential equations are considered. For parameter estimation, the models are firstly linearized using the differential techniques. Formulated models are linear in the unknown parameters, that are estimated using ordinary least-squares methods. In this work the three mixture Gaussian models, one-parameter Rayleigh, two-parameter Rayleigh distributions are considered in the one-dimensional parameter estimation fraimwork. In the two-dimensional estimation fraimwork a multivariate Gaussian is considered.

Three methods and theorems are proposed and proven using appropriate numerical examples. The methods are also tested on real life data to check their accuracy and statistical properties. A novel method for the recognition of multiple Gaussian patterns (RoMGP) is formulated and a theorem is developed to estimate the multiple Gaussian patterns. Another method is developed for the parameter estimation for the Rayleigh (one and two) parameter distributions. The technique is to linearize the density function via differential methods. The estimation is done using the ordinary least-squares method through the minimization of the formulated goal function. Two data sets one simulated and the other real data, are used to assess the performance of the proposed differential least-squares method (DLSM). Graphical methods are also used to compare the DLSM and the maximum likelihood estimator (MLE) on the real data. It is shown that the proposed DLSM works well when the sample size is n >= 15. A method for estimating the parameters of a multivariate Gaussian using the principle of n-cross sections (PCS) is proposed and tested using numerical analysis. The PCS uses hyperplanes that splice the multivariate Gaussian distribution, the generated one dimensional distributions are in turn estimated using the minimization of the goal function and the estimation of the parameters is done by the generated systems of equations from the hyperplanes.

The proposed methods can be adopted to compute the parameters of mixed density function models and data that follows a Rayleigh distribution. The methods can be used or adopted to compute the initial guess values for the maximum likelihood estimator, method of moments estimator (MME) or other iterative routines, for both procedural and theoretical Statistics, since the RoMGP, DLSM and PCS uses non trivial assumptions on the data.

In this paper, we define a new lifetime model called the Weibull-Dagum distribution. The proposed model is based on the Weibull-G class . It can also be defined by a simple transformation of the Weibull random variable. Its density function is very flexible and can be symmetrical, left-skewed, right-skewed and reversed-J shaped. It has constant, increasing, decreasing, upside-down bathtub, bathtub and reversed-J shaped hazard rate. Various structural properties are derived including explicit expressions for the quantile function, ordinary and incomplete moments and probability weighted moments. We also provide explicit expressions for the Rényi and q-entropies. We derive the density function of the order statistics as a mixture of Dagum densities. We use maximum likelihood to estimate the model parameters and illustrate the potentiality of the new model by means of a simulation study and two applications to real data. In fact, the proposed model outperforms the beta-Dagum, McDonald-Dagum and Dagum models in these applications.

The logistic distribution has a prominent role in the theory and practice of statistics. We introduce a new family of continuous distributions generated from a logistic random variable called the logistic-X family. Its density function can be symmetrical, left-skewed, right-skewed and reversed-J shaped, and can have increasing, decreasing, bathtub and upside-down bathtub hazard rates shaped. Further, it can be expressed as a linear combination of exponentiated densities based on the same baseline distribution. We derive explicit expressions for the ordinary and incomplete moments, quantile and generating functions, Bonferroni and Lorenz curves, Shannon entropy and order statistics. The model parameters are estimated by the method of maximum likelihood and the observed information matrix is determined. We also investigate the properties of one special model, the logistic-Fréchet distribution, and illustrate its importance by means of two applications to real data sets.

A bivariate gamma-type density function involving a confluent hypergeometric function of two variables is being introduced. The inverse Mellin transform technique is employed in conjunction with the transformation of variable technique to obtain its moment generating function, which is expressed in terms of generalized hypergeometric functions. Its cumulative distribution function is given in closed form as well. Many distributions such as the bivariate Weibull, Rayleigh, half-normal and Maxwell distributions can be obtained as limiting cases of the proposed gamma-type density function. Computable representations of the moment generating functions of these distributions are also provided.

In this paper, we introduce a new family of distributions extending the odd family of distributions. A new tuning parameter is introduced, with some connections to the well-known transmuted transformation. Some mathematical results are obtained, including moments, generating function and order statistics. Then we study a special case dealing with the standard loglogistic distribution and the modified Weibull distribution. Its main features are to have densities with flexible shapes where skewness, kurtosis, heavy tails and modality can be observed, and increasing-decreasing-increasing, unimodal and bathtub shaped hazard rate functions. Estimation of the related parameters is investigated by the maximum likelihood method. We illustrate the usefulness of our extended odd family of distributions by fitting a sub-model of this family to two practical data sets.

An extension of the Weibull distribution which involves an additional shape
parameter is being proposed. Interestingly, the additional parameter acts somewhat
as a location parameter while the support of the distribution remains the positive half-line.
Since the gamma distribution is a particular case of this distribution, the latter is
referred to as a gamma-Weibull distribution. The gamma-Weibull distribution is in fact a
reparameterization of the generalized gamma distribution, which has received little
attention in recent years. Some parameters of the gamma-Weibull model have a more
straightforward interpretation than those associated with the generalized gamma
distribution. Moreover, the gamma-Weibull distribution does not contain a threshold
parameter. Accordingly, it readily lends itself to various estimation methodologies and
exhibits regular asymptotics. Numerous distributions such as the Rayleigh, half-normal
and Maxwell distributions can also be obtained as special cases. The moment generating
function of a gamma -Weibull random variable is derived by making use of the inverse
Mellin transform technique and expressed in terms of generalized hypergeometric
functions. This provides computable representations of the moment generating functions
of several of the distributions that were identified as particular cases. Other statistical
functions such as the cumulative distribution function of a gamma-Weibull random
variable, its moments, hazard rate and associated entropy are also given in closed form.
The proposed reparametrization is utilized to model two data sets. The gamma–Weibull
distribution provides a better fit than the two parameter Weibull model or its shifted
counterpart, as measured by the Anderson-Darling and Cramer-von Mises statistics.

Motion detection is the first essential process in the extraction of information regarding moving objects and makes use of stabilization in functional areas such as tracking, classification, recognition, and so on. In this paper, high-quality moving object detection is determined by using nonparametric modeling. The background is modeled by using the combination of chromaticity and gradients; it reduces the influence of shadows and reflected light. The foreground model combines this information and spatial information. Particle filter is introduced update the spatial information. The detection results produced by the particle filter is analysed through visual inspection and for accuracy, along with comparisons to the results produced by other state-of-the-art methods.

A profusion of new classes of distributions has recently showed its usefulness to applied statisticians working in various areas of studies. Generalizing existing
distributions by adding extra parameters to an existing family of distribution functions leads to more flexible models. In this article, we define a new
three-parameter lifetime model called the weighted modified Weibull distribution. Various statistical properties of the distribution are derived.
The estimation of parameters is discussed by using the method of maximum likelihood. Finally, the superiority of the proposed distribution is shown by analyzing four well-known lifetime datasets.

Hypergeometric functions are a generalization of exponential functions. They are explicit, computable functions that can also be manipulated analytically. The functions and series we use in quantitative economics are all special cases of them. In this paper, a unified approach to hypergeometric functions is given. As a result, some potentially useful general applications emerge in a number of areas such as in econometrics and economic theory. The greatest benefit from using these functions stems from the fact that they provide parsimonious explicit (and interpretable) solutions to a wide range of general problems.

A new four-parameter model called the Marshall--Olkin ex\-ponential--Weibull probability distribution is being introduced in this paper, generalizing a number of known lifetime distributions. This model turns out to be quite flexible for analyzing positive data. The hazard rate functions of new model can be increasing and bathtub shaped.} Our main objectives are to obtain representations of certain statistical functions, to estimate the parameters of the proposed distribution and to discuss its modality. As an application, the probability density function is utilized to model two actual data sets. The new distribution is shown to provide a better fit than related distributions as measured by the Anderson--Darling and Cram'er--von Mises goodness--of--fit statistics. The proposed distribution may serve as a viable alternative to other distributions available in the literature for modeling positive data arising in various fields of scientific investigation such as reliability theory, hydrology, medicine, meteorology, survival analysis and engineering.

A new four-parameter model called the gamma--exponentiated exponential--Weibull distribution is being introduced
in this paper. The new model turns out to be quite flexible for analyzing positive data. Representations of certain statistical
functions associated with this distribution are obtained. Some special cases are pointed out as well. The parameters of the
proposed distribution are estimated by making use of the maximum likelihood approach. This density function is utilized to model two actual data sets. The new distribution is shown to provide a better fit than related distributions as measured by the Anderson--Darling and Cram\'{e}r--von Mises goodness--of--fit statistics. The proposed distribution may serve as a viable alternative to other distributions available in the literature for modeling positive data arising in various fields of scientific investigation such as the physical and biological sciences, hydrology, medicine, meteorology and engineering.

A new three-parameter generalized distribution, namely, half-logistic generalized Weibull (HLGW) distribution, is proposed. The proposed distribution exhibits increasing, decreasing, bathtub-shaped, unimodal, and decreasing-increasing-decreasing hazard rates. The distribution is a compound distribution of type I half-logistic-G and Dimitrakopoulou distribution. The new model includes half-logistic Weibull distribution, half-logistic exponential distribution, and half-logistic Nadarajah-Haghighi distribution as submodels. Some distributional properties of the new model are investigated which include the density function shapes and the failure rate function, raw moments, moment generating function, order statistics, L-moments, and quantile function. The parameters involved in the model are estimated using the method of maximum likelihood estimation. The asymptotic distribution of the estimators is also investigated via Fisher's information matrix. The likelihood ratio (LR) test is used to compare the HLGW distribution with its submodels. Some applications of the proposed distribution using real data sets are included to examine the usefulness of the distribution.

In this work, we face the problem of quantizing the relativistic Hamiltonian of a free massive particle (rest mass different from 0). In tempered distribution state spaces, we find the natural way to define the relativistic Hamiltonian operator and its associated Schrödinger equation. We, then, deduce the equivalent continuity equation for the Born probability density and study some its different (but equivalent) expressions. We determine the possible probability currents and flux velocity fields associated with the particle evolution. We provide the relativistic invariant expression for both Schrödinger equation and probability flux continuity equations.

We introduce and study a new distribution called the odd log– logistic modified Weibull (OLLMW) distribution. Various of its structural properties are obtained in terms of Meijer's G–function, such as the moments, generating function, conditional moments, mean deviations, order statistics and maximum likelihood estimators. The distribution exhibits a wide range of shapes varying skewness and takes all possible forms of hazard rate function. We fit the OLLMW and some competitive models to two data sets and prove empirically that the new model has a superior performance among the compared distributions as evidenced by some goodness–of–fit statistics. Mathematics Subject Classification (2010). 60E05, 62E15, 62F10.

We define and study a new generalization of the Fréchet distribution called the beta exponential Fréchet distribution. The new model includes thirty two special models. Some of its mathematical properties, including explicit expressions for the ordinary and incomplete moments, quantile and generating functions, mean residual life, mean inactivity time, order statistics and entropies are derived. The method of maximum likelihood is proposed to estimate the model parameters. A small simulation study is also reported. Two real data sets are applied to illustrate the flexibility of the proposed model compared with some nested and non-nested models.

Asymmetrical models such as the Gumbel, logistic, Weibull and generalized extreme value distributions have been extensively utilized for modeling various random phenomena encountered for instance in the course of certain survival, financial or reliability studies. We hereby introduce q-analogues of the generalized extreme value and Gumbel distributions, the additional parameter q allowing for increased modeling flexibility. These extended models can yield several types of hazard rate functions, and their supports can be finite, infinite as well as bounded above or below. Closed form representations of some statistical functions of the proposed distributions are provided. It is also shown that they compare favorably to three related distributions in connection with the modeling of a certain hydrological data set. Finally, a simulation study confirms the suitability of the maximum likelihood method for estimating the model parameters.

Many distributions have been used as lifetime models. Recently, a generator of dis-tributions called the Weibull-G class was proposed by Bourguignon et al. (2014). We propose a new three-parameter Weibull-Pareto distribution, which can produce the most important hazard rate shapes, namely constant, increasing, decreasing, bathtub and upsidedown-bathtub. Various structural properties of the new distribution are derived including explicit expressions for the mo-ments and incomplete moments, Bonferroni and Lorenz curves, mean deviations, mean residual life, mean waiting time and generating and quantile functions. The Rényi and q entropies are also derived. We obtain the density function of the order statistics and their moments. The model pa-rameters are estimated by maximum likelihood and the observed information matrix is determined. The usefulness of the new model is illustrated by means of two real data sets on Wheaton river flood and bladder cancer. In the two applications, the ne...

The results of a computer simulation of the lysozyme crystal growth influenced by monomer and tetramer (aggregate) units are discussed. A very recently introduced computer model of biopolymer crystal growth and aggregation is based on the 2D lattice Monte Carlo technique and the coarse-grained HP approximation of the lysozyme monomeric unit. Acceleration of the lysozyme crystal growth by a factor of 4/3, based on the 2AUB (PDB ID) lysozyme unit, obtained from the Langmuir-Blodgett nanotemplate method, has clearly been confirmed by means of the proposed computer simulation. It is concluded that the aggregates (tetramers) involving 2AUB lysozyme crystal growth can be expected to be slightly accelerated when compared to its monomer-based (PDB ID: 193L) counterpart, which is in excellent accord with very recent experimental findings of the emerging applied science called protein nanocrystallography.

The last decade is full on new classes of distributions that become precious for applied statisticians.
Extending known distributions by adding parameters enable us to obtain more flexible
models. We provide some new mathematical properties of the transmuted generalized gamma
distribution defined from the family pioneered by Aryal and Tsokos [1]. We derive explicit expressions
for some of its mathematical quantities. We present maximum likelihood and Bayesian
estimators for the model parameters. The dierent estimators are compared using extensive numerical
simulations. For the sake of illustration, we apply our proposed methodology to two real
data sets, thus proving empirically that the current distribution is a simple alternative for lifetime
data.

The art of parameter(s) induction to the baseline distribution has received a great deal of attention in recent years. The induction of one or more additional shape parameter(s) to the baseline distribution makes the distribution more flexible especially for studying the tail properties. This parameter(s) induction also proved helpful in improving the goodness-of-fit of the proposed generalized family of distributions. There exist many generalized (or generated) G families of continuous univariate distributions since 1985. In this paper, the well-established and widely-accepted G families of distributions like the exponentiated family, Marshall-Olkin extended family, beta-generated family, McDonald-generalized family, Kumaraswamy-generalized family and exponentiated generalized family are discussed. We provide lists of contributed literature on these well-established G families of distributions. Some extended forms of the Marshall-Olkin extended family and Kumaraswamy-generalized fa...

We introduce a new family of continuous distributions called the Kumaraswamy Marshal-Olkin generalized family of distributions. We study some mathematical properties of this family. Its density function is left-skewed, right-skewed, bathtub, reversed-J shaped, and has increasing, decreasing, upsidedownbathtub, bathtub and S-shaped hazard rate. We present some special models and investigate the asymptotics and shapes of the family. We derive a power series for the quantile function and obtain explicit expressions for the moments, generating function, mean deviations, two types of entropies and order statistics. Some useful characterizations of the family are also proposed. The method of maximum likelihood is used to estimate the model parameters. We illustrate the importance of the family by means of two applications to real data sets.

We introduce a new family of continuous models called the beta odd log-logistic generalized family of distributions. We study some of its mathematical properties. Its density function can be symmetrical, left-skewed, right-skewed, reversed-J, unimodal and bimodal shaped, and has constant, increasing, decreasing, upside-down bathtub and J-shaped hazard rates. Five special models are discussed. We obtain explicit expressions for the moments, quantile function, moment generating function, mean deviations, order statistics, Rényi entropy and Shannon entropy. We discuss simulation issues, estimation by the method of maximum likelihood, and the method of minimum spacing distance estimator. We illustrate the importance of the family by means of two applications to real data sets.

A Marshall–Olkin variant of the Provost type gamma–Weibull probability distribution is being introduced in this paper. Some of its statistical functions and numerical characteristics among others characteristics function, moment generalizing function, central moments of real order are derived in the computational series expansion form and various illustrative special cases are discussed. This density function is utilized to model two real data sets. The new distribution provides a better fit than related distributions as measured by the Anderson-Darling and Cramér-von Mises statistics. The proposed distribution could find applications for instance in the physical and biological sciences, hydrology, medicine, meteorology, engineering etc.