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Analogous to the way wind blows single grains of sand and the subsequent settling back atop sand dunes, we find statistical evidence to claim that the prices of cryptocurrencies exhibit similar unpredicted patterns, characterized by positive or negative jumps. Motivated by extant evidence of asset returns’ non-normality, we capture distributional properties of the log-returns of the Bitcoin and the following three cryptocurrencies in terms of market capitalization (Ethereum, Ripple and Bitcoin cash). The total error induced by the fitted distribution is remarkably decreased when the generalized hyperbolic distribution is used, a finding further validated by a series of goodness-of-fit type statistical tests. A complementary analysis for the foreign exchange market is conducted, with inherent similarities to that of cryptocurrencies. We reveal that the generalized hyperbolic distribution can also be used to model very widely traded currency pairs significantly more accurately than the log-normal.

Sampling within a given interval with a constraint has not been previously considered. Standard parametric simulation engines require knowledge of the parameters of the distribution from which a sa...

We identify key characteristics of a correlated point process which include moments of the time between arrivals as well as measures of variability and correlation obtained from the counting process over different time intervals. A computational framework to calculate these characteristics is presented for the general MAP(n). We present simple closed-form-expressions for the key characteristics and an efficient algorithm to fit a MAP(2) to a point process. The contributions of this article include 1) developing a computational framework, in the form of partial-moment differential equations (PMDEs) and linear equations to derive a compact matrix exponential expression for the count process moments of a MAP(n), 2) developing an efficient and accurate algorithm to fit a MAP(2) based on count and inter-arrival moments.

The relationship between the norm square of the standardized cumulative distribution and the chi-square statistic is examined using the form of the covariance matrix as well as the projection perspective. This investigation enables us to give uncorrelated components of the chi-square statistic and to provide interpretation of these components as innovations standardizing the cumulative distribution values. The norm square of the standardized difference between empirical and theoretical cumulative distributions is also examined as an objective function for parameter estimation. Its relationship to the chi-square distance enables us to discuss the large sample properties of these estimators and a difference in their properties in the cases that the distribution is evaluated at fixed and random points.

The logistic distribution and the S-shaped pattern of its cumulative distribution and quantile functions have been extensively used in many different spheres affecting human life. By far the most well-known application of logistic distribution is in the logistic regression which is used for modeling categorical response variables. The exponentiated-exponential logistic distribution, a generalization of the logistic distribution is obtained using the technique by Alzaatreh et al. (2013) of mixing two distributions hereafter called as the EEL distribution. This distribution subsumes various types of logistic distribution. The structural analysis of the distribution in this paper includes limiting behavior, quantiles, moments, mode, skewness, kurtosis, order statistics, the large sample distributions of the sample maximum and the sample minimum and the distribution of the sample median. For illustrative purposes, a real life data set is considered as an application of the EEL distribution.

In the general risk model (or the Sparre-Andersen model), it is well-known that the following assertion holds: if the claim size is exponentially distributed then the non-ruin probability distribution is a mixture of exponential distributions. In this paper, under some general conditions, we prove that the converse statement of the previous assertion is also true. Besides, we define a new non-ruin measure associated with the aggregate logarithms of the claim-over-profit ratios and obtain a result on Pareto-type distributions.

A new three-parameter distribution with decreasing, increasing, and bathtub-shaped hazard rates obtained by compounding geometric, power series, and exponential distributions is introduced. It includes some well-known distributions as particular cases. Various mathematical properties of the new distribution as well as details of the maximum likelihood estimation and a sensitivity analysis for its parameters are presented. Finally, two real data applications are presented.

In this article, we derive a new generalized geometric distribution through a weight function, which can also be viewed as a discrete analog of weighted exponential distribution introduced by Gupta...

In this paper, we propose a new compounding distribution, named the complementary Lindley-geometric distribution. It arises on a latent complementary risks scenarios where only the maximum lifetime value among all risks instead of a particular risk is observed. Its characterization and statistical properties are investigated. The maximum likelihood inference using EM algorithm is developed. Asymptotic properties of the MLEs are discussed and simulation studies are performed to assess the performance of parameter estimation. We illustrate the proposed model with a real application and it shows that the new distribution is appropriate and potential for lifetime analyses.

A new member from the T − X family of distribution called the Gumbel-Burr XII (GUBXII) distribution is introduced. This was realized by adopting the logit transformation of the cumulative distribution function of the Burr XII random variable, while taking the Gumbel distribution as the generator. An account of some mathematical properties of the new distribution including the maximum likelihood estimation of its parameters is presented. A real data set is further used to test the applicability of the new distribution.

We explore the consistency of a new type of frequency distribution, where the corresponding rank distribution is Lavalette distribution. Empirical data of journal impact factors can be well described. This distribution is distinct from Poisson distribution and negative binomial distribution, which were suggested by previous study. By a log transformation, we obtain a bell-shaped distribution, which is then compared to Gaussian and catenary curves. Possible mechanisms behind the shape of impact factor distribution are suggested.

The Frechet distribution is an absolutely continuous model which has wide applicability in extreme value theory. In this paper, we propose a new three-parameter model, so-called the modified Frechet distribution, to extend the Frechet distribution. By using the Lambert function, we obtain some properties of the new distribution. We provide a simulation study to illustrate the performance of the maximum likelihood estimates. The flexibility of the introduced distribution is illustrated by means of a real data set. We use some goodness-of-fit statistics to verify the adequacy of the proposed model. We prove empirically that it is appropriate for lifetime applications.

This article derives closed-form solutions for fifth-ordered power method polynomial transformations based on the Method of Percentiles (MOP). A proposed MOP univariate procedure is compared with the Method of Moments (MOM) in the context of distribution fitting and estimating the shape functions. The MOP is also extended from univariate to multivariate data generation. The MOP procedure has an advantage because it does not require numerical integration to compute intermediate correlations and can be applied to distributions, where conventional moments do not exist. Simulation results demonstrate that the proposed MOP procedure is superior in terms of estimation, bias, and error.

We propose a three-parameter distribution referred to as the reflected- shifted-truncated gamma (RSTG) distribution to model negatively skewed data. Various properties of the proposed distribution ...

In this article, we estimate the parameters of exponential Pareto II distribution by two new methods. The first one is based on the principle of maximum entropy (POME) and the second is by Kullback-Leibler divergence of survival function (KLS). Monte Carlo simulated data are used to evaluate these methods and compare them with the maximum likelihood method. Finally, we fit this distribution to a set of real data by estimation procedures.

In this paper, we introduce the inverted exponentiated Weibull (IEW) distribution which contains exponentiated inverted Weibull distribution, inverse Weibull (IW) distribution, and inverted exponentiated distribution as submodels. The proposed distribution is obtained by the inverse form of the exponentiated Weibull distribution. In particular, we explain that the proposed distribution can be interpreted by Marshall and Olkin`s book (Lifetime Distributions: Structure of Non-parametric, Semiparametric, and Parametric Families, 2007, Springer) idea. We derive the cumulative distribution function and hazard function and calculate expression for its moment. The hazard function of the IEW distribution can be decreasing, increasing or bathtub-shaped. The maximum likelihood estimation (MLE) is obtained. Then we show the existence and uniqueness of MLE. We can also obtain the Bayesian estimation by using the Gibbs sampler with the Metropolis-Hastings algorithm. We also give applications with a simulated data set and two real data set to show the flexibility of the IEW distribution. Finally, conclusions are mentioned.

The new modified Weibull distribution named exponentiated modified Weibull distribution introduced by many authors. In this study, we define and study a bivariate modified Weibull distribution. Some joint probability functions and conditional probability density function are obtained. Also, joint survival function is expressed in compact form. Several statistical properties of this distribution are discussed. In the area of stress-strength models, we estimate the reliability follow a bivariate exponentiated modified Weibull distribution with dependence between stress and strength variables. Parameters estimators using the maximum likelihood method are obtained. Moreover, a numerical illustration is used to obtain Maximum Likelihood Estimators (MLEs).

To model growth curves in survival analysis and biological studies the logistic distribution has been widely used. In this article, we propose a goodness-of-fit test for the logistic distribution based on an estimate of the Gini index. The exact distribution of the proposed test statistic and also its asymptotic distribution are presented. In order to compute the proposed test statistic, parameters of the logistic distribution are estimated by approximate maximum likelihood estimators (AMLEs), which are simple explicit estimators. Through Monte Carlo simulations, power comparisons of the proposed test with some known competing tests are carried. Finally, an illustrative example is presented and analyzed.

Let P(x) be a polynomial monotone increasing on ( − ∞, +∞). The probability distribution possessing the distribution function is called the polynomial logistic distribution with associated polynomial P. This has recently been introduced by Koutras et al., who have also demonstrated its importance for modeling financial data. In this article, the properties of the polynomial logistic distribution with an associated polynomial of degree 3 have been investigated in detail. An example of polynomial logistic distribution describing daily exchange rate fluctuations for the US dollar versus the Turkish lira is provided.

The logistic distribution is one of the fundamental distribution and is widely used for describing model growth curves in survival analysis and biological studies. Applications of this distribution are presented in statistical literature. In this article, goodness of fit tests for the logistic distribution based on the empirical distribution function (EDF) are considered. In order to compute the test statistics, because the MLEs cannot be obtained explicitly, we use the approximate maximum likelihood estimates (AMLEs) suggested by Balakrishnan and Cohen (1990), which are simple explicit estimators. Power comparisons of the considered tests are carried out via simulations. Finally, two illustrative examples are presented and analyzed.

In this paper we introduce a new distribution, called the modified slash Lindley distribution, which can be seen as an extension of the Lindley distribution. We show that this new distribution provides more flexibility in terms of kurtosis and skewness than the Lindley distribution. We derive moments and some basic properties for the new distribution. Moment estimators and maximum likelihood estimators are calculated using numerical procedures. We carry out a simulation study for the maximum likelihood estimators. A fit of the proposed model indicates good performance when compared with other less flexible models.

The degree distribution of a simulated Barabasi–Albert graph under linear preferential attachment is investigated. Specifically, the parameters of the power law distribution are estimated and compared against the theoretical values derived using mean field theory. Least squares method and MLE-nonparametric method were utilized to estimate the distribution parameters on 1000 simulated Barabasi–Albert graphs for edge parameter m ∈ { 2 , 4 , 6 } and size n ∈ { 2 k : k = 5 , 6 , … , 14 , 15 } . Goodness of fit metrics were computed on a second set of simulated graphs for the median of the estimated parameters and other hypothetical values for the distribution parameters. The results suggest that the distribution of the parameters from simulated graphs are significantly different from the theoretical distribution and is also dependent on m . Further results confirm the finding that the parameter of the power law distribution, β , increases as m increases.

We have proposed a new Inverse Weibull distribution by using the generalized Beta distribution of McDonald (1984). Basic properties of the proposed distribution has been studied. Parameter estimation has been discussed alongside an illustrative example.

Particulate matter (PM) is one of the criteria pollutants which has been determined as harmful to public health and the environment. For this reason the ability to recognize its emission sources is very important. There are a number of measurement methods which allow to characterize PM in terms of concentration, particles size distribution, and chemical composition. All these information are useful to establish a link between the dust found in the air, its emission sources and influence on human as well as the environment. However, the methods are typically quite sophisticated and not applicable outside laboratories. In this work, we considered PM emission source discrimination method which is based on continuous measurements of PM concentration with a relatively cheap instrument and stochastic analysis of the obtained data. The stochastic analysis is focused on the temporal variation of PM concentration and it involves two steps: (1) recognition of the category of distribution for the data i.e. stable or the domain of attraction of stable distribution and (2) finding best matching distribution out of Gaussian, stable and normal-inverse Gaussian (NIG). We examined six PM emission sources. They were associated with material processing in industrial environment, namely machining and welding aluminum, forged carbon steel and plastic with various tools. As shown by the obtained results, PM emission sources may be distinguished based on statistical distribution of PM concentration variations. Major factor responsible for the differences detectable with our method was the type of material processing and the tool applied. In case different materials were processed by the same tool the distinction of emission sources was difficult. For successful discrimination it was crucial to consider size-segregated mass fraction concentrations. In our opinion the presented approach is very promising. It deserves further study and development.

In this paper, we introduce a generalization of the slash distribution via the gamma-normal distribution. We define the new slash distribution by relation of a gamma-normal random variable with respect to a power of a uniform random variable. The newly defined distribution generalizes the slash distribution and is more flexible in terms of its kurtosis and skewness than the slash distribution. Basic properties of the new distribution are studied. We derive the maximum likelihood estimators of its parameters and apply the distribution to a real data set.

The concept of length-biased distribution can be employed in the development of proper models for the lifetime data. Length-biased distribution is a special case of the more general form known as the weighted distribution. This article introduces a new class of Maxwell length-biased distribution. The statistical properties of the proposed distribution are determined, including shape, kth moment, reliability function, hazard function, reverse hazard function, and so on. The maximum likelihood estimate (MLE), method of moments (MM) estimate, and Bayesian estimate of the unknown parameter are derived. An illustrative example demonstrates the application of the proposed distribution in real life.

A life distribution function (DF) F is said to be star-shaped if is nondecreasing on its support. This generalises the model of a convex DF, even allowing for jumps. The nonparametric maximum likelihood estimation is known to be inconsistent. We provide a uniformly strongly consistent least-squares estimator. We also derive the convergence in distribution of the estimator at a point where is increasing using the arg max theorem.

In this article, we propose a new class of distribution which is based on the concept of exponentiated generalization with some modification so as to provide a better result in terms of flexibility. Our proposed distribution accommodates various shapes of hazard rate including the bathtub. Exponential distribution has been taken as the baseline distribution. Various statistical properties of the proposed distribution have been studied. We have used the method of maximum likelihood for estimation of the parameters of the proposed model. Last, we have analyzed four real datasets to illustrate the flexibility of the model in comparison to eight existing well-known distributions.

In this paper, a new lifetime distribution is defined and studied. We refer to the new distribution as alpha power Weibull distribution. The importance of the new distribution comes from its ability to model monotone and non monotone failure rate functions, which are quite common in reliability studies. Various properties of the proposed distribution are obtained including moments, quantiles, entropy, order statistics, mean residual life function, and stress-strength parameter. The maximum likelihood estimation method is used to estimate the parameters. Two real data sets are used to illustrate the importance of the proposed distribution.

Many of studies have suggested the modifications on Weibull distribution to model the non-monotone hazards. In this paper, we combine two cumulative hazard functions and propose a new modifiedWeibull distribution function. The newly suggested distribution will be named as a new flexible Weibull distribution. Corresponding hazard function of the proposed distribution shows flexible (monotone or non-monotone) shapes. We study the characteristics of the proposed distribution that includes ageing behavior, moment, and order statistic. We also discuss an estimation method for its parameters. The performance of the proposed distribution is compared with existing modifiedWeibull distributions using various types of hazard functions. We also use real data example to illustrate the efficiency of the proposed distribution.

In this article, exponential distribution, two-parameter Weibull distribution, log-logistic distribution, log-log-logistic distribution, and Lomax distribution are characterized through selected functions of reliability theory: failure rate, aging intensity function, and log-odds rate.

In this paper, we approximate the ultimate ruin probability in the Cramer-Lundberg risk model when claim sizes have an arbitrary continuous distribution. We propose two approximation methods, based on Erlang Mixtures, which can be used for claim sizes distribution both light and heavy tailed. Additionally, using a continuous version of the empirical distribution, we develop a third approximation which can be used when the claim sizes distribution is unknown and paves the way for a statistical application. Numerical examples for the gamma, Weibull and truncated Pareto distributions are provided.

A new Pareto-type distribution is introduced and studied. This new model is a generalization of the well-known Pareto distribution. We derive some of its probabilistic and inferential properties. We deduce the mathematical form of the Lorenz curve and the Gini index associated with the new model. The maximum likelihood estimators are derived and their performance are evaluated through a Monte Carlo simulation study. Finally, we illustrate the flexibility of the new distribution by means of three applications to real data sets.

Following the work of Azzalini (1985 and 1986) on the skew-normal distribution, we propose an extension of the generalized extreme value (GEV) distribution, the SGEV. This new distribution allows for a better fit of maxima and can be interpreted as both the distribution of maxima when maxima are taken on dependent data and when maxima are taken over a random block size. We propose to estimate the parameters of the SGEV distribution via the probability-weighted moment method. A simulation study is presented to provide an application of the SGEV on block maxima procedure and return level estimation. The proposed method is also implemented on a real-life data.

In this paper, a new mixed bivariate geometric-exponential distribution is introduced. We present the properties and numerical characteristics of the distribution. In addition, the applications of this new bivariate distribution are given in a two-unit series system. Through an intensive Monte-Carlo simulation study, we assess the precision of the proposed estimation methods for the parameter inference. A real data is analysed for illustrative purpose.

The concept of length-biased distribution can be employed in development of proper models for lifetime data. Length-biased distribution is a special case of the more general form known as weighted distribution. In this paper we introduce a new class of length-biased weighted Lomax distribution, (LBWLD). The statistical properties of this distribution are derived and the model parameters are estimated by maximum likelihood estimation and the observed information matrix is determined. An application to real data set is finally presented for illustration.

In this article, we develop an algorithm for working out the distribution of proposed test statistics. The computed distribution is compared with its existing asymptotic normal approximation using convergence rates and disparity measures. Simulation study is made to estimate the level of significance, and power to assess the performance of algorithm based procedure in comparison to asymptotic normal approximation. Both power comparisons and convergence rates suggest that the distribution computed through algorithm procedure be used for smaller sample sizes.

In this paper, the researchers attempt to introduce a new generalization of the Weibull-geometric distribution. The failure rate function of the new model is found to be increasing, decreasing, upside-down bathtub, and bathtub-shaped. The researchers obtained the new model by compounding Weibull distribution and discrete generalized exponential distribution of a second type, which is a generalization of the geometric distribution. The new introduced model contains some previously known lifetime distributions as well as a new one. Some basic distributional properties and moments of the new model are discussed. Estimation of the parameters is illustrated and the model with two known real data sets is examined.

A new four-parameter distribution called the exponentiated power Lindley–Poisson distribution which is an extension of the power Lindley and Lindley–Poisson distributions is introduced. Statistical properties of the distribution including the shapes of the density and hazard functions, moments, entropy measures, and distribution of order statistics are given. Maximum likelihood estimation technique is used to estimate the parameters. A simulation study is conducted to examine the bias, mean square error of the maximum likelihood estimators, and width of the confidence interval for each parameter. Finally, applications to real data sets are presented to illustrate the usefulness of the proposed distribution.

In this paper, we introduce a bivariate weighted exponential distribution based on the generalized exponential distribution. This class of distributions generalizes the bivariate distribution with weighted exponential marginals (BWE). We derive different properties of this new distribution. It is a four-parameter distribution, and the maximum-likelihood estimator of unknown parameters cannot be obtained in explicit forms. One data set has been re-analyzed and it is observed that the proposed distribution provides better fit than the BWE distribution.

Data with multiple responses is ubiquitous in modern applications. However, few tools are available for regression analysis of multivariate counts. The most popular multinomial-logit model has a very restrictive mean-variance structure, limiting its applicability to many data sets. This article introduces an R package MGLM, short for multivariate response generalized linear models, that expands the current tools for regression analysis of polytomous data. Distribution fitting, random number generation, regression, and sparse regression are treated in a unifying framework. The algorithm, usage, and implementation details are discussed.

In this article a continuous distribution, the so-called transmuted additive Weibull distribution, that extends the additive Weibull distribution and some other distributions is proposed and studied. We will use the quadratic rank transmutation map proposed by Shaw and Buckley (2009) in order to generate the transmuted additiveWeibull distribution. Various structural properties of the new distribution including explicit expressions for the moments, random number generation and order statistics are derived. Maximum likelihood estimation of the unknown parameters of the new model for completesample is also discussed. It will be shown that the analytical results are applicable to model real world data.

A new distribution with increasing, decreasing, bathtub-shaped and unimodal failure rate forms called as the generalized modified Weibull power series (GMWPS) distribution is proposed. The new distribution is constructed based on a latent complementary risk problem and is obtained by compounding generalized modified Weibull (GMW) and power series distributions. The new distribution contains, as special submodels, several important distributions which are discussed in the literature, such as generalized modified Weibull Poisson (GMWP) distribution, generalized modified Weibull Geometric (GMWG) distribution, generalized modified Weibull Logarithmic (GMWL) distribution, generalized modified Weibull Binomial (GMWB) distribution, among others.A comprehensive mathematical treatment of the new distribution is provided. We provide closed-form expressions for the density, cumulative distribution, survival function, failure rate function, the r th raw moment, and also the moments of order statistics. Expressions for the Renyi and Shannon entropies are also given. Moreover, we discuss maximum likelihood estimation and provide formula for the elements of the Fisher information matrix. We consider the EM-algorithm for computing the estimates. Simulation studies and two real data set applications are also given for illustration of the flexibility and potentiality of the new distribution.

Here we introduce a new class of distributions namely the generalized hyper-Poisson distribution of order k (GHPD(k)) as an order k version of the alpha-generalized hyper-Poisson distribution of Kumar and Nair (Statistica, 2014b). Several properties of the GHPD(k) are derived and the estimation of the parameters of the distribution by the method of mixed moments and the method of maximum likelihood is discussed. Certain testing procedures are suggested and all these estimation and testing procedures are illustrated with the help of a real-life data set. Further a simulation study is conducted.

Financial data are usually studied via low flexible distributions, independently of the frequency of the data, due to their simplicity and analytical tractability. In this paper we analyze two highly flexible five-parameter distributions into fitting financial returns, these are the skewed generalized t (SGT) and the generalized hyperbolic (GH). Applications carried on two exchange rates (Euro–Dollar and Dollar–Yen), and two indexes (S&P 500 and Nikkei 225) over four frequencies: weekly, daily, 30-min and 5-min, confirm the superiority of the SGT and GH in approximating the distribution of a given data at a remarkable precision. Moreover, as we move from higher to lower frequency, the distribution’s overall shape does indeed change radically, and the estimated parameters refute the tendency to normality, which calls into question the aggregational Gaussianity’s stylized fact.

The beta distribution has traditionally been employed in the PERT methodology and generally used for modeling bounded continuous random variables based on expert’s judgment. The impossibility of estimating four parameters from the three values provided by the expert when the beta distribution is assumed to be the underlying distribution has been widely debated. This paper presents the generalized bicubic distribution as a good alternative to the beta distribution since, when the variance depends on the mode, the generalized bicubic distribution approximates the kurtosis of the Gaussian distribution better than the beta distribution. In addition, this distribution presents good properties in the PERT methodology in relation to moderation and conservatism criteria. Two empirical applications are presented to demonstrate the adequateness of this new distribution.