Well-known Distribution Research Articles

A sustainable generalization of inverse Lindley distribution for wind speed analysis in certain regions of Pakistan

Information on wind speed and wind power distribution is significant for a few reasons, for example, surveying wind assets, arranging wind cultivates, and limiting the liabilities for wind power improvement. This study provided an application of a new generalization of two-parameter generalized inverse Lindley distribution using the Marshall–Olkin family for analyzing wind speed and wind power characteristics. Some mathematical properties of the new distribution were studied. We had observed the suitability of new distribution as compared to the other well-known wind speed distributions such as Weibull, inverse exponential, inverted Kumaraswamy, inverse Weibull, inverse Lindley, and generalized inverse Lindley distribution. For this purpose, the time-based wind speed data is taken from the four stations of Pakistan as a case study. We conclude based on certain goodness of fit criteria that the newly developed distribution has a better fit as compared to the other wind speed distributions. Therefore, the new model can be used as an alternative distribution for the assessment of wind speed energy potential.

Moments of order statistics and k-record values arising from the complementary beta distribution with application

The order statistics (OSs) arise in both practical and theoretical aspects including goodness-of-fit tests, characterizations of probability distributions and some estimation approaches such as the L-moments method. Most of these instances are connected with moments of OSs. The records and k-record statistics, also called kth record values in the literature, are other important topics related to the OSs. This kind of data appears in some practical circumstances, e.g. shock models, industrial stress testing and meteorological data analysis. This paper deals with deriving recurrence relations for the single and the product moments of OSs as well as k-record values from the complementary beta (CB) distribution defined by Jones (2002). The CB model, as a distribution on the unit interval, is more flexible than the well-known beta distribution in problems involving moments of OSs. This fact suggests that the former distribution can be employed rather than the latter when OSs are of the main interest. The results obtained are applied on four real data sets for the issues of parameter estimation as well as providing a visual inspection for goodness-of-fit of the CB model.

Participation in wiki communities: reconsidering their statistical characterization.

Peer production online communities are groups of people that collaboratively engage in the building of common resources such as wikis and open source projects. In such communities, participation is highly unequal: few people concentrate the majority of the workload, while the rest provide irregular and sporadic contributions. The distribution of participation is typically characterized as a power law distribution. However, recent statistical studies on empirical data have challenged the power law dominance in other domains. This work critically examines the assumption that the distribution of participation in wikis follows such distribution. We use statistical tools to analyse over 6,000 wikis from Wikia/Fandom, the largest wiki repository. We study the empirical distribution of each wiki comparing it with different well-known skewed distributions. The results show that the power law performs poorly, surpassed by three others with a more moderated heavy-tail behavior. In particular, the truncated power law is superior to all competing distributions, or superior to some and as good as the rest, in 99.3% of the cases. These findings have implications that can inform a better modeling of participation in peer production, and help to produce more accurate predictions of the tail behavior, which represents the activity and frequency of the core contributors. Thus, we propose to consider the truncated power law as the distribution to characterize participation distribution in wiki communities. Furthermore, the truncated power law parameters provide a meaningful interpretation to characterize the community in terms of the frequency of participation of occasional contributors and how unequal are the group of core contributors. Finally, we found a relationship between the parameters and the productivity of the community and its size. These results open research venues for the characterization of communities in wikis and in online peer production.

Topp-Leone odd log-logistic exponential distribution: Its improved estimators and applications.

In this paper, a new three-parameter lifetime model called the Topp-Leone odd log-logistic exponential distribution is proposed. Its density function can be expressed as a linear mixture of exponentiated exponential densities and can be reversed-J shaped, skewed to the left and to the right. Further, the hazard rate function of the new model can be monotone, unimodal, constant, J-shaped, constant-increasing-decreasing and decreasing-increasing-decreasing and bathtub-shaped. Our main focus is on estimation from a frequentist point of view, yet, some statistical and reliability characteristics for the proposed model are derived. We briefly describe different estimators namely, the maximum likelihood estimators, ordinary least-squares estimators, weighted least-squares estimators, percentile estimators, maximum product of spacings estimators, Cramér-von-Mises minimum distance estimators, Anderson-Darling estimators and right-tail Anderson-Darling estimators. Monte Carlo simulations are performed to compare the performance of the proposed methods of estimation for both small and large samples. We illustrate the performance of the proposed distribution by means of two real data sets and both the data sets show the new distribution is more appropriate as compared to some other well-known distributions.

The New Extension of Inverse Weibull Distribution with Applications of Medicine Data

This paper introduced and studied a new extension of inverse Weibull distribution with three-parameter named as the X-Gamma inverse Weibull (XGIW) distribution. Reliability and hazard rate properties of this distribution are discussed. Maximum likelihood estimation (MLE), maximum product spacing (MPS) Method of the XGIW distribution parameters are discussed. A numerical study using real data analysis and Monte-Carlo simulation are performed to compare between MLE and MPS methods of estimation. The flexibility and potentiality of the XGIW distribution are examined using two real data sets. The cancer data represents remission times (in months) of a random sample of 128 bladder cancer patients and the second data set of leukemia represents 40 patient suffering from leukemia from one of the Ministry of Health Hospitals in Saudi Arabia. The XGIW model can produce better fits than some well-known distributions as generalized inverse Weibull, Kumaraswamy–inverse Weibull, exponentiated generalized inverse Weibull distribution and inverse Weibull distributions.

Statistical Studies Using Goodness-of-Fit Techniques With Dynamic Underwater Visible Light Communication Channel Modeling

In this paper, we studied the probability distribution function (PDF) of the channel gain of dynamic underwater visible light communication (UVLC) model for different types of water using goodness-of-fit (GoF). We used different water channels at different system parameters with dynamic scenarios. First, the Zemax Optics Studio simulator is used to simulate dynamic UVLC channels. UVLC links are examined using Monte Carlo Ray-Tracing (MCRT) simulation for three different water channels; namely, pure sea water channel, clear ocean water channel and coastal ocean water channel at different configuration types. With the presence of blocking divers, we added a dynamic movement in a single input multiple users (SIMU) scenario. Our simulation is based on Zemax Programming Language (ZPL) in sequence with the Zemax Optics Studio. The GoF tests are used to get the degree of fitness between the simulation data and the set of well-known candidate distributions to determine the best fit. We used the R statistical programming language and applied predefined algorithms to determine the optimum degree of fitting for each statistical result. The Kolmogorov-Smirnov (KS), Chi-Square (CS), Cramer-Von-Mises (CVM) and Anderson-Darling (AD) tests are used to represent the four GoF statistical computation techniques for each channel scenario. The received power is enhanced by 35% when the detector movement area increases from 25 m2 to 100 m2 in clear ocean water channel. The obtained results reveal that the UVLC is best represented by Weibull, Gamma or Lognormal distributions.

Bayesian estimation of sparse precision matrices in the presence of Gaussian measurement error

Estimation of sparse, high-dimensional precision matrices is an important and challenging problem. Existing methods all assume that observations can be made precisely but, in practice, this often is not the case; for example, the instruments used to measure the response may have limited precision. The present paper incorporates measurement error in the context of estimating a sparse, high-dimensional precision matrix. In particular, for a Gaussian graphical model with data corrupted by Gaussian measurement error with unknown variance, we establish a general result which gives sufficient conditions under which the posterior contraction rates that hold in the no-measurement-error case carry over to the measurement-error case. Interestingly, this result does not require that the measurement error variance be small. We apply our general result to several cases with well-known prior distributions for sparse precision matrices and also to a case with a newly-constructed prior for precision matrices with a sparse factor-loading form. Two different simulation studies highlight the empirical benefits of accounting for the measurement error as opposed to ignoring it, even when that measurement error is relatively small.

Probabilistic graphlets capture biological function in probabilistic molecular networks.

Molecular interactions have been successfully modeled and analyzed as networks, where nodes represent molecules and edges represent the interactions between them. These networks revealed that molecules with similar local network structure also have similar biological functions. The most sensitive measures of network structure are based on graphlets. However, graphlet-based methods thus far are only applicable to unweighted networks, whereas real-world molecular networks may have weighted edges that can represent the probability of an interaction occurring in the cell. This information is commonly discarded when applying thresholds to generate unweighted networks, which may lead to information loss. We introduce probabilistic graphlets as a tool for analyzing the local wiring patterns of probabilistic networks. To assess their performance compared to unweighted graphlets, we generate synthetic networks based on different well-known random network models and edge probability distributions and demonstrate that probabilistic graphlets outperform their unweighted counterparts in distinguishing network structures. Then we model different real-world molecular interaction networks as weighted graphs with probabilities as weights on edges and we analyze them with our new weighted graphlets-based methods. We show that due to their probabilistic nature, probabilistic graphlet-based methods more robustly capture biological information in these data, while simultaneously showing a higher sensitivity to identify condition-specific functions compared to their unweighted graphlet-based method counterparts. Our implementation of probabilistic graphlets is available at https://github.com/Serdobe/Probabilistic_Graphlets. Supplementary data are available at Bioinformatics online.

The Extended Beta Generator of Distributions: Properties and Applications

We define the extended beta family of distributions to generalize the beta generator pioneered by Eugene et al. [10]. This paper is cited in at least 970 scientific articles and extends more than fifty well-known distributions. Any continuous distribution can be generalized by means of this family. The proposed family can present greater flexibility to model skewed data. Some of its mathematical properties are investigated and maximum likelihood is adopted to estimate its parameters. Further, for different parameter settings and sample sizes, some simulations are conducted. The superiority of the proposed family is illustrated by means of two real data sets.

Distribution Problems, Their Modifications and Applications

In this paper, we deal with well-known distribution problems and discuss their restrictions, extensions and modifications including a possible application in agriculture. We show that the transportation problem can be transformed to an assignment problem using special constraints, but because of NPhardness it needs quite different methods of its solving. Another modification of the transportation problem, the crop problem, has an application in agriculture, but we must deal with uncertain data. We propose a genetic algorithm and fuzzy logic approach for solving these problems.

A software reliability growth model with Gompertz-logarithmic failure time distribution

PurposeThe Gompertz curve has been used in industry to estimate the number of remaining software faults. This paper aims to introduce a family of distributions for fitting software failure times which subsumes the Gompertz distribution.Design/methodology/approachThe mean value function of the corresponding non-homogenous Poisson process software reliability growth model is presented. Model parameters are estimated by the method of maximum likelihood. A comparison of the new model with eight models that use well-known failure time distributions of exponential, gamma, Rayleigh, Weibull, Gompertz, half normal, log-logistic and lognormal is performed according to the several statistical and informational criteria. Moreover, a Shannon entropy approach is used for ranking and model selection.FindingsNumerical experiments are implemented on five real software failure datasets varying from small to large datasets. The results exhibit that the proposed model is promising and particularly outperforms the Gompertz model in all considered datasets.Originality/valueThe proposed model provides optimized reliability estimation.

A New Probability Distribution Family Arising from Truncated Power Lomax Distribution with Application to Weibull Model

The truncated distributions have been widely studied, mainly in life-testing and reliability analysis. In this paper, we introduce a new right truncated generator related to power Lomax distribution, referred to right truncated power Lomax--G family. The proposed family is a generalization of recently [0, 1] truncated Lomax-G family. Statistical properties like; moments, moment generating function, probability weighted moments, quantile function, mean deviation, order statistics and Rényi entropy are derived. Five new sub-models from the truncated family are presented. Maximum likelihood estimation is investigated and simulation issues are discussed for truncated power Lomax Weibull model as particular case from the family. The flexibility of the truncated power Lomax Weibull is assessed by applying it to a real data set. The application indicates that the truncated power Lomax Weibull distribution model can give better fits than other well-known lifetime distributions.

New Directional Residuals to Treat Shape Changes Using Spherical Regression Models

One of the important problems in the statistical shape analysis context is to predict the shape change. Rather than taking the change in terms of time, we confine ourselves to the deformation of the objects represented by a regression-type model. We properly combine the shape definition in terms of the Kendall and Bookstein shape coordinate systems to break down the problem on the sphere. This is achieved through the triangulation of objects; a very popular technique in geometrical mathematics. A novel idea on tracing the residuals of the spherical regression is then proposed, enabling us to invoke the well-known spherical distributions, including von Mises–Fisher density, to make the statistical inference. New directional residuals not only lie on the sphere but also help to easily simulate the spherical responses. The performance of our proposed method is highlighted via running a simulation study as well as analysing a real data set.

The Reflected-Shifted-Truncated Lindley Distribution with Applications

AbstractIn this paper, a new probability density function with bounded domain is presented. The new distribution arises from the Lindley distribution proposed in 1958. It presents the advantage of not including any special function in its formulation. The new transformed model, called the reflected-shifted-truncated Lindley distribution can be used to model left-skewed data. We provide a comprehensive treatment of general mathematical and statistical properties of this distribution. We estimate the model parameters by maximum likelihood methods based on complete and right-censored data. To assess the performance and consistency of the maximum likelihood estimators, we conduct a simulation study with varying sample sizes. Finally, we use the distribution to model left-skewed survival and failure data from two real data sets. For the real data sets containing complete data and right-censored data, this distribution is superior in its ability to sufficiently model the data as compared to the power Lindley, exponentiated power Lindley, generalized inverse Lindley, generalized weighted Lindley and the well-known Gompertz distributions.

A Note on the Time to Failure of a Two-Unit Parallel Redundant System with Deterioration on a Lattice

In this paper, we consider a two-unit parallel redundant system with deterioration on a lattice, where each unit has multi-stage deterioration levels, say, n levels. The transition from one deterioration level to the subsequent level occurs following the well-known Marshall-Olkin bivariate exponential distribution. We derive the closed form of the Laplace transform of the time to system failure in the two-unit parallel redundant system with deterioration on n×n lattice without repair and simultaneous failure, as well as the simple system on 3×3 lattice.

On the Lifetime and Signature of Constrained (k, d)-out-of-n: F Reliability Systems

In the present paper we carry out a reliability study of the constrained (k, d)-out-of-n: F systems with exchangeable components. The signature vector is computed by the aid of the proposed algorithm. In addition, explicit signature-based expressions for the corresponding mean residual lifetime and the conditional mean residual lifetime of the aforementioned reliability system are also provided. For illustration purposes, a well-known multivariate distribution for modelling the lifetimes of the components of the constrained (k, d)-out-of-n: F structure is considered.

Bounded Weighted Exponential Distribution with Applications

In this article, we try to supplement the distribution theory literature by incorporating a new bounded distribution, called the bounded weighted exponential (BWE) distribution in the (0, 1) intervals by transformation method. The proposed distribution exhibits decreasing and left-skewed unimodal density while the hazard rate can have increasing and bathtub shaped. Although our main focus is on the estimation from the frequentist point of view, in addition, we derive some useful structural and statistical properties of the proposed BWE distribution. We briefly describe three classical estimators namely, the maximum likelihood estimators (MLE), the ordinary least-squares estimators (LSE) and the weighted least-squares estimators (WLSE). Monte Carlo simulations are performed to compare performances of the proposed methods of estimation for both moderate and large samples. An application of the model is presented by re-analyzing a real data set and comparisons are made with the fit attained by some other well-known distributions for illustrative purposes.

On the Statistical GARCH Model for Managing the Risk by Employing a Fat-Tailed Distribution in Finance

The Conditional Value-at-Risk (CVaR) is a coherent measure that evaluates the risk for different investing scenarios. On the other hand, since the extreme value distribution has been revealed to furnish better financial and economical data adjustment in contrast to the well-known normal distribution, we here employ this distribution in investigating explicit formulas for the two common risk measures, i.e., VaR and CVaR, to have better tools in risk management. The formulas are then employed under the generalized autoregressive conditional heteroskedasticity (GARCH) model for risk management as our main contribution. To confirm the theoretical discussions of this work, the daily returns of several stocks are considered and worked out. The simulation results uphold the superiority of our findings.

On an Absolute Autoregressive Model and Skew Symmetric Distributions

By exploiting the connection between a popular construction of a well-known skew-normal distribution and an absolute autoregressive process, we show how the stochastic process approach can lead to other skew symmetric distributions, including a skew-Cauchy distribution and some singular distributions. In so doing, we also correct an erroneous skew-Cauchy-distribution in the literature. We discuss the estimation, for dependent data, of the key parameter relating to the skewness.

Efficient Slantwise Aligned Dion-Jacobson Phase Perovskite Solar Cells Based on Trans-1,4-Cyclohexanediamine.

The crystalline orientation and phase distribution are two important parameters for high-performance 2D perovskite solar cells. Therefore, it is essential to understand how the structure of spacer ligands influences the orientation and phase distribution of resulting 2D perovskite films. In this work, a new member of Dion-Jacobson (DJ) phase 2D perovskites based on trans-1,4-cyclohexanediamine (CHDA) is demonstrated and it is found that the crystalline orientation is self-aligned spontaneously, which is different from the well-known graded distribution in controlled sample with its isomer 1,6-diaminohexane (HDA) as spacer ligand. Grazing incident X-ray scattering suggests that the exact alignment is strongly slantwise to the substrate while it is still beneficial for charge transfer along the vertical structure of devices. The devices can achieve high efficiency up to 15.01% for (CHDA)MA3 Pb4 I13 (n= 4), one of the highest efficiencies reported by now. The encapsulated (CHDA)MA3 Pb4 I13 (n= 4) devices can retain 80.7% efficiency for 270 min under continuous maximum power point tracking. (CHDA)MA3 Pb4 I13 (n= 4) devices can retain 96.5% efficiency under 60°C and 74.4% efficiency under 70°C heating for 68 h. The results demonstrate the slantwise aligned DJ phase perovskite solar cells with excellent stability.