Bivariate Beta–Inverse Weibull Distribution: Theory and Applications

Probability distributions have been in use for modeling of random phenomenon in various areas of life. Generalization of probability distributions has been the area of interest of several authors in the recent years. Several situations arise where joint modeling of two random phenomenon is required. In such cases the bivariate distributions are needed. Development of the bivariate distributions necessitates certain conditions, in a field where few work has been performed. This paper deals with a bivariate beta-inverse Weibull distribution. The marginal and conditional distributions from the proposed distribution have been obtained. Expansions for the joint and conditional density functions for the proposed distribution have been obtained. The properties, including product, marginal and conditional moments, joint moment generating function and joint hazard rate function of the proposed bivariate distribution have been studied. Numerical study for the dependence function has been implemented to see the effect of various parameters on the dependence of variables. Estimation of the parameters of the proposed bivariate distribution has been done by using the maximum likelihood method of estimation. Simulation and real data application of the distribution are presented.     

A New Generalized Weibull Model: Classical and Bayesian Estimation

Statistical distributions play a prominent role in applied sciences, particularly in biomedical sciences. The medical data sets are generally skewed to the right, and skewed distributions can be used quite effectively to model such kind of data sets. In the present study, therefore, we propose a new family of distributions suitable for modeling right-skewed medical data sets. The proposed family may be called a new generalized-X family. A special sub-model of the proposed family called a new generalized-Weibull distribution is discussed in detail. The maximum likelihood estimators of the model parameters are obtained. A brief Monte Carlo simulation study is conducted to evaluate the performance of these estimators. Finally, the proposed model is applied to the remission times of the stomach cancer patient’s data. The comparison of the goodness of fit results of the proposed model is made with the other competing models such as Weibull, Kumaraswamy Weibull, and exponentiated Weibull distributions. Certain analytical measures such as Akaike information criterion, Bayesian information criterion, Anderson Darling statistic, and Kolmogorov–Smirnov test statistic are considered to show which distribution provides the best fit to data. Based on these measures, it is showed that the proposed distribution is a reasonable candidate for modeling data in medical sciences and other related fields.     

A compound class of Weibull and power series distributions

In this paper we introduce the Weibull power series (WPS) class of distributions which is obtained by compounding Weibull and power series distributions, where the compounding procedure follows same way that was previously carried out by Adamidis and Loukas (1998). This new class of distributions has as a particular case the two-parameter exponential power series (EPS) class of distributions (Chahkandi and Ganjali, 2009), which contains several lifetime models such as: exponential geometric (Adamidis and Loukas, 1998), exponential Poisson (Kus, 2007) and exponential logarithmic (Tahmasbi and Rezaei, 2008) distributions. The hazard function of our class can be increasing, decreasing and upside down bathtub shaped, among others, while the hazard function of an EPS distribution is only decreasing. We obtain several properties of the WPS distributions such as moments, order statistics, estimation by maximum likelihood and inference for a large sample. Furthermore, the EM algorithm is also used to determine the maximum likelihood estimates of the parameters and we discuss maximum entropy characterizations under suitable constraints. Special distributions are studied in some detail. Applications to two real data sets are given to show the flexibility and potentiality of the new class of distributions.     

Plotting and checking the bivariate distributions of multiple Gaussian data

The geostatistical modeling of continuous variables relies heavily on the multivariate Gaussian distribution. It is remarkably tractable. The multivariate Gaussian distribution is adopted for K multiple variables (often K is between 2 and 10) and for N multiple locations (often N is in the tens of millions). Our focus is on the relationship between the K variables. Each variable is transformed to be univariate Gaussian, but the multivariate nature of the data is not necessarily Gaussian after univariate transformation. If multiple data variables are deemed non-Gaussian, then additional steps need to be taken such as linearization by alternating conditional expectation (ACE) or multivariate transformation by the stepwise conditional transformation (SCT). Although all L-variate distributions (1<L≤K) should be checked, the bivariate distributions are practically important; there are relatively few data in practice to investigate higher order distributions. A quantitative measure of departure from the bivariate Gaussian distribution is established based on quadrants and the distribution of differences from the theoretically expected distribution. Although approximate, the measure of departure is useful for comparing different distributions and guiding the geostatistician to look closer at some data variables. A scatnscores program is shown that will plot all K(K–1)/2 bivariate cross plots associated with K variables. The correlation coefficients, number of data, degree of departure from the bivariate Gaussian distribution, and bivariate Gaussian probability contours associated with specified cumulative probabilities are shown. The data ID numbers can also be shown to help identify outlier or problematic data.     

Smooth semiparametric and nonparametric Bayesian estimation of bivariate densities from bivariate histogram data

Penalized B-splines combined with the composite link model are used to estimate a bivariate density from a histogram with wide bins. The goals are multiple: they include the visualization of the dependence between the two variates, but also the estimation of derived quantities like Kendall’s tau, conditional moments and quantiles. Two strategies are proposed: the first one is semiparametric with flexible margins modeled using B-splines and a parametric copula for the dependence structure; the second one is nonparametric and is based on Kronecker products of the marginal B-spline bases. Frequentist and Bayesian estimations are described. A large simulation study quantifies the performances of the two methods under different dependence structures and for varying strengths of dependence, sample sizes and amounts of grouping. It suggests that Schwarz’s BIC is a good tool for classifying the competing models. The density estimates are used to evaluate conditional quantiles in two applications in social and in medical sciences.     

Further properties and new applications of the nested Dirichlet distribution

Recently, Ng et al. (2009) studied a new family of distributions, namely the nested Dirichlet distributions. This family includes the traditional Dirichlet distribution as a special member and can be adopted to analyze incomplete categorical data. However, other important aspects of the family, such as marginal and conditional distributions and related properties are not yet available in the literature. Moreover, diverse applications of the family to the real world need to be further explored. In this paper, we first obtain the marginal and conditional distributions and other related properties of the nested Dirichlet distribution. We then present new applications of the family in fitting competing-risks model, analyzing incomplete categorical data and evaluating cancer diagnosis tests. Three real data involving failure times of radio transmitter receivers, attitude toward the death penalty and ultrasound ratings for breast cancer metastasis are provided.     

Schema di prove ripetute «senza rimessa» a due variabili ed a due alternative per ciascuna variabile

Sommario In questo articolo è stata studiata la distribuzione bivariata che si ottiene considerando simultaneamente i successi di due modalità di due caratteri di una popolazione quando dalla stessa si estraggono ?senza rimessa?n unità. Di detta distribuzione si sono determinati la legge di probabilità ed i momenti. Inoltre è stata rivolta particolare attenziono alle distribuzioni subordinate che equivalgono alla somma stocastica di dne distribuzioni ipergeometriche. Tra l'altro si è determinato che le medie subordinate giacciono su rette e che le varianze subordinate giacciono su parabole. Completano la memoria esempi numerici ed una esperienza tratta dalle estrazioni dei numeri del lotto la quale mostra molta aderenza alla distribuzione studiata. In appendice si è riportato il flow-chart per il calcolo della probabilità di una determinazione della distribuzione esaminata.

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Summary In this article we have examined the bivariate distribution that results when we consider jointly the succes of two qualities of a population when we draw ?withont replacement?n elements from it. Of this distribution we have examined the probability law and its moments. Moreover we have considered the conditional distributions that are the stocastique sum of two binomial distributions. In particular we have determined that the conditional averages lies on straight lines. This paper is completed by numerical examples and an experience that we have derived from the ?State Lottery? which conforms to the examined bivariate distribution. The flow-chart which is in the appendix can be used to calculate the probability of a determination of the bivariate distribution shown in this note.

     

A generalized modified Weibull distribution for lifetime modeling

A four parameter generalization of the Weibull distribution capable of modeling a bathtub-shaped hazard rate function is defined and studied. The beauty and importance of this distribution lies in its ability to model monotone as well as non-monotone failure rates, which are quite common in lifetime problems and reliability. The new distribution has a number of well-known lifetime special sub-models, such as the Weibull, extreme value, exponentiated Weibull, generalized Rayleigh and modified Weibull distributions, among others. We derive two infinite sum representations for its moments. The density of the order statistics is obtained. The method of maximum likelihood is used for estimating the model parameters. Also, the observed information matrix is obtained. Two applications are presented to illustrate the proposed distribution.     

Using algebraic software to compute the moments of order statistics

Here we examine using the algebraic software Mathematica for computing the moments of the order statistics of several frequently used distribution functions, specifically those for the extreme-value, Weibull, double Weibull, logistic and Cauchy distributions. These are illustrated in obtaining the generalised least-squares estimates of the location and scale parameters of the extreme-value distribution.     

Stochastic volatility models with leverage and heavy-tailed distributions: A Bayesian approach using scale mixtures

This paper studies a heavy-tailed stochastic volatility (SV) model with leverage effect, where a bivariate Student-t distribution is used to model the error innovations of the return and volatility equations. Choy et al. (2008) studied this model by expressing the bivariate Student-t distribution as a scale mixture of bivariate normal distributions. We propose an alternative formulation by first deriving a conditional Student-t distribution for the return and a marginal Student-t distribution for the log-volatility and then express these two Student-t distributions as a scale mixture of normal (SMN) distributions. Our approach separates the sources of outliers and allows for distinguishing between outliers generated by the return process or by the volatility process, and hence is an improvement over the approach of Choy et al. (2008). In addition, it allows an efficient model implementation using the WinBUGS software. A simulation study is conducted to assess the performance of the proposed approach and its comparison with the approach by Choy et al. (2008). In the empirical study, daily exchange rate returns of the Australian dollar to various currencies and daily stock market index returns of various international stock markets are analysed. Model comparison relies on the Deviance Information Criterion and convergence diagnostic is monitored by Geweke’s convergence test.     

The beta generalized half-normal distribution

For the first time, we propose the so-called beta generalized half-normal distribution, which contains some important distributions as special cases, such as the half-normal and generalized half-normal (Cooray and Ananda, 2008) distributions. We derive expansions for the cumulative distribution and density functions which do not depend on complicated functions. We obtain formal expressions for the moments of the new distribution. We examine the maximum likelihood estimation of the parameters and provide the expected information matrix. The usefulness of the new distribution is illustrated through a real data set by showing that it is quite flexible in analyzing positive data instead of the generalized half-normal, half-normal, Weibull and beta Weibull distributions.     

Sufficient Conditions of Extremality of Extreme Distributions F5 - F7 in Generalized Chebyshev Inequalities. III

An optimization problem for a linear functional on the distribution set with two fixed power moments is considered. New sufficient conditions of extremality of extreme distributions F₅ - F₇ in this problem are proposed.     

RDWT domain statistical watermark detector using FRHFMs magnitudes and bivariate Cauchy-Rayleigh distribution

Imperceptibility, robustness and data payload, which are complimentary to each other, are widely considered as the three main properties vital for any image watermarking systems. It is a challenging work to design a statistical model-based multiplicative watermarking scheme for achieving the tradeoff among three main properties. In this paper, we propose a novel statistical image watermarking scheme by modeling local redundant discrete wavelet transform (RDWT) and fast Radial harmonic Fourier moments (FRHFMs) magnitudes with bivariate Cauchy-Rayleigh distribution. Our image watermarking scheme consists of two parts, namely, embedding and detection. In the embedding process, RDWT is firstly performed on the host image and RDWT highpass subbands are divided into non-overlapping blocks. Then FRHFMs are computed on RDWT coefficient blocks. And finally, the watermark signal is inserted into robust RDWT-FRHFMs magnitudes through a non-linear multiplicative approach. In the detection process, robust local RDWT-FRHFMs magnitudes are firstly modeled by employing bivariate Cauchy-Rayleigh distribution, which can capture accurately both marginal distributions and strong dependencies of local RDWT-FRHFMs magnitudes. Statistical model parameters are then estimated effectively by the method of logarithmic cumulants (MoLC) approach. And finally, an image watermark detector for multiplicative watermarking is developed using bivariate Cauchy-Rayleigh model and locally most powerful (LMP) test. Also, we utilize the bivariate Cauchy-Rayleigh model to derive the closed-form expressions for the watermark detector. After performance testing and comparison with the experimental results of existing methods, the proposed statistical image watermarking method has achieved relatively ideal results in terms of robustness, imperceptibility and data payload.

     

A Copula‐Based BRDF Model

In this paper, we introduce a novel approach for modeling surface reflection. We focus on using a family of probability distributions called Archimedean copulas as BRDF models. The Archimedean representation has an attractive property in that the multivariate distributions are characterized by their marginal distributions through a single univariate function only. It is shown that the proposed model meets the reciprocity property of reflection. Based on measured BRDF data, we demonstrate that the proposed approach provides a good approximation to BRDF. Empirical comparisons are made with some classically used BRDF models.     

Exponentiated Weibull–Poisson distribution: Model,properties and applications

In this paper we propose a new four-parameters distribution with increasing, decreasing, bathtub-shaped and unimodal failure rate, called as the exponentiated Weibull–Poisson (EWP) distribution. The new distribution arises on a latent complementary risk problem base and is obtained by compounding exponentiated Weibull (EW) and Poisson distributions. This distribution contains several lifetime sub-models such as: generalized exponential-Poisson (GEP), complementary Weibull–Poisson (CWP), complementary exponential-Poisson (CEP), exponentiated Rayleigh–Poisson (ERP) and Rayleigh–Poisson (RP) distributions.We obtain several properties of the new distribution such as its probability density function, its reliability and failure rate functions, quantiles and moments. The maximum likelihood estimation procedure via an EM-algorithm is presented in this paper. Sub-models of the EWP distribution are studied in details. In the end, applications to two real data sets are given to show the flexibility and potentiality of the new distribution.     

Lifetime analysis based on the generalized Birnbaum-Saunders distribution

In this paper, we consider a family of generalized Birnbaum-Saunders distributions and present a lifetime analysis based mainly on the hazard function of this model. In addition, we carry out maximum likelihood estimation by using an iterative algorithm, which produces robust estimates. Asymptotic inference is also presented. Next, the quality of the estimation method is examined by means of Monte Carlo simulations. We then provide a practical example to illustrate the obtained results. From this example and based on goodness-of-fit methods, we show that the GBS distribution results in a more appropriate model for modeling fatigue data than other models commonly used to model this type of data. Finally, we estimate the hazard function and the critical point of this function.     

Estimating the parameters of a generalized lambda distribution

The method of moments is a popular technique for estimating the parameters of a generalized lambda distribution (GLD), but published results suggest that the percentile method gives superior results. However, the percentile method cannot be implemented in an automatic fashion, and automatic methods, like the starship method, can lead to prohibitive execution time with large sample sizes. A new estimation method is proposed that is automatic (it does not require the use of special tables or graphs), and it reduces the computational time. Based partly on the usual percentile method, this new method also requires choosing which quantile u to use when fitting a GLD to data. The choice for u is studied and it is found that the best choice depends on the final goal of the modeling process. The sampling distribution of the new estimator is studied and compared to the sampling distribution of estimators that have been proposed. Naturally, all estimators are biased and here it is found that the bias becomes negligible with sample sizes n?2×10³. The .025 and .975 quantiles of the sampling distribution are investigated, and the difference between these quantiles is found to decrease proportionally to . The same results hold for the moment and percentile estimates. Finally, the influence of the sample size is studied when a normal distribution is modeled by a GLD. Both bounded and unbounded GLDs are used and the bounded GLD turns out to be the most accurate. Indeed it is shown that, up to n=10⁶, bounded GLD modeling cannot be rejected by usual goodness-of-fit tests.     

Inverse Length Biased Maxwell Distribution: Statistical Inference with an Application

In this paper, we suggested and studied the inverse length biased Maxell distribution (ILBMD) as a new continuous distribution of one parameter. The ILBMD is obtained by considering the inverse transformation technique of the Maxwell length biased distribution. Statistical characteristics of the ILBMD such as the moments, moment generating function, mode, quantile function, the coefficient of variation, coefficient of skewness, Moors and Bowley measures of kurtosis and skewness , stochastic ordering, stress-strength reliability, and mean deviations are obtained. In addition, the Bonferroni and Lorenz curves, Gini index, the reliability function, the hazard rate function, the reverse hazard rate function, the odds function, and the distributions of order statistics for the ILBMD, are presented. The ILBMD parameter is estimated using the maximum likelihood method, the method of moments, the maximum product of spacing technique, the ordinary and weight least square procedures, and the Cramer-Von-Mises methods. The Fishers information, as well as the Rényi and q-entropies, are derived. To investigate the usefulness of the proposed lifetime distribution and to illustrate the purpose of the study, a real dataset of the relief times of 20 patients receiving an analgesic is used.     

Flexible Random Intercept Models for Binary Outcomes Using Mixtures of Normals

Random intercept models for binary data are useful tools for addressing between-subject heterogeneity. Unlike linear models, the non-linearity of link functions used for binary data force a distinction between marginal and conditional interpretations. This distinction is blurred in probit models with a normally distributed random intercept because the resulting model implies a probit marginal link as well. That is, this model is closed in the sense that the distribution associated with the marginal and conditional link functions and the random effect distribution are all of the same family. It is shown that the closure property is also attained when the distributions associated with the conditional and marginal link functions and the random effect distribution are mixtures of normals. The resulting flexible family of models is demonstrated to be related to several others present in the literature and can be used to synthesize several seemingly disparate modeling approaches. In addition, this family of models offers considerable computational benefits. A diverse series of examples is explored that illustrates the wide applicability of this approach.     

Exact Inference Using Variable Integrating Constant Importance Distributions

In this paper a new class of importance distributions that are based on normaland t-distributions are introduced. They are labeled variable integratingconstant distributions or VIC distributions. The main innovation in thisclass of distributions is that the standard error associated with eachprameter is expanded in an exponential polynomial function of the scaleddistance from its mode, with a different polynomial used in each direction andfor each parameter. Resulting variances differ at each point in the parameterspace, thereby breaking the flexibility limitations that are inherent innormal and t-distributions allowing them to be closely matched to theposterior distribution whose moments are the goal of the inference exercise.Two illustrations are presented in which the performance of normal, t,normal-VIC, and t-VIC importance distributions are compared. Results show theVIC distributions to produce globally stable weights and substantialimprovements in the numerical efficiency of importance sampling andindependence Metropolis estimates.