Marcelo Bourguignon

General Results for the Transmuted Family of Distributions and New Models

The transmuted family of distributions has been receiving increased attention over the last few years. For a baseline G distribution, we derive a simple representation for the transmuted-G family density function as a linear mixture of the G and exponentiated-G densities. We investigate the asymptotes and shapes and obtain explicit expressions for the ordinary and incomplete moments, quantile and generating functions, mean deviations, Renyi and Shannon entropies, and order statistics and their moments. We estimate the model parameters of the family by the method of maximum likelihood. We prove empirically the flexibility of the proposed model by means of an application to a real data set.

A new compounding family of distributions

We propose a new four-parameter family of distributions by compounding the generalized gamma and power series distributions. The compounding procedure is based on the work by Marshall and Olkin (1997) and defines 76 sub-models. Further, it includes as special models the Weibull power series and exponential power series distributions. Some mathematical properties of the new family are studied including moments and generating function. Three special models are investigated in detail. Maximum likelihood estimation of the unknown parameters for complete sample is discussed. Two applications of the new models to real data are performed for illustrative purposes.

Extended Poisson INAR(1) processes with equidispersion, underdispersion and overdispersion

ABSTRACT Real count data time series often show the phenomenon of the underdispersion and overdispersion. In this paper, we develop two extensions of the first-order integer-valued autoregressive process with Poisson innovations, based on binomial thinning, for modeling integer-valued time series with equidispersion, underdispersion, and overdispersion. The main properties of the models are derived. The methods of conditional maximum likelihood, Yule–Walker, and conditional least squares are used for estimating the parameters, and their asymptotic properties are established. We also use a test based on our processes for checking if the count time series considered is overdispersed or underdispersed. The proposed models are fitted to time series of the weekly number of syphilis cases and monthly counts of family violence illustrating its capabilities in challenging the overdispersed and underdispersed count data.

General Mathematical Properties, Regression and Applications of the Log-Gamma-Generated Family

ABSTRACT The construction of some wider families of continuous distributions obtained recently has attracted applied statisticians due to the analytical facilities available for easy computation of special functions in programming software. We study some general mathematical properties of the log-gamma-generated (LGG) family defined by Amini, MirMostafaee, and Ahmadi (2014). It generalizes the gamma-generated class pioneered by Ristić and Balakrishnan (2012). We present some of its special models and derive explicit expressions for the ordinary and incomplete moments, generating and quantile functions, mean deviations, Bonferroni and Lorenz curves, Shannon entropy, Rényi entropy, reliability, and order statistics. Models in this family are compared with nested and non nested models. Further, we propose and study a new LGG family regression model. We demonstrate that the new regression model can be applied to censored data since it represents a parametric family of models and therefore can be used more effectively in the analysis of survival data. We prove that the proposed models can provide consistently better fits in some applications to real data sets.

A new generalized gamma distribution with applications

SYNOPTIC ABSTRACT The modeling and analysis of lifetimes is an important aspect of statistical work in a wide variety of scientific and technological fields. We introduce and study the gamma-Nadarajah–Haghighi model, which can be interpreted as a truncated generalized gamma distribution (Stacy, 1962). It can have a constant, decreasing, increasing, upside-down bathtub or bathtub-shaped hazard rate function depending on the parameter values. We demonstrate that the new density function can be expressed as a mixture of exponentiated Nadarajah–Haghighi densities. Various of its structural properties are derived, including explicit expressions for the moments, quantile and generating functions, skewness, kurtosis, mean deviations, Bonferroni and Lorenz curves, probability weighted moments, and two types of entropy. We also investigate the order statistics. The method of maximum likelihood is used for estimating the model parameters and the observed information matrix is derived. We illustrate the flexibility of the new distribution by means of two applications to real datasets.

The effects of additive outliers in INAR(1) process and robust estimation

ABSTRACT In this paper, methods based on ranks and signs for estimating the parameters of the first-order integer-valued autoregressive model in the presence of additive outliers are proposed. In particular, we use the robust sample autocorrelations based on ranks and signs to obtain estimators for the parameters of the Poisson INAR(1) process. The effects of additive outliers on the estimates of parameters of integer-valued time series are examined. Some numerical results of the estimators are presented with a discussion of the obtained results. The proposed methods are applied to a dataset concerning the number of different IP addresses accessing the server of the pages of the Department of Statistics of the University of Würzburg. The results presented here give motivation to use the methodology in practical situations in which Poisson INAR(1) process contains additive outliers.

A skew integer valued time series process with generalized Poisson difference marginal distribution

In this article, we propose a new integer-valued autoregressive process with generalized Poisson difference marginal distributions based on difference of two quasi-binomial thinning operators. This model is suitable for data sets on ℤ = {..., -2, -1, 0, 1, 2,...} and can be viewed as a generalization of the Poisson difference INAR(1) process. An advantage of the difference of two generalized Poisson random variables is it can have longer or shorter tails compared to the Poisson difference distribution. We present some basic properties of the process like mean, variance, skewness, and kurtosis, and conditional properties of the process are derived. Yule-Walker estimators are considered for the unknown parameters of the model and a Monte Carlo simulation is presented to study the performance of estimators. An application to a real data set is discussed to show the potential for practice of our model.

A new geometric INAR(1) process based on counting series with deflation or inflation of zeros

ABSTRACT In this paper, we introduce a new non-negative integer-valued autoregressive time series model based on a new thinning operator, so called generalized zero-modified geometric (GZMG) thinning operator. The first part of the paper is devoted to the distribution, GZMG distribution, which is obtained as the convolution of the zero-modified geometric (ZMG) distributed random variables. Some properties of this distribution are derived. Then, we construct a thinning operator based on the counting processes with ZMG distribution. Finally, an INAR(1) time series model is introduced and its properties including estimation issues are derived and discussed. A small Monte Carlo experiment is conducted to evaluate the performance of maximum likelihood estimators in finite samples. At the end of the paper, we consider an empirical illustration of the introduced INAR(1) model.

Zero-Inflated NGINAR(1) process

ABSTRACT In this paper, we develop a zero-inflated NGINAR(1) process as an alternative to the NGINAR(1) process (Ristić, Nastić, and Bakouch 2009) when the number of zeros in the data is larger than the expected number of zeros by the geometric process. The proposed process has zero-inflated geometric marginals and contains the NGINAR(1) process as a particular case. In addition, various properties of the new process are derived such as conditional distribution and autocorrelation structure. Yule-Walker, probability based Yule-Walker, conditional least squares and conditional maximum likelihood estimators of the model parameters are derived. An extensive Monte Carlo experiment is conducted to evaluate the performances of these estimators in finite samples. Forecasting performances of the model are discussed. Application to a real data set shows the flexibility and potentiality of the new model.

Some simple estimators for the two-parameter gamma distribution

ABSTRACT In this paper, we propose two new simple estimation methods for the two-parameter gamma distribution. The first one is a modified version of the method of moments, whereas the second one makes use of some key properties of the distribution. We then derive the asymptotic distributions of these estimators. Also, bias-reduction methods are suggested to reduce the bias of these estimators. The performance of the estimators are evaluated through a Monte Carlo simulation study. The probability coverages of confidence intervals are also discussed. Finally, two examples are used to illustrate the proposed methods.