The exponentiated-log-logistic geometric distribution: Dual activation

ABSTRACT

The log-logistic distribution is commonly used to model lifetime data. We propose a wider distribution, named the exponentiated log-logistic geometric distribution, based on a double activation approach. We obtain the quantile function, ordinary moments, and generating function. The method of maximum likelihood is used to estimate the model parameters. We propose a new extended regression model based on the logarithm of the exponentiated log-logistic geometric distribution. This regression model can be very useful in the analysis of real data and could provide better fits than other special regression models. The potentiality of the new models is illustrated by means of two applications to real lifetime data sets.     

A bimodal flexible distribution for lifetime data

ABSTRACT

A four-parameter extended bimodal lifetime model called the exponentiated log-sinh Cauchy distribution is proposed. It extends the log-sinh Cauchy and folded Cauchy distributions. We derive some of its mathematical properties including explicit expressions for the ordinary moments and generating and quantile functions. The method of maximum likelihood is used to estimate the model parameters. We implement the fit of the model in the GAMLSS package and provide the codes. The flexibility of the model is illustrated by means of three real data sets.     

The Kumaraswamy normal linear regression model with applications

ABSTRACT

For any continuous baseline G distribution, Cordeiro and Castro pioneered the Kumaraswamy-G family of distributions with two extra positive parameters, which generalizes both Lehmann types I and II classes. We study some mathematical properties of the Kumaraswamy-normal (KwN) distribution including ordinary and incomplete moments, mean deviations, quantile and generating functions, probability weighted moments, and two entropy measures. We propose a new linear regression model based on the KwN distribution, which extends the normal linear regression model. We obtain the maximum likelihood estimates of the model parameters and provide some diagnostic measures such as global influence, local influence, and residuals. We illustrate the potentiality of the introduced models by means of two applications to real datasets.     

Inference for IZAWA’S Bivariate Gamma Distribution

Abstract

In this article, we aim to establish some theoretical properties of Izawa’s bivariate gamma distribution having equal shape parameters. First, we propose a procedure to obtain the maximum likelihood estimates and derive an expression for the Fisher information. Simulation studies illuminate the properties of maximum likelihood estimators. We also establish an asymptotic test for independence based on the limiting distribution of maximum likelihood estimators.     

The logistic-X family of distributions and its applications

ABSTRACT

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.     

The Weibull–Dagum distribution: Properties and applications

ABSTRACT

In this article, 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.     

Influence diagnostics in the capital asset pricing model under elliptical distributions

In this paper we consider the Capital Asset Pricing Model under Elliptical (symmetric) Distributions. This class of distributions, which contains the normal distribution, t, contaminated normal and power exponential, among others, offers a more flexible framework for modelling asset prices or returns. In order to analyze the sensibility to possible outliers and/or atypical returns of the maximum likelihood estimators, the local influence method was implemented. The results are illustrated by using a set of shares from companies who trade in the Chilean Stock Market. Our main conclusion is that symmetric distributions having heavier tails than those of the normal distribution, especially the t distribution with small degrees of freedom, show a better fit and allow the reduction of the influence of atypical returns in the maximum likelihood estimators.     

The Poisson-X family of distributions

ABSTRACT

Recently, Risti? and Nadarajah [A new lifetime distribution. J Stat Comput Simul. 2014;84:135–150] introduced the Poisson generated family of distributions and investigated the properties of a special case named the exponentiated-exponential Poisson distribution. In this paper, we study general mathematical properties of the Poisson-X family in the context of the T-X family of distributions pioneered by Alzaatreh et al. [A new method for generating families of continuous distributions. Metron. 2013;71:63–79], which include quantile, shapes of the density and hazard rate functions, asymptotics and Shannon entropy. We obtain a useful linear representation of the family density and explicit expressions for the ordinary and incomplete moments, mean deviations and generating function. One special lifetime model called the Poisson power-Cauchy is defined and some of its properties are investigated. This model can have flexible hazard rate shapes such as increasing, decreasing, bathtub and upside-down bathtub. The method of maximum likelihood is used to estimate the model parameters. We illustrate the flexibility of the new distribution by means of three applications to real life data sets.     

Inference on proportional hazard rate model parameter under Type-I progressively hybrid censoring scheme

ABSTRACT

In this paper, under Type-I progressive hybrid censoring sample, we obtain maximum likelihood estimator of unknown parameter when the parent distribution belongs to proportional hazard rate family. We derive the conditional probability density function of the maximum likelihood estimator using moment-generating function technique. The exact confidence interval is obtained and compared by conducting a Monte Carlo simulation study for burr Type XII distribution. Finally, we obtain the Bayes and posterior regret gamma minimax estimates of the parameter under a precautionary loss function with precautionary index k = 2 and compare their behavior via a Monte Carlo simulation study.     

Modified skew-slash distribution

Abstract

In this work, we introduce a new skewed slash distribution. This modification of the skew-slash distribution is obtained by the quotient of two independent random variables. That quotient consists on a skew-normal distribution divided by a power of an exponential distribution with scale parameter equal to two. In this way, the new skew distribution has a heavier tail than that of the skew-slash distribution. We give the probability density function expressed by an integral, but we obtain some important properties useful for making inferences, such as moment estimators and maximum likelihood estimators. By way of illustration and by using real data, we provide maximum likelihood estimates for the parameters of the modified skew-slash and the skew-slash distributions. Finally, we introduce a multivariate version of this new distribution.     

Estimation of exponential-polynomial distribution by holonomic gradient descent

ABSTRACT

We study the holonomic gradient decent for maximum likelihood estimation of exponential-polynomial distribution, whose density is the exponential function of a polynomial in the random variable. We first consider the case that the support of the distribution is the set of positive reals. We show that the maximum likelihood estimate (MLE) can be easily computed by the holonomic gradient descent, even though the normalizing constant of this family does not have a closed-form expression, and discuss the determination of the degree of the polynomial based on the score test statistic. Then, we present extensions to the whole real line and to the bivariate distribution on the positive orthant.     

Extreme value index estimator using maximum likelihood and moment estimation

ABSTRACT

When a distribution function is in the max domain of attraction of an extreme value distribution, its tail can be well approximated by a generalized Pareto distribution. Based on this fact we use a moment estimation idea to propose an adapted maximum likelihood estimator for the extreme value index, which can be understood as a combination of the maximum likelihood estimation and moment estimation. Under certain regularity conditions, we derive the asymptotic normality of the new estimator and investigate its finite sample behavior by comparing with several classical or competitive estimators. A simulation study shows that the new estimator is competitive with other estimators in view of average bias, average MSE, and coefficient of variance of the new device for the optimal selection of the threshold.     

Analysis of left censored data from the generalized exponential distribution

The generalized exponential distribution proposed by Gupta and Kundu [Gupta, R.D and Kundu, D., 1999, Generalized exponential distributions. Australian and New Zealand Journal of Statistics, 41(2), 173–188.] is an important lifetime distribution in survival analysis. In this paper, we consider the maximum likelihood estimation procedure of the parameters of the generalized exponential distribution when the data are left censored. We obtain the maximum likelihood estimators of the unknown para-meters and the Fisher information matrix. Simulation studies are carried out to observe the performance of the estimators in small sample.     

A Three-Parameter Asymmetric Laplace Distribution and Its Extension

ABSTRACT

In this article, a new three-parameter asymmetric Laplace distribution and its extension are introduced. This includes as special case the symmetric Laplace double-exponential distribution. The distribution has established a direct link to estimation of quantile and quantile regression. Properties of the new distribution are presented. Application is made to a flood data modeling example.     

A new generalized odd log-logistic family of distributions

We introduce and study general mathematical properties of a new generator of continuous distributions with three extra parameters called the new generalized odd log-logistic family of distributions. The proposed family contains several important classes discussed in the literature as submodels such as the proportional reversed hazard rate and odd log-logistic classes. Its density function can be expressed as a mixture of exponentiated densities based on the same baseline distribution. Some of its mathematical properties including ordinary moments, quantile and generating functions, entropy measures, and order statistics, which hold for any baseline model, are presented. We also present certain characterization of the proposed distribution and derive a power series for the quantile function. We discuss the method of maximum likelihood to estimate the model parameters. We study the behavior of the maximum likelihood estimator via simulation. The importance of the new family is illustrated by means of two real data sets. These applications indicate that the new family can provide better fits than other well-known classes of distributions. The beauty and importance of the new family lies in its ability to model real data.     

Quantile regression in longitudinal studies with dropouts and measurement errors

ABSTRACT

Quantile regression models, as an important tool in practice, can describe effects of risk factors on the entire conditional distribution of the response variable with its estimates robust to outliers. However, there is few discussion on quantile regression for longitudinal data with both missing responses and measurement errors, which are commonly seen in practice. We develop a weighted and bias-corrected quantile loss function for the quantile regression with longitudinal data, which allows both missingness and measurement errors. Additionally, we establish the asymptotic properties of the proposed estimator. Simulation studies demonstrate the expected performance in correcting the bias resulted from missingness and measurement errors. Finally, we investigate the Lifestyle Education for Activity and Nutrition study and confirm the effective of intervention in producing weight loss after nine month at the high quantile.     

Quantile inference for heteroscedastic regression models

Consider the nonparametric heteroscedastic regression model Y=m(X)+σ(X)?, where m(·) is an unknown conditional mean function and σ(·) is an unknown conditional scale function. In this paper, the limit distribution of the quantile estimate for the scale function σ(X) is derived. Since the limit distribution depends on the unknown density of the errors, an empirical likelihood ratio statistic based on quantile estimator is proposed. This statistics is used to construct confidence intervals for the variance function. Under certain regularity conditions, it is shown that the quantile estimate of the scale function converges to a Brownian motion and the empirical likelihood ratio statistic converges to a chi-squared random variable. Simulation results demonstrate the superiority of the proposed method over the least squares procedure when the underlying errors have heavy tails.     

Extreme Quantile Estimation for Autoregressive Models

ABSTRACT

A quantile autoregresive model is a useful extension of classical autoregresive models as it can capture the influences of conditioning variables on the location, scale, and shape of the response distribution. However, at the extreme tails, standard quantile autoregression estimator is often unstable due to data sparsity. In this article, assuming quantile autoregresive models, we develop a new estimator for extreme conditional quantiles of time series data based on extreme value theory. We build the connection between the second-order conditions for the autoregression coefficients and for the conditional quantile functions, and establish the asymptotic properties of the proposed estimator. The finite sample performance of the proposed method is illustrated through a simulation study and the analysis of U.S. retail gasoline price.     

Quantile estimation in the proportional hazards model of random censorship

The asymptotic theory is given for quantile estimation in the proportional hazards model of random censorship. In this model, the tail of the censoring distribution function is some power of the tail of the survival distribution function. The quantile estimator is based on the maximum likelihood estimator for the survival time distribution, due to Abdushukurov, Cheng and Lin.     

The first-order random coefficient integer valued autoregressive process with the occasional level shift random noise based on dual empirical likelihood

Abstract

This paper investigates the first-order random coefficient integer valued autoregressive process with the occasional level shift random noise based on dual empirical likelihood. The limiting distribution of log empirical likelihood ratio statistic is constructed. Asymptotic convergence and confidence region results of empirical likelihood ratio are given. Hypothesis testing is considering, and maximum empirical likelihood estimation for parameter is acquired. Simulations are given to show that the maximum empirical likelihood estimation is more efficient than the conditional least squares estimation.