In 2010, the Samejima–Bolfarine–Bazán (SBB) Item Response Theory (IRT) models were introduced by (Journal of Educational and Behavioral Statistics 35 (2010) 693–713) under a Bayesian approach. These models extend the regular Bayesian One and Two Parameter Logistic IRT models by incorporating a parameter accounting for asymmetry of the Item Characteristic Curve (ICC) which is named the complexity of the item. It includes the Logistic Positive Exponent (LPE) IRT model formulated initially by (Psychometrika 65 (2000) 319–335) and the Reflection of the LPE (RLPE). In the present work, new properties of the SBB models are developed including a random effect for testlet structures with a Bayesian inference through a Markov chain Monte Carlo (MCMC) algorithm which includes the parameter estimation and model comparison. The asymmetric behavior of the Item Characteristic Curve (ICC) is detected using a marginal item information function. Two simulation studies are developed to analyze the sensitiveness of the penalized parameter in the asymmetric behavior of the ICC and to evaluate the parameter recovery of the proposed model. A real data set, with a testlet structure and empirical evidence of asymmetric behavior of the ICCs, is used to apply the models.

We develop here a multivariate generalization of Birnbaum–Saunders (BS) distribution based on the multivariate skew-normal distribution. Some distributional characteristics and properties are presented, as well as a simple and efficient EM algorithm for the iterative computation of the maximum likelihood (ML) estimates of model parameters, through the hierarchical representation of the proposed model. The standard errors of the maximum likelihood estimates are calculated from the observed Fisher information matrix. Moreover, by using the tools, we present a log-linear regression model, where the the ML estimates are once again obtained using an EM algorithm. Finally, simulation studies and two applications to real data sets are presented for illustrating the model and the inferential results developed here.

The present study introduces a new bivariate distribution based on the Sushila distribution to model bivariate lifetime data in presence of a cure fraction, right- censored data and covariates. The new bivariate probability distribution was obtained using a methodology used in the reliability theory based on fatal shocks, usually used to build new bivariate models. Additionally, the cure rate was introduced in the model based on a generalization of standard mixture models extensively used for the univariate lifetime case. The inferences of interest for the model parameters are obtained under a Bayesian approach using MCMC (Markov Chain Monte Carlo) simulation methods to generate samples of the joint posterior distribution for all parameters of the model. A simulation study was developed to study the inferential properties of the new methodology.The proposed methodology also was applied to analyze a set of real medical data obtained from a retrospective cohort study that aimed to assess specific clinical conditions that affect the lives of patients with diabetic retinopathy. For the discrimination of the proposed model with other usual models used in the analysis of bivariate survival data, some Bayesian techniques of model discrimination were used and the model validation was verified from usual Cox-Snell residuals, which allowed us to identify the adequacy of the proposed bivariate cure rate model.

Suppose that ${\mathit{X}}_{\mathit{n}}$ is a sample of size n with log likelihood $\mathit{n}\mathit{l}(\mathit{\theta })$, where θ is an unknown parameter in ${\mathit{R}}^{\mathit{p}}$ having a prior distribution $\mathit{\xi }(\mathit{\theta })$. We need not assume that the sample values are independent or even stationary. Let $\hat{\mathit{\theta }}$ be the maximum likelihood estimate (MLE). We show that $\mathit{\theta }|{\mathit{X}}_{\mathit{n}}$ is asymptotically normal with mean $\hat{\mathit{\theta }}$ and covariance $-{\mathit{n}}^{-1}{\mathit{l}}_{․,․}{(\hat{\mathit{\theta }})}^{-1}$, where ${\mathit{l}}_{․,․}(\mathit{\theta })={\partial }^2\mathit{l}(\mathit{\theta })/\partial \mathit{\theta }\partial {\mathit{\theta }}^{\prime }$. In contrast, $\hat{\mathit{\theta }}|\mathit{\theta }$ is asymptotically normal with mean θ and covariance ${\mathit{n}}^{-1}{[\mathit{I}(\mathit{\theta })]}^{-1}$, where $\mathit{I}(\mathit{\theta })=-\mathit{E}[{\mathit{l}}_{․,․}(\hat{\mathit{\theta }})|\mathit{\theta }]$ is Fisher’s information. So, frequentist inference conditional on θ cannot be used to approximate Bayesian inference, except for exponential families. However, under mild conditions $-{\mathit{l}}_{․,․}(\hat{\mathit{\theta }})|\mathit{\theta }\to \mathit{I}(\mathit{\theta })$ in probability. So, Bayesian inference (that is, conditional on ${\mathit{X}}_{\mathit{n}}$) can be used to approximate frequentist inference.

For $\mathit{t}(\mathit{\theta })$ any smooth function, we obtain posterior cumulant expansions, posterior Edgeworth–Cornish–Fisher (ECF) expansions and posterior tilted Edgeworth expansions for $\mathcal{L}\mathit{t}(\mathit{\theta })|{\mathit{X}}_{\mathit{n}}$, as well as confidence regions for $\mathit{t}(\mathit{\theta })|{\mathit{X}}_{\mathit{n}}$ of high accuracy. We also give expansions for the Bayes estimate (estimator) of $\mathit{t}(\mathit{\theta })$ about $\mathit{t}(\hat{\mathit{\theta }})$, and for the maximum a posteriori estimate about $\hat{\mathit{\theta }}$, as well as their relative efficiencies with respect to squared error loss.

The problem of estimating location (scale) parameters ${\mathit{\theta }}_1$ and ${\mathit{\theta }}_2$ of two distributions when the ordering between them is known apriori (say, ${\mathit{\theta }}_1\le {\mathit{\theta }}_2$) has been extensively studied in the literature. Many of these studies are centered around deriving estimators that dominate the best location (scale) equivariant estimators, for the unrestricted case, by exploiting the prior information that ${\mathit{\theta }}_1\le {\mathit{\theta }}_2$. Several of these studies consider specific distributions such that the associated random variables are statistically independent. In this paper, we consider a general bivariate model and a general loss function, and unify various results proved in the literature. We also consider applications of these results to a bivariate normal and a Cheriyan and Ramabhadran’s bivariate gamma model. A simulation study is also considered to compare the risk performances of various estimators under bivariate normal and Cheriyan and Ramabhadran’s bivariate gamma models.

In this article, we propose a simple method to study high concentrations in spherical data, in particular in the directional perspective. To this end, we define a distance-based distribution in the interval $(0,1)$, called the T-statistic distance ($\mathit{DT}$) law, to describe the scattering of points on the unit sphere. Our model is derived from the von Mises–Fisher (vMF) distribution, which is one of the most known directional laws. We show that if the data are vMF distributed, their concentration can be modeled by our distribution. Some of its properties are derived and discussed: Moment generating function, kurtosis and skewness. Likelihood-based inference procedures are provided for both points and hypotheses concerning the $\mathit{DT}$ concentration. Further, we propose a new test statistic in terms of the $\mathit{DT}$ distribution to deal with scattering within spherical data and derive its exact density. Numerical studies show that maximum likelihood estimates behave asymptotically well even for small sample sizes and that the likelihood ratio test for the $\mathit{DT}$ distribution often performs better than Wald tests. We apply our model to paleomagnetic data to illustrate how it is used to analyze spherical data concentration. Results show that the distance-based approach works well to identify high concentration on the unit sphere.

The supremum of the standardized empirical process is a promising statistic for testing whether the distribution function F of i.i.d. real random variables is either equal to a given distribution function ${\mathit{F}}_0$ (hypothesis) or $\mathit{F}\ge {\mathit{F}}_0$ (one-sided alternative). Since (The Annals of Statistics 7 (1979) 108–115) it is well-known that an affine-linear transformation of the suprema converge in distribution to the Gumbel law as the sample size tends to infinity. This enables the construction of an asymptotic level-α test. However, the rate of convergence is extremely slow. As a consequence the probability of the type I error is much larger than α even for sample sizes beyond 10.000. Now, the standardization consists of the weight-function $1/\sqrt{{\mathit{F}}_0(\mathit{x})(1-{\mathit{F}}_0(\mathit{x}))}$. Substituting the weight-function by a suitable random constant leads to a new test-statistic, for which we can derive the exact distribution (and the limit distribution) under the hypothesis. A comparison via a Monte-Carlo simulation shows that the new test is uniformly better than the Smirnov-test and an appropriately modified test due to (The Annals of Statistics 11 (1983) 933–946). Our methodology also works for the two-sided alternative $\mathit{F}\ne {\mathit{F}}_0$.

The class of locally stationary processes assumes a time-varying (tv) spectral representation and finite second moment. Different areas have observed phenomena with heavy tail distributions or infinite variance. Using stable distribution as a heavy-tailed innovation is an attractive option. However, its estimation is difficult due to the absence of a closed expression for the density function and the non-existence of second moment. In this paper, we propose the tvARMA model with tempered stable innovations, which have lighter tails than the stable distribution and have finite moments. A two-step method is proposed to estimate this parametric model. In the first step, we use the blocked Whittle estimation to estimate the time-varying structure of the process. In the second step, we recover residuals from the first step and use the maximum likelihood method to estimate the rest of the parameters related to the standardized classical tempered stable (stdCTS) innovations. We perform simulation studies to evaluate the consistency of the maximum likelihood estimation of independent stdCTS samples. Then, we execute simulations to study the two-step estimation of our model. Finally, an empirical application is illustrated.

In this paper, we propose a new regression model based on an extension of the Rice distribution to model linear and nonlinear effects for correlated data in the presence of bimodality. The new model is referred to as the odd log-logistic Rice distribution and we provide general mathematical properties, including the event risk and moments. We discuss parameter estimation by the penalized maximum likelihood method. We also present several simulations with different parameter configurations and sample sizes to analyze the behavior of the maximum likelihood estimators, as well as to study the empirical distribution of the quantile residuals. The usefulness of the proposed regression model is proved empirically through analysis of a real dataset.

In this paper, we are concerned with a family of N-urn branching processes, where some particles are initially placed in N urns, and then each particle gives birth to several new particles in some urn when dies. This model includes the N-urn Ehrenfest model and N-urn branching random walk as special cases. We show that the scaling limit of the process is driven by a $\mathit{C}(\mathbb{T})$-valued linear ordinary differential equation and the fluctuation of the process is driven by a generalized Ornstein–Uhlenbeck process in the dual of ${\mathit{C}}^{\infty }(\mathbb{T})$, where $\mathbb{T}=(0,1]$ is the one-dimensional torus. A crucial step for the proofs of the above main results is to show that numbers of particles in different urns are approximately independent. As applications of our main results, the limit theorems of the hitting times of the process are also discussed.

General conditions on smooth real valued random processes and fields are given to ensure the finiteness of the moments of the measure of their level sets. As a by product, a new generalized Kac–Rice formula for the expectation of the measure of these level sets in the one-dimensional case is obtained when the second moment is uniformly bounded. The conditions involve: (i) the differentiability of the trajectories up to a certain order k; (ii) the finiteness of the moments of the k-th partial derivatives of the field up to another order; and (iii) the boundedness of the field’s joint density and some of its derivatives. Particular attention is given to the shot noise processes and fields. Other applications include stationary Gaussian processes, Chi-square processes and regularized diffusion processes. We also sketch the application of these tools to study critical points of random fields.