Mostrando recursos 1 - 20 de 42

  1. Products of normal, beta and gamma random variables: Stein operators and distributional theory

    Gaunt, Robert E.
    In this paper, we extend Stein’s method to products of independent beta, gamma, generalised gamma and mean zero normal random variables. In particular, we obtain Stein operators for mixed products of these distributions, which include the classical beta, gamma and normal Stein operators as special cases. These operators lead us to closed-form expressions involving the Meijer $G$-function for the probability density function and characteristic function of the mixed product of independent beta, gamma and central normal random variables.

  2. Some unified results on stochastic properties of residual lifetimes at random times

    Misra, Neeraj; Naqvi, Sameen
    The residual life of a random variable $X$ at random time $\Theta$ is defined to be a random variable $X_{\Theta}$ having the same distribution as the conditional distribution of $X-\Theta$ given $X>\Theta$ (denoted by $X_{\Theta}=(X-\Theta|X>\Theta)$). Let $(X,\Theta_{1})$ and $(Y,\Theta_{2})$ be two pairs of jointly distributed random variables, where $X$ and $\Theta_{1}$ (and, $Y$ and $\Theta_{2}$) are not necessarily independent. In this paper, we compare random variables $X_{\Theta_{1}}$ and $Y_{\Theta_{2}}$ by providing sufficient conditions under which $X_{\Theta_{1}}$ and $Y_{\Theta_{2}}$ are stochastically ordered with respect to various stochastic orderings. These comparisons have been made with respect to hazard rate, likelihood ratio and...

  3. Nonlinear filtering with correlated Lévy noise characterized by copulas

    Fernando, B. P. W.; Hausenblas, E.
    The objective in stochastic filtering is to reconstruct the information about an unobserved (random) process, called the signal process, given the current available observations of a certain noisy transformation of that process. ¶ Usually $X$ and $Y$ are modeled by stochastic differential equations driven by a Brownian motion or a jump (or Lévy) process. We are interested in the situation where both the state process $X$ and the observation process $Y$ are perturbed by coupled Lévy processes. More precisely, $L=(L_{1},L_{2})$ is a $2$-dimensional Lévy process in which the structure of dependence is described by a Lévy copula. We derive the associated...

  4. Identifiability of structural characteristics: How relevant is it for the Bayesian approach?

    San Martín, Ernesto
    The role of identification in the Bayesian approach is still debatable. Since Lindley [Bayesian Statistics. A Review (1971) Philadelphia], most Bayesian statisticians pretend that unidentifiabiity causes no real difficulty in their approach. Recently, Wechsler, Izbicki and Esteves [Amer. Statist. 67 (2013) 90–93] provide a simple example illustrating this perspective. By critically reading Wechsler, Izbicki and Esteves [Amer. Statist. 67 (2013) 90–93], we intend to show that the Bayesian approach is far from being free of the identification problems, provided that the interest is focused on the interpretation of the parameters. It is written using a rather ancient style, the so-called...

  5. Mixture models applied to heterogeneous populations

    Cavalcante, Carolina V.; Gonçalves, Kelly C. M.
    Mixture models provide a flexible representation of heterogeneity in a finite number of latent classes. From the Bayesian point of view, Markov Chain Monte Carlo methods provide a way to draw inferences from these models. In particular, when the number of subpopulations is considered unknown, more sophisticated methods are required to perform Bayesian analysis. The Reversible Jump Markov Chain Monte Carlo is an alternative method for computing the posterior distribution by simulation in this case. Some problems associated with the Bayesian analysis of these class of models are frequent, such as the so-called “label-switching” problem. However, as the level of...

  6. On the number of unobserved and observed categories when sampling from a multivariate hypergeometric population

    Kim, Sungsu; Park, Chong Jin
    Consider taking a random sample of size $n$ from a finite population that consists of $N$ categories with $M_{i}$ copies in the $i$th category for $i=1,\dots,N$. Each observed unit in a sample is presumed to have a probability $1-p$ ($0

  7. The exponentiated logarithmic generated family of distributions and the evaluation of the confidence intervals by percentile bootstrap

    Marinho, Pedro Rafael Diniz; Cordeiro, Gauss M.; Peña Ramírez, Fernando; Alizadeh, Morad; Bourguignon, Marcelo
    We study some mathematical properties of a new generator of continuous distributions with three additional parameters, called the exponentiated logarithmic generated family, to extend the normal, Weibull, gamma and Gumbel distributions, among other well-known models. Some special models are discussed. Many properties of this family are studied, some inference procedures developed and a simulation study performed to verify the adequacy of the estimators of the model parameters. We prove empirically the potentiality of the new family by means of two real data sets. The simulation study for different samples sizes assesses the performance of the maximum likelihood estimates obtained by...

  8. Poisson–Lindley INAR(1) model with applications

    Mohammadpour, M.; Bakouch, Hassan S.; Shirozhan, M.
    The paper focuses on a new stationary integer-valued autoregressive model of first order with Poisson–Lindley marginal distribution. Several statistical properties of the model are established, like spectral density function, multi-step ahead conditional measures, stationarity, ergodicity and irreducibility. We consider several methods for estimating the unknown parameters of the model and investigate properties of the estimators. The performances of these estimators are compared via simulation. The model is motivated by some applications to two real count time series data.

  9. Bayesian analysis of multiple-inflation Poisson models and its application to infection data

    Ryu, Duchwan; Bilgili, Devrim; Ergönül, Önder; Ebrahimi, Nader
    In this article we propose a multiple-inflation Poisson regression to model count response data containing excessive frequencies at more than one non-negative integer values. To handle multiple excessive count responses, we generalize the zero-inflated Poisson regression by replacing its binary regression with the multinomial regression, while Su et al. [Statist. Sinica 23 (2013) 1071–1090] proposed a multiple-inflation Poisson model for consecutive count responses with excessive frequencies. We give several properties of our proposed model, and do statistical inference under the fully Bayesian framework. We perform simulation studies and also analyze the data related to the number of infections collected in...

  10. Truncated sequential Monte Carlo test with exact power

    Silva, Ivair; Assunção, Renato
    Monte Carlo hypothesis testing is extensively used for statistical inference. Surprisingly, despite the many theoretical advances in the field, statistical power performance of Monte Carlo tests remains an open question. Because the last assertion may sound questionable for some, the first goal in this paper is to show that the power performance of truncated Monte Carlo tests is still an unsolved question. The second goal here is to present a solution for this issue, that is, we introduce a truncated sequential Monte Carlo procedure with statistical power arbitrarily close to the power of the theoretical exact test. The most significant...

  11. A Skellam GARCH model

    Alomani, Ghadah A.; Alzaid, Abdulhamid A.; Omair, Maha A.
    This paper considers the modeling of nonstationary integer valued time series with conditional heteroskedasticity using Skellam distribution. Two approaches of estimation of the model’s parameters are treated and discussed. The obtained results are verified through some numerical simulation. In addition, the proposed model is applied to real time series.

  12. Abrupt convergence for a family of Ornstein–Uhlenbeck processes

    Barrera, Gerardo
    We consider a family of Ornstein–Uhlenbeck processes. Under some suitable assumptions on the behaviour of the drift and diffusion coefficients, we prove profile cut-off phenomenon with respect to the total variation distance in the sense of the definition given by Barrera and Ycart [ALEA Lat. Am. J. Probab. Math. Stat. 11 (2014) 445–458]. We compute explicitly the cut-off time, the window time, and the profile function. Moreover, we prove that the average process satisfies a profile cut-off phenomenon with respect to the total variation distance. Also, a sample of $N$ Ornstein–Uhlenbeck processes has a window cut-off with respect to the...

  13. Hölderian weak invariance principle under the Maxwell and Woodroofe condition

    Giraudo, Davide
    We investigate the weak invariance principle in Hölder spaces under some reinforcement of the Maxwell and Woodroofe condition. Optimality of the obtained condition is established.

  14. Noise-indicator nonnegative integer-valued autoregressive time series of the first order

    Stojanović, Vladica; Randjelović, Dragan; Kuk, Kristijan
    This paper presents a modification and, at the same time, a generalization of the linear first order nonnegative integer-valued autoregressive processes, well-known as INAR(1) processes. By using the so-called Noise-Indicator, a nonlinear model with the threshold regime and with more complex structure than the appropriate linear models was obtained. The new model, named NIINAR(1) process, has been investigated in terms of the most general, the power series distribution of its innovations. Basic stochastic properties of the NIINAR(1) model (e.g., correlation structure, over-dispersion conditions and distributional properties) are given. Also, besides of some standard parameters estimators, a novel estimation techniques, together...

  15. On the exit time from an orthant for badly oriented random walks

    Garbit, Rodolphe
    It was recently proved that the exponential decreasing rate of the probability that a random walk stays in a $d$-dimensional orthant is given by the minimum on this orthant of the Laplace transform of the random walk increments, provided that this minimum exists. In other cases, the random walk is “badly oriented” and the exponential rate may depend on the starting point $x$. We show here that this rate is nevertheless asymptotically equal to the infimum of the Laplace transform, as some selected coordinates of $x$ tend to infinity.

  16. Nonlinear measurement errors models subject to partial linear additive distortion

    Zhang, Jun; Zhou, Nanguang; Chen, Qian; Chu, Tianyue
    We study nonlinear regression models when the response and predictors are unobservable and distorted in a multiplicative fashion by partial linear additive models (PLAM) of some observed confounding variables. After approximating the additive nonparametric components in the PLAM via polynomial splines and calibrating the unobserved response and unobserved predictors, we develop a semi-parametric profile nonlinear least squares procedure to estimate the parameters of interest. The resulting estimators are shown to be asymptotically normal. To construct confidence intervals for the parameters of interest, an empirical likelihood-based statistic is proposed to improve the accuracy of the associated normal approximation. We also show...

  17. Improved inference for the generalized Pareto distribution

    Pires, Juliana F.; Cysneiros, Audrey H. M. A.; Cribari-Neto, Francisco
    The generalized Pareto distribution is commonly used to model exceedances over a threshold. In this paper, we obtain adjustments to the generalized Pareto profile likelihood function using the likelihood function modifications proposed by Barndorff-Nielsen (Biometrika 70 (1983) 343–365), Cox and Reid (J. R. Stat. Soc. Ser. B. Stat. Methodol. 55 (1993) 467–471), Fraser and Reid (Utilitas Mathematica 47 (1995) 33–53), Fraser, Reid and Wu (Biometrika 86 (1999) 249–264) and Severini (Biometrika 86 (1999) 235–247). We consider inference on the generalized Pareto distribution shape parameter, the scale parameter being a nuisance parameter. Bootstrap-based testing inference is also considered. Monte Carlo simulation...

  18. Improved estimation in a general multivariate elliptical model

    Melo, Tatiane F. N.; Ferrari, Silvia L. P.; Patriota, Alexandre G.
    The problem of reducing the bias of maximum likelihood estimator in a general multivariate elliptical regression model is considered. The model is very flexible and allows the mean vector and the dispersion matrix to have parameters in common. Many frequently used models are special cases of this general formulation, namely: errors-in-variables models, nonlinear mixed-effects models, heteroscedastic nonlinear models, among others. In any of these models, the vector of the errors may have any multivariate elliptical distribution. We obtain the second-order bias of the maximum likelihood estimator, a bias-corrected estimator, and a bias-reduced estimator. Simulation results indicate the effectiveness of the...

  19. Weighted Weibull distribution: Bivariate and multivariate cases

    Al-Mutairi, D. K.; Ghitany, M. E.; Kundu, Debasis
    Gupta and Kundu (Statistics 43 (2009) 621–643) introduced a new class of weighted exponential distribution and established its several properties. The probability density function of the proposed weighted exponential distribution is unimodal and it has an increasing hazard function. Following the same line Shahbaz, Shahbaz and Butt (Pak. J. Stat. Oper. Res. VI (2010) 53–59) introduced weighted Weibull distribution, and we derive several new properties of this weighted Weibull distribution. The main aim of this paper is to introduce bivariate and multivariate distributions with weighted Weibull marginals and establish their several properties. It is shown that the hazard function of...

  20. Effects of prior distributions: An application to pipedwater demand

    Ramírez Hassan, Andrés; Pericchi, Luis
    In this paper, we analyze the effect on posterior parameter distributions of four possible alternative prior distributions, namely Normal-Inverse Gamma, Normal-Scaled Beta two, Student’s $t$-Inverse Gamma and Student’s $t$-Scaled Beta two. We show the effects of these prior distributions when there is apparently conflict between the sample information and the elicited hyperparameters. In particular, we show that there is not systematic differences of posterior parameter distributions associated with these four priors using data of piped water demand in a linear model with autoregressive errors. To test the hypothesis that this result is due to using a moderate sample size and...

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