THE LOG-EIG DISTRIBUTION: A NEW PROBABILITY MODEL FOR LIFETIME DATA

ABSTRACT

In this paper, we propose a new probability model called the log-EIG distribution for lifetime data analysis. Some important properties of the proposed model and maximum likelihood estimation of its parameters are discussed. Its relationship with the exponential inverse Gaussian distribution is similar to that of the lognormal and the normal distributions. Through applications to well-known datasets, we show that the log-EIG distribution competes well, and in some instances even provides a better fit than the commonly used lifetime models such as the gamma, lognormal, Weibull and inverse Gaussian distributions. It can accommodate situations where an increasing failure rate model is required as well as those with a decreasing failure rate at larger times.     

The Likelihood Ratio Tests for Homogeneity of Inverse Gaussian Means Under Simple Order and Simple Tree Order

Abstract

The inverse Gaussian (IG) family is now widely used for modeling non negative skewed measurements. In this article, we construct the likelihood-ratio tests (LRTs) for homogeneity of the order constrained IG means and study the null distributions for simple order and simple tree order cases. Interestingly, it is seen that the null distribution results for the normal case are applicable without modification to the IG case. This supplements the numerous well known and striking analogies between Gaussian and inverse Gaussian families     

Multiplicative Algorithms for Maximum Penalized Likelihood Inversion with Non Negative Constraints and Generalized Error Distributions

In many linear inverse problems the unknown function f (or its discrete approximation Θ _p×1), which needs to be reconstructed, is subject to the non negative constraint(s); we call these problems the non negative linear inverse problems (NNLIPs). This article considers NNLIPs. However, the error distribution is not confined to the traditional Gaussian or Poisson distributions. We adopt the exponential family of distributions where Gaussian and Poisson are special cases. We search for the non negative maximum penalized likelihood (NNMPL) estimate of Θ. The size of Θ often prohibits direct implementation of the traditional methods for constrained optimization. Given that the measurements and point-spread-function (PSF) values are all non negative, we propose a simple multiplicative iterative algorithm. We show that if there is no penalty, then this algorithm is almost sure to converge; otherwise a relaxation or line search is necessitated to assure its convergence.     

Poisson-Driven Stationary Markov Models

ABSTRACT

We propose a simple yet powerful method to construct strictly stationary Markovian models with given but arbitrary invariant distributions. The idea is based on a Poisson-type transform modulating the dependence structure in the model. An appealing feature of our approach is the possibility to control the underlying transition probabilities and, therefore, incorporate them within standard estimation methods. Given the resulting representation of the transition density, a Gibbs sampler algorithm based on the slice method is proposed and implemented. In the discrete-time case, special attention is placed to the class of generalized inverse Gaussian distributions. In the continuous case, we first provide a brief treatment of the class of gamma distributions, and then extend it to cover other invariant distributions, such as the generalized extreme value class. The proposed approach and estimation algorithm are illustrated with real financial datasets. Supplementary materials for this article are available online.     

A General Class of Distributions: Properties and Applications

ABSTRACT

In this article, we derive a general class of distributions and establish its relationship to χ² distribution. The proposed class includes normal, inverse Gaussian, lognormal, gamma, Rayleigh, and Maxwell distributions. Various statistical properties of the class are discussed. Some applications of the class are given.     

Selection of bandwidth for kernel regression

ABSTRACT

The most important factor in kernel regression is a choice of a bandwidth. Considerable attention has been paid to extension the idea of an iterative method known for a kernel density estimate to kernel regression. Data-driven selectors of the bandwidth for kernel regression are considered. The proposed method is based on an optimally balanced relation between the integrated variance and the integrated square bias. This approach leads to an iterative quadratically convergent process. The analysis of statistical properties shows the rationale of the proposed method. In order to see statistical properties of this method the consistency is determined. The utility of the method is illustrated through a simulation study and real data applications.     

A Multivariate Extension of Inverse Gaussian Distribution Derived from Inverse Relationship

Abstract

We propose a new multivariate extension of the inverse Gaussian distribution derived from a certain multivariate inverse relationship. First we define a multivariate extension of the inverse relationship between two sets of multivariate distributions, then define a reduced inverse relationship between two multivariate distributions. We derive the multivariate continuous distribution that has the reduced multivariate inverse relationship with a multivariate normal distribution and call it a multivariate inverse Gaussian distribution. This distribution is also characterized as the distribution of the location of a multivariate Brownian motion at some stopping time. The marginal distribution in one direction is the inverse Gaussian distribution, and the conditional distribution in the space perpendicular to this direction is a multivariate normal distribution. Mean, variance, and higher order cumulants are derived from the multivariate inverse relationship with a multivariate normal distribution. Other properties such as reproductivity and infinite divisibility are also given.     

Lévy‐based Modelling in Brain Imaging

Abstract. A substantive problem in neuroscience is the lack of valid statistical methods for non‐Gaussian random fields. In the present study, we develop a flexible, yet tractable model for a random field based on kernel smoothing of a so‐called Lévy basis. The resulting field may be Gaussian, but there are many other possibilities, including random fields based on Gamma, inverse Gaussian and normal inverse Gaussian (NIG) Lévy bases. It is easy to estimate the parameters of the model and accordingly to assess by simulation the quantiles of test statistics commonly used in neuroscience. We give a concrete example of magnetic resonance imaging scans that are non‐Gaussian. For these data, simulations under the fitted models show that traditional methods based on Gaussian random field theory may leave small, but significant changes in signal level undetected, while these changes are detectable under a non‐Gaussian Lévy model.     

Semiparametric multiple kernel estimators and model diagnostics for count regression functions

Abstract

This study concerns semiparametric approaches to estimate discrete multivariate count regression functions. The semiparametric approaches investigated consist of combining discrete multivariate nonparametric kernel and parametric estimations such that (i) a prior knowledge of the conditional distribution of model response may be incorporated and (ii) the bias of the traditional nonparametric kernel regression estimator of Nadaraya-Watson may be reduced. We are precisely interested in combination of the two estimations approaches with some asymptotic properties of the resulting estimators. Asymptotic normality results were showed for nonparametric correction terms of parametric start function of the estimators. The performance of discrete semiparametric multivariate kernel estimators studied is illustrated using simulations and real count data. In addition, diagnostic checks are performed to test the adequacy of the parametric start model to the true discrete regression model. Finally, using discrete semiparametric multivariate kernel estimators provides a bias reduction when the parametric multivariate regression model used as start regression function belongs to a neighborhood of the true regression model.     

Testing Equality of Order Constrained Inverse Gaussian Means

The inverse Gaussian family of non negative, skewed random variables is analytically simple, and its inference theory is well known to be analogous to the normal theory in numerous ways. Hence, it is widely used for modeling non negative positively skewed data. In this note, we consider the problem of testing homogeneity of order restricted means of several inverse Gaussian populations with a common unknown scale parameter using an approach based on the classical methods, such as Fisher's, for combining independent tests. Unlike the likelihood approach which can only be readily applied to a limited number of restrictions and the settings of equal sample sizes, this approach is applicable to problems involving a broad variety of order restrictions and arbitrary sample size settings, and most importantly, no new null distributions are needed. An empirical power study shows that, in case of the simple order, the test based on Fisher's combination method compares reasonably with the corresponding likelihood ratio procedure.     

BIAS REDUCTION AND ELIMINATION WITH KERNEL ESTIMATORS

A great deal of research has focused on improving the bias properties of kernel estimators. One proposal involves removing the restriction of non-negativity on the kernel to construct “higher-order” kernels that eliminate additional terms in the Taylor's series expansion of the bias. This paper considers an alternative that uses a local approach to bandwidth selection to not only reduce the bias, but to eliminate it entirely. These so-called “zero-bias bandwidths” are shown to exist for univariate and multivariate kernel density estimation as well as kernel regression. Implications of the existence of such bandwidths are discussed. An estimation strategy is presented, and the extent of the reduction or elimination of bias in practice is studied through simulation and example.     

A taxonomy of mixing and outcome distributions based on conjugacy and bridging

Abstract

The generalized linear mixed model (GLMM) is commonly used for the analysis of hierarchical non Gaussian data. It combines an exponential family model formulation with normally distributed random effects. A drawback is the difficulty of deriving convenient marginal mean functions with straightforward parametric interpretations. Several solutions have been proposed, including the marginalized multilevel model (directly formulating the marginal mean, together with a hierarchical association structure) and the bridging approach (choosing the random-effects distribution such that marginal and hierarchical mean functions share functional forms). Another approach, useful in both a Bayesian and a maximum-likelihood setting, is to choose a random-effects distribution that is conjugate to the outcome distribution. In this paper, we contrast the bridging and conjugate approaches. For binary outcomes, using characteristic functions and cumulant generating functions, it is shown that the bridge distribution is unique. Self-bridging is introduced as the situation in which the outcome and random-effects distributions are the same. It is shown that only the Gaussian and degenerate distributions have well-defined cumulant generating functions for which self-bridging holds.     

Conditional and marginal estimates in case-control family data – extensions and sensitivity analyses

This work considers two specific estimation techniques for the family-specific proportional hazards model and for the population-averaged proportional hazards model. So far, these two estimation procedures were presented and studied under the gamma frailty distribution mainly because of its simple interpretation and mathematical tractability. Modifications of both procedures for other frailty distributions, such as the inverse Gaussian, positive stable and a specific case of discrete distribution, are presented. By extensive simulations, it is shown that under the family-specific proportional hazards model, the gamma frailty model appears to be robust to frailty distribution mis-specification in both bias and efficiency loss in the marginal parameters. The population-averaged proportional hazards model, is found to be robust under the gamma frailty model mis-specification only under moderate or weak dependency within cluster members.     

Lindley first-order autoregressive model with applications

ABSTRACT

A new stationary first-order autoregressive process with Lindley marginal distribution, denoted as LAR(1) is introduced. We derive the probability function for the innovation process. We consider many properties of this process, involving spectral density, some multi-step ahead conditional measures, run probabilities, stationary solution, uniqueness and ergodicity. We estimate the unknown parameters of the process using three methods of estimation and investigate properties of the estimators with some numerical results to illustrate them. Some applications of the process are discussed to two real data sets and it is shown that the LAR(1) model fits better than other known non Gaussian AR(1) models.     

Local linear double and asymmetric kernel estimation of conditional quantiles

Abstract

In this work, we propose and investigate a family of non parametric quantile regression estimates. The proposed estimates combine local linear fitting and double kernel approaches. More precisely, we use a Beta kernel when covariate’s support is compact and Gamma kernel for left-bounded supports. Finite sample properties together with asymptotic behavior of the proposed estimators are presented. It is also shown that these estimates enjoy the property of having finite variance and resistance to sparse design.     

A permutation-based Bayesian approach for inverse covariance estimation

Abstract

Covariance estimation and selection for multivariate datasets in a high-dimensional regime is a fundamental problem in modern statistics. Gaussian graphical models are a popular class of models used for this purpose. Current Bayesian methods for inverse covariance matrix estimation under Gaussian graphical models require the underlying graph and hence the ordering of variables to be known. However, in practice, such information on the true underlying model is often unavailable. We therefore propose a novel permutation-based Bayesian approach to tackle the unknown variable ordering issue. In particular, we utilize multiple maximum a posteriori estimates under the DAG-Wishart prior for each permutation, and subsequently construct the final estimate of the inverse covariance matrix. The proposed estimator has smaller variability and yields order-invariant property. We establish posterior convergence rates under mild assumptions and illustrate that our method outperforms existing approaches in estimating the inverse covariance matrices via simulation studies.     

A Sequential Multi-Hypothesis Test for the Mean Function of a Discrete-Time Gaussian Process

Abstract

A sequential multi-hypothesis test for the mean function of a discrete-time Gaussian process with known covariance kernel is developed. It is obtained by applying the Bechhofer-Kiefer-Sobel generalized sequential probability ratio test GSPRT, and its properties are studied analytically. Selected applications to i.i.d. normal random variables, observation in a time series AR(1) model, and Wiener processes are given.     

Seemingly Unrelated Ridge Regression in Semiparametric Models

This article is concerned with the problem of multicollinearity in the linear part of a seemingly unrelated semiparametric (SUS) model. It is also suspected that some additional non stochastic linear constraints hold on the whole parameter space. In the sequel, we propose semiparametric ridge and non ridge type estimators combining the restricted least squares methods in the model under study. For practical aspects, it is assumed that the covariance matrix of error terms is unknown and thus feasible estimators are proposed and their asymptotic distributional properties are derived. Also, necessary and sufficient conditions for the superiority of the ridge-type estimator over the non ridge type estimator for selecting the ridge parameter K are derived. Lastly, a Monte Carlo simulation study is conducted to estimate the parametric and nonparametric parts. In this regard, kernel smoothing and cross validation methods for estimating the nonparametric function are used.     

Kolmogorov-Smirnov Type Test-Statistics For The Gamma,Erlang-2 And The Inverse Gaussian Distributions When The Parameters Are Unknown.

Critical values are presented for the Kolmogorov-Smirnov type test statistics for the following three cases: (i) the gamma distribution when both the scale and the shape parameters are not known, (ii) the scale parameter of the gamma distribution is not known and (iii) the inverse Gaussian distribution when both the parameters are unknown. This study was motivated by the necessity to fit the gamma, the Erlang-2 and the inverse Gaussian distributions to the interpurchase times of individuals for coffee in marketing research.     

Statistical inference in the partial linear models with the inverse gaussian kernel

Symmetric kernel smoothing is commonly used in estimating the nonparametric component in the partial linear regression models. In this article, we propose a new estimation method for the partial linear regression models using the inverse Gaussian kernel when the explanatory variable of the nonparametric component is non-negatively supported. As an asymmetric kernel function, the inverse Gaussian kernel is also supported on the non-negative half line. The asymptotic properties, including the asymptotic normality, uniform almost sure convergence, and the iterated logarithm laws, of the proposed estimators are thoroughly discussed for both homoscedastic and heteroscedastic cases. The simulation study is conducted to evaluate the finite sample performance of the proposed estimators.