Exponentiated Geometric Distribution: Another Generalization of Geometric Distribution

Exponentiated geometric distribution with two parameters q(0 < q < 1) and α( > 0) is proposed as a new generalization of the geometric distribution by employing the techniques of Mudholkar and Srivastava (1993). A few realistics basis where the proposed distribution may arise naturally are discussed, its distributional and reliability properties are investigated. Parameter estimation is discussed. Application in discrete failure time data modeling is illustrated with real life data. The suitability of the proposed distribution in empirical modeling of other count data is investigated by conducting comparative data fitting experiments with over and under dispersed data sets.     

Geometric consistency of principal component scores for high-dimensional mixture models and its application

In this article, we consider clustering based on principal component analysis (PCA) for high-dimensional mixture models. We present theoretical reasons why PCA is effective for clustering high-dimensional data. First, we derive a geometric representation of high-dimension, low-sample-size (HDLSS) data taken from a two-class mixture model. With the help of the geometric representation, we give geometric consistency properties of sample principal component scores in the HDLSS context. We develop ideas of the geometric representation and provide geometric consistency properties for multiclass mixture models. We show that PCA can cluster HDLSS data under certain conditions in a surprisingly explicit way. Finally, we demonstrate the performance of the clustering using gene expression datasets.     

A geometric time series model with a new dependent Bernoulli counting series

ABSTRACT

New generalized binomial thinning operator with dependent counting series is introduced. An integer valued time series model with geometric marginals based on this thinning operator is constructed. Main features of the process are analyzed and determined. Estimation of the parameters are presented and some asymptotic properties of the obtained estimators are discussed. Behavior of the estimators is described through the numerical results. Also, model is applied on the real data set and compared to some relevant INAR(1) models.     

Geometric approach to the skewed normal distribution

The representations of the skewed normal distribution given in Propositions 1–4 in Genton (Ed., 2004) are considered here from a unified geometric point of view and are, based upon this, generalized in two respects. On the one hand, the four concrete representations motivate us for a unified and much more general algebraic–geometric representation of the skewed normal distribution (Theorems 1 and 2 as well as Remarks 2 and 3); on the other hand, the mentioned representations are generalized to the elliptically contoured case (Propositions and Corollaries 1c–4c).     

混合指数几何分布

本文讨论元件的寿命分布服从双参数混合指数分布,元件个数服从几何分布的情形下,定义了混合指数几何分布,并研究了该分布的各种性质,讨论了参数的极大似然估计,并使用EM算法得到参数的近似估计. 最后,本文给出了参数的渐近方差、协方差、置信区间.     

Accelerated Degradation Models for Failure Based on Geometric Brownian Motion and Gamma Processes

Based on a generalized cumulative damage approach with a stochastic process describing degradation, new accelerated life test models are presented in which both observed failures and degradation measures can be considered for parametric inference of system lifetime. Incorporating an accelerated test variable, we provide several new accelerated degradation models for failure based on the geometric Brownian motion or gamma process. It is shown that in most cases, our models for failure can be approximated closely by accelerated test versions of Birnbaum–Saunders and inverse Gaussian distributions. Estimation of model parameters and a model selection procedure are discussed, and two illustrative examples using real data for carbon-film resistors and fatigue crack size are presented.     

Geometric ergodicity of nonlinear autoregressive models with changing conditional variances

The authors give easy‐to‐check sufficient conditions for the geometric ergodicity and the finiteness of the moments of a random process x_t = ?(x_t‐1,…, x_t‐p) + ?_tσ(x_t‐1,…, x_t‐q) in which ?: R^p → R, σ R^q → R and (?_t) is a sequence of independent and identically distributed random variables. They deduce strong mixing properties for this class of nonlinear autoregressive models with changing conditional variances which includes, among others, the ARCH(p), the AR(p)‐ARCH(p), and the double‐threshold autoregressive models.     

Geometric Anisotropic Spatial Point Pattern Analysis and Cox Processes

We consider spatial point processes with a pair correlation function, which depends only on the lag vector between a pair of points. Our interest is in statistical models with a special kind of ‘structured’ anisotropy: the pair correlation function is geometric anisotropic if it is elliptical but not spherical. In particular, we study Cox process models with an elliptical pair correlation function, including shot noise Cox processes and log Gaussian Cox processes, and we develop estimation procedures using summary statistics and Bayesian methods. Our methodology is illustrated on real and synthetic datasets of spatial point patterns.     

Geometric characterization of planar regression

The geometric characterization of linear regression in terms of the ‘concentration ellipse’ by Galton [Galton, F., 1886, Family likeness in stature (with Appendix by Dickson, J.D.H.). Proceedings of the Royal Society of London, 40, 42–73.] and Pearson [Pearson, K., 1901, On lines and planes of closest fit to systems of points in space. Philosophical Magazine, 2, 559–572.] was extended to the case of unequal variances of the presumably uncorrelated errors in the experimental data [McCartin, B.J., 2003, A geometric characterization of linear regression. Statistics, 37(2), 101–117.]. In this paper, this geometric characterization is further extended to planar (and also linear) regression in three dimensions where a beautiful interpretation in terms of the concentration ellipsoid is developed.     

The MLE of Geometric Parameter for a Geometric Process

In this article, we provide the MLE of the ratio parameter of a geometric process and discuss its consistency and asymptotic normality.     

A geometric time-series model with an alternative dependent Bernoulli counting series

In this article, we first introduce an alternative way for construction of the generalized binomial thinning operator with dependent counting series. Some properties of this thinning operator are derived and discussed. Then, by using this thinning operator, we introduce an integer-valued time-series model with geometric marginals. Some conditional and unconditional properties of this model are derived and discussed. Some estimation methods are considered and for some of them, asymptotic properties of the obtained estimates are derived. Performances of the estimates are discussed through some simulations. Finally, a real data example is considered and the goodness-of-fit of this model is compared with the models based on the binomial, negative binomial, and dependent binomial thinning operators.     

Geometric Mean of Nonnegative Random Variable

In this article, we define the geometric mean of a nonnegative random variable in different cases and study the asymptotic unbiasedness of the sample geometric mean.     

A Geometric Examination of Linear Model Assumptions

From a geometric perspective, linear model theory relies on a single assumption, that (‘corrected’) data vector directions are uniformly distributed in Euclidean space. We use this perspective to explore pictorially the effects of violations of the traditional assumptions (normality, independence and homogeneity of variance) on the Type I error rate. First, for several non‐normal distributions we draw geometric pictures and carry out simulations to show how the effects of non‐normality diminish with increased parent distribution symmetry and continuity, and increased sample size. Second, we explore the effects of dependencies on Type I error rate. Third, we use simulation and geometry to investigate the effect of heterogeneity of variance on Type I error rate. We conclude, in a fresh way, that independence and homogeneity of variance are more important assumptions than normality. The practical implication is that statisticians and authors of statistical computing packages need to pay more attention to the correctness of these assumptions than to normality.     

Negative binomial time series models based on expectation thinning operators

The study of count data time series has been active in the past decade, mainly in theory and model construction. There are different ways to construct time series models with a geometric autocorrelation function, and a given univariate margin such as negative binomial. In this paper, we investigate negative binomial time series models based on the binomial thinning and two other expectation thinning operators, and show how they differ in conditional variance or heteroscedasticity. Since the model construction is in terms of probability generating functions, typically, the relevant conditional probability mass functions do not have explicit forms. In order to do simulations, likelihood inference, graphical diagnostics and prediction, we use a numerical method for inversion of characteristic functions. We illustrate the numerical methods and compare the various negative binomial time series models for a real data example.     

Normal Deviation and Poisson Approximation of a Security Market by the Geometric Markov Renewal Processes

We consider the geometric Markov renewal processes (GMRP) as a model for a security market. Normal deviations of the geometric Markov renewal processes for ergodic averaging and double averaging schemes are derived. We introduce Poisson averaging scheme for the geometric Markov renewal processes. European call option pricing formulas for GMRP are presented.     

On the Characterization of a Bivariate Geometric Distribution

In this article it is shown that a bivariate random variable has a constant failure rate, and mixture geometric marginals, if and only if, it has the loss of memory property and the discrete Freund distribution. This characterization is achieved by extending a key lemma in this area. The mixture geometric can be collapsed to geometric marginals, thus validating the results.     

Geometric programming for optimal allocation of integrated samples in quality control

We apply geometric programming, developed by Duffin, Peterson Zener (1967), to the optimal allocation of stratified samples. As an introduction, we show how geometric programming is used to allocate samples according to Neyman (1934), using the data of Cornell (1947) and following the exposition of Cochran (1953).

Then we use geometric programming to allocate an integrated sample introduced by Schwartz (1978) for more efficient sampling of three U. S. Federal welfare quality control systems, Aid to Families with Dependent Children, Food Stamps and Medicaid.

We develop methods for setting up the allocation problem, interpreting it as a geometric programming primal problem, transforming it to the corresponding dual problem, solving that, and finding the sample sizes required in the allocation problem. We show that the integrated sample saves sampling costs.     

A Mixed Bivariate Distribution Connected with Geometric Maxima of Exponential Variables

We study the joint distribution of X and N, where N has a geometric distribution and X is the maximum of N i.i.d. exponential variables, independent of N. We present basic properties of these mixed bivariate distributions and discuss parameter estimation for this model. An example from finance, where N represents the number of consecutive positive daily log-returns of currency exchange rates, illustrates stochastic modeling potential of these laws.     

Nonparametric Estimation of the Geometric Vitality Function

Recently, Nair and Rajesh (2000 Nair , K. R. M. , Rajesh , G. ( 2000 ). Geometric vitality function and its application to reliability . IAPQR Tran. 25 ( 1 ): 1 – 8 . [Google Scholar]) proposed a measure to describe the failure pattern of components/devices in terms of the geometric mean of the residual life. This measure find applications in modeling life time data. In the present work we provide a nonparametric kernel-type estimator for the geometric vitality function, both in the case of complete and censored samples. The properties of the estimator, under certain regularity conditions, are studied. The performance of the estimator is compared with the empirical estimator using a real data set and simulation studies are carried out using the Monte Carlo method.