Bounds for the Expected Time to Extinction and the Probability of Extinction in the Galton-Watson Process

Upper bounds for the expected time to extinction in the Galton-Watson process are obtained. We also found upper and lower bounds for the probability of extinction of this process. These bounds improve some bounds previously obtained by other authors.     

Conditional Least Squares Estimators for the Offspring Mean in a Subcritical Branching Process with Immigration

Consider a Bienayme–Galton–Watson process with generation-dependent immigration, whose mean and variance vary regularly with non negative exponents α and β, respectively. We study the estimation problem of the offspring mean based on an observation of population sizes. We show that if β <2α, the conditional least squares estimator (CLSE) is strongly consistent. Conditions which are sufficient for the CLSE to be asymptotically normal will also be derived. The rate of convergence is faster than n ^?1/2, which is not the case in the process with stationary immigration.     

On the Mean Value Theorem for Estimating Functions

Abstract

Feng et al. revealed that the usual mean value theorem (MVT) should not be applied directly to a vector-valued function (e.g., the score function or a general estimating function under a multiparametric model). This note shows that the application of the Cramer–Wold’s device to a corrected version of the MVT is sufficient to obtain standard asymptotics for the estimators attained from vector-valued estimating functions.     

Testing for the Cointegrating Rank of a VAR Process With Structural Shifts

Tests for the cointegrating rank of a vector autoregressive process are considered that allow for possible exogenous shifts in the mean of the data-generation process. The break points are assumed to be known a priori. It is proposed to estimate and remove the deterministic terms such as mean, linear-trend term, and a shift in a first step. Then systems cointegration tests are applied to the adjusted series. The resulting tests are shown to have known limiting null distributions that are free of nuisance parameters and do not depend on the break point. The tests are applied for analyzing the number of cointegrating relations in two German money-demand systems.     

Moderate deviations for the total population arising from a nearly unstable sub-critical Galton-Watson process with immigration

Abstract

In this paper, we consider the moderate deviation of the nearly unstable sub-critical Galton-Waston process with immigration for the centered total population arising. We discuss the main influence factors for the form of moderate deviation in different stochastic processes. Moreover, we compute the exact rate function in every different situation of MDP.     

The Horvitz-Thompson Theorem as a Unifying Perspective for Probability Sampling: With Examples from Natural Resource Sampling

Courses in sampling often lack a coherent structure because many related sampling designs, estimators, variances, and variance estimators are presented as separate cases. The Horvitz-Thompson theorem offers a needed integrating perspective for teaching the methods and fundamental concepts of probability sampling. Development of basic concepts in sampling via this approach provides the student with tools to solve more complicated problems, and helps to avoid some common stumbling blocks of beginning students. Examples from natural resource sampling are provided to illustrate applications and insight gained from this approach.     

On a Process Capability Index for Asymmetric Specifications

Abstract

Among the process capability indices considered in the literature C _pm is one of the most widely used, despite the fact that its performance often becomes unsatisfactory in the case of processes with asymmetric specifications, i.e., processes whose target value is not located at the midpoint of the specification area. In this article, a new index that is a variant of C _pm , is introduced and shown to overcome this deficiency. In particular, the proposed index performs satisfactorily for processes with symmetric or asymmetric specifications. Moreover, the article compares the suggested index with the existing indices for asymmetric specifications, investigates the distributional properties of its estimator under the assumption of normality and deals with the assessment of confidence intervals using three bootstrap methods. The coverage achieved by each of these methods is investigated via simulation.     

Sojourning With the Homogeneous Poisson Process

In this pedagogical article, distributional properties, some surprising, pertaining to the homogeneous Poisson process (HPP), when observed over a possibly random window, are presented. Properties of the gap-time that covered the termination time and the correlations among gap-times of the observed events are obtained. Inference procedures, such as estimation and model validation, based on event occurrence data over the observation window, are also presented. We envision that through the results in this article, a better appreciation of the subtleties involved in the modeling and analysis of recurrent events data will ensue, since the HPP is arguably one of the simplest among recurrent event models. In addition, the use of the theorem of total probability, Bayes’ theorem, the iterated rules of expectation, variance and covariance, and the renewal equation could be illustrative when teaching distribution theory, mathematical statistics, and stochastic processes at both the undergraduate and graduate levels. This article is targeted toward both instructors and students.     

Modelling Heterogeneity With and Without the Dirichlet Process

We investigate the relationships between Dirichlet process (DP) based models and allocation models for a variable number of components, based on exchangeable distributions. It is shown that the DP partition distribution is a limiting case of a Dirichlet–multinomial allocation model. Comparisons of posterior performance of DP and allocation models are made in the Bayesian paradigm and illustrated in the context of univariate mixture models. It is shown in particular that the unbalancedness of the allocation distribution, present in the prior DP model, persists a posteriori . Exploiting the model connections, a new MCMC sampler for general DP based models is introduced, which uses split/merge moves in a reversible jump framework. Performance of this new sampler relative to that of some traditional samplers for DP processes is then explored.     

On Sampling Designs for Integral Estimation of a Random Process

Abstract

In this paper the problem of finding exactly optimal sampling designs for estimating the weighted integral of a stochastic process with a product covariance structure (R(s,t)=u(s)v(t), s<t) is discussed. The sampling designs for certain standard processes belonging to the product class are calculated. An asymptotic solution to the design problem also follows as a consequence.     

A Theorem on Products of Random Variables, With Application to Regression

It is shown that the product of two independent random variables in the domain of attraction of the normal distribution is also in the domain of attraction of the normal distribution, while if the product is in the domain of attraction of the normal distribution and one of the variables has finite variance, the other is in the domain of attraction of the normal distribution. This result is applied to prove the asymptotic normality of the regression coefficient in a linear regression when the error variance is not necessarily finite.     

On Bayesian D-optimum Design Criteria and the Equivalence Theorem in Non-linear Models

The concavity of some Bayesian D -optimality criteria is investigated and is found in some cases to depend on the prior distribution. In the case of a non-concave criterion, the standard equivalence theorem may fail, but a local version continues to apply.     

On Tests for Group Variation With a Small to Moderate Number of Groups

This paper considers a family of penalized likelihood score tests for group variation. The tests can be indexed by a measure of degrees of freedom. At one extreme, with degrees of freedom one less than the number of groups, is the usual score test for a fixed effects alternative using indicator variables for the groups, while at the other extreme, in the limit as the degrees of freedom 0, is a test closely related to a score test based on a random effects alternative. Asymptotic power comparisons are made for the tests in the family. As would be expected, different members of the family are more efficient for different alternatives. Generally the tests with smaller degrees of freedom appear to have better power than the standard test for alternatives focusing on differences among the larger groups, and lower power for alternatives focusing on differences among the smaller groups. Simulations indicate the asymptotic approximation to the distribution performs better for the tests with small degrees of freedom.     

On the Extinction Time of a Continuous time Markov Branching Process

Several authors have studied and derived bounds for the mean extinction time of a discrete branching process. In this note, we obtain the mean and variance of the extinction time, given extinction of a continuous time Markov branching process. We also obtain bounds for the mean extinction time in terms of moments of the offspring distribution.     

On Confidence Intervals for Process Capability Indices in a One-Way Random Model

In this article, we investigated the bootstrap calibrated generalized confidence limits for process capability indices C _pk for the one-way random effect model. Also, we derived Bissell's approximation formula for the lower confidence limit using Satterthwaite's method and calculated its coverage probabilities and expected values. Then we compared it with standard bootstrap (SB) method and generalized confidence interval method. The simulation results indicate that the confidence limit obtained offers satisfactory coverage probabilities. The proposed method is illustrated with the help of simulation studies and data sets.     

A Derivation of a Useful Consequence of the Gauss-Markov Theorem

The Gauss-Markov (GM) theorem and its many consequences are well known. It has been our experience that graduate students in linear model courses benefit substantially by deriving these consequences, all of which form natural corollaries to the GM theorem. In deriving any one lemma, the student is compelled to reuse many other GM-based lemmas. In this article we present another example of such a lemma that arises in the forward-elimination variable selection procedure. The derivation given is considered to be pedagogically useful.     

On Unit-Root Tests When the Alternative Is a Trend-Break Stationary Process

Minimum t statistics to test for a unit-root are available when the form of break under the alternative evolves according to the crash, changing growth, and mixed models. It is shown that serious power distortions occur if the form of break is misspecified, and thus the practitioner should use the mixed model as the appropriate alternative in empirical applications. The mixed model may reveal useful information regarding the location and form of break. The maximum F statistic for the joint null of a unit-root and no breaks is shown to have greater and less erratic power compared to the minimumt statistic. Stronger evidence against the unit-root is found for the Nelson-Plosser series and U.S. Postwar quarterly real gross national product.     

On the Rounded Integer-Valued Autoregressive Process

In recent years, there has been a growing interest in modelling integred-valued time series. In this article, we propose a modified and generalized version of the first order rounded integer-valued autoregressive RINAR(1) model, originally introduced by Kachour and Yao (2009 Kachour , M. , Yao , J. F. ( 2009 ). First-order rounded integer-valued autoregressive (RINAR(1)) process . Journal of Time Series Analysis 30 ( 4 ): 417 – 448 .[Crossref], [Web of Science ®] , [Google Scholar]). Indeed, this class can be considered as an alternative of classical models based on the thinning operators. Using a Markov chain method, conditions for stationarity and the existence of moments are investigated. Least squares estimator of the model parameters is considered and its consistence is established. Finally, we describe the price change data using a model of the new class.