Pólya–Aeppli of Order k Risk Model

In this study, we define the Pólya–Aeppli process of order k as a compound Poisson process with truncated geometric compounding distribution with success probability 1 ? ρ > 0 and investigate some of its basic properties. Using simulation, we provide a comparison between the sample paths of the Pólya–Aeppli process of order k and the Poisson process. Also, we consider a risk model in which the claim counting process {N(t)} is a Pólya-Aeppli process of order k, and call it a Pólya—Aeppli of order k risk model. For the Pólya–Aeppli of order k risk model, we derive the ruin probability and the distribution of the deficit at the time of ruin. We discuss in detail the particular case of exponentially distributed claims and provide simulation results for more general cases.     

Some asymptotic results for the integrated empirical process with applications to statistical tests

The main purpose of this paper is to investigate the strong approximation of the integrated empirical process. More precisely, we obtain the exact rate of the approximations by a sequence of weighted Brownian bridges and a weighted Kiefer process. Our arguments are based in part on the Komlós et al. (1975 Komlós, J., Major, P., Tusnády, G. (1975). An approximation of partial sums of independent RV's and the sample DF. I. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 32:111–131.[Crossref], [Web of Science ®] , [Google Scholar])'s results. Applications include the two-sample testing procedures together with the change-point problems. We also consider the strong approximation of the integrated empirical process when the parameters are estimated. Finally, we study the behavior of the self-intersection local time of the partial-sum process representation of the integrated empirical process.     

Non Uniform Bounds on Geometric Approximation Via Stein's Method and w-Functions

In this article, we use Stein's method and w-functions to give uniform and non uniform bounds in the geometric approximation of a non negative integer-valued random variable. We give some applications of the results of this approximation concerning the beta-geometric, Pólya, and Poisson distributions.     

Non-homogeneous Pólya-Aeppli process

Abstract

In this paper we suppose that the intensity parameter of the Pólya-Aeppli process is a function of time t and call the resulting process a non-homogeneous Pólya-Aeppli process (NHPAP). The NHPAP can be represented as a compound non-homogeneous Poisson process with geometric compounding distribution as well as a pure birth process. For this process we give two definitions and show their equivalence. Also, we derive some interesting properties of NHPAP and use simulation the illustrate the process for particular intensity functions. In addition, we introduce the standard risk model based on NHPAP, analyze the ruin probability for this model and include an example of the process under exponentially distributed claims.     

Estimation of the Memory Parameters of the Fractionally Integrated Separable Spatial Autoregressive (FISSAR(1, 1)) Model: A Simulation Study

In this article, we implement the Regression Method for estimating (d ₁, d ₂) of the FISSAR(1, 1) model. It is also possible to estimate d ₁ and d ₂ by Whittle's method. We also compute the estimated bias, standard error, and root mean square error by a simulation study. A comparison was made between the Regression Method of estimating d ₁ and d ₂ to that of the Whittle's method. It was found in this simulation study that the Regression Method of estimation was better when compare with the Whittle's estimator, in the sense that it had smaller root mean square errors (RMSE) values.     

Unification of Dirichlet Methodology

In this article we provide a unified framework for solving Dirichlet related probability and waiting time problems. We consider a Pólya sampling scheme in which each time an object is selected, it is put back into the population along with c additional objects of the same type. By considering both fixed sample size and inverse sampling procedures, we unify the Dirichlet I, J, C, and D functions with their hypergeometric counterparts by extending these functions to Pólya sampling. We then use these functions to unify and extend the corresponding expected waiting time results.     

Large Deviation Results for Wave Governed Random Motions Driven by Semi-Markov Processes

In this article, we present large deviation results for a model {ξ₁ + … + ξ _n : n ≥ 1} which is close to a random walk. More precisely, we consider independent random variables {ξ _n : n ≥ 1} such that {ξ _n : n ≥ 2} are i.i.d. and a different distribution for ξ₁ is allowed. We prove large deviation estimates for P(N _x  ≤ xT) and P(N _x < ∞) as x → ∞, where N _x : = inf {n ≥ 1: ξ₁ + … + ξ _n  ≥ x}. Moreover, we provide an asymptotically efficient simulation law for the estimation of P(N _x  ≤ xT) and P(N _x < ∞) by Monte Carlo simulation based on the importance sampling technique. These results will be adapted to wave governed random motions driven by semi-Markov processes and we present some simulations. Finally, we study the convergence of some large deviation rates for standard wave governed random motions based on a scaling presented in the literature (see Kac, 1974 Kac , M. ( 1974 ). A stochastic model related to the telegrapher's equation . Rocky Mountain Journal of Mathematics 4 : 497 – 509 .[Crossref] , [Google Scholar]; Orsingher, 1990 Orsingher , E. ( 1990 ). Probability law, flow function, maximum distribution of wave governed random motions and their connections with Kirchoff's laws . Stochastic Processes and their Applications 34 ( 1 ): 49 – 66 . [Google Scholar]).     

Asymptotic Results for the MPL Estimators of the Contact Process of Order p

Let X be a discrete time contact process (CP) of order p on Z ² as defined by Durrett and Levin (1994 Durrett , R. , Levin , S. A. ( 1994 ). Stochastic spatial models: a user's guide to ecological applications . Phil. Trans. Roy. Soc. Lond. B 343 : 329 – 350 .[Crossref], [Web of Science ®] , [Google Scholar]), that is a CP where the function of birth takes p different values. We study the maximum marginal pseudo-likelihood (MPL) estimator of the model based on space-time evolution of X, that is, T + 1 successive observations of X on a finite subset S of sites. When T → ∞, this estimator is consistent and asymptotically normal for a non vanishing supercritical CP. We also propose a statistical test for the parameters of the model and verify the asymptotic results by some simulation studies.     

Estimation of Parameters of a Clipped MA(1) Process

Let {X _t , t ∈ ?} be a sequence of iid random variables with an absolutely continuous distribution. Let a > 0 and c ∈ ? be some constants. We consider a sequence of 0-1 valued variables {ξ _t , t ∈ ?} obtained by clipping an MA(1) process X _t  ? aX _t?1 at the level c, i.e., ξ _t  = I[X _t  ? aX _t?1 < c] for all t ∈ ?. We deal with the estimation problem in this model. Properties of the estimators of the parameters a and c, the success probability p, and the 1-lag autocorrelation r ₁ are investigated. A numerical study is provided as an illustration of the theoretical results.     

Locking the Wiener Process to its Level-Crossing Time

We consider the specific transformation of a Wiener process {X(t), t ≥ 0} in the presence of an absorbing barrier a that results when this process is “time-locked” with respect to its first passage time T _a through a criterion level a, and the evolution of X(t) is considered backwards (retrospectively) from T _a . Formally, we study the random variables defined by Y(t) ≡ X(T _a  ? t) and derive explicit results for their density and mean, and also for their asymptotic forms. We discuss how our results can aid interpretations of time series “response-locked” to their times of crossing a criterion level.     

Asymptotic Tail Properties of Student's t-Distribution

We investigate the asymptotic behavior of the probability density function (pdf) and the cumulative distribution function (cdf) of Student's t-distribution with ν > 0 degrees of freedom (t _ν for short) for ν tending to infinity when the argument x = x _ν of the pdf (cdf) depends on ν and tends to ± ∞ (?∞). To this end, we consider the ratio of the pdf's (cdf's) of the t _ν- and the standard normal distribution. Depending on the choice of the argument x _ν, the pdf-ratio (cdf-ratio) tends to 1, a fixed value greater than 1, or to ∞. As a byproduct, we obtain a result for Mill' ratio when x _ν → ?∞.     

Nonparametric Prediction Intervals for Future Order Statistics in a Proportional Hazard Model

Consider k independent random samples with different sample sizes such that the ith sample comes from the cumulative distribution function (cdf) F _i  = 1 ? (1 ? F)^{α _i} , where α _i is a known positive constant and F is an absolutely continuous cdf. Also, suppose that we have observed the maximum and minimum of the first k samples. This article shows how one can construct the nonparametric prediction intervals for the order statistics of the future samples on the basis of these information. Three schemes are studied and in each case exact expressions for the prediction coefficients of prediction intervals are derived. Numerical computations are given for illustrating the results. Also, a comparison study is done while the complete samples are available.     

Locally minimax tests for multiple correlations

Let X be a normally distributed p-dimensional column vector with mean μ and positive definite covariance matrix σ. and let X α, α = 1,…, N, be a random sample of size N from this distribution. Partition X as ( X ₁, X ₍₂₎', X '₍₃₎)', where X₁ is one-dimension, X₍₂₎ is p₂- dimensional, and so 1 + p₁ + p₂ = p. Let ρ₁ and ρ be the multiple correlation coefficients of X₁ with X₍₂₎ and with ( X '₍₂₎, X '₍₃₎)', respectively. Write ρ2/2 = ρ² - ρ2/1. We shall cosider the following two problems     

Space-time estimation of a particle system model

Let X be a discrete time contact process (CP) on ?², as defined by Durrett and Levin (1994, Stochastic spatial models: a user's guide to ecological applications. Philosophical Transactions of the Royal Society of London Series B, 343, 329–350). We study the estimation of the model based on space-time evolution of X, that is, T + 1 successive observations of X on a finite subset S of sites. We consider the maximum marginal pseudo-likelihood (MPL) estimator and show that, when T→∞, this estimator is consistent and asymptotically normal for a non-vanishing supercritical CP. Numerical studies confirm these theoretical ones.     

Estimating prediction error for complex samples

With a growing interest in using non-representative samples to train prediction models for numerous outcomes it is necessary to account for the sampling design that gives rise to the data in order to assess the generalized predictive utility of a proposed prediction rule. After learning a prediction rule based on a non-uniform sample, it is of interest to estimate the rule's error rate when applied to unobserved members of the population. Efron (1986) proposed a general class of covariance penalty inflated prediction error estimators that assume the available training data are representative of the target population for which the prediction rule is to be applied. We extend Efron's estimator to the complex sample context by incorporating Horvitz–Thompson sampling weights and show that it is consistent for the true generalization error rate when applied to the underlying superpopulation. The resulting Horvitz–Thompson–Efron estimator is equivalent to dAIC, a recent extension of Akaike's information criteria to survey sampling data, but is more widely applicable. The proposed methodology is assessed with simulations and is applied to models predicting renal function obtained from the large-scale National Health and Nutrition Examination Study survey. The Canadian Journal of Statistics 48: 204–221; 2020 © 2019 Statistical Society of Canada     

Finding maximum G-criterion values for central composite designs on the hypercube

For quadratic regression on the hypercube, G—efficiencies are often used in the selection process of an experimental design. To calculate a design's G—efficiency, it is necessary to maximize the prediction variance over the experimental design region. However, it is common to approximate a G—efficiency. This is achieved by calculating the prediction variances generated from a subset of points in the design space and taking the maximum to estimate the maximum prediction variance. This estimate is then applied to approximate the G—efficiency. In this paper, it will be shown that over the class of central composite designs (CCDs) on the hypercube. the prediction variance can be expressed in a closed-form. An exact value of the maximum prediction variance can then be determined by evaluating this closed-form expression over a finite subset of barycentric points. Tables of exact G—efficiencies will be presented. Design optimality criteria, quadratic regression on the hypercube, and the structures of the design matrix X, X'X, and (X'X)^?1 for any CCD will be discussed.     

Order-preserving Estimators and an Inequality on the Integration of Zonal Polynomial

ABSTRACT

Suppose X , p × p p.d. random matrix, has the distribution which depends on a p × p p.d. parameter matrix Σ and this distribution is orthogonally invariant. The orthogonally invariant estimator of Σ which has the eigenvalues of the same order as the eigenvalues of X is called order-preserving. We conjecture that a non-order-preserving estimator is dominated by modified order-preserving estimators with respect to the entropy (Stein's) loss function. We show that an inequality on the integration of zonal polynomial is sufficient for this conjecture. We also prove this inequality for the case p = 2.     

Modification of Pickands' Dependence Function for Ordered Bivariate Extreme Distribution

We study the characteristics of the Pickands' dependence function for bivariate extreme distribution for minima, BEVM, when considering the stochastics ordering of the two variables, X < Y. The existing Pickand's dependence function terminologies and theories are modified to suit the dependence functions of extreme minimum cases. The main result is the introduction of the restricted logistic dependence function, A _RL , and the restricted exponential function, V _RL (x, y).     

On constructing T 2 control charts for retrospective examination

In this paper we consider the issue of constructing retrospective T ² control chart limits so as to control the overall probability of a false alarm at a specified value. We describe an exact method for constructing the control limits for retrospective examination. We then consider Bonferroni-adjustments to Alt's control limit and to the standard x ² control limit as alternatives to the exact limit since it is computationally cumbersome to find the exact limit. We present the results of some simulation experiments that are carried out to compare the performance of these control limits. The results indicate that the Bonferroni-adjusted Alt's control limit performs better that the Bonferroni-adjusted x ² control limit. Furthermore, it appears that the Bonferroni-adjusted Alt's control limit is more than adequate for controlling the overall false alarm probability at a specified value.     

On improvement of prediction in arma processes

Necessary and sufficient conditions are derived in the paper that enable to decide whether an additional multivariate process will improve the prediction in a given multivariate discrete stationary process. The both processes are assumed to form together a process ARMAm n Further it was investigated wnen one can asser t that the both processes are uncorrelated provided the additional process did not improve the prediction in the original process, Some hints for the actual construction of predictors in a multivariate ARMA. (m n) process can be found in the paper.