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In this paper, we present a probabilistic extension of the Fubini polynomials and numbers associated with a random variable satisfying some appropriate moment conditions. We obtain the exponential generating function and an integral representation for it. The higher order Fubini polynomials and recurrence relations are also derived. A probabilistic generalization of a series transformation formula and some interesting examples are discussed. A connection between the probabilistic Fubini polynomials and Bernoulli, Poisson, and geometric random variables are also established. Finally, a determinant expression formula is presented. [ABSTRACT FROM AUTHOR]

In this paper, we develop Stein’s method for binomial approximation using the stop-loss metric that allows one to obtain a bound on the error term between the expectation of call functions. We obtain the results for a locally dependent collateralized debt obligation (CDO), under certain conditions on moments. The results are also exemplified for an independent CDO. Finally, it is shown that our bounds are sharper than the existing bounds. [ABSTRACT FROM AUTHOR]

The distribution of the sum of 1-dependent lattice vectors with supports on coordinate axes is approximated by a multivariate compound Poisson distribution and by signed compound Poisson measure. The local and ℓ α -norms are used to obtain the error bounds. The Heinrich method is used for the proofs. [ABSTRACT FROM AUTHOR]

We obtain the explicit expressions for the state probabilities of various state-dependent versions of fractional point processes. The inversion of the Laplace transforms of the state probabilities of such processes is rather cumbersome and involved. We employ the Adomian decomposition method to solve the difference-differential equations governing the state probabilities of these state-dependent processes. The distributions of some convolutions of the Mittag-Leffler random variables are derived as special cases of the obtained results. [ABSTRACT FROM AUTHOR]

In this paper, we study the fractional Poisson process (FPP) time-changed by an independent Lévy subordinator and the inverse of the Lévy subordinator, which we call TCFPP-I and TCFPP-II, respectively. Various distributional properties of these processes are established. We show that, under certain conditions, the TCFPP-I has the long-range dependence property, and also its law of iterated logarithm is proved. It is shown that the TCFPP-II is a renewal process and its waiting time distribution is identified. The bivariate distributions of the TCFPP-II are derived. Some specific examples for both the processes are discussed. Finally, we present simulations of the sample paths of these processes. [ABSTRACT FROM AUTHOR]

The analysis of incomplete contingency tables is a practical and an interesting problem. In this paper, we provide characterizations for the various missing mechanisms of a variable in terms of response and non-response odds for two and three dimensional incomplete tables. Log-linear parametrization and some distinctive properties of the missing data models for the above tables are discussed. All possible cases in which data on one, two or all variables may be missing are considered. We study the missingness of each variable in a model, which is more insightful for analyzing cross-classified data than the missingness of the outcome vector. For sensitivity analysis of the incomplete tables, we propose easily verifiable procedures to evaluate the missing at random (MAR), missing completely at random (MCAR) and not missing at random (NMAR) assumptions of the missing data models. These methods depend only on joint and marginal odds computed from fully and partially observed counts in the tables, respectively. Finally, some real-life datasets are analyzed to illustrate our results, which are confirmed based on simulation studies. [ABSTRACT FROM AUTHOR]

Assuming conditions on factorial cumulants, we estimate the closeness of distribution of a sum of nonnegative integer-valued m-dependent random variables to the class of all infinitely divisible laws. The accuracy of approximation is measured in total variation and local metrics. Our results are exemplified by an analogue of the first uniform Kolmogorov theorem for the statistic of (k1,k2) events. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a space fractional negative binomial process (SFNB) by time-changing the space fractional Poisson process by a gamma subordinator. Its one-dimensional distributions are derived in terms of generalized Wright functions and their governing equations are obtained. It is a Lévy process and the corresponding Lévy measure is given. Extensions to the case of distributed order SFNB, where the fractional index follows a two-point distribution, are investigated in detail. The relationship with space fractional Polya-type processes is also discussed. Moreover, we define and study multivariate versions, which we obtain by time-changing a d-dimensional space-fractional Poisson process by a common independent gamma subordinator. Some applications to population’s growth and epidemiology models are explored. Finally, we discuss algorithms for the simulation of the SFNB process. [ABSTRACT FROM AUTHOR]

Collapsibility deals with the conditions under which a conditional (on a covariate W) measure of association between two random variables Y and X equals the marginal measure of association. In this paper, we discuss the average collapsibility of certain well-known measures of association, and also with respect to a new measure of association. The concept of average collapsibility is more general than collapsibility, and requires that the conditional average of an association measure equals the corresponding marginal measure. Sufficient conditions for the average collapsibility of the association measures are obtained. Some interesting counterexamples are constructed and applications to linear, Poisson, logistic and negative binomial regression models are discussed. An extension to the case of multivariate covariate W is also analyzed. Finally, we discuss the collapsibility conditions of some dependence measures for survival models and illustrate them for the case of linear transformation models. [ABSTRACT FROM AUTHOR]

We prove the Simons-Johnson theorem for sums $$S_n$$ of $$m$$ -dependent random variables with exponential weights and limiting compound Poisson distribution $$\mathrm {CP}(s,\lambda )$$ . More precisely, we give sufficient conditions for $$\sum _{k=0}^\infty {\mathrm e}^{hk}\vert P(S_n=k)-\mathrm {CP}(s,\lambda )\{k\}\vert \rightarrow 0$$ and provide an estimate on the rate of convergence. It is shown that the Simons-Johnson theorem holds for the weighted Wasserstein norm as well. The results are then illustrated for $$N(n;k_1,k_2)$$ and $$k$$ -runs statistics. [ABSTRACT FROM AUTHOR]

In this paper, the problem of compound Poisson approximation to the convolution of compound negative binomial distributions, under total variation distance, is considered. First, we obtain an error bound using the method of exponents and it is compared with existing ones. It is known that Kerstan's method is more powerful in compound approximation problems. We employ Kerstan's method to obtain better estimates, using higher-order approximations. These bounds are of higher-order accuracy and improve upon some of the known results in the literature. Finally, an interesting application to risk theory is discussed. [ABSTRACT FROM AUTHOR]

Analysis of large dimensional contingency tables is rather difficult. Fienberg and Kim (1999, Journal of American Statistical Association, 94, 229–239) studied the problem of combining conditional (on single variable) log-linear structures for graphical models to obtain partial information about the full graphical log-linear model. In this paper, we consider the general log-linear models and obtain explicit representation for the log-linear parameters of the full model based on that of conditional structures. As a consequence, we give conditions under which a particular log-linear parameter is present or not in the full model. Some of the main results of Fienberg and Kim follow from our results. The explicit relationships between full model and the conditional structures are also presented. The connections between conditional structures and the layer structures are pointed out. We investigate also the hierarchical nature of the full model, based on conditional structures. Kim (2006, Computational Statistics and Data Analysis, 50, 2044–2064) analyzed graphical log-linear models based on conditional log-linear structures, when a set of variables is conditioned. For this case, we employ the Möbius inversion technique to obtain the interaction parameters of the full log-linear model, and discuss their properties. The hierarchical nature of the full model is also studied based on conditional structures. This result could be effectively used for the model selection also. As applications of our results, we have discussed several typical examples, including a real-life example. [ABSTRACT FROM AUTHOR]

For a multidimensional contingency table, we obtain several necessary and sufficient conditions for collapsibility and strict collapsibility, using the technique of Möbius inversion formula. As a consequence, the results of Whittemore ( Journal of the Royal Statistical Society B, 40, 328–340, 1978) are stated in a form which is easy to understand and the proofs are much simpler and straightforward. Several new results on collapsibility and strict collapsibility with respect to more than one interaction parameter, are established, and their relationships to conditional independence are also pointed out. As applications of our results, several typical examples on collapsibility, strict collapsibility and conditional independence are discussed. It is also shown that Bishop et al. ( Discrete Multivariate Analysis: Theory and Practice, MIT Press, Cambridge, 1975) conditions are necessary and sufficient for strict collapsibility with respect to a set of interaction factors. [ABSTRACT FROM AUTHOR]

In this paper, we consider a probabilistic model to represent some general dependent production processes and present a unified approach for designing attribute sampling plans for monitoring the ongoing production process. This model includes the classical iid model, independent model, Markov-dependent model and previous-sum dependent model, to mention a few. Some important properties of this model are established. We derive the recurrence relations for the probability distribution of the sum of n consecutive characteristics observed from the process. Using these recurrence relations, we present efficient algorithms for designing optimal single and double sampling plans for attributes, for monitoring the ongoing production process. Our algorithmic approach, which uses effectively the recurrence relations, yields a direct and an exact method, unlike many approximate methods adopted in the literature. Several interesting examples concerning specific models are discussed and a few tables for some special cases are also presented. It is demonstrated that the optimal double sampling plans lead to about 42% reduction in average sample number over the single sampling plans for process monitoring. [ABSTRACT FROM AUTHOR]

We consider the problem of designing single and the double sampling plans for monitoring dependent production processes. Based on simulated samples from the process, Nelson proposed a new approach of estimating the characteristics of single sampling plans and, using these estimates, designing optimal plans. In this paper, we extend his approach to the design of optimal double sampling plans. We first propose a simple methodology for obtaining the unbiased estimators of various characteristics of single and double sampling plans. This is achieved by defining the various characteristics of sampling plans as explicit random variables. Some of the important properties of the double sampling plans are established. Using these results, an efficient algorithm is developed to obtain optimal double sampling plans. A comparison with a crude search shows that our algorithm leads to about 90% savings, on the average, in computational timings. The procedure is also explained through a suitable example for the ARMA(l,l) model. It is observed, for instance, that an optimal double sampling plan leads to about 23% reduction in average sample number, compared to an optimal single sampling plan. Tables for choosing the optimal plans for certain auto regressive moving average processes at some practically useful values of acceptable quality level and rejectable quality level are also presented. [ABSTRACT FROM AUTHOR]

Considers the estimation of the location parameters associated with the selected populations for the squared error loss. Factor that dominates the unbiased estimator; Comparison of the unbiased estimator with natural estimator; Method used in obtaining a class of estimators that dominate the natural estimator N.