Sequences Of Discrete Random Variables Research Articles

Semibinomial conditionally nonlinear autoregressive models of discrete random sequences: probabilistic properties and statistical parameter estimation

AbstractWe introduce a new model ${\mathscr{P}}$-CNAR(s) of sequences of discrete random variables with long memory determined by semibinomial conditionally nonlinear autoregression of order s ∈ ℕ with small number of parameters. Probabilistic properties of this model are studied. For parameters of the model ${\mathscr{P}}$-CNAR a family of consistent asymptotically normal statistical FB-estimates is suggested and the existence of an efficient FB-estimate is proved. Computational advantages of FB-estimate w.r.t. maximum likelihood estimate are shown: less restrictive sufficient conditions for uniqueness, explicit form of FB-estimate, fast recursive computation algorithm under extension of the model ${\mathscr{P}}$-CNAR. Subfamily of “sparse” FB-estimates that use some subset of frequencies of s-tuples is constructed, the asymptotic variance minimization problem within this subfamily is solved.

Stochastic modeling of global risk severity

The paper formulates a stochastic model by incorporating a discrete random variable, two sequences of discrete random variables, and a sequence of positive random variables. The paper also provides an interpretation of the formulated stochastic model in investigating the severity of a global risk. Sufficient conditions for the representation of the formulated stochastic model as a random sum are also established by the paper.

Testing Dependence Among Serially Correlated Multi-Category Variables

The contingency table literature on tests for dependence among discrete multi-category variables is extensive. Existing tests assume, however, that draws are independent, and there are no tests that account for serial dependencies - a problem that is particularly important in economics and finance. This paper proposes a new test of independence based on the maximum canonical correlation between pairs of discrete variables. We also propose a trace canonical correlation test using dynamically augmented reduced rank regressions or an iterated weighting method in order to account for serial dependence. Such tests are useful, for example, when testing for predictability of one sequence of discrete random variables by means of another sequence of discrete random variables as in tests of market timing skills or business cycle analysis. The proposed tests allow for an arbitrary number of categories, are robust in the presence of serial dependencies and are simple to implement using multivariate regression methods. Monte Carlo experiments show that the proposed tests have good finite sample properties. An empirical application to survey data on forecasts of GDP growth demonstrates the importance of correcting for serial dependencies in predictability tests.

Maximum term of a particular autoregressivesequence with discrete margins

In this paper we consider a stationary sequence of discrete random variables with marginal distribution H(x), obtained by a simple transformation from the max-AR(1) sequence considered by Alpuim (1989). Because discrete distributions impose severe restrictions on the convergence of the normalized maxima to an extreme value distribution, it is seen that in this particular case, whenever H(x) belongs to the domain of attraction of any max-stable distribution, the sequence possesses an extremal index 0 = 0. Nevertheless, it, is possible to obtain a nondegenerate limiting distribution for the linearized maxima by choosing other sets of normalizing constants. Whenever H(x) does not belong to the domain of attraction of any max-stable distribution, but, satisfies adequate conditions, the maxima nearly possess an asymptotic stability with the presence of an extremal index 0 <θ<1. Motivated by the behaviour of these sequences we obtained a more general result extending the results of Anderson (1970) and Me (Jon nick and Park (1992) over the mixing conditionsD (k)(un), defined by Chermck et al (1991). Several examples, obtained after simulation, are presented in order to illustrate the different situations that may occur.

A test for randomness based on a complexity measure

A test for randomness based on a statistic related to the complexity of finite sequences is presented. Simulation of binary sequences under different stochastic models provides estimates of the power of the test. The results show that the test is sensitive to a variety of alternatives to randomness and suggest that the proposed test statistic is a reasonable measure of the stochastic complexity of a finite sequence of discrete random variables.

Waiting time problems for a sequence of discrete random variables

Let X1, X2,... be a sequence of nonnegative integer valued random variables.For each nonnegative integer i, we are given a positive integer ki. For every i = 0, 1, 2,..., Ei denotes the event that a run of i of length ki occurs in the sequence X1, X2,.... For the sequence X1, X2,..., the generalized pgf's of the distributions of the waiting times until the r-th occurrence among the events % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% WaaiWabeaacaWGfbWaaSbaaSqaaiaadMgaaeqaaaGccaGL7bGaayzF% aaWaa0baaSqaaiaadMgacqGH9aqpcaaIWaaabaGaeyOhIukaaaaa!43D8!\[\left\{ {E_i } \right\}_{i = 0}^\infty\]are obtained. Though our situations are general, the results are very simple. For the special cases that X's are i.i.d. and {0, 1}-valued, the corresponding results are consistent with previously published results.

Capacity of a pulse amplitude modulated direct detection photon channel

The classical direct detection optical channel is modelled by an observed Poisson process with intensity (rate) γ(t) + γo, where γ(t) is the information carrying input waveform and γo represents the ‘dark current’. The capacity of this channel is considered within a restricted class of peak power γ(t) ≤ A and average power E(γ(t)) ≤ σ constrained-pulse amplitude modulated- input waveforms. Within this class where γ(t) = γi, during the ith signalling interval iΔ ≤ t &lt;(i + 1)Δ the ‘symbol duration’ Δ affects the spectral properties (‘bandwidth’) of γ(t). The capacity achieving distribution of the symbols {γi} is determined by setting {γi} to be an independent identically distributed sequence of discrete random variables taking on a finite number of values. The two valued distribution of γ with mass points located at 0 and A is capacity achieving for σ = A (no average power constraint) and γo = 0, in the region 0 &lt; AΔ &lt; 3.3679. In the following region (3.3679 ≤ AΔ &lt; ε) the ternary distribution is capacity achieving with the additional mass point rising at 0.339A.

Best choice problem for diserete bandom variables

The optimal selection of a maximum of a sequence of discrete random variables is considered. Rules which maximize the probability of correctly selecting a maximum are investigated.

Normal approximations to discrete unimodal distributions

Sufficient conditions are presented for the limiting normality of sequences of discrete random variables possessing unimodal distributions. The conditions are applied to obtain normal approximations directly for the hypergeometric distribution and the stationary distribution of a special birth-death process.

Normal approximations to discrete unimodal distributions

Sufficient conditions are presented for the limiting normality of sequences of discrete random variables possessing unimodal distributions. The conditions are applied to obtain normal approximations directly for the hypergeometric distribution and the stationary distribution of a special birth-death process.

Normal approximations to discrete unimodal distributions

Sufficient conditions are presented for the limiting normality of sequences of discrete random variables possessing unimodal distributions. The conditions are applied to obtain normal approximations directly for the hypergeometric distribution and the stationary distribution of a special birth-death process.

Response-time simulation of multivariate point process models for multiprogrammed jobstreams

We consider simulation methods for particular marked multivariate point processes, calledR-processes, which have recently been proposed as models for multiprogrammed jobstreams. Using “workload marks” on events, such models facilitate the incorporation of realistic workload characteristics into computer system performance predictions. We consider R-processes in which workload marks for an individual jobstream form a stationary sequence of discrete random variables having a (generally non-Markovian) mixed moving average-autoregressive dependency structure. For such models we provide a method for obtaining from a single simulation run point and interval estimates for general characteristics of job response times.

A storage process by semi-Markov input

A sequence of random variables η 0, η 1, …, ηn , … is defined by the recurrence formula ηn = max (η n–1 + ξn , 0) where η 0 is a discrete random variable taking on non-negative integers only and ξ 1, ξ 2, … ξn , … is a semi-Markov sequence of discrete random variables taking on integers only. Define Δ as the smallest n = 1, 2, … for which ηn = 0. The random variable ηn can be interpreted as the content of a dam at time t = n(n = 0, 1, 2, …) and Δ as the time of first emptiness. This paper deals with the determination of the distributions of ηn and Δ by using the method of matrix factorisation.

A storage process by semi-Markov input

A sequence of random variables η0, η1, …, ηn, … is defined by the recurrence formula ηn = max (ηn–1 + ξn, 0) where η0 is a discrete random variable taking on non-negative integers only and ξ1, ξ2, … ξn, … is a semi-Markov sequence of discrete random variables taking on integers only. Define Δ as the smallest n = 1, 2, … for which ηn = 0. The random variable ηn can be interpreted as the content of a dam at time t = n(n = 0, 1, 2, …) and Δ as the time of first emptiness. This paper deals with the determination of the distributions of ηn and Δ by using the method of matrix factorisation.