Non-negative Integer-valued Random Variables Research Articles

Statistical Inference for Damaged Poisson Distribution

For non-negative integer-valued random variables, the concept of “damaged” observations was introduced, for the first time, by Rao and Rubin [Rao, C. R., Rubin, H. (1964). On a characterization of the Poisson distribution. Sankhya 26:295–298] in 1964 on a paper concerning the characterization of Poisson distribution. In 1965, Rao [Rao, C. R. (1965). On discrete distribution arising out of methods of ascertainment. Sankhya Ser. A. 27:311–324] discusses some results related with inferences for parameters of a Poisson Model when it has occurred partial destruction of observations. A random variable is said to be damaged if it is unobservable, due to a damage mechanism which randomly reduces its magnitude. In subsequent years, considerable attention has been given to characterizations of distributions of such random variables that satisfy the “Rao–Rubin” condition. This article presents some inference aspects of a damaged Poisson distribution, under reasonable assumption that, when an observation on the random variable is made, it is also possible to determine whether or not some damage has occurred. In other words, we do not know how many items are damaged, but we can identify the existence of damage. Particularly it is illustrated the situation in which it is possible to identify the occurrence of some damage although it is not possible to determine the amount of items damaged. Maximum likelihood estimators of the underlying parameters and their asymptotic covariance matrix are obtained. Convergence of the estimates of parameters to the asymptotic values are studied through Monte Carlo simulations.

On limit theorems for the generalised allocation scheme

For sums of independent identically distributed non-negative integer-valued random variables we give a series of limit theorems which may be of use in probabilistic combinatorics in the framework of the generalised allocation scheme.

Extremes of Integer-Valued Moving Average Models with Exponential Type Tails

In this paper we study the limiting distribution of the maximum term of non-negative integer-valued moving average sequences of the form X n = Σ∞ i= - ∞β i o Z n - i where {Z n } is an iid sequence of non-negative integer-valued random variables with exponential type tails of the form 1 - F(n) ∼ Kn ζ (1 + λ) -n when n → ∞, ζ ∈ R, K, λ > 0, and o denotes binomial thinning. Several models are considered allowing different dependence structures of the thinning operations. For these models results are established which present similarities with those obtained for the classic linear Fmoving average: {X n } behaves as if it was iid regarding the limiting distribution of the maximum term. The paper concludes with some examples which apply the results to a particular model, the INAR(1), and with a simulation study to illustrate the results.

Large deviations for heavy-tailed random sums of independent random variables with dominatedly varying tails

We prove large deviation results on the partial and random sums s n =Σ i=1 n X i , n⩾1; S(t)=Σ i=1 N(t) X i , t⩾0, where {N(t);t⩾0} are non-negative integer-valued random variables and {X n ;n⩾1} are independent non-negative random variables with distribution, F n , of X n , independent of {N(t);t⩾0}. Special attention is paid to the distribution of dominated variation.

Moment Inequalities for Discrete Ageing Families

Some moment inequalities are well known in the literature to be valid for several continuous ageing families. In this article, we establish moment inequalities for D-ILR [D-DLR], DS-IFR, DS-DMRL [DS-IMRL], DW-NBU [DS-NWU], DW-NBUE [DS-NBUE] and DW-HNBUE [DS-HNWUE] families of discrete non-negative integer-valued random variables, and also give some minor results about moment inequalities of random variables that are ordered in the mean residual life order.

On limit theorems for the generalised allocation scheme

For sums of independent identically distributed non-negative integer-valued random variables we give a series of limit theorems which may be of use in probabilistic combinatorics in the framework of the generalised allocation scheme.

A contribution to large deviations for heavy-tailed random sums

In this paper we consider the large deviations for random sums S ( t )=Σ i =1 N(t) X i , t ≥0, where { X n , n ≥1｝ are independent, identically distributed and non _ negative random variables with a common heavy-tailed distribution function F , and ｛ N ( t ), t ≥0｝ is a process of non-negative integer-valued random variables, independent of { X n , n ≥1｝. Under the assumption that the tail of F is of Pareto's type (regularly or extended regularly varying), we investigate what reasonable condition can be given on ｛ N ( t ), t ≥0｝ under which precise large deviation for S ( t ) holds. In particular, the condition we obtain is satisfied for renewal counting processes.

Extremes of Integer-Valued Moving Average Models with Regularly Varying Tails

In this paper we study the limiting distribution of the maximum term of non-negative integer-valued moving average sequences of the form X n = ∑ = −∞ ∞ βi ○ Z n − i where { Z n} is an iid sequence of non-negative integer-valued random variables with regularly varying tails, β i○Z n−i denotes binomial thinning. Several models are considered allowing different dependence structures on the thinning operations. For these models we manage to establish results which are similar to the ones obtained for the classic linear moving average: { X n} > has an extremal index θ = \maxi{ β α } / (∑β α ), where α is the index of regular variation, irrespectively of the dependence structure considered. The paper is concluded with a simulation study to illustrate the results.

Large deviations of heavy-tailed random sums with applications in insurance and finance

We prove large deviation results for the random sum , , where are non-negative integer-valued random variables and are i.i.d. non-negative random variables with common distribution function F, independent of . Special attention is paid to the compound Poisson process and its ramifications. The right tail of the distribution function F is supposed to be of Pareto type (regularly or extended regularly varying). The large deviation results are applied to certain problems in insurance and finance which are related to large claims.

Large deviations of heavy-tailed random sums with applications in insurance and finance

We prove large deviation results for the random sum , , where are non-negative integer-valued random variables and are i.i.d. non-negative random variables with common distribution function F, independent of . Special attention is paid to the compound Poisson process and its ramifications. The right tail of the distribution function F is supposed to be of Pareto type (regularly or extended regularly varying). The large deviation results are applied to certain problems in insurance and finance which are related to large claims.

A kind of strong deviation theorem for the sequences of nonnegative integer-valued random variables

Using the notion of likelihood ratio, the limit properties of the sequences of dependent nonnegative integer-valued ndom variables are studied, and a kind of strong limit theorem represented by inequalities, or the strong deviation eorem, is obtained. In the proof an approach of applying the tool of generating function together with the method of litting intervals to the study of the strong laws is proposed.

Comparison between the Chernoff and Factorial Moment Bounds for Discrete Random Variables

The purpose of this note is to establish the superiority of the factorial moment bound over Chernoff's bound for non-negative integer-valued random variables.

A CLASS OF SMALL DEVIATION THEOREMS FOR THE SEQUENCES OF NONNEGATIVE INTEGER-VALUED RANDOM VARIABLES

In virtue of the notion of likelihood ratio the limit properties of the sequences of dependent nonnegative integer-valued random variables are studied, and a class of small deviation theorems are obtained. In the proof an approach of applying the tool of generating function together with the method of splitting intervals to the study of the strong laws is proposed.

Poisson and compound Poisson approximations for random sums of random variables

We derive upper bounds for the total variation distance, d, between the distributions of two random sums of non-negative integer-valued random variables. The main results are then applied to some important random sums, including cluster binomial and cluster multinomial distributions, to obtain bounds on approximating them to suitable Poisson or compound Poisson distributions. These bounds are generally better than the known results on Poisson and compound Poisson approximations. We also obtain a lower bound for d and illustrate it with an example.

Poisson and compound Poisson approximations for random sums of random variables

We derive upper bounds for the total variation distance, d, between the distributions of two random sums of non-negative integer-valued random variables. The main results are then applied to some important random sums, including cluster binomial and cluster multinomial distributions, to obtain bounds on approximating them to suitable Poisson or compound Poisson distributions. These bounds are generally better than the known results on Poisson and compound Poisson approximations. We also obtain a lower bound for d and illustrate it with an example.

Precision Calculation of Distributions for Trimmed Sums

Recursive methods are described for computing the frequency and distribution functions of trimmed sums of independent and identically distributed nonnegative integer-valued random variables. Surprisingly, for fixed arguments, these can be evaluated with just a finite number of arithmetic operations (and whatever else it takes to evaluate the common frequency function of the original summands). These methods give rise to very accurate computational algorithms that permit a delicate numerical investigation, herein described, of Feller's weak law of large numbers and its trimmed version for repeated St. Petersburg games. The performance of Stigler's theorem for the asymptotic distribution of trimmed sums is also investigated on the same example.

A characteriszation of the geometric distribution related to random sums

In this paper, we consider the sum Z of randomly truncated non-negative integer-valued random variables, where the summation index depends on the truncating process. We derive properties of the variable Z. These are closely related to some new properties of the geometric distribution and in fact provide an another characterization of the geometric law.

Hoeffding's inequalities: A counterexample

A sequence ψ: ℕ0 → ℝ satisfiesHoeffding's inequality of order n if wheneverX1,...,Xn are independent nonnegative integer-valued elementary random variables and\(\tilde X_1 , \ldots ,\tilde X_n \) are independent identically distributed nonnegative integer-valued elementary random variables, the common distribution of which is the average of those ofX1,...,Xn. We show that for each integerm greater than 2 there exists a sequence ψ satisfying Hoeffding's inequality of every order greater thanm but not that of orderm. This answers a question raised by Berg, Christensen, and Ressel.

On characterization of power series distributions by a marginal distribution and a regression function

The conditional distribution of Y given X = x, where X and Y are non-negative integer-valued random variables, is characterized in terms of the regression function of X on Y and the marginal distribution of X which is assumed to be of a power series form. Characterizations are given for a binomial conditional distribution when X follows a Poisson, binomial or negative binomial, for a hypergeometric conditional distribu- tion when X is binomial and for a negative hypergeometric conditional distribution when X follows a negative binomial.

A necessary and sufficient condition for convergence in law of random sums of random variables under nonrandom centering

Let { X n } \{X_n\} be a sequence of independent, identically distributed (i.i.d.) random variables with common mean μ ≠ 0 \mu \ne 0 and variance σ 2 > 0 \sigma ^2 > 0 . Let { S n } \{S_n\} be a sequence of nonnegative integer-valued random variables such that for each n n the random variables S n S_n , X 1 X_1 , X 2 X_2 , … are independent. Then ( X 1 + ⋯ + X S n − n μ ) / n σ 2 → L (X_1 + \dots + X_{S_n} - n \mu ) / \sqrt {n\sigma ^2} \stackrel {\mathcal {L}}{\rightarrow } {} (some) Z Z if and only if ( S n − n ) / n → L (S_n - n)/\sqrt {n} \stackrel {\mathcal {L}}{\rightarrow } {} (some) U U , in which case the distribution of Z Z is that of X + Y X+Y , where X X and Y Y are independent random variables, X X being N ( 0 , 1 ) \mathcal {N}(0, 1) and Y Y having the same distribution as μ U / σ \mu U/\sigma .