On characterization of certain probability distributions

The problem of identifying solutions of general convolution equations relative to a group has been studied in two classical papers by Choquet and Deny (1960) and Deny (1961). Recently, Lau and Rao (1982) have considered the analogous problem relative to a certain semigroup of the real line, which extends the results of Marsaglia and Tubilla (1975) and a lemma of Shanbhag (1977). The extended versions of Deny&s theorem contained in the papers by Lau and Rao, and Shanbhag (which we refer to as LRS theorems) yield as special cases improved versions of several characterizations of exponential, Weibull, stable, Pareto, geometric, Poisson and negative binomial distributions obtained by various authors during the last few years. In this paper we review some of the recent contributions to characterization of probability distributions (whose authors do not seem to be aware of LRS theorems or special cases existing earlier) and show how improved versions of these results follow as immediate corollaries to LRS theorems. We also give a short proof of Lau–Rao theorem based on Deny&s theorem and thus establish a direct link between the results of Deny (1961) and those of Lau and Rao (1982). A variant of Lau–Rao theorem is proved and applied to some characterization problems.

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Recent results on characterization of probability distributions: a unified approach through extensions of Deny&s theorem

Advances in Applied Probability ◽

10.2307/1427182 ◽

1986 ◽

Vol 18 (3) ◽

pp. 660-678 ◽

Cited By ~ 49

Author(s):

C. Radhakrishna Rao ◽

D. N. Shanbhag

Keyword(s):

Negative Binomial ◽

Short Proof ◽

Probability Distributions ◽

Direct Link ◽

Unified Approach ◽

Binomial Distributions ◽

Characterization Of Probability Distributions ◽

Special Cases ◽

Convolution Equations

The problem of identifying solutions of general convolution equations relative to a group has been studied in two classical papers by Choquet and Deny (1960) and Deny (1961). Recently, Lau and Rao (1982) have considered the analogous problem relative to a certain semigroup of the real line, which extends the results of Marsaglia and Tubilla (1975) and a lemma of Shanbhag (1977). The extended versions of Deny&s theorem contained in the papers by Lau and Rao, and Shanbhag (which we refer to as LRS theorems) yield as special cases improved versions of several characterizations of exponential, Weibull, stable, Pareto, geometric, Poisson and negative binomial distributions obtained by various authors during the last few years. In this paper we review some of the recent contributions to characterization of probability distributions (whose authors do not seem to be aware of LRS theorems or special cases existing earlier) and show how improved versions of these results follow as immediate corollaries to LRS theorems. We also give a short proof of Lau–Rao theorem based on Deny&s theorem and thus establish a direct link between the results of Deny (1961) and those of Lau and Rao (1982). A variant of Lau–Rao theorem is proved and applied to some characterization problems.

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Characterization of the logistic and loglogistic distributions by extreme value related stability with random sample size

Journal of Applied Probability ◽

10.1017/s0021900200116729 ◽

1987 ◽

Vol 24 (04) ◽

pp. 838-851 ◽

Cited By ~ 2

Author(s):

W. J. Voorn

Keyword(s):

Positive Integer ◽

Sample Size ◽

Random Variable ◽

Logistic Distribution ◽

Size Distributions ◽

Random Sample Size ◽

Maximum Value ◽

Maximum Stability ◽

Independent Observations

Maximum stability of a distribution with respect to a positive integer random variable N is defined by the property that the type of distribution is not changed when considering the maximum value of N independent observations. The logistic distribution is proved to be the only symmetric distribution which is maximum stable with respect to each member of a sequence of positive integer random variables assuming value 1 with probability tending to 1. If a distribution is maximum stable with respect to such a sequence and minimum stable with respect to another, then it must be logistic, loglogistic or ‘backward' loglogistic. The only possible sample size distributions in these cases are geometric.

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A characterization of stable processes

Journal of Applied Probability ◽

10.1017/s0021900200032915 ◽

1969 ◽

Vol 6 (02) ◽

pp. 409-418 ◽

Cited By ~ 5

Author(s):

Eugene Lukacs

Keyword(s):

Stochastic Process ◽

Distribution Function ◽

Interior Point ◽

Random Variable ◽

Stable Processes ◽

Process With Independent Increments ◽

Independent Increments

Let X(t) be a stochastic process whose parameter t runs over a finite or infinite n terval T. Let t 1 , t 2 ɛ T, t 1 〈 t2; the random variable X(t 2) – X(t 1) is called the increment of the process X(t) over the interval [t 1, t 2]. A process X(t) is said to be homogeneous if the distribution function of the increment X(t + τ) — X(t) depends only on the length τ of the interval but is independent of the endpoint t. Two intervals are said to be non-overlapping if they have no interior point in common. A process X(t) is called a process with independent increments if the increments over non-overlapping intervals are stochastically independent. A process X(t) is said to be continuous at the point t if plimτ→0 [X(t + τ) — X(t)] = 0, that is if for any ε > 0, limτ→0 P(| X(t + τ) — X(t) | > ε) = 0. A process is continuous in an interval [A, B] if it is continuous in every point of [A, B].

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Characterization of matrix probability distributions by mean residual lifetime

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100066305 ◽

1986 ◽

Vol 100 (3) ◽

pp. 583-589

Author(s):

P. E. Jupp

Keyword(s):

Symmetric Matrix ◽

Probability Distributions ◽

Random Variable ◽

Residual Lifetime ◽

Regularity Conditions ◽

Mean Residual Lifetime ◽

The Mean ◽

Image Position ◽

Vector Valued

The mean residual lifetime of a real-valued random variable X is the function e defined byOne of the more important properties of the mean residual lifetime function is that it determines the distribution of X. See, for example, Swartz [10]. References to related characterizations are given by Galambos and Kotz [3], pages 30–35. It was established by Jupp and Mardia[6] that this property holds also for vector-valued X. As (1·1) makes sense if X is a random symmetric matrix, it is natural to ask whether the property holds in this case also. The purpose of this note is to show that, under certain regularity conditions, the distributions of such matrices are indeed determined by their mean residual lifetimes.

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On a conjecture of Seneta on regular variation of truncated moments

Publications de l Institut Mathematique ◽

10.2298/pim2123077k ◽

2021 ◽

Vol 109 (123) ◽

pp. 77-82

Author(s):

Péter Kevei

Keyword(s):

Distribution Function ◽

Relative Stability ◽

Regular Variation ◽

Domain Of Attraction ◽

Random Variable ◽

Nonnegative Random Variable ◽

Equivalent Characterization ◽

Regularly Varying ◽

Normal Law

We prove that h?(x) = ??x0 y??1F?(y)dy is regularly varying with index ? [0, ?) if and only if V?(x) = ?[0,x] y?dF(y) is regularly varying with the same index, where ? > 0, F(x) is a distribution function of a nonnegative random variable, and F?(x) = 1?F(x). This contains at ? = 0, ?= 1 a result of Rogozin [8] on relative stability, and at ? = 0, ? = 2 a new, equivalent characterization of the domain of attraction of the normal law. For ? = 0 and ? > 0 our result implies a recent conjecture by Seneta [9].

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Approximating gamma distributions by normalized negative binomial distributions

Journal of Applied Probability ◽

10.1017/s0021900200044910 ◽

1994 ◽

Vol 31 (02) ◽

pp. 391-400 ◽

Cited By ~ 1

Author(s):

José A. Adell ◽

Jesús De La Cal

Keyword(s):

Distribution Function ◽

Negative Binomial ◽

Upper And Lower Bounds ◽

Exact Order ◽

Bernstein Type ◽

Birth Process ◽

Charged Multiplicity ◽

Binomial Distributions ◽

Gamma Distributions ◽

Multiplicity Distributions

Let F be the gamma distribution function with parameters a > 0 and α > 0 and let Gs be the negative binomial distribution function with parameters α and a/s, s > 0. By combining both probabilistic and approximation-theoretic methods, we obtain sharp upper and lower bounds for . In particular, we show that the exact order of uniform convergence is s–p , where p = min(1, α). Various kinds of applications concerning charged multiplicity distributions, the Yule birth process and Bernstein-type operators are also given.

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Characterizations of certain multivariate distributions

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s030500410004843x ◽

1974 ◽

Vol 75 (2) ◽

pp. 219-234 ◽

Cited By ~ 2

Author(s):

Y. H. Wang

Keyword(s):

Distribution Function ◽

Normal Distribution ◽

Random Variable ◽

Sample Variance ◽

Sufficient Condition ◽

Multivariate Distributions ◽

Sample Mean ◽

One Dimensional ◽

Independent Observations ◽

Image Position

Let X1, X2, …, Xn, be n (n ≥ 2) independent observations on a one-dimensional random variable X with distribution function F. Letbe the sample mean andbe the sample variance. In 1925, Fisher (2) showed that if the distribution function F is normal then and S2 are stochastically independent. This property was used to derive the student's t-distribution which has played a very important role in statistics. In 1936, Geary(3) proved that the independence of and S2 is a sufficient condition for F to be a normal distribution under the assumption that F has moments of all order. Later, Lukacs (14) proved this result assuming only the existence of the second moment of F. The assumption of the existence of moments of F was subsequently dropped in the proofs given by Kawata and Sakamoto (7) and by Zinger (27). Thus the independence of and S2 is a characterizing property of the normal distribution.

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Best constants for tensor products of Bernstein, Szász and Baskakov operators

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700018682 ◽

2000 ◽

Vol 62 (2) ◽

pp. 211-220 ◽

Cited By ~ 6

Author(s):

Jesús de la Cal ◽

Ana M. Valle

Keyword(s):

Tensor Product ◽

Negative Binomial ◽

Probability Distributions ◽

Best Constants ◽

Moduli Of Continuity ◽

Lower And Upper Bounds ◽

Baskakov Operators ◽

Binomial Distributions ◽

Families Of Operators ◽

Crucial Ingredient

We consider tensor product operators and discuss their best constants in preservation inequalities concerning the usual moduli of continuity. In a previous paper, we obtained lower and upper bounds on such constants, under fairly general assumptions on the operators. Here, we concentrate on the l∞-modulus of continuity and three celebrated families of operators. For the tensor product of k identical copies of the Bernstein operator Bn, we show that the best uniform constant coincides with the dimension k when k ≥ 3, while, in case k = 2, it lies in the interval [2, 5/2] but depends upon n. Similar results also hold when Bn is replaced by a univariate Szász or Baskakov operator. The three proofs follow the same pattern, a crucial ingredient being some special properties of the probability distributions involved in the mentioned operators, namely: the binomial, Poisson, and negative binomial distributions.

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A characterization of the gamma distribution by the negative binomial distribution

Journal of Applied Probability ◽

10.1017/s0021900200097473 ◽

1980 ◽

Vol 17 (04) ◽

pp. 1138-1144 ◽

Cited By ~ 4

Author(s):

Jan Engel ◽

Mynt Zijlstra

Keyword(s):

Poisson Process ◽

Exponential Distribution ◽

Gamma Distribution ◽

Negative Binomial Distribution ◽

Binomial Distribution ◽

Negative Binomial ◽

Random Variable ◽

One To One

It is proved that for a Poisson process there exists a one-to-one relation between the distribution of the random variable N(Y) and the distribution of the non-negative random variable Y. This relation is used to characterize the gamma distribution by the negative binomial distribution. Furthermore it is applied to obtain some characterizations of the exponential distribution.

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