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1.
Sigeo Aki Hisataka Kuboki Katuomi Hirano 《Annals of the Institute of Statistical Mathematics》1984,36(1):431-440
Summary This paper gives some results on calculation of probabilities and moments of the discrete distributions of orderk. Further, a new distribution of orderk, which is called the logarithmic series distribution of orderk, is investigated. Finally, we discuss the meaning of theorder of the distributions.
The Institute of Statistical Mathematics 相似文献
2.
In this paper, we investigate the exact distribution of the waiting time for ther-th ℓ-overlapping occurrence of success-runs of a specified length in a sequence of two state Markov dependent trials. The
probability generating functions are derived explicitly, and as asymptotic results, relationships of a negative binomial distribution
of orderk and an extended Poisson distribution of orderk are discussed. We provide further insights into the run-related problems from the viewpoint of the ℓ-overlapping enumeration
scheme. We also study the exact distribution of the number of ℓ-overlapping occurrences of success-runs in a fixed number
of trials and derive the probability generating functions. The present work extends several properties of distributions of
orderk and leads us a new type of geneses of the discrete distributions. 相似文献
3.
Andreas N. Philippou 《Annals of the Institute of Statistical Mathematics》1988,40(3):467-475
A multiparameter negative binomial distribution of order k is obtained by compounding the extended (or multiparameter) Poisson distribution of order k by the gamma distribution. A multiparameter logarithmic series distribution of order k is derived next, as the zero truncated limit of the first distribution. Finally a few genesis schemes and interrelationships are established for these three multiparameter distributions of order k. The present work extends several properties of distributions of order k. 相似文献
4.
Sigeo Aki 《Annals of the Institute of Statistical Mathematics》1985,37(1):205-224
Summary This paper considers discrete distributions of orderk based on a binary sequence which is defined as an extension of independent trials with a constant success probability and
is more practical than the independent trials. Some results on calculation of probabilities and characteristics of the distributions
are obtained as well as their formal expressions. Examples and an application are also given.
The Institute of Statistical Mathematics 相似文献
5.
Exact distributions of the numbers of failures, successes and successes with indices no less thanl (1lk–1) until the first consecutivek successes are obtained for some {0, 1}-valued random sequences such as a sequence of independent and identically distributed (iid) trials, a homogeneous Markov chain and a binary sequence of orderk. The number of failures until the first consecutivek successes follows the geometric distribution with an appropriate parameter for each of the above three cases. When the {0, 1}-sequence is an iid sequence or a Markov chain, the distribution of the number of successes with indices no less thanl is shown to be a shifted geometric distribution of orderk - l. When the {0, 1}-sequence is a binary sequence of orderk, the corresponding number follows a shifted version of an extended geometric distribution of orderk - l.This research was partially supported by the ISM Cooperative Research Program (92-ISM-CRP-16) of the Institute of Statistical Mathematics. 相似文献
6.
This paper considers estimating parameters in the discrete distributions of order k such as the binomial, the geometric, the Poisson and the logarithmic series distributions of order k. It is discussed how to calculate maximum likelihood estimates of parameters of the distributions based on independent observations. Further, asymptotic properties of estimators by the method of moments are investigated. In some cases, it is found that the values of asymptotic efficiency of the moment estimators are surprisingly close to one. 相似文献
7.
N. Andreas Filippu 《Journal of Mathematical Sciences》1984,27(6):3294-3297
One investigates the Poisson distribution of orderk. One finds its generating function and with its aid one establishes that the sum of independent random variables, distributed according to the Poisson law of orderk, is distributed in the same manner. In addition, one considers a generalized compound Poisson distribution.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 130, pp. 175–180, 1983. 相似文献
8.
S. G. Mohanty 《Annals of the Institute of Statistical Mathematics》1994,46(4):777-796
The geometric type and inverse Polýa-Eggenberger type distributions of waiting time for success runs of lengthk in two-state Markov dependent trials are derived by using the probability generating function method and the combinatorial method. The second is related to the minimal sufficient partition of the sample space. The first two moments of the geometric type distribution are obtained. Generalizations to ballot type probabilities of which negative binomial probabilities are special cases are considered. Since the probabilities do not form a proper distribution, a modification is introduced and new distributions of orderk for Markov dependent trials are developed. 相似文献
9.
Summary A new ideal metric of orderr>1 is introduced on
k
and a thorough analysis of its metric properties is given. In comparison to the known ideal metric of Zolotarev this new metric allows estimates from above by pseudo difference moments and thus allows applications to stable limit theorems. As applications we give the right order Berry-Esséen type result in the stable case, obtain the limiting behaviour of multivariate summability methods and discuss the approximation problem by compound Poisson distributions.Research supported by NATO GRANT CRG 900 798 and by a DFG Grant 相似文献
10.
Dimitris Bertsimas 《Queueing Systems》1988,3(4):305-320
A closed form expression for the waiting time distribution under FCFS is derived for the queueing system MGEk/MGEm/s, where MGEn is the class of mixed generalized Erlang probability density functions (pdfs) of ordern, which is a subset of the Coxian pdfs that have rational Laplace transform. Using the calculus of difference equations and based on previous results of the author, it is proved that the waiting time distribution is of the form 1-
, under the assumption that the rootsU
j
are distinct, i.e. belongs to the Coxian class of distributions of order
. The present approach offers qualitative insight by providing exact and asymptotic expressions, generalizes and unifies the well known theories developed for the G/G/1,G/M/s systems and leads to an
algorithm, which is polynomial if only one of the parameterss orm varies, and is exponential if both parameters vary. As an example, numerical results for the waiting time distribution of the MGE2/MGE2/s queueing system are presented. 相似文献
11.
Joint distributions of numbers of success-runs and failures until the first consecutivek successes 总被引:3,自引:3,他引:0
Joint distributions of the numbers of failures, successes and success-runs of length less thank until the first consecutivek successes are obtained for some random sequences such as a sequence of independent and identically distributed integer valued random variables, a {0, 1}-valued Markov chain and a binary sequence of orderk. There are some ways of counting numbers of runs with a specified length. This paper studies the joint distributions based on three ways of counting numbers of runs, i.e., the number of overlapping runs with a specified length, the number of non-overlapping runs with a specified length and the number of runs with a specified length or more. Marginal distributions of them can be derived immediately, and most of them are surprisingly simple.This research was partially supported by the ISM Cooperative Research Program (93-ISM-CRP-8). 相似文献
12.
Shun-Hwa Li Wen-Jang Huang Mong-Na Lo Huang 《Annals of the Institute of Statistical Mathematics》1994,46(2):351-360
Given two independent positive random variables, under some minor conditions, it is known that fromE(XrX+Y)=a(X+Y)r andE(XsX+Y)=b(X+Y)s, for certain pairs ofr ands, wherea andb are two constants, we can characterizeX andY to have gamma distributions. Inspired by this, in this article we will characterize the Poisson process among the class of renewal processes via two conditional moments. More precisely, let {A(t), t0} be a renewal process, with {S
k, k1} the sequence of arrival times, andF the common distribution function of the inter-arrival times. We prove that for some fixedn andk, kn, ifE(S
k
r
A(t)=n)=atr andE(S
k
s
A(t)=n)=bts, for certain pairs ofr ands, wherea andb are independent oft, then {A(t), t0} has to be a Poisson process. We also give some corresponding results about characterizingFto be geometric whenF is discrete.Support for this research was provided in part by the National Science Council of the Republic of China, Grant No. NSC 81-0208-M110-06. 相似文献
13.
T. Royen 《Journal of multivariate analysis》1991,38(2)
Three classes of expansions for the distribution function of the χk2(d, R)-distribution are given, where k denotes the dimension, d the degree of freedom, and R the “accompanying correlation matrix.” The first class generalizes the orthogonal series with generalized Laguerre polynomials, originally given by Krishnamoorthy and Parthasarathy [12]. The second class contains always absolutely convergent representations of the distribution function by univariate chi-square distributions and the third class provides also the probabilities for any unbounded rectangular regions. In particular, simple formulas are given for the three-variate case including singular correlation matrices R, which simplify the computation of third order Bonferroni inequalities, e.g., for the tail probabilities of max{χi2|1 ≤ i ≤ k} (k > 3). 相似文献
14.
G. M. Fel'dman 《Journal of Mathematical Sciences》1983,23(3):2335-2351
The article deals with the arithmetic of distributions on groups. Let X be a locally compact separable abelian metric group, \(e\left( F \right) = e^{ - F\left( X \right)} \left( {\varepsilon _o + F + \frac{{F*2 - }}{{2!}} + ...} \right)\) the generalized Poisson distribution associated with a completely finite measure F, I0 the class of distributions without indecomposable or idempotent divisors. Some conditions under which the generalized Poisson distributions belong and do not belong to the class I0 are derived. Some other topics related to the class I0 are considered. 相似文献
15.
Ghislain R. Franssens 《Mathematical Methods in the Applied Sciences》2013,36(9):1014-1027
The set of associated homogeneous distributions (AHDs) with support in R is an important subset of the tempered distributions because it contains the majority of the (one‐dimensional) distributions typically encountered in physics applications (including the δ distribution). In a previous work of the author, a convolution and multiplication product for AHDs on R was defined and fully investigated. The aim of this paper is to give an easy introduction to these new distributional products. The constructed algebras are internal to Schwartz’ theory of distributions and, when one restricts to AHDs, provide a simple alternative for any of the larger generalized function algebras, currently used in non‐linear models. Our approach belongs to the same class as certain methods of renormalization, used in quantum field theory, and are known in the distributional literature as multi‐valued methods. Products of AHDs on R, based on this definition, are generally multi‐valued only at critical degrees of homogeneity. Unlike other definitions proposed in this class, the multi‐valuedness of our products is canonical in the sense that it involves at most one arbitrary constant. A selection of results of (one‐dimensional) distributional convolution and multiplication products are given, with some of them justifying certain distributional products used in quantum field theory. Copyright © 2012 John Wiley & Sons, Ltd. 相似文献
16.
Namir Ghoraf 《TOP》2008,16(1):62-72
An “m-consecutive-k-out-of-n:F system” consists of n components ordered on a line; the system fails if and only if there are at least m nonoverlapping runs of k consecutive failed components. In this paper, we give a recursive formula to compute the reliability of such a system. Thereafter,
we state two asymptotic results concerning the failure time Z
n
of the system. The first result concerns a limit theorem for Z
n
when the failure times of components are not necessarily with identical failure distributions. In the second one, we prove
that, for an arbitrary common failure distribution of components, the limit system failure distribution is always of the Poisson
class.
相似文献
17.
Henry Braun 《Probability Theory and Related Fields》1978,42(1):13-21
Summary Consider the set of proper probability distributions on the nonnegative integers having the first k moments (fixed) in common. It is desired to find the element of this set whose corresponding probability generating function (p.g.f.) lies entirely above or below the others. Using convexity arguments, it is shown that the extremal distribution exists, is unique, and is necessarily an element of a certain subclass of the class of all (k + 1)-point distributions. This subclass is entirely characterized by the geometrical properties of its set of support. The alternation of upper and lower bounds with the parity of k is also explained. There is mention of how the techniques developed here apply to more general discrete optimization problems, and the connection with the theory of cyclic polytopes is also discussed.This work was partially supported by Army Research Office Grant #DAHCO 04-74-G-0178 awarded to the Department of Statistics, Princeton University 相似文献
18.
Y. -F. S. Pétermann 《manuscripta mathematica》1990,69(1):305-318
For a distribution functionD we define itsabsolute andsigned moments of orderk∈R, which generalise in a natural way the Hamburger moments of orders an even and an odd natural number. Similarly, for a real
functionh we define itsabsolute andsigned asymptotic means of orderk∈R. We show that if the means exist on an infinite and bounded set of values ofk, then they exist on an intervalI and coincide onI
o with the moments ofD=D
h, the distribution function of the values ofh, which is shown to exist (in the sense of Wintner). We also give a sufficient condition forD
h to be symmetric. These results apply to a class of functionsh that contain in particular error terms related to the Euler phi function and to the sigma divisor function. A further application
on a certain class of converging trigonometrical series implies in particular classical results of A. Wintner establishing
the existence for such functions of a distribution function as well as Hamburger moments of arbitrarily large orders. The
remainder term of the prime number theorem belongs to this class provided the Riemann hypothesis holds, and the distribution
function of its values is shown to be “almost” symmetric. 相似文献
19.
Anatole Joffe Éric Marchand François Perron Paul Popadiuk 《Journal of Theoretical Probability》2004,17(1):285-292
Let {X
k
}
k1 be independent Bernoulli random variables with parameters p
k
. We study the distribution of the number or runs of length 2: that is
. Let S=lim
n
S
n
. For the particular case p
k
=1/(k+B), B being given, we show that the distribution of S is a Beta mixture of Poisson distributions. When B=0 this is a Poisson(1) distribution. For the particular case p
k
=p for all k we obtain the generating function of S
n
and the limiting distribution of S
n
for
. 相似文献
20.
Approximation with generalized hyperexponential distributions: Weak convergence results 总被引:1,自引:0,他引:1
Generalized hyperexponential (GH) distributions are linear combinations of exponential CDFs with mixing parameters (positive and negative) that sum to unity. The denseness of the class GH with respect to the class of all CDFs defined on [0, ) is established by showing that a GH distribution can be found that is as close to a given CDF as desired, with respect to a suitably defined metric. The metric induces the usual topology of weak convergence so that, equivalently, there exists a sequence of GH CDFs that converges weakly to a given CDF. This result is established by using a similar result for weak convergence of Erlang mixtures. Various set inclusion relations are also obtained relating the GH distributions to other commonly used classes of approximating distributions, including generalized Erlang (GE), mixed generalized Erlang (MGE), those with reciprocal polynomial Laplace transforms (K
n
), those with rational Laplace transforms (R
n
), and phase-type (PH) distributions. A brief survey of the history and use of approximating distributions in queueing theory is also included.This research was partially supported by the Office of Naval Research under Contract No. N00014-86-K0029. Much of this work is taken from the first-named author's doctoral dissertation, accepted by the faculty at the University of Virginia. 相似文献