Gregory A. Tripsiannis

Multivariate distributions of order κ

Three multivariate distributions of order ? are introduced and studied. A multivariate negative binomial distribution of order ? is derived first, by means of an urn scheme, and two limiting cases of it are obtained next. They are, respectively, a multivariate Poisson distribution of order ? and a multivariate logarithmic series distribution of the same order. The probability generating functions, means variances and covariances of these distributions are obtained, and some further genesis schemes of them and interrelationships among them are also established. The present paper extends to the multivariate case the work of Philippou (1987) on multiparameter distributions of order ?. At the same time, several results of Aki (1985) on extended distributions of order ? are also generalized to the multivariate case.

Multivariate distributions of order k, part II

Several additional properties and genesis schemes of the multivariate distributions of order k of Philippou et al. (1989) are obtained. Some of them deal with two new distributions introduced presently, the multivariate k-point distribution and the modified multivariate logarithmic series distribution of order k.

New polya and inverse polya distributions of order k

New Polya and inverse Polya distributions of order k are derived by means of generalized urn models and by compounding the binomial and negative binomial distributions of order k of Philippou (1986, 1983) with the beta distribution. It i s noted that the present Polpa distribution of order k includes as special cases a new hypergeometric distribution of order k, a negative one,an inverse one, and a discrete uniform of the same order. The probability generating functions, means and variances of the new distributions are obtained, and five asymptotic results are established relating them to the abovedmentioned binomial and negative binomial distributions of order k, and to the Poisson distribution of the same order of Philippou (1983).Moment estimates are also given and applications are indicated.

A new multivariate inverse polya distribution of order k

A new multivariate inverse Polya distribution of order k, type I, is derived by means of a generalized urn scheme and by compounding the multivariate negative binomial distribution of order k, type I, of Philippou, Antzoulakos and Tripsiannis (1988) with the Dirichlet distribution. It is noted that this new distribution includes as special cases a new multivariate inverse hypergeometric distribution of order k and a new multivariate negative inverse one of the same order. The mean and variance-covariance of the multivariate inverse Polya distribution of order k, type I, are derived, and two known distributions of the same order are shown to be limiting cases of it.

Multivariate Generalized Distributions of Order k

Abstract The study of multivariate distributions of order k, two of which are the multivariate negative binomial of order k and the multinomial of the same order, was introduced in Philippou et al. (Philippou, A. N., Antzoulakos, D. L., Tripsiannis, G. A. (1988). Multivariate distributions of order k. Statistics and Probability Letters 7(3):207–216.), and Philippou et al. (Philippou, A. N., Antzoulakos, D. L., Tripsiannis, G. A. (1990). Multivariate distributions of order k, part II. Statistics and Probability Letters 10(1):29–35.). Recently, an order k (or cluster) generalized negative binomial distribution and a multivariate negative binomial distribution were derived in Sen and Jain (Sen, K., Jain, R. (1996). Cluster generalized negative binomial distribution. In: Borthakur et al. A. C., Eds.; Probability Models and Statistics Medhi Festschrift, A. J., on the Occasion of his 70th Birthday. New Age International Publishers: New Delhi, 227–241.) and Sen and Jain (Sen, K., Jain, R. (1997). A multivariate generalized Polya-Eggenberger probability model-first passage approach. Communications in Statistics-Theory and Methods 26:871–884.), respectively. In this paper, all four distributions are generalized to a multivariate generalized negative binomial distribution of order k by means of an appropriate sampling scheme and a first passage event. This new distribution includes as special cases several known and new multivariate distributions of order k, and gives rise in the limit to multivariate generalized logarithmic, Poisson and Borel-Tanner distributions of the same order. Applications are indicated.

CIRCULAR POLYA DISTRIBUTIONS OF ORDER k

Two circular Polya distributions of order k are derived by means of generalized urn models and by compounding, respectively, the type I and type II circular binomial distributions of order k of Makri and Philippou (1994) with the beta distribution. It is noted that the above two distributions include, as special cases, new circular hypergeometric, negative hypergeometric, and discrete uniform distributions of the same order and type. The means of the new distributions are obtained and two asymptotic results are established relating them to the above-mentioned circular binomial distributions of order k.

MULTIVARIATE GENERALIZED POLYA DISTRIBUTION OF ORDER k

ABSTRACT An order k (or cluster) generalized Polya distribution and a multivariate generalized Polya-Eggenberger one where derived in (Sen, K.; Jain, R. Cluster Generalized Negative Binomial Distribution. In Probability Models and Statistics, A. J. Medhi Festschrift on the Occasion of his 70th Birthday; Borthakur, A.C. et al., Eds.; New Age International Publishers: New Delhi, 1996; 227–241 and Sen, K.; Jain, R. A Multivariate Generalized Polya-Eggenberger Probability Model-First Passage Approach. Communications in Statistics—Theory and Methods 1997, 26, 871–884). Presently, both distributions are generalized to a multivariate generalized Polya distribution of order k by means of an appropriate sampling scheme and a first passage event. This new distribution includes as special cases new multivariate Polya and inverse Polya distributions of order k and the multivariate generalized negative binomial distribution of the same order derived recently in (Tripsiannis, G.A.; Philippou, A.N.; Papathanasiou, A.A. Multivariate Generalized Distributions of Order k. Medical Statistics Technical Report #41: Democritus University of Thrace, Greece, 2001). Limiting cases are considered and applications are indicated.

A Multivariate Inverse Polya Distribution of Order k Arising in the Case of Overlapping Success Runs

The study of multivariate distributions of order k was initiated by Philippou et al. [20], [21], who introduced and studied the multivariate negative binomial, Poisson, k-point, logarithmetic series and modified logarithmic series distributions of order k. These two papers generalized several results of Sibuya [24], Patil and Bildikar [13], Johnson, Kotz and Kemp [7], Philippou [15, [16], [17], Philippou et al. [22], Aki et al. [3], Aki [1] and Hirano and Aki [5] on multivariate distributions and distributions of order k.

A Multivariate Negative Binomial Distribution of Order k Arising When Success Runs are Allowed to Overlap

Ling (1989) introduced and studied a negative binomial distribution of order k, type III, which he denoted by NB k III(r, p), as the probability distribution of the number of Bernoulli trials M r (k) until the occurrence of r possibly overlapping success runs of length k [see also Hirano et al. (1991)]. In the present paper, independent trials are considered with m + 1 possible outcomes and the multivariate negative binomial distribution of order k, type III, say \(\overline {MNB} _{k,III} (r;q_1 \ldots ,q_m ),\) is introduced as the distribution of a random vector Y which is a multivariate analogue of Y r (k) − (k + r − 1). The probability generating function, mean and variance-covariance, and several distributional properties of Y are established. The present paper generalizes to the multivariate case shifted versions of results of Ling (1989) and Hirano et al. (1991) on NB k,III (r, p). Three new results on NB k,III (r, p) or/and its shifted version are derived first; another one arises as a corollary of a proposition on \(\overline {MNB} _{k,III} (r;q_1 \ldots ,q_m ),\).

GENERALIZED DISTRIBUTIONS OF ORDER k ASSOCIATED WITH SUCCESS RUNS IN BERNOULLI TRIALS

In a sequence of independent Bernoulli trials, by counting multidimensional lattice paths in order to compute the probability of a first-passage event, we derive and study a generalized negative binomial distribution of order k, type I, which extends to distributions of order k, the generalized negative binomial distribution of Jain and Consul (1971), and includes as a special case the negative binomial distribution of order k, type I, of Philippou et al. (1983). This new distribution gives rise in the limit to generalized logarithmic and Borel-Tanner distributions and, by compounding, to the generalized Pólya distribution of the same order and type. Limiting cases are considered and an application to observed data is presented.