© Hindawi Publishing Corp. 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.


Introduction.
In five pioneering papers, Philippou and Muwafi [22], Philippou et al. [21], Philippou [17], and Philippou et al. [19,20] introduced the study of univariate and multivariate distributions of order k.Since then, the subject matter received a lot of attention from many researchers.For comprehensive reviews at the time of publication, we refer to Johnson et al. [9,10].
Consider a sequence of Bernoulli trials with success (S) probability p (0 < p < 1), and let T k,r denote the number of trials until r (r ≥ 1) nonoverlapping success runs of length k (k ≥ 1) appear.Philippou et al. [21] (see also Philippou [18]) derived, for x = kr , kr + 1,..., the following exact formula for the probability distribution of T k,r : x 1 ,...,x k ,r − 1 p x q p Σ j x j . (1.1) The probability distribution (1.1) is known as negative binomial distribution of order k, type I, with parameters r and p and it is denoted by NB k,I (r ; p).A different sampling derivation of the (suitably shifted) negative binomial distribution of order k, type I, say NB k,I (r ; p), was given by Antzoulakos and Philippou [2] (see also Tripsiannis and Philippou [25]).The Poisson and the logarithmic series distributions of order k, say type I, were obtained as limiting cases of NB k,I (r ; p), by Philippou et al. [21] and Aki et al. [1], respectively.Panaretos and Xekalaki [15] derived and studied a hypergeometric, a negative hypergeometric, and a generalized Waring distribution of order k by means of the methodology of Philippou and Muwafi [22], and Philippou et al. [21].They also stated, without further details, the derivation of a Polya and an inverse Polya distribution of order k, of which the preceding three are proper special cases.Ling [11] rederived the above-mentioned inverse Polya distribution of order k, say type I, and introduced, allowing runs of length k to overlap, a new inverse Polya distribution of order k, say type III.A different sampling derivation of the (shifted) inverse Polya distribution of order k, type I, was given by Tripsiannis and Philippou [25].Jain and Consul [6] introduced and studied the generalized negative binomial distribution (see also Mohanty [13]).The generalized logarithmic series distribution and the Borel-Tanner or generalized Poisson distribution were obtained as limiting cases of the generalized negative binomial distribution by Jain and Consul [6], and Jain and Singh [8], respectively (see also Jain [5], Jain and Gupta [7], and Haight and Breuer [4]).Recently, Sen and Mishra [23] obtained the generalized Polya distribution, which unifies the usual Polya and inverse Polya distributions.
In this paper, we extend the above-mentioned generalized distributions to distributions of order k.In Section 2, we derive a generalized negative binomial distribution of order k, type I, say GNB k,I (•), which includes as a special case the negative binomial distribution of order k, type I, of Philippou et al. [21].We do it by counting multidimensional lattice paths in a generalized sampling scheme employing a first passage approach (see Theorem 2.1 and Definition 2.2).Another genesis scheme of GNB k,I (•) is given next (see Proposition 2.3), which indicates potential applications and provides its probability generating function (PGF), mean and variance (see Proposition 2.4).We next obtain two limiting cases of GNB k,I (•) (see Propositions 2.5 and 2.7), which provide, respectively, a generalized logarithmic series distribution of order k, type I, say GLS k,I (•), and a Borel-Tanner distribution of the same order and type, say BT k,I (•) (see Definitions 2.6 and 2.8).By means of a generalized sampling scheme and a first-passage approach (see Theorem 2.9), and by compounding the GNB k,I (•) with the Beta distribution (see Proposition 2.11), we introduce a generalized Polya distribution of order k, type I, say GP k,I (•) (see Definition 2.10), which includes as a special case the inverse Polya distribution of order k, type I, of Panaretos and Xekalaki [15].In Section 3, we introduce, as special cases of GP k,I (•), several distributions of order k, most of which are new.In Section 4, we relate asymptotically GNB k,I (•) to the Poisson distribution (P k,I (λ)) of order k, type I, of Philippou et al. [21] (see Proposition 4.1), and GP k,I (•) to GNB k,I (•), BT k,I (•), P k,I (λ) and to the negative binomial distribution (NB k,I (n; p)) of order k, type I, of Philippou et al. [21] (see Propositions 4.2,4.3,4.4,and 4.5).An application to observed data is also presented.
In order to avoid unnecessary repetitions, we mention here that in this paper, x 1 ,...,x k are nonnegative integers as specified.In addition, whenever sums and products are taken over j, ranging from 1 to k, we will omit these limits for notational simplicity.
2. Generalized negative binomial distribution of order k, type I.In this section, we introduce a new distribution of order k, type I, by means of a generalized sampling scheme and a first-passage approach, and we obtain its PGF, mean and variance.Furthermore, we derive three new distributions of order k: the generalized logarithmic series, Polya distributions of order k, type I, and the Borel-Tanner distribution of the same order and type.First, we consider the following theorem.
Theorem 2.1.In a sequence of independent Bernoulli trials with success (S) probability p (0 < p < 1), consider the random variables X j (1 ≤ j ≤ k) and L k (k ≥ 1) denoting, respectively, the number of events Let X be a random variable denoting the number of occurrences of failures and the total number of successes which precede directly the occurrences of failures but do not belong to any success run of length k, that is, X = Σ j jX j .Trials are continued until n + µΣ j X j (n > 0 and µ ≥ −1) nonoverlapping success runs of length k appear for the first time, that is, at any trial t (1 ≤ t ≤ Σ j jX j + k(n + µΣ j X j ) − 1), the condition A = {L j , where X [t] j and L [t] k are the numbers of events e j and e k , respectively, in the first t trials}, is satisfied.Then, for x = 0, 1,..., where q = 1 − p.
Proof.For any fixed nonnegative integer x, a typical element of the event (X = x) is a sequence of outcomes Σ j jx j +k(n+µΣ j x j ) of the letters F and S, such that the event e j appears x j (1 ≤ j ≤ k) times and the event e k appears n + µΣ j x j times, satisfying the condition A and Σ j jx j = x.
Fix x j (1 ≤ j ≤ k) (n and µ are fixed), and denote the event e j (1 ≤ j ≤ k) by a unit step in Z j -direction and the event e k by a unit step in Z 0 -direction.Therefore, we represent a sequence of x j events e j (1 ≤ j ≤ k) and n+µΣ j x j events e k by a (k+1)-dimensional lattice path from the origin to (n+µΣ j x j ,x 1 ,...,x k ), which does not touch the hyperplane z 0 = n + µΣ j x j except at the point (n + µΣ j x j ,x 1 ,...,x k ).Then, the number of such lattice paths is (see [14,Example 10,page 25]) and each one of them has probability p k(n+µΣ j x j )+x q p Σ j x j . (2.4) Then the theorem follows since the nonnegative integers x 1 ,...,x k may vary subject to Σ j jx j = x.
By means of the transformations x j = r j (1 ≤ j ≤ k) and x = r +Σ j (j −1)r j , and by the multinomial theorem and relation (2.3) of Jain and Consul [6], it may be seen that the above derived probability function is a proper probability distribution.
For k = 1, this distribution reduces to the generalized negative binomial distribution of Jain and Consul [6]; with β = µ + 1, and for µ = 0, it reduces to the (shifted) negative binomial distribution of order k, type I, of Philippou et al. [21].We, therefore, introduce the following definition.Definition 2.2.A random variable (RV) X is said to have the generalized negative binomial distribution of order k, type I, with parameters n, µ, and p (n>0 and µ ≥−1, both integers, and 0 <p <1) to be denoted by GNB k,I (n; µ; p) if, for x = 0, 1,..., P (X = x) is given by (2.2).
The following proposition, which is a direct consequence of Definition 2.2, indicates that GNB k,I (•) may have potential applications whenever there are multiple groups of items and we are interested in the distribution of the total number of items.Proposition 2.3.Let X j , 1 ≤ j ≤ k, be random variables and set X = Σ j jX j .Then, X is distributed as GNB k,I (n; µ; p) if and only if X 1 ,...,X k are jointly distributed as multivariate generalized negative binomial distribution with parameters n,µ,...,µ and Q The PGF, mean and variance, of the generalized negative binomial distribution of order k, type I, may be readily obtained by means of Proposition 2.3 or by means of Definition 2.2, the transformations x j = r j (1 ≤ j ≤ k) and x = r + Σ j (j − 1)r j , and simple expectation properties.Proposition 2.4.Let X be a RV following the generalized negative binomial distribution of order k, type I.Then, (i) the PGF of X is given by: where (2.6) (ii) The mean and variance of X are given by: (2.7) For k = 1, Proposition 2.4(i) reduces to the PGF of the generalized negative binomial distribution (see Jain [5]) and (ii) reduces to its mean and variance (see Jain and Consul [6]).
Jain and Consul [6] obtained the generalized logarithmic series distribution as a limit of the generalized negative binomial distribution.We extend this result to the generalized negative binomial distribution of order k, type I, and we name the limit distribution accordingly.Proposition 2.5.Let X n (n > 0) be a RV distributed as GNB k,I (n; µ; p), and assume that n → 0.Then, for x = 1, 2,..., ) Proof.For x = 1, 2,..., we have which establishes the proposition.
We proceed now to derive a generalized Polya distribution of order k, type I, using a first-passage approach in a generalized sampling scheme, which follows along the same lines as those in the proof of Theorem 2.1.
Theorem 2.9.An urn contains c 0 + c 1 (= c) balls of which c 1 bear the letter F and c 0 bear the letter S. A ball is drawn at random from the urn, its letter is recorded, and it is replaced into the urn, together with s balls bearing the same letter.Consider the random variables X j (1 ≤ j ≤ k), L k , and X as in Theorem 2.1.Then, for x = 0, 1,..., For k = 1, this distribution reduces to the generalized Polya distribution of Sen and Mishra [23] and for µ = 0, it reduces to the inverse Polya distribution of order k, type I, of Panaretos and Xekalaki [15].We, therefore, introduce the following definition.Definition 2.10.A RV X is said to have the generalized Polya distribution of order k, type I, with parameters n, µ, s, c, and c 1 (s, µ ≥ −1, both integers, and n, c, and c 1 , positive integers) to be denoted by GP k,I (n; µ; s; c, c 1 ) if, for x = 0, 1,..., P (X = x) is given by (2.14).
The next proposition provides another derivation of the generalized Polya distribution of order k, type I, by compounding the generalized negative binomial distribution of the same order and type.Proposition 2.11.Let X and P be two RVs such that (X | P = p) is distributed as GNB k,I (n; µ; p), and P is distributed as B(α, β) (the Beta distribution with positive real parameters α and β).Then, for x = 0, 1,..., x j ,β+ Σ j x j B(α, β) . (2.15) Proof.For x = 0, 1,..., we get which establishes the proposition.
For k = 1, the above proposition provides a new derivation of the generalized Polya distribution of Sen and Mishra [23].
Case 3. The IP k,I (n; s; c, c 1 ), for s = 0, reduces to the negative binomial distribution of order k, type I, of Philippou et al. [21] with p = (c − c 1 )/c.Class 3.3.The GP k,I (n; µ; s; c, c 1 ), for s = 0, reduses to the generalized negative binomial distribution of order k, type I, with p = (c − c 1 )/c.Case 1.In GNB k,I (n; µ; p), let µ = 1 and q/p = P so that q = P /Q and p = 1/Q, where Q = 1 + P , and replace n by nk and x j by x j − n.Then, for x = k(k + 1)n/2, k(k + 1)n/2 + 1,..., which reduces to the Haight distribution (see Haight [3]), for k = 1.We say that the RV X has the Haight distribution of order k, type I, with parameters n and P .
Case 2. In GNB k,I (n; µ; p), let q/p = P so that q = P /Q and p = 1/Q, where Q = 1 + P , and replace n by kµ and x j by x j − 1.Then, for (3.9) which reduces to the Takács distribution (see Takács [24]), for k = 1.We say that the RV X has the Takács distribution of order k, type I, with parameters µ and P .
Case 4. In GNB k,I (n; µ; p), let q/p = P so that q = P /Q and p = 1/Q, where Q = 1 + P , and replace n by nkµ and x j by x j − n.Then, for x = k(k + 1)n/2, k(k + 1)n/2 + 1,..., /2] , (3.11) which reduces to the negative binomial-delta distribution (see Johnson et al. [10,page 144]), for k = 1.We say that the RV X has the negative binomial-delta distribution of order k, type I, with parameters n, µ, and P .
Case 5.In GNB k,I (n; µ; p), let q/p = P so that q = P /Q and p = 1/Q, where Q = 1 + P .Then, for x = 0, 1,..., x 1 ,...,x k ,n+ µΣ j x j Q −k(n+µΣ j x j )−x P Σ j x j , (3.12) which reduces to the negative binomial-negative binomial distribution (see Johnson et al. [10, page 145]), for k = 1.We say that the RV X has the negative binomial-negative binomial distribution of order k, type I, with parameters n, µ, and P .
Proposition 4.3.Let X n,µ,c,c 1 (µ ≥ −1 is an integer and n, c, and c 1 are positive integers) and X be two RVs distributed as in (2.14) and in the Borel-Tanner distribution of order k, type I, respectively, and assume that c 1 /c → 0, nc 1 /c → r aβ (a > 0, β > 0, and r = 1, 2,...), and µc Proof.We observe that from which the proof follows.
Proposition 4.5.Let X n (n is a positive integer) and X be two RVs distributed as in (2.14) and NB k,II (β, c/(c + k)), respectively, and assume that n −1 α = c n → c (0 < c < ∞) as n → ∞.Then, P X n = x → P (X = x), x = 0, 1,.... (4.9) Proof.We observe that n n + (1 + µ)Σ j x j − 1 !n + µΣ j x j !B α + k n + µΣ j x j + Σ j (j − 1)x j ,β+ Σ j x j B(a, β) Finally, to illustrate the fit of the generalized distributions of order k, type I, to observed data, we consider the following application of the generalized Poisson (or Borel-Tanner) distribution of order k, type I.
McKendrick [12] considered the distribution of the sum of two correlated Poisson variables, which he applied to the counts of bacteria in leucocytes.Table 4.1 shows the expected frequencies of the counts of bacteria in leucocytes estimated by the generalized Poisson distribution of order 2, type I (GP 2,I (•)), using the following moment estimators of the parameters θ and λ: with x = 0,2 and s 2 = 0,374582.An ordinary Poisson (P (•)) and a generalized Poisson distribution (GP(•)) (see, e.g., Jain [5]) have been fitted for comparison.We observe that the generalized Poisson distribution of order 2, type I, gives a good fit to the data.

Table 4 .
1. Distribution of the counts of bacteria in leucocytes.