© Hindawi Publishing Corp. 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.


Introduction.
In five pioneering papers, Philippou and Muwafi [18], Philippou et al. [17], Philippou [14], and Philippou et al. [15,16] 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. [7,8].
Makri and Philippou [11] introduced two circular binomial distributions of order k, as the distribution of nonoverlapping and possibly overlapping success runs of length k in n Bernoulli trials arranged on a circle.They also derived two exact formulas for the reliability of a consecutive-k-out-of-n : F system whose components are ordered circularly.The distribution of circular nonoverlapping or possibly overlapping success runs of length k was also studied by Chryssaphinou et al. [5], Koutras et al. [9], Charalambides [4], and Koutras et al. [10].
In the present paper, we introduce two circular Polya distributions of order k, type I and II, say P c k,I (•) and P c k,II (•), respectively (see Definitions 2.4 and 3.2), as the distribution of nonoverlapping (N c n,k,s ) and possibly overlapping (M c n,k,s ) success runs of length k in a generalized sampling scheme (see Theorems 2.1 and 3.1).We also derive P c k,I (•) and P c k,II (•) by compounding, respectively, the circular binomial distribution of order k, type I (B c k,I (•)) and type II (B c k,II (•) ) of Makri and Philippou [11] with the beta distribution (see Remarks 2.5 and 3.3).We introduce as special cases of P c k,I (

Circular
Polya distribution of order k, type I.In this section, we derive the circular Polya distribution of order k, type I, by means of a generalized urn model and by compounding the circular binomial distribution of order k, type I, of Makri and Philippou [11].The factorial moment of this new distribution is also obtained.
Theorem 2.1.Consider an urn containing c 0 +c 1 (= c) balls of which c 0 bear the letter S (success) and c 1 bear the letter F (failure).A ball is drawn at random and then it is replaced together with s balls bearing the same letter.We repeat this procedure n (≥ 1) times and we assume that the outcomes are bent into a circle.Let N c n,k,s be a random variable denoting the number of nonoverlapping success runs of length k (≥ 1).Then, for x = 0, 1,...,[n/k], where the summation 1 is taken over all nonnegative integers x 1 ,...,x k such that k j=1 jx j = n − i − kx and δ is the Kronecker delta function.
First, we will establish two preliminary lemmas.The first one gives a formula for Lemma 2.2.Let N c n,k,s , 1 , and δ as in Theorem 2.1.Then, for x = 0, 1,... , (c/s) [n] or a circular arrangement ) such that x j (1 ≤ j ≤ k) and x − r − 1 of the α's are as above, and Fix r , x j (1 ≤ j ≤ k), α, and β.Then, the number of the circular arrangements of type (2.3) is and each one of them has probability Furthermore, the number of the circular arrangements of type (2.5) is and each one of them has probability (c/s) [n]  . (2.10) Finally, we observe that (c/s) [n] . (2.11) But r , x j (1 ≤ j ≤ k), α, and β may vary subject to (2.4) and (2.6) for the elements of type (2.3) and (2.5), respectively.Denote by 1 and 1 the summation over all nonnegative integers x 1 ,...,x k satisfying (2.4) and (2.6), respectively.Then, for x = 0, 1,...,[n/k] and 1 as in the theorem, we have (2.12) which establishes the lemma.
The next lemma (see Makri and Philippou [11]) provides a formula for executing the summation with respect to r appearing in Lemma 2.2.
Definition 2.4.A random variable X is said to have the circular Polya distribution of order k, type I, with parameters n, s, c, and c 0 (s integer and n, c, and c 0 positive integers), to be denoted by P c k,I (n; s; c, c 0 ) if for x = 0, 1,...,[n/k], P (X = x) is given by (2.1).
Remark 2.5.It may be noted that if X and P are two random variables such that (X | P = p) is distributed as B c k,I (n, p) and P is distributed as B(α, β) (the beta distribution with positive real parameters α and β), then, for x = 0, 1,...,[n/k], (2.17) where 1 and δ are as in Theorem 2.1.Relation (2.17) reduces to (2.1) if α = c 0 /s and β = c 1 /s (s ≠ 0), which indicates that (2.17) may be considered as another form of P c k,I (n; s; c, c 0 ).
The r th factorial moment of the circular Polya distribution of order k, type I, is given next.Proposition 2.6.Let X be a random variable following the circular Polya distribution of order k, type I, with probability function as given by (2.17) and let (2.18) be the r th factorial moment of X.Then, where m r = min{r ,n− kr }.Case 4. For s = 0, P c k,I (n; s; c, c 0 ) reduces to the circular binomial distribution of order k, type I, of Makri and Philippou [11], with p = c 0 /c.

Proof. It follows by means of Charalambides
Next, we relate P c k,I (•) asymptotically to the circular binomial distribution of the same order and type.Proposition 2.7.Let X c 0 ,c 1 and X be two random vectors distributed as P c k,I (n; s; c, c 0 ) and B c k,I (n, p), respectively, and assume that c 0 /c → p (0 < p < 1) (2.20) Proof.We observe that from which the proof follows.

Circular Polya distribution of order k, type II.
In this section, we derive the circular Polya distribution of order k, type II, by means of a generalized urn model and by compounding the circular binomial distribution of order k, type II, of Makri and Philippou [11].The mean of this new distribution is also obtained.
Theorem 3.1.Consider an urn containing c 0 +c 1 (= c) balls of which c 0 bear the letter S (success) and c 1 bear the letter F (failure).A ball is drawn at random and then it is replaced together with s balls bearing the same letter.We repeat this procedure n (≥ 1) times and we assume that the outcomes are bent into a circle.Let M c n,k,s be a random variable denoting the number of overlapping success runs of length k (≥ 1).Then, P (M c n,k,s = x) = (c 0 /s) [n] /(c/s) [n] (c/s) [n]  , ( where the summation 2 are taken over all nonnegative integers x 1 ,...,x k such that Proof.We first observe that, for [n] /(c/s) [n] .For 0 ) and β.Then, the number of the above arrangements is and each one of them has probability But α and β may vary subject to (3.3) and the nonnegative integers x j (1 ≤ j ≤ k) may vary subject to (3.4).Therefore, for 0 ≤ x ≤ n − k and 2 as in the theorem, we have (c/s) [n]   (3.7) which establishes the theorem.
Definition 3.2.A random variable X is said to have the circular Polya distribution of order k, type II, with parameters n, s, c, and c 0 (s integer and n, c, and c 0 positive integers), to be denoted by P c k,II (n; s; c, c 0 ) if (c/s) [n] , if x = n, (3.8)where the summation 2 are taken over all nonnegative integers x 1 ,...,x k such that Remark 3.3.It may be noted that if X and P are two random variables such that (X | P = p) is distributed as B c k,II (n, p) and P is distributed as B(α, β), then (3.9) where the summation 2 is as in Theorem 3.1.Relation (3.9) reduces to (3.8) if α = c 0 /s and β = c 1 /s (s ≠ 0), which indicates that (3.9) may be considered as another form of P c k,II (n; s; c, c 0 ).
theory (introducing the circular-m-consecutive-k-out-of-n : F system, which fails whenever at least m nonoverlapping failure runs of length k occur, see Alevizos et al. [2], Chao et al. [3], Papastavridis and Koutras [13], and Makri and Philippou [12]).In these cases, when p is allowed to vary from trial to trial according to the beta distribution, the circular Polya distribution of order k arises.Agin and Godbole [1] proposed a nonparametric test for randomness based on the number of success runs of length k in a linear sequence of n Bernoulli trials, for an alternative hypothesis of clustering.Furthermore, tests based on the number of success runs of length k in a linear sequence of trials in the sampling scheme described in Theorem 2.1 are valid even for finite populations (see Godbole [6]).An analogous test (which is expected to be more sensitive for the alternative hypothesis of circular clustering) can be established on the number N c n,k,s .Other potential applications are indicated by Theorems 2.1 and 3.1.

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7 . 1 .
[4, relation (3.10)] and Remark 2.5.The circular Polya distribution of order k, type I, includes as special cases four distributions of order k, of which three are new.Each one arises when success runs of length k are not allowed to overlap by means of Theorem 2.1.Their moments may be easily obtained from Proposition 2.Case The P c k,I (n; s; c, c 0 ), for s = −1, reduces to a new distribution, which we call circular hypergeometric distribution of order k, type I, with parameters n, c, and c 0 , and denote it by H c k,I (n; c, c 0 ).Case 2. The P c k,I (n; s; c, c 0 ), for s = +1, reduces to a new distribution, which we call circular negative hypergeometric distribution of order k, type I, with parameters n, c, and c 0 , and denote it by NH c k,I (n; c, c 0 ).Case 3. The P c k,I (n; s; c, c 0 ), for c = c 0 = s, reduces to a new distribution, which we call circular Bose-Einstein uniform distribution of order k, type I.