The Modes of the Poisson Distribution of Order 3 and 4

In this article, new properties of the Poisson distribution of order k with parameter λ are found. Based on them, the modes of the Poisson distributions of order k=3 and 4 are derived for λ in (0,1). They are 0, 3, 5, and 0, 4, 7, 8, respectively, for λ in specified subintervals of (0, 1). In addition, using Mathematica, computational results for the modes of the Poisson distributions of order k=2,3, and 4 are presented for λ in specified subintervals of (0,2).

Let m k,λ denote the mode(s) of f k (x; λ), i.e., the value(s) of x for which f k (x; λ) attains its maximum. It is well known that m 1,λ = λ or λ − 1 if λ ∈ N and m 1,λ = λ , if λ does not belong to N, where α denotes the greatest integer part of α. Philippou [3] derived some properties of f k (x; λ) and posed the problem of finding its mode(s) for k ≥ 2.
Hirano et al. [28] presented several graphs of f k (x; λ) for λ ∈ (0, 1) and 2 ≤ k ≤ 8, and Luo [20] derived the following lower bound inequality for m k,λ , Georghiou et al. [21] employed the probability generating function of the Poisson distribution of order k to improve the lower bound of Luo [20] and also to give an upper bound for m k,λ where δ k,1 denotes the Kronecker delta. With the bounds of m k,λ in (2), they showed that Using the upper bound u k,λ of (2) and the definition of m k,λ , Philippou [22] found that: (a) For any integer k ≥ 1 and 0 < λ < 2/k(k + 1), the Poisson distribution of order k has a unique mode m k,λ = 0. (b) The Poisson distribution of order 2 has a unique mode m k,λ = 0 if 0 < λ < √ 3 − 1; it has two modes m k,λ = 0 and 2 if 0 < λ = √ 3 − 1, and it has a unique mode

Remark 1.
Since the modes of the Poisson distribution of order k with parameter λ are defined as the values of x ∈ {0, 1, 2, . . .}, which maximize f k (x; λ), they are its most probable values and they may be obtained numerically for any given positive integer k and positive λ, from f k (m k,λ ; λ) = max f k (x; λ) x ∈ {0, 1, 2, . . . , u k (λ)} .
In the present short note, we derive some additional properties of f k (x; λ) and find the modes of the Poisson distribution of order k = 3 and k = 4 for 0 < λ < 1. Furthermore, Section 3 presents computational results for the modes of the Poisson distributions of order k = 2, 3, and 4 for λ ∈ (0, 2). Finally, in Section 4, we briefly discuss our results, give the moment estimator of λ (> 0) for k ≥ 1, and indicate further research.

Main and Preliminary Results
The mode(s) of a discrete probability mass function is (are) its most probable value(s). In this section, we derive the modes of the Poisson distributions of order 3 and 4, respectively, when 0 < λ < 1 (see Propositions 1 and 2). In order to do so, we first state and prove three lemmas, regarding h k (x; λ) = e kλ f k (x; λ), which we use, along with relation (2), to prove the propositions.
Because of (1), where the summation is taken over all k-tuples of non-negative integers x 1 , x 2 , . . . , x k such that Georghiou et al. [21] provided a recursive form of f k (x; λ) as It can be restated, in terms of h k (x; λ), as with h k (0; λ) = 1.

More Computational Resutls
This section provides more computational results using the computer algebra system Mathematica. Table 2 shows the modes of Poisson distribution of order k = 2, 3, and 4 for 0 < λ < 2. For λ > 1, we observe that the mode values frequently change as λ increases. However, for every value of k, the first two modes are 0 and k for some subintervals of λ between zero and one. We also note that for k = 2, 3, and 4, m k,1 = 2, 5, and 8, (the modes of the Poisson distribution of order k with λ = 1) as it should, in accordance with (3).

Moment Estimation of the Parameter λ of P k (λ), Discussion, and Further Research
Despite the upsurge of the study of distributions of order k or runs since the early 1980s, their modes, due to the difficulty of obtaining them, are not known, except for the mode of the geometric distribution of order k and partial results for the modes of the negative binomial and Poisson distributions of order k. Their probability generating functions, however, and moments are well known.
In the present article, in addition to the above paragraph regarding P k (λ), we derived a few new properties of the Poisson distribution of order k, and using them, along with a result of Georghiou et al. [21], we found the modes or most probable values of the Poisson distributions of order 3 and 4 for λ in the interval (0, 1). In addition, using Mathematica and a personal computer, we found the modes of the Poisson distributions of order 2, 3, and 4 for λ ∈ (0, 2). We observe that for k = 2, 3, and 4, the first two modes are 0 for 0 < λ < r k , and k for r k < λ < r k + l k , where l k stands for the length of the interval on which k is the mode of the Poisson distribution of order k. Further research may include several interesting problems: Is it generally true that m k,λ = 0 for k ≥ 2 and 0 < λ < r k , and m k,λ = k for k ≥ 2 and r k < λ < r k + l k ? Does r k decrease as k increases? If it does, how fast is r k decreasing? What positive integers cannot be modes of the Poisson distribution of order k?