On Absolute Central Moments of Poisson Distribution
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On Absolute Central Moments of Poisson Distribution Pavel S. Ruzankin1,2
© Grace Scientific Publishing 2020
Abstract A recurrence formula for absolute central moments of Poisson distribution is suggested. Keywords Poisson distribution · Absolute central moment · Central moment Let X be a Poisson random variable with mean m. In this paper, we study absolute central moments 𝐄|X − a|r about a for naturals r. Explicit representations for such moments may be useful in situations where we want to test whether observations in a large sample are independent Poisson with given means, which is the case, for instance, for some image reconstruction techniques in emission tomography (e.g., see [1, 4]). An explicit representation for the mean deviation was obtained independently in [3, 7]:
𝐄�X − m� = 2e−m
m⌊m⌋+1 , ⌊m⌋!
where ⌊⋅⌋ denotes the floor function. Kendall ([6], relations (5.21) and (5.22) on p. 121) has showed that for all integers r ≥ 2,
) r−2 ( ∑ r−1 𝐄(X − m)k , 𝐄(X − m) = m k k=0 r
𝐄(X − m)r+1 = rm𝐄(X − m)r−1 + m
d 𝐄(X − m)r . dm
(1)
(2)
* Pavel S. Ruzankin [email protected] 1
Sobolev Institute of Mathematics, Novosibirsk, Russia
2
Novosibirsk State University, Novosibirsk, Russia
13
Vol.:(0123456789)
56
Page 2 of 6
Journal of Statistical Theory and Practice
(2020) 14:56
Denote by 1F1 the confluent hypergeometric function of the first kind 1F1 (𝛼, 𝛽, z) =
∞ n n−1 ∑ z ∏𝛼+j , n! j=0 𝛽 + j n=0
∏−1 where j=0 = 1.Katti [5] has derived the following representation for absolute central moments of X about a for all odd r ≥ 1: 𝐄�X − a�r = −𝐄(X − a)r +
e−m m⌊a⌋+1 (r) G (0, 0), (⌊a⌋ + 1)!
(3)
where
G(𝛽, t) = exp{t(⌊a⌋ − a + 𝛽 + 1)} 1F1 (𝛽 + 1, 𝛽 + ⌊a⌋ + 2, met ),
G(r) (𝛽, t) is its r-th partial derivative with respect to t, which, for t = 0 , can be computed by the recurrence formula G(s+1) (𝛽, 0) = (⌊a⌋ − a + 𝛽 + 1)G(s) (𝛽, 0) + for s = 1, 2, … , where
m(𝛽 + 1) (s) G (𝛽 + 1, 0) 𝛽 + ⌊a⌋ + 2
G(0) (𝛽, 0) ∶= G(𝛽, 0) = 1F1 (𝛽 + 1, 𝛽 + ⌊a⌋ + 2, m).
The main goal of this paper is to suggest a simpler recurrence formula for absolute central moments of X about a. Define the sign function { −1 if y ≤ 0, sign(y) = 1 if y > 0 and put
F(b) ∶ = 𝐏(X ≤ b), C(r, a) ∶ = 𝐄(X − a)r , D(r, a, b) ∶ = 𝐄(X − a)r sign(X − b), B(r, a, f ) ∶ = 𝐄(X − a)r f (X), where 00 = 1 by definition, f is a real-valued function. Here, F(b) is the cumulative distribution function of X, and C(r, a) is the r-th central moment about a. The above definition of the sign function is not common for y = 0 , but is chosen here for the sake of convenience, to provide the equality
D(0, a, b) = 1 − 2F(b). We have r
𝐄|X − a| =
13
{
C(r, a) D(r, a, a)
if r is even, if r is odd .
(4)
Journal of Statistical Theory and Practice
(2020) 14:56
Page 3 of 6
Theorem 1 For all integers r ≥ 1, reals b ≥ 0 and a, and functions f such that 𝐄X r |f (X)| < ∞, the following recurrence relations are valid:
C(r, a) = (m − a)C(r − 1, a) + m
) r−2 ( ∑ r−1 C(k, a) k k=0
56
(5)
= mC(r − 1, a − 1) − aC(r − 1, a),
(6)
� r−2 � � r−1 D(k, a,
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