Edgeworth Expansions for the Number of Distinct Components Associated with the Ewens Sampling Formula

Abstract

The Ewens sampling formula is well-known as a distribution of a random partition of the positive integer n. For the number of distinct components of the Ewens sampling formula, we derive its Edgeworth expansion. It is different from the Edgeworth expansion for the sum of independent and identically distributed random variables. It contains the digamma function of the parameter of the Ewens sampling formula. Especially, for the random permutation, the Edgeworth expansion contains Euler's constant. The Edgeworth expansion is examined numerically using its graph.

Estimating the large mutation parameter of the Ewens sampling formula

“…In this case, the left-hand side of (18) is approximately equal to (K n − θ(log n − ψ(θ)))/(θ (log n − ψ(θ ))) 1/2 . Yamato [8] showed that the approximation accuracy of this statistic is better than (11) when θ is a constant, which corresponds to β = 0.…”

On Poisson approximations for the Ewens sampling formula when the mutation parameter grows with the sample size

“…as n → ∞. Moreover, Yamato (2013) showed the approximation for K n by a Poisson variable with the approximate mean: For fixed θ > 0,…”

“…Later, in order to improve the approximation accuracy, Yamato (2013) provided the following CLT which adopts another standardization: For…”

“…In Proposition 4.1, we have considered Poisson variables with rigorous means E[K n ] and n − E[K n ]. Next, let us discuss centerings by approximate means presented by Yamato (2013) and Tsukuda (2017a) from the viewpoint of Poisson approximation. Introduce three Poisson variables Y A , Y a and Y C whose means are given…”

Error bounds for the normal approximation to the length of a Ewens partition

“…In particular, when θ = 1 Goncharov (1944) proved that Fn,1 (x) → Φ(x) for any x ∈ R. From a theoretical perspective, it is important to derive error bounds for the approximation. Yamato (2013) discussed the first-order Edgeworth expansion of Fn,θ (•) via the Poisson approximation (Arratia and Tavaré, 1992, Remark after Theorem 3) and proved that Fn,θ Kabluchko, Marynych and Sulzbach (2016) derived the Edgeworth expansion of the probability function of K, and provided the firstorder Edgeworth expansion of Fn,θ (•).…”