Let $X_{nr}$ be the $r$th largest of a random sample of size $n$ from a distribution $F (x) = 1 - \sum_{i = 0}^\infty c_i x^{-\alpha - i \beta}$ for $\alpha > 0$ and $\beta > 0$. An inversion theorem is proved and used to derive an expansion for the quantile $F^{-1} (u)$ and powers of it. From this an expansion in powers of $(n^{-1}, n^{-\beta/\alpha})$ is given for the multivariate moments of the extremes $\{X_{n, n - s_i}, 1 \leq i \leq k \}/n^{1/\alpha}$ for fixed ${\bf s} = (s_1, ..., s_k)$, where $k \geq 1$. Examples include the Cauchy, Student $t$, $F$, second extreme distributions and stable laws of index $\alpha < 1$.
Deep Dive into Expansions for Quantiles and Multivariate Moments of Extremes for Distributions of Pareto Type.
Let $X_{nr}$ be the $r$th largest of a random sample of size $n$ from a distribution $F (x) = 1 - \sum_{i = 0}^\infty c_i x^{-\alpha - i \beta}$ for $\alpha > 0$ and $\beta > 0$. An inversion theorem is proved and used to derive an expansion for the quantile $F^{-1} (u)$ and powers of it. From this an expansion in powers of $(n^{-1}, n^{-\beta/\alpha})$ is given for the multivariate moments of the extremes $\{X_{n, n - s_i}, 1 \leq i \leq k \}/n^{1/\alpha}$ for fixed ${\bf s} = (s_1, ..., s_k)$, where $k \geq 1$. Examples include the Cauchy, Student $t$, $F$, second extreme distributions and stable laws of index $\alpha < 1$.
For 1 ≤ r ≤ n, let X nr be the rth largest of a random sample of size n from a continuous distribution F on R, the real numbers. Let f denote the density of F when it exists. The study of the asymptotics of the moments of X nr has been of considerable interest. McCord (1964) gave a first approximation to the moments of X n1 for three classes. This showed that a moment of X n1 can behave like any positive power of n or n 1 = log n. (Here log is to the base e.) Pickands (1968) explored the conditions under which various moments of (X n1 -b n )/a n converge to the corresponding moments of the extreme value distribution. It was proved that this is indeed true for all F in the domain of attraction of an extreme value distribution provided that the moments are finite for sufficiently large n. For other work, we refer the readers to Polfeldt (1970), Ramachandran (1984) and Resnick (1987).
The asymptotics of the quantiles of X nr have also been studied. Note that U nr = F (X nr ) is the rth order statistics from U (0, 1). For 1 ≤ r 1 < r 2 < • • • < r k ≤ n set U n,r = {U nr i , 1 ≤ i ≤ k}. By Section 14.2 of Stuart and Ord (1987), U n has the multivariate beta density
, where u 0 = 0, u k+1 = 1, r 0 = 0, r k+1 = n + 1 and
David and Johnson (1954) expanded X nr i = F -1 (U nr i ) about u ni = EU nr i = r i /(n + 1):
where G(u) = F -1 (u), and using the properties of (1.1) showed that if r depends on n in such a ways that r/n → p ∈ (0, 1) as n → ∞ then the mth order cumulants of X n,r = {X nr i , 1 ≤ i ≤ k} have magnitude O(n 1-m ) -at least for n ≤ 4, so that the distribution of X n,r has a multivariate Edgeworth expansion in powers of n -1/2 . (Alternatively one can use James and Mayne (1962) to derive the cumulants of X n,r from those of U n,r .) The method requires the derivatives of F at {F -1 (p i ), 1 ≤ i ≤ k} so breaks down if p i = 0 or p k = 1 -which is the situation we study here. For definiteness, we confine ourselves to F -1 (u) having a power singularity at 1, say
as x → ∞. For a nonparametric estimate of α see Novak and Utev (1990).
Distributions satisfying (1.3) are known as Pareto type distributions. These distributions arise in many areas of the sciences, engineering and medicine. Some of these areas -where publications involving Pareto type distributions have appeared -are: hydrology, physics, wind engineering and industrial aerodynamics, computer science, water resources, insurance mathematics and economics, structural safety, material science, performance evaluation, queueing systems, geophysical research, ironmaking and steelmaking, banking and finance, atmospheric environment, civil engineering, communications, information processing and management, high speed networks, lightwave technology, solar energy engineering, supercomputing, natural hazards and earth system sciences, ocean engineering, optics communications, reliability engineering, signal processing and urban studies.
In Withers and Nadarajah (2007a) we showed that for fixed r when (1.3) holds the distribution of X n,n1-r (where 1 is the vector of ones in ℜ k ), suitably normalized tends to a certain multivariate extreme value distribution as n → ∞, and so obtained the leading terms of the expansions of its moments in inverse powers of n. Here we show how to extend those expansions when Hall (1978) considered (1.4) with α i = i -1/α, but did not give the corresponding expansion for F (x) or expansions in inverse powers of n. He applied it to the Cauchy. In Section 2, we demonstrate the method when
where α > 0 and β > 0. In this case, (1.4) holds with α i = (iβ -1)/α. In Section 3, we apply it to the Student t, F and second extreme value distribution and to stable laws of exponent α < 1. Appendix A gives the inverse theorem needed to pass from (1.5) to (1.4), and expansions for powers and logs of series.
We use the following notation and terminology. Let (x) i = Γ(x + i)/Γ(x) and < x > i = Γ(x + 1)/Γ(x -i + 1). An inequality in ℜ k consists of k inequalities. For example, for x in C k , where C is the set of complex numbers, Re(x) < 0 means that Re(x i ) < 0 for 1 ≤ i ≤ k. Also I(A) = 1 or 0 for A true or false and δ ij = I(i = j). For θ ∈ C k let θ denote the vector with θi = k j=1 θ j .
For
Here, we show how to obtain expansions in inverse powers of n for the moments of the X n,s for fixed r when (1.4) holds, and in particular when the upper tail of F satisfies (1.5).
Theorem 2.1 Suppose (1.5) holds with c 0 , α, β > 0. Then F -1 (u) is given by (1.4) with
, and so on. Also for any θ in ℜ,
Note 2.1 On those rate occasions where the coefficients
, where
and so on. So, for S of (3.29),
where
which is the RHS (2.7) with denominator replaced by the RHS (2.8). Putting θ = 0 gives (2.7), (2.9) for k = 2. Now use induction.
Lemma 2.2 In Lemma 2.1, the restriction
(2.10)
, the ith factor is 1 and the product of the others is E k j=1,j =i (1 -U nr j ) θ * j , where θ * j = θ j for j = i -1 and θ * j = θ i-1 + θ i for j =
…(Full text truncated)…
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