Abstract
We present a new method for estimating the endpoint of a unidimensional sample when the distribution function decreases at a polynomial rate to zero in the neighborhood of the endpoint. The estimator is based on the use of high-order moments of the variable of interest. It is assumed that the order of the moments goes to infinity, and we give conditions on its rate of divergence to get the asymptotic normality of the estimator. The good performance of the estimator is illustrated on some finite sample situations.
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Acknowledgements
The authors are indebted to the anonymous referees for their helpful comments and suggestions that have contributed to an improved presentation of the results of this paper.
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Communicated by Domingo Morales.
Appendices
Appendix A: Auxiliary results
Let us set \(\overline{F}_{1}(y):=\overline{F}(\theta y)\) and \(\mu_{1,\,p_{n}}:=\mu_{p_{n}}/\theta^{p_{n}}\). The first result deals with the behavior of the moment \(\mu_{1,\, p_{n}}\).
Lemma 1
If (A0) holds, then \(\mu_{1,\, p_{n}} / \mu_{1, p_{n}+1} \to1\) as n→∞.
The next lemma establishes some consequences of the property (A2):
Lemma 2
Let η be a continuously derivable function on (1,∞) such that |η′| is regularly varying at infinity with index ν−1, where ν≤0, and xη′(x)→0 as x→∞. Then,
-
(i)
tsup x≥1|η(tx)−η((t+1)x)|→0 as t→∞.
-
(ii)
For all q>−ν, tsup x∈(0,1] x q|η(tx+1)−η((t+1)x+1)|→0 as t→∞.
Before proceeding, let us introduce some more notations. For all k∈ℝ, let P k be the set of collections of Borel functions (f p ) p≥1 on (0,1] such that
-
1.
∃p k ≥1, ∃C k ≥0, ∀p≥p k , ∀x∈(0,1], |f p (x)|≤C k x k,
-
2.
∃p k ≥1, ∃C k ≥0, ∀p≥p k , ∀x∈(0,1], p 2|f p+1−f p |(x)≤C k x k,
-
3.
∀x∈(0,1], p 2|f p+2−2f p+1+f p |(x)→0 as p→∞.
Let P=⋂ k≥0 P k . Besides, let U be the set of collections of Borel functions (f p ) p≥1 on [1,∞) such that
-
1.
sup x≥1|f p (x)|=O (1) as p→∞,
-
2.
p 2sup x≥1|f p+1−f p |(x)=O (1) as p→∞,
-
3.
p 2sup x≥1|f p+2−2f p+1+f p |(x)→0 as p→∞.
These sets will reveal useful to study the asymptotic properties of \(\widehat{\theta}_{n}\) since this estimator is based on increments of sequences of functions. A stability property of the set P is given in the next lemma.
Lemma 3
Let (f p ), (g p ) be two collections of Borel functions. If for some k∈ℝ, (f p )∈P k and (g p )∈P, then (f p g p )∈P.
We now give a continuity property of some integral transforms defined on P and U.
Lemma 4
Let (f p )∈P, (g p )∈U and (u p ), (v p ) be two collections of Borel functions such that f p (x)→f(x) for all x∈(0,1],
where f,g,u,v are four Borel functions such that f and u (resp. g and v) are defined on (0,1] (resp. [1,∞)). Assume further that u and v are bounded. Then, for all k>1,
as p→∞.
The following lemma provides sufficient conditions on collections of functions to belong to the previous sets.
Lemma 5
Let (f p ), (g p ) be two collections of Borel functions. Assume that there exist Borel functions F i and Borel bounded functions G i , 0≤i≤2, such that
Then, for all x∈(0,1], p 2|f p+2−2f p+1+f p |(x)→0 as p→∞, and (g p )∈U.
We are now in position to exhibit two particular elements of P and U:
Lemma 6
Let (f p ) and (g p ), p≥1 be two collections of Borel functions defined by
Then (f p )∈P, (g p )∈U and
Lemma 7 is the key tool for establishing precise expansions of the moments μ p and M p .
Lemma 7
Let (f p )∈P and (g p )∈U such that (15) holds and define
where L is a Borel slowly varying function at infinity. Then, for all i=1, 2,
-
(i)
E i (p)→0 as p→∞,
-
(ii)
p 2(E i (p+1)−E i (p))=O (1),
-
(iii)
p 2(E i (p+2)−2E i (p+1)+E i (p))→0 as p→∞,
-
(iv)
δ i (p)→0 as p→∞.
Moreover, if L satisfies (A2), then
-
(v)
There exists a slowly varying function \(\mathcal{L}\) such that \(\delta_{1}(p) = \operatorname{O} ( |\eta(p)| \mathcal{L}(p))\),
-
(vi)
δ 2(p)=O (|η(p)|),
-
(vii)
For all i=1,2, δ i (p+1)−δ i (p)=O (|η(p)|/p),
-
(viii)
For all i=1,2, p 2(δ i (p+2)−2δ i (p+1)+δ i (p))→0 as p→∞.
Sometimes, a first order expansion of the moment μ p is sufficient:
Lemma 8
If (A1) holds then, as p→∞,
The next result consists of linearizing the quantity ξ n appearing in the proof of Theorem 2:
Lemma 9
Let p n →∞ and \(\nu_{p}=\widehat{\mu}_{p}-\mu_{p}\). If (A1) is satisfied, then
where
with
and
The final lemma of this section provides an asymptotic bound of the third-order moments appearing in the proof of Theorem 2.
Lemma 10
Let k∈ℕ and p n →∞. Let (H n,j )0≤j≤k be sequences of Borel uniformly bounded functions on [0,1] and
If Y is a random variable with survival function \(\overline {G}(x)=(1-x)^{\alpha} L((1-x)^{-1})\) where α>0 and L is a Borel slowly varying function at infinity, then
Appendix B: Proofs
Proof of Lemma 1
Let \(I_{p_{n}}:={\mu_{1,\, p_{n}}}/{p_{n}}\) and ε>0. The integral \(I_{p_{n}}\) is expanded as
where
Since \(\bigl[ \frac{1-\varepsilon/2}{1-\varepsilon} \bigr]^{p_{n}-1} \to \infty\) as n→∞, it follows that
In view of
and (16), one thus has \({I_{p_{n}}}/{I_{p_{n}+1}} \to1\) as n→∞ and Lemma 1 is proved. □
Proof of Lemma 2
Let us consider (i) and (ii) separately.
(i) Let t,x≥1. The mean value theorem shows that there exists h 1(t,x)∈(0,1) such that
uniformly in x≥1, as t→∞.
(ii) Pick t≥1 and x∈(0,1]: applying the mean value theorem again shows that there exists h 2(t,x)∈(0,1) such that
Now, for all h∈(0,1), one has
Since x↦x q+1|η′(x+1)| is regularly varying with index q+ν>0, Bingham et al. (1987), Theorem 1.5.2 yields
as t→∞. Using the hypothesis xη′(x)→0 as x→∞ then gives
as t→∞, uniformly in x∈(0,1] and h∈(0,1), which concludes the proof of Lemma 2. □
Proof of Lemma 3
This result easily follows from the identities
and from the properties of (f p ) and (g p ). □
Proof of Lemma 4
Remark that, for p large enough,
where r is a bounded Borel function on (0,1]. The upper bound is an integrable function on (0,1], so that the dominated convergence theorem yields
as p→∞, which proves the first part of the lemma.
Since v is bounded on [1,∞), (g p v p ) converges uniformly to gv on [1,∞). The function x↦x −k being integrable on [1,∞), the dominated convergence theorem yields
as p→∞, which concludes the proof of Lemma 4. □
Proof of Lemma 5
Remark that
to obtain the result. □
Proof of Lemma 6
It is clear that for all x∈(0,1], f p (x)→e −1/x as p→∞.
In order to prove that (f p )∈P, let us rewrite f p (x) as f p (x)=σ p φ p (x)ψ p (x) where
for all x∈(0,1], and prove that (σ p )∈P 0, (φ p )∈P −1 and (ψ p )∈P. First, note that
so that the collection of constant functions (σ p ) lies in P 0. Second, we have
Moreover,
and since ∀x∈(0,1], x/(px+1)≤1/p, Taylor expansions yield, uniformly in x∈(0,1],
It follows that there exists a positive constant C (1) such that for p large enough,
Third, let x∈(0,1], and consider a pointwise Taylor expansion of φ p to get
Using (17), (18) and applying Lemma 5 therefore shows that (φ p )∈P −1.
Let x∈(0,1], k≥0, Ψ x (p)=(1−1/(px+1))p, so that ψ p (x)=Ψ x (p−1). Routine calculations show that Ψ x (p) is a positive non-increasing function of p. Consequently, for all p≥k+1 and for all x∈(0,1], ψ p (x)≤ψ k+1(x). Remarking that ψ k+1(x)≤k k x k for all x∈(0,1], it follows that
Recall that Ψ x is non-increasing and write
Taylor expansions of the logarithm function at 1 and of the exponential function at 0 imply that, uniformly in x∈(0,1],
Since for all x∈(0,1], 0≤1/(px+1)≤1, applying the mean value theorem to the function h↦(1−h)e h gives
A Taylor expansion of \([ 1+ \frac{1}{p-1} ]^{p-1}\) then yields, uniformly in x∈(0,1],
Therefore, there exists C (2)≥0 such that, for all p large enough,
Taking (19) into account, this entails
A pointwise Taylor expansion of ψ p finally gives
Using (19), (20) and applying Lemma 5 shows that (ψ p )∈P. Lemma 3 therefore shows that (f p )∈P.
Finally, a Taylor expansion entails
as p→∞. It follows that g p (x)→e −1/x as p→∞ uniformly on [1,∞). Lemma 5 then shows that (g p )∈U. □
Proof of Lemma 7
(i), (ii) and (iii) are simple consequences of (f p )∈P, (g p )∈U, (15) and of the dominated convergence theorem.
(iv) Let us introduce
so that
First, remark that x↦L(x+1) is a slowly varying function, so that L 1 is regularly varying with index 1. Bingham et al. (1987), Theorem 1.5.2 thus entails \(Q_{p}^{(1)}(x)\to0\) uniformly in x∈(0,1] as p→∞. Applying Lemma 4 yields δ 1(p)→0 as p→∞. Second, since L 2 is regularly varying with index −1, using again Bingham et al. (1987), Theorem 1.5.2 leads to \(Q_{p}^{(2)}(x) \to0\) as p→∞ uniformly in x≥1. Applying Lemma 4 again entails δ 2(p)→0 as p→∞.
(v) Let p be large enough so that |η| is non-increasing in [p,∞). Pick s>1−ν and let \(Q^{(1, 1)}_{p}(x):=x^{s} Q_{p}^{(1)}(x)\). Using the ideas of the proof of Lemma 4, one has
Introducing
(A2) and the well-known inequality |e u−1|≤|u|e |u| for all u∈ℝ yield
Letting \(\widetilde{\eta}(t)=(t+1)^{1-\nu} \eta(t+1)\), we get
Remarking that \(\widetilde{\eta}\) is regularly varying with index 1, Bingham et al. (1987), Theorem 1.5.2 implies that for p large enough,
Moreover, for all u>0 and x∈(0,1], one has x/(ux+1)≤1/u, so that, for p large enough,
uniformly in x∈(0,1] since x↦x s−(1−ν)lnx is bounded on (0,1]. Let us now consider \(\mathcal{L}(y)=\exp ( \int_{1}^{y} |\eta(t)| t^{-1} dt )\). Clearly, \(\mathcal{L}\) is slowly varying at infinity and \(\exp|R^{(1)}_{p}(x)| \leq\mathcal{L}(p)\). Consequently, in view of (21) and (22), it follows that
and therefore \(\delta_{1}(p)=\operatorname{O}(|\eta(p)| \mathcal{L}(p))\).
(vi) Similarly, for all x≥1 and large p, we have
Let p be so large that |η(p)|≤1. Since x↦x −α−1lnx is integrable on [1,∞), the arguments of the proof of Lemma 4 entail
(vii) Keeping in mind that s>1−ν, the following expansion holds:
Let us first focus on (25). In view of (A2), and considering
for x∈(0,1], one obtains
Mimicking the proof of (v), we thus get, for p large enough,
uniformly in x∈(0,1]. A Taylor expansion of the exponential function at 0 then entails
where ρ is locally bounded on ℝ. Since
it follows that
Applying Bingham et al. (1987), Theorem 1.5.2 to L 1 yields
and consequently,
Focusing on (26), for all 0<x≤1, because (f p )∈P, we get for all sufficiently large p
which is integrable on (0,1]. Consequently, in view of (23),
Collecting (29) and (30) yields δ 1(p+1)−δ 1(p)=O (|η(p)|/p). Let us remark that
and consider first (31). From (A2), we have
Since for all x≥1, we have \(p \vert \int_{p}^{p+1}\eta(tx) t^{-1}\, dt \vert \leq|\eta(p)|\), and recalling that, as p→∞
a Taylor expansion of the exponential function at 0 yields
Taking into account that (g p )∈U and using (24), it follows that
Moreover, from (33), the uniform convergence of (g p ) to x↦e −1/x on [1,∞) and the dominated convergence theorem, we get
This eventually leads to δ 2(p+1)−δ 2(p)=O (|η(p)|/p) and establishes (vii).
(viii) Let q>1−ν and \(Q_{p}^{(1, 2)}(x)=x^{2q+1} Q_{p}^{(1)}(x)\) so that
and the following expansion holds:
Considering (34), arguments given in the proof of (vii) show that
Let us now focus on (35). From (27), (28), and (A2), a Taylor expansion yields
uniformly in x∈(0,1]. Let x∈(0,1]: using the inequality x/(tx+1)≤1/t, we obtain
Moreover, since t↦t q+1 η(t+1) is regularly varying with index q+1+ν>0, Theorem 1.5.2 in Bingham et al. (1987) yields
uniformly in x∈(0,1] as t→∞. Lemma 2(ii) therefore entails, as p→∞,
The dominated convergence theorem then yields
Let us finally consider (36). Since (f p )∈P and in view of the triangular inequality, we have, for p large enough,
Because (f p )∈P, the dominated convergence theorem yields
Collecting (37), (38) and (39), it follows that p 2(δ 1(p+2)−2δ 1(p+1)+δ 1(p))→0 as p→∞. Similarly,
and the three terms are considered separately. First, ideas similar to those developed in the proof of (vii) allow us to control (40):
Second, since \(p \vert \int_{p}^{p+1} \eta(tx) t^{-1} dt\vert \to0\) as p→∞ uniformly in x≥1, (A2) entails
uniformly in x≥1. Remarking that
Lemma 2(i) implies that \(p^{2} |Q_{p+2}^{(2)}- 2Q_{p+1}^{(2)}+Q_{p}^{(2)}|(x) \to0\), uniformly in x≥1 as p→∞, in view of Bingham et al. (1987), Theorem 1.5.2. The uniform convergence of (g p ) to x↦e −1/x on [1,∞) and the dominated convergence theorem yield the following bound for (41):
Finally, recalling that (g p )∈U and the uniform convergence of \((Q_{p}^{(2)})\) to 0 on [1,∞), (42) is controlled as
Collecting (43), (44) and (45), it follows that p 2(δ 2(p+2)−2δ 2(p+1)+δ 2(p))→0 as p→∞ and the lemma is proved. □
Proof of Lemma 8
It is a direct consequence of the expansion (4) and Lemma 7(i), (iv). □
Proof of Lemma 9
Let us remark that, from Lemma 8,
and consider the expansion
with
Replacing in (46), Lemma 9 follows. □
Proof of Lemma 10
Hölder’s inequality yields
It then suffices to prove that ∀j∈{0,…,k},
Let λ,μ≥0. The function
being Lebesgue⊗ℙ-integrable, Fubini’s theorem entails
Finally, if (s n ) is a positive real sequence tending to ∞ and d≥0, we have, from Lemma 8,
Replacing in the inequality above and recalling that L is slowly varying at infinity, it follows that \(\mathbb {E}[ Y^{p_{n}} (1-Y)^{k-j}]^{3} = \operatorname{O}(p_{n}^{-\alpha-(3k-3j)} L(p_{n}))\), which establishes Lemma 10. □
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Girard, S., Guillou, A. & Stupfler, G. Estimating an endpoint with high-order moments. TEST 21, 697–729 (2012). https://doi.org/10.1007/s11749-011-0277-8
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DOI: https://doi.org/10.1007/s11749-011-0277-8