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Estimating an endpoint with high-order moments

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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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Correspondence to Armelle Guillou.

Additional information

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 ′(x)→0 as x→∞. Then,

  1. (i)

    tsup x≥1|η(tx)−η((t+1)x)|→0 as t→∞.

  2. (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. 1.

    p k ≥1, ∃C k ≥0, ∀pp k , ∀x∈(0,1], |f p (x)|≤C k x k,

  2. 2.

    p k ≥1, ∃C k ≥0, ∀pp k , ∀x∈(0,1], p 2|f p+1f p |(x)≤C k x k,

  3. 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. 1.

    sup x≥1|f p (x)|=O (1) as p→∞,

  2. 2.

    p 2sup x≥1|f p+1f p |(x)=O (1) as p→∞,

  3. 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

$$ \forall x\in(0, 1], \quad f_p(x)\to e^{-1/x}\quad \mbox{\textit{and}}\quad \sup_{x\geq 1} \bigl \vert g_p(x)-e^{-1/x}\bigr \vert \to0 \quad \mbox{\textit{as}}\ p\to\infty.$$
(15)

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,

  1. (i)

    E i (p)→0 as p→∞,

  2. (ii)

    p 2(E i (p+1)−E i (p))=O (1),

  3. (iii)

    p 2(E i (p+2)−2E i (p+1)+E i (p))→0 as p→∞,

  4. (iv)

    δ i (p)→0 as p→∞.

Moreover, if L satisfies (A2), then

  1. (v)

    There exists a slowly varying function \(\mathcal{L}\) such that \(\delta_{1}(p) = \operatorname{O} ( |\eta(p)| \mathcal{L}(p))\),

  2. (vi)

    δ 2(p)=O (|η(p)|),

  3. (vii)

    For all i=1,2, δ i (p+1)−δ i (p)=O (|η(p)|/p),

  4. (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→∞,

$$\mu_p = p^{-\alpha} \theta^{p} L(p) \Gamma(\alpha+1)\bigl(1+\operatorname{o}(1)\bigr).$$

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

$$\zeta_n^{(1)} = \zeta_n^{(2)} +\biggl[ 1+ \frac{a p_n}{p_n+1} \biggr] \zeta_n^{(3)},$$

with

and

$$u_{n, a} = \frac{1}{a \Gamma(\alpha+1)} \sqrt{ \frac {1}{V(\alpha, a)}} \frac{p_n^{\alpha} v_n}{ \theta^{p_n} L(p_n)}.$$

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≤jk be sequences of Borel uniformly bounded functions on [0,1] and

$$\forall u \in[0, 1], \quad h_n(u)=\sum_{j=0}^k\frac{H_{n,j}(u)}{p_n^j} (1-u)^{k-j}.$$

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

$$\mathbb {E}\big|Y^{p_n} h_n(Y)\big|^3 = \operatorname{O}\bigl(p_n^{-\alpha-3k} L(p_n)\bigr).$$

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

$$I_{p_n} = \int_{1-\varepsilon}^1 y^{p_n-1}\overline{F_1}(y) dy \biggl[1+ \frac{ \int_0^{1-\varepsilon} y^{p_n-1} \overline {F_1}(y)\,dy }{\int_{1-\varepsilon}^1 y^{p_n-1} \overline {F_1}(y)\, dy } \biggr],$$

where

Since \(\bigl[ \frac{1-\varepsilon/2}{1-\varepsilon} \bigr]^{p_{n}-1} \to \infty\) as n→∞, it follows that

$$ I_{p_n} = \int_{1-\varepsilon}^1y^{p_n-1} \overline {F_1}(y)\, dy \bigl(1+\operatorname{o}(1)\bigr).$$
(16)

In view of

$$1\leq {\int_{1-\varepsilon}^1 y^{p_n-1}\overline {F_1}(y) \,dy} \Big/ \int_{1-\varepsilon}^1y^{p_n} \overline{F_1}(y) \,dy \leq \frac{1}{1-\varepsilon}$$

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

$$(t+h)x^{q+1} \eta'\bigl((t+h)x+1\bigr) = \biggl\{\frac{ [ (t+h)x]^{q+1} \eta'((t+h)x+1)}{(t+h)^{q+1} \eta'(t+h+1)} \biggr\} (t+h) \eta'(t+h+1).$$

Since xx q+1|η′(x+1)| is regularly varying with index q+ν>0, Bingham et al. (1987), Theorem 1.5.2 yields

$$\sup_{\substack{x\in(0, 1] \\ h\in(0, 1)}} \biggl \vert \frac{ [ (t+h)x]^{q+1} \eta'((t+h)x+1)}{(t+h)^{q+1} \eta'(t+h+1)}-x^{q+\nu} \biggr \vert \to0$$

as t→∞. Using the hypothesis ′(x)→0 as x→∞ then gives

$$(t+h)x^{q+1} \eta'\bigl((t+h)x+1\bigr) \to0$$

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,

$$\forall x\in(0, 1], \quad x^{-k} \big|f_p(x)\big|\big|u_p(x)\big| \leq C_k \bigl\{ \big|u(x)\big|+r(x) \bigr\}$$

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

$$\int_{0}^{1} x^{-k}f_p(x) u_p(x) \,dx \to\int_{0}^{1}x^{-k} f(x) u(x)\, dx$$

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 xx k being integrable on [1,∞), the dominated convergence theorem yields

$$\int_{1}^{\infty} x^{-k} g_p(x)v_p(x) \,dx \to\int_{1}^{\infty}x^{-k} g(x) v(x)\, dx$$

as p→∞, which concludes the proof of Lemma 4. □

Proof of Lemma 5

Remark that

$$\frac{1}{p+1}- \frac{1}{p}=\operatorname{O}\biggl( \frac{1}{p^2} \biggr)\quad \mbox{and}\quad \frac{1}{p+2}- \frac{2}{p+1}+\frac{1}{p} = \operatorname{O} \biggl( \frac {1}{p^3} \biggr)$$

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

$$\sigma_p=1+ \frac{\alpha+1}{p}+ \frac{(\alpha+1)(\alpha+2)}{2} \frac {1}{p^2}+\operatorname{o} \biggl( \frac{1}{p^2} \biggr)$$

so that the collection of constant functions (σ p ) lies in P 0. Second, we have

$$ \forall p>1, \ \forall x\in(0, 1], \quad\big|\varphi_p(x)\big| \leq1 \leq x^{-1}.$$
(17)

Moreover,

$$[\varphi_{p+1}-\varphi_{p}](x) = \varphi_p(x)\biggl[ \biggl( 1- \frac {1}{p} \biggr)^{-\alpha-2} \biggl( 1- \frac{x}{p x +1} \biggr)^{\alpha+2} -1 \biggr],$$

and since ∀x∈(0,1], x/(px+1)≤1/p, Taylor expansions yield, uniformly in x∈(0,1],

$$[\varphi_{p+1}-\varphi_{p}](x) = \varphi_p(x)\biggl[ \frac{\alpha +2}{p(px+1)}+\operatorname{O} \biggl( \frac{1}{p^2} \biggr)\biggr].$$

It follows that there exists a positive constant C (1) such that for p large enough,

$$ p^2 |\varphi_{p+1}-\varphi_{p}|(x)\leq C^{(1)} x^{-1}.$$
(18)

Third, let x∈(0,1], and consider a pointwise Taylor expansion of φ p to get

$$\varphi_p(x)=1- \frac{\alpha+2}{px}+ \frac{\alpha+2}{p^2 x} \biggl( -1+\frac{\alpha+3}{2x} \biggr) +\operatorname{o} \biggl( \frac{1}{p^2}\biggr).$$

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 pk+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

$$ \forall k\geq0, \ \exists p_k\geq1, \ \exists C_k \geq0, \ \forall p\geq p_k, \ \forall x\in(0,1], \quad \big|\psi_p(x)\big| \leq C_k x^k.$$
(19)

Recall that Ψ x is non-increasing and write

$$|\psi_{p+1}-\psi_{p}|(x) = \psi_{p}(x) \biggl[ 1-\biggl( 1- \frac{1}{px+1} \biggr) \biggl( 1+ \frac{1}{p-1}\biggr)^{p-1} \biggl( 1- \frac{x}{px+1} \biggr)^{p-1}\biggr].$$

Taylor expansions of the logarithm function at 1 and of the exponential function at 0 imply that, uniformly in x∈(0,1],

$$e \biggl(1- \frac{x}{p x+1} \biggr)^{p-1}= \exp \biggl( \frac{1}{p x+1} \biggr) \biggl[ 1+ \frac{x}{p x+1} - \frac{p}{2} \biggl(\frac{x}{px+1} \biggr)^2 +\operatorname{O} \biggl( \frac{1}{p^2} \biggr) \biggr].$$

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

$$\biggl \vert \biggl[ 1- \frac{1}{p x+1} \biggr] \exp \biggl[ \frac{1}{p x+1} \biggr]-1 \biggr \vert \leq \frac{e}{(px+1)^2}.$$

A Taylor expansion of \([ 1+ \frac{1}{p-1} ]^{p-1}\) then yields, uniformly in x∈(0,1],

$$|\psi_{p+1}-\psi_{p}|(x) \leq\psi_{p}(x) \biggl[\biggl( e+ \frac{1}{2p} \biggr) \frac{1}{(p x+1)^2} + \operatorname{O}\biggl( \frac{1}{p^2} \biggr) \biggr].$$

Therefore, there exists C (2)≥0 such that, for all p large enough,

$$p^2 \vert \psi_{p+1} - \psi_p \vert (x)\leq\psi_p(x) C^{(2)} x^{-2}.$$

Taking (19) into account, this entails

$$ \forall k\geq0, \ \exists p_k\geq1, \ \exists C_k \geq0, \ \forall p\geq p_k, \ \forall x\in(0,1], \quad p^2 |\psi_{p+1}-\psi_p|(x) \leq C_k x^k.$$
(20)

A pointwise Taylor expansion of ψ p finally gives

$$\psi_p(x)=e^{-1/x}\biggl[ 1+ \frac{1}{2 p x^2}+ \frac{1}{p^2 x^2} \biggl( \frac{1}{2}- \frac{1}{3 x}+ \frac{1}{8 x^2}\biggr) +\operatorname{o} \biggl( \frac{1}{p^2} \biggr) \biggr].$$

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

$$p^2 \sup_{x\geq1} \biggl \vert g_p(x)-e^{-1/x}\biggl[ 1+ \frac{1}{p x} \biggl( 1- \frac{1}{2x} \biggr) +\frac{1}{p^2 x^2} \biggl( \frac{1}{2}- \frac {5}{6x} +\frac{1}{8x^2} \biggr) \biggr] \biggr \vert \to0$$

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 xL(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

$$I_1 \delta_1(p) = \int_{0}^1f_{p}(x) Q_p^{(1, 1)}(x) x^{-\alpha-s-3} \,dx =\operatorname{O} \Bigl( \sup_{0<x\leq1} x^{-1} \bigl \vert Q^{(1,1)}_p(x)\bigr \vert \Bigr).$$

Introducing

$$R^{(1)}_p(x):=\int_{(p-1) x+1}^{p}\frac{\eta(t)}{t}\, dt,$$

(A2) and the well-known inequality |e u−1|≤|u|e |u| for all u∈ℝ yield

(21)

Letting \(\widetilde{\eta}(t)=(t+1)^{1-\nu} \eta(t+1)\), we get

$$x^s \bigl \vert R^{(1)}_p(x) \bigr \vert \leq x^{s+1} \int_{p-1}^{(p-1)/x} \biggl[ \biggl \vert \frac{\widetilde{\eta}(ux)}{\widetilde{\eta}(u)}- x \biggr \vert + x \biggr] \big|\widetilde{\eta}(u)\big|\frac{du}{(ux+1)^{2-\nu}}.$$

Remarking that \(\widetilde{\eta}\) is regularly varying with index 1, Bingham et al. (1987), Theorem 1.5.2 implies that for p large enough,

$$\sup_{\substack{0<x\leq1 \\ u\geq p-1}}\biggl \vert \frac{\widetilde{\eta }(ux)}{\widetilde{\eta}(u)} - x \biggr \vert \leq1.$$

Moreover, for all u>0 and x∈(0,1], one has x/(ux+1)≤1/u, so that, for p large enough,

(22)

uniformly in x∈(0,1] since xx 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

$$ \sup_{0<x\leq1} x^{-1} \bigl \vert Q^{(1, 1)}_p(x) \bigr \vert =\operatorname {O}\bigl(\big|\eta(p)\big|\mathcal{L}(p)\bigr),$$
(23)

and therefore \(\delta_{1}(p)=\operatorname{O}(|\eta(p)| \mathcal{L}(p))\).

(vi) Similarly, for all x≥1 and large p, we have

$$ \bigl \vert Q_{p}^{(2)}(x) \bigr \vert \leq x^{-1} \biggl \vert \int_{p}^{px}\frac {\eta(t)}{t} dt \biggr \vert \exp\biggl \vert \int _{p}^{px} \frac {\eta(t)}{t}\,dt \biggr \vert \leq\big|\eta(p)\big| x^{|\eta(p)|-1} \ln x.$$
(24)

Let p be so large that |η(p)|≤1. Since xx 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:

(25)
(26)

Let us first focus on (25). In view of (A2), and considering

$$R^{(2)}_p(x):=\int_{(p-1) x+1}^{px+1}\frac{\eta(t)}{t}\, dt,$$

for x∈(0,1], one obtains

$$\bigl[Q_{p+1}^{(1, 1)}- Q_{p}^{(1, 1)}\bigr](x)= x^s \frac{L_1(p x)}{L_1(p-1)} \frac{p-1}{p} \bigl\{ \exp \bigl(-R^{(2)}_p(1) \bigr) - \exp \bigl( -R^{(2)}_p(x)\bigr) \bigr\}.$$

Mimicking the proof of (v), we thus get, for p large enough,

(27)

uniformly in x∈(0,1]. A Taylor expansion of the exponential function at 0 then entails

$$\exp \bigl( - R^{(2)}_p(x) \bigr) = 1- \frac{1}{x^s}\bigl\{ x^s R^{(2)}_p(x) \bigr\} \bigl[1+\rho \bigl( R^{(2)}_p(x) \bigr) \bigr],$$

where ρ is locally bounded on ℝ. Since

$$ \sup_{\substack{0<x\leq1 \\ p \geq1}} \bigl \vert R^{(2)}_p(x)\bigr \vert \leq \sup_{t\geq1} \big|\eta(t)\big| \quad\Rightarrow\quad \sup_{\substack {0<x\leq1 \\ p \geq1}} \bigl \vert \rho \bigl( R^{(2)}_p(x)\bigr) \bigr \vert < \infty,$$
(28)

it follows that

$$\sup_{0<x\leq1} \bigl\{ x^s \bigl \vert \exp \bigl(-R^{(2)}_p(x) \bigr)-1 \bigr \vert \bigr\} =\operatorname{O} \bigl( \big|\eta(p)\big|/{p} \bigr).$$

Applying Bingham et al. (1987), Theorem 1.5.2 to L 1 yields

$$\sup_{0<x\leq1} \big|Q_{p+1}^{(1, 1)}- Q_p^{(1, 1)}\big|(x)= \operatorname {O} \bigl( \big|\eta(p)\big|/{p} \bigr)$$

and consequently,

$$ \int_{0}^{1}f_p(x) \bigl[Q_{p+1}^{(1, 1)}-Q_p^{(1, 1)}\bigr](x) x^{-\alpha-s-3} \,dx = \operatorname{O} \bigl( \big|\eta(p)\big|/{p} \bigr).$$
(29)

Focusing on (26), for all 0<x≤1, because (f p )∈P, we get for all sufficiently large p

$$p^2 x^{-\alpha-s-2} |f_{p+1}- f_p|(x) \leq C_{\alpha+s+2}$$

which is integrable on (0,1]. Consequently, in view of (23),

$$ \int_{0}^{1}[f_{p+1}-f_p](x) Q_{p+1}^{(1, 1)}(x)x^{-\alpha-s-3} \,dx =\operatorname{O} \biggl( \frac{|\eta(p)| \mathcal{L}(p)}{p^2} \biggr) =\operatorname{O} \bigl( \big|\eta(p)\big|/p \bigr).$$
(30)

Collecting (29) and (30) yields δ 1(p+1)−δ 1(p)=O (|η(p)|/p). Let us remark that

(31)
(32)

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→∞

$$\sup_{x\geq1} \biggl \vert \frac{L_2((p+1) x)}{L_2(p)}- \frac{1}{x}\biggr \vert \to0,$$

a Taylor expansion of the exponential function at 0 yields

$$ \sup_{x\geq1}\big|Q_{p+1}^{(2)}- Q_p^{(2)}\big|(x) =\operatorname{O} \bigl( \big|\eta(p)\big|/{p} \bigr).$$
(33)

Taking into account that (g p )∈U and using (24), it follows that

$$\int_{1}^{\infty} [g_{p+1}-g_p](x)Q_{p+1}^{(2)}(x) x^{-\alpha-1}\, dx =\operatorname{O} \bigl(\big|\eta(p)\big|/p \bigr).$$

Moreover, from (33), the uniform convergence of (g p ) to xe −1/x on [1,∞) and the dominated convergence theorem, we get

$$\int_{1}^{\infty} g_p(x)\bigl[Q_{p+1}^{(2)}-Q_p^{(2)}\bigr](x)x^{-\alpha-1}\, dx =\operatorname{O} \bigl(\big|\eta(p)\big|/p \bigr).$$

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

$$I_1 \delta_1(p) = \int_{0}^{1}f_p(x) Q_p^{(1, 2)}(x) x^{-\alpha-2q-4}\,dx$$

and the following expansion holds:

(34)
(35)
(36)

Considering (34), arguments given in the proof of (vii) show that

$$ \int_{0}^{1}[f_{p+1}-f_p](x) \bigl[Q_{p+2}^{(1, 2)}-Q_p^{(1, 2)}\bigr](x) x^{-\alpha-2q-4}\, dx = \operatorname{o}\bigl({1}/{p^2} \bigr).$$
(37)

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 tt q+1 η(t+1) is regularly varying with index q+1+ν>0, Theorem 1.5.2 in Bingham et al. (1987) yields

$$x^{q+1} \bigl \vert \eta(tx+1) \bigr \vert = \biggl \vert \frac{(tx)^{q+1} \eta (tx+1)}{t^{q+1} \eta(t+1)} \biggr \vert \bigl \vert \eta(t+1) \bigr \vert \to0$$

uniformly in x∈(0,1] as t→∞. Lemma 2(ii) therefore entails, as p→∞,

$$p^2 \sup_{0<x\leq1} \big|Q_{p+2}^{(1, 2)}- 2Q_{p+1}^{(1, 2)} +Q_p^{(1,2)}\big|(x) \to0.$$

The dominated convergence theorem then yields

$$ \int_{0}^{1} f_{p+1}(x)\bigl[Q_{p+2}^{(1, 2)}- 2 Q_{p+1}^{(1, 2)}+Q_p^{(1,2)}\bigr](x) x^{-\alpha-2q-4} \,dx =\operatorname{o}\bigl( {1}/{p^2} \bigr).$$
(38)

Let us finally consider (36). Since (f p )∈P and in view of the triangular inequality, we have, for p large enough,

$$p^2 x^{-\alpha-2q-4} |f_{p+2}- 2 f_{p+1}+f_p|(x) \leq C_{\alpha+2q+4}.$$

Because (f p )∈P, the dominated convergence theorem yields

$$ \int_{0}^1 [f_{p+2}-2 f_{p+1}+f_p](x) Q_{p+2}^{(1, 2)}(x)x^{-\alpha-2q-4} \,dx =\operatorname{o} \bigl( {1}/{p^2} \bigr).$$
(39)

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,

(40)
(41)
(42)

and the three terms are considered separately. First, ideas similar to those developed in the proof of (vii) allow us to control (40):

$$ \int_{1}^{\infty} [g_{p+1}-g_p](x)\bigl[Q_{p+2}^{(2)}- Q_p^{(2)}\bigr](x)x^{-\alpha-1}\, dx = \operatorname{o} \bigl( {1}/{p^2} \bigr).$$
(43)

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 xe −1/x on [1,∞) and the dominated convergence theorem yield the following bound for (41):

$$ \int_{1}^{\infty} g_{p+1}(x)\bigl[Q_{p+2}^{(2)}-2 Q_{p+1}^{(2)}+Q_p^{(2)}\bigr](x) x^{-\alpha-1}\, dx=\operatorname{o} \bigl( {1}/{p^2}\bigr).$$
(44)

Finally, recalling that (g p )∈U and the uniform convergence of \((Q_{p}^{(2)})\) to 0 on [1,∞), (42) is controlled as

$$ \int_{1}^{\infty} [g_{p+2}-2g_{p+1}+g_p](x) Q_{p+2}^{(2)}(x)x^{-\alpha-1}\, dx =\operatorname{o} \bigl( {1}/{p^2} \bigr).$$
(45)

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,

$$ \xi_n= \frac{\mu_{p_n+1}}{p_n+1} u_{n, a}\cdot ap_n \biggl( \frac {1}{\widehat{\theta}_n} - \frac{1}{\varTheta _n} \biggr)\bigl(1+\operatorname{o}(1)\bigr),$$
(46)

and consider the expansion

with

Replacing in (46), Lemma 9 follows. □

Proof of Lemma 10

Hölder’s inequality yields

$$\mathbb {E}\big|Y^{p_n} h_n(Y)\big|^3 \leq(k+1)^2\Biggl[ \sum_{j=0}^{k} \frac {1}{p_n^{3j}} \sup_{\substack{[0, 1] \\ n \in \mathbb {N}\setminus\{ 0 \}}} |H_{n, j}|^3 \mathbb {E}\bigl[Y^{p_n} (1-Y)^{k-j} \bigr]^3 \Biggr].$$

It then suffices to prove that ∀j∈{0,…,k},

$$\mathbb {E}\bigl[ Y^{p_n} (1-Y)^{k-j} \bigr]^3 = \operatorname{O}\bigl(p_n^{-\alpha -(3k-3j)} L(p_n)\bigr).$$

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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