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Density expansions of extremes from general error distribution with applications
Journal of Inequalities and Applications volume 2015, Article number: 356 (2015)
Abstract
In this paper the higher-order expansions of density of normalized maximum with parent following general error distribution are established. The main results are applied to derive the higher-order expansions of the moments of extremes.
1 Introduction
In extreme value theory, the quality of convergence of normalized partial maximum of a sample has been studied in recent literature. For the convergence rate of distribution of normalized maximum, we refer to Smith [1], Leadbetter et al. [2], de Haan and Resnick [3] for general cases, and specific cases were studied by Hall [4], Nair [5], Peng et al. [6] and Jia and Li [7]. Nair [5] derived the higher-order expansions of moments of normalized maximum with parent following normal distribution. Liao et al. [8] and Jia et al. [9] extended Nair’s results to skew-normal distribution and general error distribution, respectively.
The main objective of this paper is to derive the higher-order expansions of density of normalized maximum with parent following the general error distribution. To the best of our knowledge, there are few studies on the rate of convergence of density of normalized maximum except the work of de Haan and Resnick [10] for local limit theorems and Omey [11] for rates of convergence of densities with regular variation with remainders excluding the case we will study in this paper, i.e., the general error distribution.
Let \(\{X_{n},n\geq1\}\) be a sequence of independent and identically distributed (i.i.d.) random variables with marginal cumulative distribution function (cdf) \(F_{v}\) following the general error distribution (\(F_{v} \sim\operatorname{GED}(v)\) for short), and let \(M_{n}=\max_{1\leq k\leq n}X_{k}\) denote its partial maximum. The probability density function (pdf) of the \(\operatorname{GED}(v)\) is given by
where \(v >0\) is the shape parameter, \(\lambda={[2^{-2/v}\Gamma(1/v)/\Gamma(3/v)]}^{1/2} \) and \(\Gamma(\cdot)\) denotes the gamma function (Nelson [12]). Note that the \(\operatorname{GED}(2)\) reduces to the standard normal distribution.
For the \(\operatorname{GED}(v)\), the limiting distribution of maximum \(M_{n}\) and its associated higher-order expansions are given by Peng et al. [13] and Jia and Li [7]. Peng et al. [6] showed that
provided the norming constants \(a_{n}\) and \(b_{n}\) satisfy the following equations:
where \(f(x)=2v^{-1}\lambda^{v} x^{1-v}\). In the sequel, let
denote the density of normalized maximum, and
with \(\Lambda^{\prime}(x)=e^{-x}\Lambda(x)\). By Proposition 2.5 in Resnick [14], \(\Delta_{n}(g_{n},\Lambda^{\prime})\to0\) as \(n\to\infty\). For both applications and theoretical analysis, it is of interest to know the convergence rate of (1.4). This paper focuses on this topic and applies the main results to derive the high-order expansions of moments of extremes.
The paper is organized as follows. Section 2 provides the main results and all proofs are deferred to Section 4. Auxiliary lemmas with proofs are given in Section 3.
2 Main results
In this section, we present the asymptotic expansions of density for the normalized maximum formed by the \(\operatorname{GED}(v)\) random variables and its applications to the higher-order expansions of moments of extremes.
Theorem 2.1
Let \(F_{v}(x)\) denote the cdf of \(\operatorname {GED}(v)\) with \(v > 0\), then for \(v\neq1\), with norming constants \(a_{n}\) and \(b_{n}\) given by (1.2), we have
as \(n\to\infty\), where \(k_{v}(x)\) and \(\omega_{v}(x)\) are respectively given by
with
Remark 2.1
If we choose the norming constants \(a_{n}\) and \(b_{n}\) such that
with \(v\ne1\), then
as \(n\to\infty\), where \(\bar{k}_{v}(x)\) and \(\bar{\omega}_{v}(x)\) are respectively given by
and
with
For the case of \(v=1\), we have the following results.
Theorem 2.2
For \(v = 1\), with norming constants \(a_{n}=2^{-1/2}\) and \(b_{n}=2^{-1/2}\log(n/2)\), we have
as \(n\to\infty\), where \(k_{1}(x)\) and \(\omega_{1}(x)\) are respectively given by
To end this section, we apply the higher-order expansions of densities to derive the asymptotic expansions of the moments of extremes. Methods used here are different from those in Nair [5] and Jia et al. [9].
In the sequel, for nonnegative integers r, let
denote respectively the rth moments of \((M_{n}-b_{n})/a_{n}\) and its limits.
Theorem 2.3
Let \(\{X_{n},n\geq1\}\) be an iid sequence with marginal distribution \(F_{v} \sim\operatorname{GED}(v)\), then
-
(i)
for \(v \neq1\), with norming constants \(a_{n}\) and \(b_{n}\) given by (1.2), we have
$$\begin{aligned}& b_{n}^{v} \bigl[b_{n}^{v} \bigl(m_{r}(n)-m_{r}\bigr)+\bigl(1-v^{-1}\bigr) \lambda^{v}r(m_{r+1}+2m_{r}) \bigr] \\& \quad \to 2r\lambda^{2v}\bigl(1-v^{-1}\bigr) \biggl[\bigl( \bigl(1-v^{-1}\bigr) (r+1)+2\bigr)m_{r}+\bigl( \bigl(1-v^{-1}\bigr) (r+1)+1\bigr)m_{r+1} \\& \qquad {} + \biggl(\frac{1}{4}\bigl(1-v^{-1}\bigr) (r-1)+ \frac{1}{3}\bigl(2-v^{-1}\bigr) \biggr)m_{r+2} \biggr] \end{aligned}$$(2.10)as \(n\to\infty\);
-
(ii)
for \(v = 1\), with norming constants \(a_{n}=2^{-1/2}\) and \(b_{n}=2^{-1/2}\log(n/2)\), we have
$$ n \biggl[n\bigl(m_{r}(n)-m_{r}\bigr)+ (-1)^{r}\frac{r}{2}\Gamma^{(r-1)}(2) \biggr] \to(-1)^{r-1}\frac {r}{24} \bigl[8\Gamma^{(r-1)}(3)-3 \Gamma^{(r-1)}(4) \bigr] $$(2.11)as \(n\to\infty\), where \(\Gamma^{(r-1)}(t)\) denote the \((r-1)\) th derivative of the gamma function at \(x=t\).
3 Auxiliary lemmas
In this section we provide auxiliary lemmas which are needed to prove the main results.
Lemma 3.1
Let \(F_{v}(x)\) and \(f_{v}(x)\) respectively denote the cdf and pdf of \(\operatorname{GED}(v)\) with \(v\neq1\), for large x, we have
Furthermore, with the norming constants \(a_{n}\) and \(b_{n}\) given by (1.2), we have
-
(i)
for \(v\neq1\),
$$ b^{v}_{n} \bigl[b^{v}_{n} \bigl(F^{n}_{v}(a_{n}x+b_{n})- \Lambda(x) \bigr)-\tilde {k}_{v}(x)\Lambda(x) \bigr] \to \biggl(\tilde{ \omega}_{v}(x)+\frac{\tilde{k}^{2}_{v}(x)}{2} \biggr)\Lambda(x) $$(3.2)as \(n\to\infty\), where \(\tilde{k}_{v}(x)\) and \(\tilde{\omega}_{v}(x)\) are respectively given by
$$\begin{aligned}& \tilde{k}_{v}(x)=\bigl(1-v^{-1}\bigr) \lambda^{v}\bigl(x^{2}+2x\bigr)e^{-x}, \end{aligned}$$(3.3)$$\begin{aligned}& \tilde{\omega}_{v}(x)=\bigl(v^{-1}-1\bigr) \lambda^{2v} \biggl(4x+2x^{2}+\frac {2}{3} \bigl(2-v^{-1}\bigr)x^{3} +\frac{1}{2} \bigl(1-v^{-1}\bigr)x^{4} \biggr)e^{-x}; \end{aligned}$$(3.4) -
(ii)
for \(v = 1\), with norming constants \(a_{n}=2^{-1/2}\) and \(b_{n}=2^{-1/2}\log(n/2)\), we have
$$ n \bigl[n \bigl(F^{n}_{1}(a_{n}x+b_{n})- \Lambda(x) \bigr)-{k}_{1}(x)\Lambda (x) \bigr] \to \biggl({ \omega}_{1}(x)+\frac{{k}^{2}_{1}(x)}{2} \biggr)\Lambda(x) $$(3.5)as \(n\to\infty\), where \({k}_{1}(x)\) and \({\omega}_{1}(x)\) are those given by (2.9).
Proof
See Lemma 1 and Theorem 1 in Jia and Li [7]. □
Lemma 3.2
Let \(F_{v}(x)\) denote the cdf of \(\operatorname {GED}(v)\) with \(v > 0\), then
-
(i)
for \(v\neq1\), with norming constants given by (1.2), we have
$$ F^{n-1}_{v}(a_{n}x+b_{n}) = \biggl(1+\tilde{k}_{v}(x)b^{-v}_{n}+ \biggl( \tilde{\omega}_{v}(x)+\frac {\tilde{k}^{2}_{v}(x)}{2} \biggr) b^{-2v}_{n} \bigl(1+o(1)\bigr) \biggr)\Lambda(x) $$(3.6)as \(n\to\infty\), where \(\tilde{k}_{v}(x)\) and \(\tilde{\omega}_{v}(x)\) are respectively given by (3.3) and (3.4);
-
(ii)
for \(v=1\), with norming constants \(a_{n}=2^{-1/2}\) and \(b_{n}=2^{-1/2}\log(n/2)\), we have
$$ F^{n-1}_{1}(a_{n}x+b_{n}) = \biggl(1+\frac{1}{n}{k}_{1}(x)+\frac{1}{n^{2}} \biggl({ \omega}_{1}(x)+\frac {{k}^{2}_{1}(x)}{2} \biggr) \bigl(1+o(1)\bigr) \biggr) \Lambda(x) $$(3.7)as \(n\to\infty\), where \({k}_{1}(x)\) and \({\omega}_{1}(x)\) are given by (2.9).
Proof
(i) It follows from (3.2) and (3.3) that
Noting that
by taking logarithms, we have
Thus
since \(b_{n}\sim2^{1/v}\lambda(\log n)^{1/v}\), which implies
The desired result (3.6) follows by (3.8) and (3.9). The proof of (ii) is similar and details are omitted here. □
Lemma 3.3
Let \(f_{v}(x)\) denote the pdf of \(\operatorname {GED}(v)\) with \(v \neq1\), then
for large x, and
as \(n\to\infty\).
Proof
The desired results follow directly by (3.1). □
Lemma 3.4
Let
and the norming constants \(a_{n}\) and \(b_{n}\) be given by (1.2), then
as \(n\to\infty\), where \(k_{v1}(x)\) is given by (2.2) and \(\omega^{\circ}_{v}(x)\) is given by
Proof
Let
it is easy to check that \(\lim_{n\to\infty}B_{n}(x)=1\) and
Hence,
and
By (3.1), we have
Combining (3.13)-(3.15) together, we have
Hence,
The proof is complete. □
Lemma 3.5
Let \(C_{n}(x)\) be given by (3.11) and \(D_{n}(x)\) be denoted by
For \(v\neq1\) and \(-d\log b_{n}< x< cb_{n}^{\frac{3}{v}}\) with \(0< c, d<1\), we have
for large n, where \(k_{v1}(x)\) and \(k_{v2}(x)\) are given by (2.2).
Proof
The desired results follow from Lemmas 3.2 and 3.3. □
The following Mills’ inequalities are from the \(\operatorname{GED}(v)\) in Jia et al. [9], which will be used later.
Lemma 3.6
Let \(F_{v}(x)\) and \(f_{v}(x)\) denote the cdf and pdf of \(\operatorname{GED}(v)\), respectively. Then
-
(i)
for \(v>1\) and all \(x>0\), we have
$$ \frac{2\lambda^{v}}{v}x^{1-v} \biggl( 1+ \frac{2(v-1)\lambda ^{v}}{v}x^{-v} \biggr)^{-1} < \frac{1-F_{v}(x)}{f_{v}(x)} < \frac{2\lambda^{v}}{v}x^{1-v}; $$(3.16) -
(ii)
for \(0< v<1\) and all \(x>\lambda[2(1/v-1)]^{1/v}\), we have
$$ \frac{2\lambda^{v}}{v}x^{1-v} < \frac{1-F_{v}(x)}{f_{v}(x)} < \frac{2\lambda^{v}}{v}x^{1-v} \biggl( 1+\frac{2(v-1)\lambda ^{v}}{v}x^{-v} \biggr)^{-1}. $$(3.17)
Lemma 3.7
Let the norming constant \(b_{n}\) be given by (1.2), for any constant \(0< c<1\) and arbitrary nonnegative integers i, j and k, we have
if \(v\neq1\).
Proof
By arguments similar to Lemma 3.3 in Jia et al. [9], we can get (3.18). The rest is to prove (3.19). By (3.16) and (3.17), and Lemma 3.5, we have
as \(n\to\infty\). The proof is complete. □
Lemma 3.8
Assume that the shape parameter \(v\neq1\), then for any constant \(0< d<1\) and arbitrary nonnegative integers i, j and k, we have
Proof
By arguments similar to that of Lemma 3.7, we have
as \(n\to\infty\) since \(\int_{1}^{\infty} x^{j}e^{kx}\exp(-\frac{e^{x}}{2})\,dx<\infty\).
For assertion (3.21), we only consider the case of \(v>1\) since the proof of the case of \(0< v<1\) is similar. Rewrite
First note that \(\int_{\mathbb{R}}|x|^{j}f_{v}(x)\,dx<\infty\) and the symmetry of \(f_{v}\) implies \(F_{v}(-x)+F_{v}(x)=1\). By using (1.2) and (3.16) we have
as \(n\to\infty\), where \(c=(1-v-1/v)\log2+(v-1)\log\lambda-\log \Gamma(1/v)\).
To show \(\mathit{II}_{n}\to0\) and \(\mathit{III}_{n}\to0\), we consider the case of \(v>2\) first. By using the following inequalities
we can get
and
for large n, which implies that
as \(n\to\infty\).
Similarly, for \(-\frac{v\lambda^{-v}b_{n}^{\frac {v-1}{2}}}{2}< x<-d\log b_{n}\), we have
and
for large n by using (3.23). Then
as \(n\to\infty\).
Combining with (3.22)-(3.25), the assertion (3.21) is derived for \(v>2\). Similar proofs for the case of \(1< v\leq2\) and details are omitted here. The proof is complete. □
Lemma 3.9
Let \(\alpha=\min(1,v)\) as \(v\neq1\). For large n and \(-d\log b_{n}< x< cb_{n}^{\frac{v}{3}}\), both \(x^{r}b_{n}^{v}\Delta_{n}(g_{n}, \Lambda^{\prime};x)\) and \(x^{r}b_{n}^{v}[b_{n}^{v}\Delta_{n}(g_{n},\Lambda^{\prime};x)-k_{v}(x)\Lambda'(x)]\) are bounded by integrable functions independent of n, with \(r>0\), \(0< c<1\) and \(0< d<\alpha\), where \(a_{n}\) and \(b_{n}\) are given by (1.2), and \(k_{v}(x)\) is given by (2.2).
Proof
We only consider the case of \(v>1\). For the case of \(0< v<1\), the proofs are similar and details are omitted here. Rewrite
where \(C_{n}(x)\) is given by (3.11), \(H_{v}(b_{n};x)\) and \(D_{n}(x)\) are respectively defined in Lemma 3.4 and Lemma 3.5. Note that \(\int_{-\infty}^{\infty}x^{k}e^{-tx}\exp(-e^{-x})\,dx=(-1)^{k}\Gamma^{(k)}(t)\) is finite for \(t>0\) and nonnegative integers k. Lemma 3.5 shows that \(b_{n}^{v}(C_{n}(x)D_{n}(x)-1)e^{-x}\Lambda(x)\) is bounded by integrable function independent of n. The rest is to prove that \(b_{n}^{v} {H_{v}(b_{n};x)}\) is bounded by \(m(x)\), where \(m(x)\) is a polynomial on x. Rewrite
where
For \(-d\log b_{n}< x< cb_{n}^{\frac{v}{3}}\), from Lemma 3.4 it follows that
and
Hence, the desired result (3.26) follows by combining (3.16), (3.27) and (3.28) together.
Rewrite
By Lemma 3.5, we only need to estimate the bound of \(b^{v}_{n} [b^{v}_{n} C_{n}(x)D_{n}(x) (H_{v}(b_{n};x)-b^{-v}_{n}k_{v1}(x) ) ]\). Rewrite
where
For \(0< x< cb_{n}^{\frac{v}{3}}\), by using \(1-vy<(1+y)^{-v}<1\) for \(v>0\) and \(y>0\), we have
due to Lemma 3.4. If \(-d\log b_{n} < x<0\), by using \(1+vy<(1+y)^{v}<1\) for \(v>1\) and \(-1< y<0\), Lemma 3.4 shows that
for large n. Similarly,
as \(-d\log b_{n}< x< cb_{n}^{\frac{v}{3}}\). Hence, we derive the desired result by combining (3.16), (3.29) and (3.30) together.
The proof is complete. □
For \(v=1\), note that the \(\operatorname{GED}(1)\) is the Laplace distribution with pdf given by
and its distributional tail can be written as
with \(f(t)=2^{-\frac{1}{2}}\). For the Laplace distribution, similar to the case of \(v>1\), we have the following two results.
Lemma 3.10
For \(0< d<1\) and an arbitrary nonnegative real number j, we have
Lemma 3.11
For \(x>-db_{n}^{\frac{1}{2}}\), both \(x^{r}n ((F_{1}^{n}(a_{n}x+b_{n}))'-\Lambda'(x) )\) and \(x^{r}n[n ((F_{1}^{n}(a_{n}x+b_{n}))'-\Lambda'(x) )+\frac {1}{2}e^{-2x}\Lambda'(x)]\) are bounded by integrable functions independent of n, where \(r>0\) and \(0< d<1\).
4 Proofs of the main results
Proof of Theorem 2.1
From Lemma 3.3 and Lemma 3.5 it follows that
where \(k_{v2}(x)\) is given by (2.2). Note that (3.12) shows
Hence,
implying that
The proof is complete. □
Proof of Theorem 2.3
For \(v\neq1\), by Lemmas 3.5-3.9 and the dominated convergence theorem, we have
and
as \(n\to\infty\).
For the case of \(v=1\), note that
Combining with Lemmas 3.10 and 3.11 and the dominated convergence theorem, we can derive the desired results.
The proof is complete. □
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Acknowledgements
We would like to appreciate the reviewers for reading the paper and making helpful comments that improved the original paper. This work was supported by the National Natural Science Foundation of China (11171275), the Natural Science Foundation Project of CQ (cstc2012jjA00029), the Fundamental Research Funds for the Central Universities (XDJK2013C021) and the Doctoral Grant of Southwest University (SWU113012).
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CL drafted the manuscript and TL revised the whole paper critically. Both authors were involved in the proof of the main results of the paper. All authors read and approved the final manuscript.
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Li, C., Li, T. Density expansions of extremes from general error distribution with applications. J Inequal Appl 2015, 356 (2015). https://doi.org/10.1186/s13660-015-0881-3
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DOI: https://doi.org/10.1186/s13660-015-0881-3