Abstract
The problem of optimal estimation of location and scale parameters of distributions, by means of two-dimensional confidence regions based on L-statistics, is considered. The case, when the sample size tends to infinity, is analyzed.
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References
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The authors are grateful to the referees for useful suggestions improving the paper.
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Appendix
Appendix
Proof of Lemma 1
Denote
Let \(j=j^*\) be arbitrarily but fixed. Evidently, \(B_{j^*+1}\rightarrow 0,\ A_{j^*+1}\rightarrow 0\) if \(u_{j^*}\rightarrow u_{j^*+1}\) for \(1\le j^*<k\) and, therefore, \(Q(i,j^*+1)\rightarrow 0\) for all \(i<j^*+1\) and \(Q(j^*+1,i)\rightarrow 0\) for all \(i>j^*+1.\) Thus, we obtain as \(u_{j^*}\rightarrow u_{j^*+1}\) given \(1\le j^*<k\): \(\square \)
A similar result we get letting \(u_{j^*}\rightarrow u_{j^*-1}\) given \(1<j^*\le k.\)
Now we prove the second part of the statement concerning the derivatives. The derivative \(\varDelta '_{u_{j^*}}\) can be written as
We obtain
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If \(u_{j^*}\rightarrow u_{j^*\!+\!1},\) then
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for \(i<j^*:\ A_iB_{j^*}-B_iA_{j^*}\sim A_i\left( B_{j^*\!+\!1}+B_{j^*}\right) -B_i\left( A_{j^*\!+\!1}+A_{j^*}\right) \ \) and
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\(A_iB_{j^*\!+\!1}-B_iA_{j^*\!+\!1}\sim \left( u_{j^*\!+\!1}-u_{j^*}\right) \left[ A_i\left( f(u_{j^*\!+\!1})+u_{j^*\!+\!1}f'(u_{j^*\!+\!1})\right) -B_if'(u_{j^*\!+\!1})\right] ;\)
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\(A_{j^*}B_{j^*\!+\!1}-B_{j^*}A_{j^*\!+\!1}\sim \left( u_{j^*\!+\!1}-u_{j^*}\right) \left[ \left( A_{j^*\!+\!1}+A_{j^*}\right) \left( f(u_{j^*\!+\!1})+ u_{j^*\!+\!1}f'(u_{j^*\!+\!1})\right) -\left( B_{j^*\!+\!1}+B_{j^*}\right) f'(u_{j^*+\!1})\right] ;\)
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for \(i>j^*\!+\!1:\ A_{j^*}B_i-B_{j^*}A_i\sim B_i\left( A_{j^*\!+\!1}+A_{j^*}\right) -A _i(B_{j^*\!+\!1}+B_{j^*})\ \) and
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\(A_{j^*\!+\!1}B_i-B_{j^*\!+\!1}A_i\sim \left( u_{j^*\!+\!1}-u_{j^*}\right) \left[ B_if'(u_{j^*\!+\!1})-A_i\left( f(u_{j^*\!+\!1})+ u_{j^*\!+\!1}f'(u_{j^*\!+\!1})\right) \right] .\) Therefore, as \(u_{j^*}\rightarrow u_{j^*\!+\!1},\)
Finally,
where
and \(\varDelta '_{u_{j^*}}=\varDelta '_1+\varDelta '_2+\varDelta '_3+\varDelta '_4+\varDelta '_5<0\) as \(u_{j^*}\rightarrow u_{j^*\!+\!1}.\)
Similarly, one can prove that if \(u_{j^*}\rightarrow u_{j^*\!-\!1},\) then \(\varDelta '_{u_{j^*}}=\varDelta '_1+\varDelta '_2+ \varDelta '_3 +\varDelta '_4+\varDelta '_5>0.\)
Proof of Theorem 2
Consider \(\varDelta \) as a function of \(u_k\) and use the representation from the proof of Lemma 1. The derivative \(\varDelta '_{u_k}\) can be written as
\(\square \)
As in the proof of Lemma 1, we obtain
First, assume that (20) holds. Then, as \(u_k\rightarrow u^+=\infty ,\)
where \(h(u)=f(u)/(1-F(u)).\) Since \(f'(u_k)u_k+f(u_k)\sim -\beta f(u_k),\) we get
and \(\varDelta \) is a decreasing function as \(u_k\rightarrow \infty .\)
Now assume that (21) holds. When \(u_k\rightarrow u^+,\) we have (see, e.g. Zaigraev and Alama-Bućko 2013)
Thus,
For \(\beta >2\) we obtain (without loss in generality we do not take into account the slowly varying function L):
Thus, \(\varDelta '_1<<\varDelta '_2+\varDelta '_3,\) that is \(\varDelta '_{u_k}\sim \varDelta '_2+\varDelta '_3<0,\) and \(\varDelta \) is a decreasing function as \(u_k\rightarrow u^+.\)
For \(\beta <2\) we have \(\varDelta \rightarrow \infty \) as \(u_k\rightarrow u^+,\) since \(\varDelta _1>0\) while
At last, assume that (22) holds, that is \(f'(u_k)\sim -f(u_k)h(u_k)\) and \(h(u_k)u_k\rightarrow \infty \) (see Zaigraev and Alama-Bućko 2013) as \(u_k\rightarrow u^+.\) Then,
Since \(f(u_k)=o(h(u_k)),\) we get \(\varDelta '_1=o(f(u_k)h^2(u_k)u^2_k)=o(\varDelta '_2+\varDelta '_3)\) and, therefore, \(\varDelta \) is a decreasing function as \(u_k\rightarrow u^+.\)
Lemma 2
Let \(\mathbf{\xi _n}=(\xi ^{(n)}_1,\ldots ,\xi ^{(n)}_k),\ n\ge 1,\) be random vectors and assume that there exist sequences of positive numbers \(\{c'_n\}\) and \(\{c''_n\}\) and real numbers \(\{h^{(n)}_j\},\ j=1,\ldots ,k,\) such that the sequence of random vectors
as \(n\rightarrow \infty ,\) converges in distribution to a random vector with a continuous density function \(\rho .\) Then
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(i)
under the conditions \(c'_n=c''_n=\sqrt{n}, \rho =\varphi _V,\) where \(\varphi _V\) stands for the density corresponding to \(\mathcal{N}(0, V),\) the sequence of random vectors
$$\begin{aligned} \sqrt{n}\Bigl (-\frac{\mathbf{a}\xi _\mathbf{n}^T}{\mathbf{b}\xi _\mathbf{n}^T}, \frac{1}{\mathbf{b}\xi _\mathbf{n}^T}-1\Bigr ), \end{aligned}$$(26)where \(\mathbf{a},\mathbf{b}\in R^k\) are given vectors such that \(\mathbf{a}{} \mathbf{h}^T_n=0, \mathbf{b}{} \mathbf{h}^T_n=1, \mathbf{h}_n=(h^{(n)}_1,\ldots , h^{(n)}_k),\) converges in distribution, as \(n\rightarrow \infty ,\) to the normal random vector with the density \(\varphi _W,\)
$$\begin{aligned} W=\left[ \begin{array}{cc} \mathbf{a}V\mathbf{a}^T &{} \mathbf{a}V\mathbf{b}^T\\ \mathbf{a}V\mathbf{b}^T &{} \mathbf{b}V\mathbf{b}^T\end{array}\right] ; \end{aligned}$$ -
(ii)
under the conditions \(c'_n>>\sqrt{n},\ c''_n=\sqrt{n},\ h^{(n)}_1=0,\ \rho (u_1,\ldots ,u_k)=\rho _0(u_1)\varphi _V(u_2,\ldots ,u_k),\) the sequence of random vectors
$$\begin{aligned} \left( -c'_n\frac{\xi ^{(n)}_1}{\mathbf{b}\mathbf{\xi _n}^T}, \sqrt{n}\Bigl (\frac{1}{\mathbf{b}\mathbf{\xi _n}^T}-1\Bigr )\right) , \end{aligned}$$
where \(\mathbf{b}\in R^k\) is a given vector such that \(\mathbf{b}\mathbf{h}^T_n=1,\) converges in distribution, as \(n\rightarrow \infty ,\) to the random vector with the density \(\rho _0(-v_1)\varphi _{\overline{\mathbf{b}}V\overline{\mathbf{b}}^T}(v_2),\) \(\overline{\mathbf{b}}=(b_2,\ldots ,b_k);\) (iii) under the conditions \(c'_n=\sqrt{n},\ c''_n>>\sqrt{n},\ h^{(n)}_k=0,\ \rho (u_1,\ldots ,u_k)=\varphi _V(u_1,\ldots ,u_{k-1})\rho _0(u_k),\) the sequence of random vectors
where \(\mathbf{b}\in R^k\) is a given vector such that \(\mathbf{b}\mathbf{h}^T_n=1,\) converges in distribution, as \(n\rightarrow \infty ,\) to the random vector with the density \(\rho _0(-v_1)\varphi _{\overline{\mathbf{b}}V\overline{\mathbf{b}}^T}(v_2),\) \(\overline{\mathbf{b}}=(b_1,\ldots ,b_{k-1}).\)
Proof
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(i)
Obviously, the limit distribution of \(\sqrt{n}(\mathbf{a}\mathbf{\xi _n}^T, \mathbf{b}\mathbf{\xi _n}^T-1)\) is \(\mathcal{N}(0, W).\) The transformation \((u_1,u_2)\mapsto (v_1,v_2)\) of
$$\begin{aligned} \sqrt{n}(\mathbf{a}\mathbf{\xi _n}^T, \mathbf{b}\mathbf{\xi _n}^T-1)\quad \text{ onto } \sqrt{n}\left( -\frac{\mathbf{a}\mathbf{\xi _n}^T}{\mathbf{b}\mathbf{\xi _n}^T}, \frac{1}{\mathbf{b}\mathbf{\xi _n}^T}-1\right) \end{aligned}$$is given by
$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle {v_1=-\frac{u_1}{1+u_2/\sqrt{n}}}\\ \displaystyle {v_2=-\frac{u_2}{1+u_2/\sqrt{n}}} \end{array}\right. \quad \Longrightarrow \quad \left\{ \begin{array}{l} \displaystyle {u_1=-\frac{v_1}{1+v_2/\sqrt{n}}}\\ \displaystyle {u_2=-\frac{v_2}{1+v_2/\sqrt{n}}.} \end{array}\right. \end{aligned}$$This transformation has the Jakobian \(J(v_1,v_2)=(1+v_2/\sqrt{n})^{-3}.\) Therefore, the density of the random vector \(\sqrt{n}(-\mathbf{a}\mathbf{\xi _n}^T/\mathbf{b}\mathbf{\xi _n}^T, 1/\mathbf{b}\mathbf{\xi _n}^T-1)\) is of the form
$$\begin{aligned} \frac{1}{(1+v_2/\sqrt{n})^3}\varphi _W\Bigl (-\frac{v_1}{1+v_2/\sqrt{n}}, -\frac{v_2}{1+v_2/\sqrt{n}}\Bigr ) \end{aligned}$$and, as \(n\rightarrow \infty ,\) statement (i) follows.
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(ii)
Obviously, the limit distribution of \(\sqrt{n}(\sum ^k_{i=2}b_i\xi ^{(n)}_i-1)\) is \(\mathcal{N}(0, \overline{\mathbf{b}}V\overline{\mathbf{b}}^T).\) Due to the asymptotical independency of the components, the random vector \((c'_n\xi ^{(n)}_1,\sqrt{n}(\sum ^k_{i=2}b_i\xi ^{(n)}_i-1)\) has the limit distribution with the density \(\rho _0(u_1)\varphi _{\overline{\mathbf{b}}V\overline{\mathbf{b}}^T}(u_2).\) The transformation \((u_1,u_2)\mapsto (v_1,v_2)\) of
$$\begin{aligned} (c'_n\xi ^{(n)}_1,\sqrt{n}(\sum ^k_{i=2}b_i\xi ^{(n)}_i-1)\quad \text{ onto } \; \Bigl (-c'_n\frac{\xi ^{(n)}_1}{\mathbf{b}\mathbf{\xi _n}^T}, \sqrt{n}\Bigl (\frac{1}{\mathbf{b}\mathbf{\xi _n}^T}-1\Bigr )\Bigr ) \end{aligned}$$is given by
$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle {v_1=-\frac{u_1}{1+b_1u_1/c'_n+u_2/\sqrt{n}}}\\ \\ \displaystyle {v_2=-\frac{b_1\sqrt{n}u_1/c'_n+u_2}{1+b_1u_1/c'_n+u_2/\sqrt{n}}} \end{array}\right. \quad \Longrightarrow \quad \left\{ \begin{array}{l} \displaystyle {u_1=-\frac{v_1}{1+v_2/\sqrt{n}}}\\ \\ \displaystyle {u_2=-\frac{v_2-b_1\sqrt{n}v_1/c'_n}{1+v_2/\sqrt{n}}.} \end{array}\right. \end{aligned}$$This transformation has the Jakobian \(J(v_1,v_2)=(1+v_2/\sqrt{n})^{-3}.\) Thus, the density of the random vector \((-c'_n\xi ^{(n)}_1/\mathbf{b}\mathbf{\xi _n}^T, \sqrt{n}(1/\mathbf{b}\mathbf{\xi _n}^T-1))\) is of the form
$$\begin{aligned} \frac{1}{(1+v_2/\sqrt{n})^3}\rho _0\Bigl (-\frac{v_1}{1+v_2/\sqrt{n}}\Bigr )\varphi _{\overline{\mathbf{b}} V\overline{\mathbf{b}}^T}\Bigl (-\frac{v_2-b_1\sqrt{n}v_1/c'_n}{1+v_2/\sqrt{n}}\Bigr ). \end{aligned}$$As \(n\rightarrow \infty ,\) statement (ii) follows.
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(iii)
This case is treated similarly.
\(\square \)
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Zaigraev, A., Alama-Bućko, M. Optimal choice of order statistics under confidence region estimation in case of large samples. Metrika 81, 283–305 (2018). https://doi.org/10.1007/s00184-018-0643-6
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DOI: https://doi.org/10.1007/s00184-018-0643-6