On optimal choice of order statistics in large samples for the construction of confidence regions for the location and scale
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 First Online:
 Received:
DOI: 10.1007/s0018401204059
 Cite this article as:
 Zaigraev, A. & AlamaBućko, M. Metrika (2013) 76: 577. doi:10.1007/s0018401204059
Abstract
Given a large sample from a locationscale population we estimate the unknown parameters by means of confidence regions constructed on the basis of two order statistics. The problem of the best choice of those statistics to obtain good estimates, as \(n\rightarrow \infty ,\) is considered.
Keywords
Order statistics Optimal confidence regions Pivot Locationscale population1 Introduction
A problem of optimal choice of order statistics in large samples for the best estimation of the location and scale is not new. For example, Subsection 10.4 of David and Nagaraja (2003) is devoted to such a problem in case of the point estimation (see also the references cited therein). However, the same problem for the confidence region estimation has not attracted the attention so far, as far as we know. This paper is an attempt to fill the gap.
Let \(x=(x_1, x_2, \ldots , x_n)\) be a sample from a distribution \(P_{\theta },\ \theta =(\theta _1, \theta _2),\) that is \(\{x_i\}\) are independent realvalued random variables having the distribution \(P_{\theta }.\) We deal with the case where \(\theta _1\in R\) is a location parameter and \(\theta _2>0\) is a scale parameter. As the estimators of \(\theta =(\theta _1, \theta _2),\) let us consider twodimensional confidence regions.
Of course, the optimal confidence region depends on the choice of \(t_1\) and \(t_2.\)
For the natural interpretation of confidence region (5) it is reasonable to take as \(t_1(x)\) and \(t_2(x)\) the estimators of the location and scale parameters, respectively. Then \((t_1(x), t_2(x))\) is the center of the region, while the set \(A\) defines the shape of the region and \(t_2(x)\) is responsible for its rescaling.
In this paper we consider the case, where \(t_1\) and \(t_2\) are linear functions of two order statistics. Some other cases were considered in AlamaBućko et al. (2006) and Czarnowska and Nagaev (2001).
Let \(x_{k:n}\) and \(x_{m:n}\) be the \(k\)th and the \(m\)th order statistic of the sample \(x,\) respectively, \(k<m.\) The main goal of the paper is to make the best possible choice of \(k=k_n\) and \(m=m_n\) to minimize risk function (6), as \(n\rightarrow \infty ,\) under the assumption that \(k/n\rightarrow p,\ m/n\rightarrow q,\ p<q.\)
Asymptotics of the optimal confidence region in case \(0<p<q<1\) is obtained in Sect. 2. Our main results are established in Sect. 3, while Sect. 4 contains examples. In “Appendix” we prove three useful auxiliary lemmas.
2 Asymptotics of the optimal confidence region
We start with the classical result on limit distribution for central order statistics (see, for example, Theorem 10.3 of David and Nagaraja 2003 or Theorem 4.1.3 of Reiss 1989).
Proposition 1
Making use of statement (i) of Lemma 1 from “Appendix” with \(a_n=c_n=n^{1/2},\ b_n=F^{1}(p),\ d_n=F^{1}(q),\ \xi _n=y_{k:n},\ \eta _n=y_{m:n},\ f(u_1,u_2)=\varphi _V(u_1,u_2),\) we immediately obtain the following result.
Corollary 1
The problem of interest is to search for \(p^*\) and \(q^*\) to minimize (11). In other words, this is the problem of choice the order statistics \(x_{k:n}, x_{m:n}\) to obtain the optimal confidence region for \(\theta \) with the smallest risk function.
3 Optimal choice of order statistics

(Fréchet) \(\ H_1(u; \beta )=\left\{ \begin{array}{ll} 0,&u\le 0\\ \exp (u^{\beta }),&u>0, \end{array}\right.\)

(Weibull) \(\ H_2(u; \beta )=\left\{ \begin{array}{ll} \exp ((u)^{\beta }),&u\le 0\\ 1,&u>0,\end{array}\right.\)

(Gumbel) \(\ H_3(u)=\exp (\exp (u)),\ u\in R.\)
Proposition 2

\(F\in D(H_1),\) if \(u_F^+=\infty \) and for some \(\beta >0,\)here \(d_n=0,\ c_n=(F^{1}(11/n))^{1}=n^{1/\beta }L(n);\)$$\begin{aligned} \lim _{u\rightarrow \infty }\ uh(u)=\beta ; \end{aligned}$$(14)

\(F\in D(H_2),\) if \(u_F^+<\infty \) and for some \(\beta >0,\)here \(d_n=u^+_F,\ c_n=(u^+_FF^{1}(11/n))^{1}=n^{1/\beta }L(n);\)$$\begin{aligned} \lim _{u\rightarrow u_F^+}\ (u_F^+u)h(u)=\beta ; \end{aligned}$$(15)

\(F\in D(H_3),\) if \(f(u)\) is differentiable for all \(u>u_0\) andhere \(d_n=F^{1}(11/n),\ c_n=h(d_n)=nf(d_n).\)$$\begin{aligned} \lim _{u\rightarrow u_F^+}\ (1/h(u))^{\prime }=0; \end{aligned}$$(16)
Remark 1.
Comparing the norming sequences \(\{c_n\}\) from all the above cases to \(n^{1/2},\) one can conclude that: \(c_n\gg n^{1/2}\) if \(F\in D(H_2)\) with \(\beta <2,\) while \(c_n\ll n^{1/2}\) if \(F\in D(H_1),\) or \(F\in D(H_3),\) or \(F\in D(H_2)\) with \(\beta >2.\) In what follows, we exclude the case \(F\in D(H_2)\) with \(\beta =2\) from the consideration since uncertainty remains here.
Now we are able to establish the crucial result for optimal choice of order statistics.
Theorem 1.
 (i)
if condition (14) holds, then \(G(u,v)\uparrow \infty \) as \(v\uparrow u^+_F=\infty ;\)
 (ii)
if condition (16) holds, then \(G(u,v)\) is a nondecreasing function for all \(v>v_0;\)
 (iii)if condition (15) holds, then$$\begin{aligned}&\beta >2\ \Longrightarrow \ G(u,v)\uparrow \infty ,\ v\uparrow u^+_F,\\&\beta <2\ \Longrightarrow \ G(u,v)\downarrow 0,\ v\uparrow u^+_F. \end{aligned}$$
Proof
Statement (i) is a direct consequence of (14).
Statement (iii) follows from Theorem 3.3.12 of Embrechts et al. (1997) and properties of slowly varying functions.
Theorem 1 immediately implies the following result.
Corollary 2
If condition (15) holds with \(\beta <2,\) then \(v^*=u^+_F<\infty \ (q^*=1)\) and \(\inf G(u,v)=0.\) In other cases (condition (14), or condition (16), or condition (15) with \(\beta >2\) holds), \(v^*<u^+_F\ (q^*<1)\) and \(\ \inf G(u,v)>0.\)
As it is known, similar results hold also for the minimal order statistic \(y_{1:n}.\) More precisely, if there exist \(a_n>0\) and \(b_n\in R\) such that the limit distribution of the sequence \(a_n(y_{1:n}b_n)\) exists, as \(n\rightarrow \infty ,\) then the limit distribution function is one of just three types: \(H^*_i(u;\gamma )=1H_i(u;\gamma ),\ i=1, 2, 3.\) So, with the small evident modifications one can establish for \(y_{1:n}\) the similar results as for \(y_{n:n}.\) We have gathered them in the following theorem.
Theorem 2
 1.If \(u_F^=\infty \) and for some \(\gamma >0,\)then \(F\in D(H^*_1)\) and \(G(u,v)\uparrow \infty \) as \(u\downarrow u^_F=\infty .\)$$\begin{aligned} \lim _{u\rightarrow \infty }\ uh^*(u)=\gamma , \end{aligned}$$(19)
 2.If \(f(u)\) is differentiable for all \(u<u^*_1\) andthen \(F\in D(H^*_3)\) and \(G(u,v)\) is a nonincreasing function for \(u<u^*_0.\)$$\begin{aligned} \lim _{u\rightarrow u_F^}\ (1/h^*(u))^{\prime }=0, \end{aligned}$$(20)
 3.If \(u_F^>\infty \) and for some \(\gamma >0,\)then \(F\in D(H^*_2)\) and$$\begin{aligned} \lim _{u\rightarrow u_F^}\ (uu_F^)h^*(u)=\gamma , \end{aligned}$$(21)$$\begin{aligned}&\gamma >2\ \Longrightarrow \ G(u,v)\uparrow \infty ,\ u\downarrow u^_F,\\&\gamma <2\ \Longrightarrow \ G(u,v)\downarrow 0,\ u\downarrow u^_F. \end{aligned}$$
Summing up, assuming that the underlying distribution in a neighborhood of \(u_F^+\) satisfies one of von Mises conditions (14)–(16) and in a neighborhood of \(u_F^\) satisfies one of von Mises conditions (19)–(21), we can formulate the results on optimal choice of order statistics distinguishing between four cases.
Case I
The important note: the order of the risk function for the optimal confidence region equals to the reciprocal of the product of norming sequences of the components of \(T_n(y),\) that is \(1/n=1/(n^{1/2}\cdot n^{1/2}).\)
It remains to consider the cases when in a neighborhood of \(u^+_F\ (u^+_F<\infty )\) (15) with \(\beta <2\) holds and/or in a neighborhood of \(u^_F\ (u^_F>\infty )\) (21) with \(\gamma <2\) holds. Here, the order of the corresponding risk function is evidently \(o(1/n)\) and \(q^*=1\) and/or \(p^*=0.\) In this case we need to change the vector \(T_n(y)\) and norming sequences according to statements (ii) or (iii) of Lemma 1 from “Appendix”. Again the reciprocal of the product of norming sequences of the components of \(T_n(y)\) determines the order of the risk function.
 (a)
\(m_n=n;\) in this case \(y_{m_n:n}=y_{n:n},\) i. e. we deal with the extreme order statistics;
 (b)
\(m_n=nj+1,\ j>1\) is fixed; in this case \(y_{m_n:n}=y_{nj+1:n},\) i. e. we deal with other extreme order statistics;
 (c)
\(m_n=nj+1,\ j=j_n\rightarrow \infty ,\ j_n/n\rightarrow 0,\ n\rightarrow \infty ;\) in this case \(y_{m_n:n}=y_{nj_n+1:n},\) i. e. we deal with intermediate order statistics.
First of all, note that according to the end of Subsection 10.8 of David and Nagaraja (2003), lower extremes are asymptotically independent of upper extremes and both are asymptotically independent of central order statistics as well as of intermediate order statistics.
In situation (a) the possible limit laws and corresponding norming sequences are given in Proposition 2. In situation (b), as it follows from Theorem 8.4.1 of Arnold et al. (1992), \(F\in D(H_i),\ i=1, 2, 3,\) iff the limit distribution function of an extreme order statistic \(y_{nj+1:n},\) as \(n\rightarrow \infty ,\) where \(j\) is fixed, is of the form \(\sum ^{j1}_{r=0}H_i(u)[\ln (H_i(u))]^r/r!,\) \(i=1, 2, 3;\) the sequences \(\{c_n, d_n\}\) are the same as in Proposition 2. Therefore, comparing the choice of \(y_{n:n}\) with that of \(y_{nj+1:n},\) where \(j>1\) is fixed, we conclude that the norming sequences are the same, but the first choice is better since it gives the shorter interval for the appropriate coordinate (see Lemma 2 from “Appendix”).
At last, in situation (c), as it follows from Theorem 8.5.3 of Arnold et al. (1992), if von Mises conditions (14)–(16) hold, then the limit law for \(y_{nj+1:n},\ n\rightarrow \infty ,\ j\rightarrow \infty ,\) \(j/n\rightarrow 0,\) is standard normal and \(d_n=F^{1}(1j/n),\ c_n=nf(d_n)/j^{1/2}.\) Note that in the case of interest (when in a neighborhood of \(u^+_F\) (15) with \(\beta <2\) holds) this norming sequence \(\{c_n\}\) is less than that for \(y_{n:n}\) given in Proposition 2 (see Lemma 3 from “Appendix”); in all other cases it is less than \(n^{1/2}.\)
Case II
Case III
Case IV
At last, if in a neighborhood of \(u^+_F\) (15) with \(\beta <2\) holds and in a neighborhood of \(u^_F\) (21) with \(\gamma <2\) holds, then \(p^*=0,\ q^*=1.\) In this case the construction repeats one of the previous cases depending on the relation between \(\beta \) and \(\gamma \) (see Examples).
At last, it is worth to note that if the distribution \(F\) is symmetric, that is its density \(f\) satisfies the condition \(f(u)=f(u),\) and, moreover, if \(f\) is a differentiable infinitely many times function such that \(f^{\prime }(u)=f^{\prime }(u),\) then \(p^*=1q^*\) (see Theorem 10.4 of David and Nagaraja 2003 and also Ogawa 1998 for the proof and discussion).
4 Examples
Here we consider three examples of distributions. In all the cases we calculate the values of the risk function, according to (6): firstly, for \((p,q)=(0.25,0.75),\) and secondly, for \((p^*,q^*).\) Even for the realistic sample sizes, the risk is smaller in the second case.
Example 1.
Uniform distribution \(U(\theta _1\theta _2/2,\theta _1+\theta _2/2).\)
The optimal confidence region has the risk of order \(1/n^2.\)
\(n\) 
\(k\) 
\(m\) 
\(\lambda _2(A)\) 
\( \mathrm{E}_{(0,1)}t_2^2(y)\) 
\(R(\theta ,B_A)/\theta _2^2\) 

30 
8 
23 
1.076282 
0.241935 
0.260390 
40 
10 
31 
0.640971 
0.243902 
0.156334 
50 
13 
38 
0.599687 
0.245098 
0.146982 
60 
15 
46 
0.429006 
0.245901 
0.105493 
70 
18 
53 
0.414028 
0.246478 
0.102049 
80 
20 
61 
0.326711 
0.246913 
0.080669 
90 
23 
68 
0.312016 
0.247252 
0.077147 
100 
25 
76 
0.262212 
0.247524 
0.064904 
500 
125 
376 
0.052771 
0.249500 
0.013166 
\(n\) 
\(k\) 
\(m\) 
\(\lambda _2(A)\) 
\( \mathrm{E}_{(0,1)}t_2^2(y)\) 
\(R(\theta ,B_A)/\theta _2^2\) 

30 
1 
30 
0.015230 
0.877016 
0.013357 
40 
1 
40 
0.008145 
0.905923 
0.007379 
50 
1 
50 
0.005059 
0.923831 
0.004674 
60 
1 
60 
0.003444 
0.936012 
0.003224 
70 
1 
70 
0.002495 
0.944835 
0.002357 
80 
1 
80 
0.001890 
0.951520 
0.001798 
90 
1 
90 
0.001481 
0.956760 
0.001417 
100 
1 
100 
0.001192 
0.960978 
0.001145 
500 
1 
500 
0.000050 
0.992039 
0.000049 
Example 2.
Exponential distribution \(E(\theta _1,\theta _2).\)
It is the case III since in a neighborhood of \(u^_F=0\) (21) with \(\gamma =1\) holds, while in a neighborhood of \(u^+_F=\infty \) (16) holds.
The optimal confidence region has the risk of order \(1/n^{3/2}.\)
\(n\) 
\(k\) 
\(m\) 
\(\lambda _2(A)\) 
\(\mathrm{E}_{(0,1)} t_2^2(y)\) 
\(R(\theta ,B_A)/\theta _2^2\) 

30 
8 
23 
0.636060 
1.294206 
0.823192 
40 
10 
31 
0.361046 
1.431981 
0.517011 
50 
13 
38 
0.332177 
1.259720 
0.418450 
60 
15 
46 
0.237054 
1.354291 
0.321040 
70 
18 
53 
0.224124 
1.244763 
0.278981 
80 
20 
61 
0.177264 
1.316461 
0.233361 
90 
23 
68 
0.171444 
1.236410 
0.211975 
100 
25 
76 
0.141810 
1.294084 
0.183514 
500 
125 
376 
0.027746 
1.224075 
0.033963 
\(n\) 
\(j\) 
\(k\) 
\(\lambda _2(A)\) 
\(\mathrm{E}_{(0,1)} t_2^2(y)\) 
\(R(\theta ,B_A)/\theta _2^2\) 

30 
1 
24 
0.059575 
2.783466 
0.165825 
40 
1 
32 
0.036114 
2.973219 
0.107374 
50 
1 
40 
0.025097 
3.149164 
0.079034 
60 
1 
48 
0.018263 
3.320204 
0.060636 
70 
1 
56 
0.014387 
3.490169 
0.050213 
80 
1 
64 
0.011406 
3.660958 
0.041756 
90 
1 
72 
0.009464 
3.833603 
0.036281 
100 
1 
80 
0.008152 
4.008691 
0.032678 
500 
1 
399 
0.000517 
12.822575 
0.006629 
Example 3.
Normal distribution \(N(\theta _1,\theta _2).\)
It is the case I since in a neighborhood of \(u^_F=\infty \) (20) holds, while in a neighborhood of \(u^+_F=\infty \) (16) holds.
The optimal confidence region has the risk of order \(1/n.\)
\(n\) 
\(k\) 
\(m\) 
\(\lambda _2(A)\) 
\( \mathrm{E}_{(0,1)}t_2^2(y)\) 
\(R(\theta ,B_A)/\theta _2^2\) 

30 
8 
23 
0.520142 
1.869376 
0.972341 
40 
10 
31 
0.338648 
2.079427 
0.704193 
50 
13 
38 
0.294872 
1.850483 
0.545655 
60 
15 
46 
0.219608 
1.991273 
0.437299 
70 
18 
53 
0.204000 
1.841996 
0.375767 
80 
20 
61 
0.166246 
1.947792 
0.323812 
90 
23 
68 
0.156422 
1.837179 
0.287375 
100 
25 
76 
0.133028 
1.921895 
0.255665 
500 
125 
376 
0.026954 
1.840690 
0.049613 
\(n\) 
\(k\) 
\(m\) 
\(\lambda _2(A)\) 
\( \mathrm{E}_{(0,1)}t_2^2(y)\) 
\(R(\theta ,B_A)/\theta _2^2\) 

30 
5 
26 
0.182234 
4.335487 
0.790073 
40 
6 
35 
0.114722 
4.794549 
0.550040 
50 
7 
44 
0.086356 
5.096922 
0.440149 
60 
9 
52 
0.079592 
4.625095 
0.368120 
70 
10 
61 
0.063934 
4.855632 
0.310440 
80 
11 
70 
0.053892 
5.037201 
0.271464 
90 
13 
78 
0.050402 
4.726140 
0.238206 
100 
14 
87 
0.043930 
4.879758 
0.214367 
500 
67 
434 
0.008606 
4.952811 
0.042623 
Acknowledgments
The authors are grateful to the referee for useful suggestions improving the paper.
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