On optimal choice of order statistics in large samples for the construction of confidence regions for the location and scale

Given a large sample from a location-scale 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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\rightarrow \infty ,$$\end{document} is considered.

Let α ∈ (0, 1) be a given confidence level. A strong confidence region of level α is a mapping B : R n → B 2 such that P θ (θ ∈ B(x)) = α ∀θ, where B 2 is the σ -algebra of Borel subsets of R 2 . The quality of a confidence region can be characterized by the risk function defined as where λ 2 is the Lebesgue measure on B 2 . Among strong confidence regions we distinguish those having the minimal risk and call them optimal.
The method for construction of an optimal confidence region is well-known (see, for example, Alama-Bućko et al. 2006 or Czarnowska andNagaev 2001) and is based on using a pivot. Let t 1 (x) and t 2 (x) be a couple of statistics satisfying the following conditions: for any a ∈ R, b > 0, where 1 n = (1, 1, . . . , 1) ∈ R n . Let from now on y = (y 1 , y 2 , . . . , y n ) be a sample from the standard distribution P (0,1) . Taking a set A ∈ B 2 such that one can obtain due to (1) That is, is a pivot. Thus, the set is a strong confidence region for (θ 1 , θ 2 ). In this case, that is the risk function is proportional to the area of the set A, and the problem is to choose the set A with the smallest area. Assume that the density function g of the random vector (−t 1 (y)/t 2 (y), 1/t 2 (y)−1) exists, continuous and such that λ 2 ({u ∈ R 2 : g(u) = z}) = 0 ∀z ≥ 0.
The confidence region is optimal among all the confidence regions of the form (5), if where z α is defined by the equation This is a corollary of Proposition 2.1 of Einmahl and Mason (1992).
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  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 → ∞, under the assumption that k/n → p, m/n → 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.

Asymptotics of the optimal confidence region
Let F = F (0,1) be the continuous distribution function corresponding to P (0,1) and be the so-called quantile function. We assume that the distribution F is absolutely continuous and denote by f its density function. Let ϕ V be the density corresponding to the normal distribution with zero mean vector and covariance matrix V.
We start with the classical result on limit distribution for central order statistics (see, for example, Theorem 10.3 of David andNagaraja 2003 or Theorem 4.1.3 of Reiss 1989).
Corollary 1 Under the conditions of Proposition 1, the limit distribution of the vector n 1/2 (−t 1 (y)/t 2 (y), 1/t 2 (y) − 1), where t 1 and t 2 are defined by (7), as n → ∞, is normal with zero mean vector and the covariance matrix Applying the method of construction of optimal confidence regions described in Sect. 1 (see formulae (2)-(5)), one can obtain the following optimal confidence region based on the vector n 1/2 (−t 1 (y)/t 2 (y), 1/t 2 (y) − 1): where the set A n is defined by and g n is the density corresponding to n 1/2 (−t 1 (y)/t 2 (y), 1/t 2 (y) − 1). The corresponding risk function has the form Let us investigate the behaviour of R(θ, B A n ) as n → ∞. From Proposition 1 it follows that Moreover, basing on Proposition 1, as it was shown in Theorem 2 of , one can obtain the asymptotic expansion of the set A n as n → ∞. Namely, the set A n , as n → ∞, approximates the ellipse A 0 of the form where z α is defined by the equation In other words, Therefore, Summing up, R(θ, B A n ) is of order 1/n as n → ∞, if 0 < p < q < 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 θ with the smallest risk function.

Optimal choice of order statistics
After changing the notation u = F −1 ( p), v = F −1 (q), the right-hand side of (11) is rewritten as are the lower and the upper end of the support of the distribution F, respectively.
Note that for any fixed u ∈ By simple calculations, and In the sequel we need some well-known facts from the extreme value theory (see, for example, Subsection 10.5 of David and Nagaraja 2003).
If there exist c n > 0 and d n ∈ R such that the limit distribution of the sequence c n (y n:n − d n ) exists, as n → ∞, then the limit distribution function is one of just three types (β > 0): In this case it is said that F belongs to the domain of attraction of the distribution Let h be the hazard rate function, that is It turns out that the possible limit laws for the properly centered and normed maximal order statistics x n:n are determined by the behaviour of the function h in a neighborhood of the right endpoint of F. The following result (see, for example, Theorems 8.3.3 and 8.3.4 of Arnold et al. 1992) contains the well-known sufficient von Mises conditions of attraction to D(H i ), i = 1, 2, 3, and description of sequences {c n , d n }. In what follows, L(v) denotes a slowly varying function as v → ∞.

Proposition 2
The following statements hold: Remark 1 Comparing the norming sequences {c n } from all the above cases to n 1/2 , one can conclude that: In what follows, we exclude the case F ∈ D(H 2 ) with β = 2 from the consideration since uncertainty remains here. Now we are able to establish the crucial result for optimal choice of order statistics. (14).

Theorem 1 The following statements hold for any fixed u
To prove the statement (ii), it is enough to show that From (12) it follows that In view of condition (16), it is enough to prove that For this purpose one can use the arguments from the proof of Remark 2 of Subsection 3.3.3 of Embrechts et al. (1997).
Theorem 1 immediately implies the following result. (15) holds with β < 2, then v * = u + F < ∞ (q * = 1) and inf G(u, v) = 0. In other cases (condition (14), or condition (16), or condition (15) with β > 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 ∈ R such that the limit distribution of the sequence a n (y 1:n − b n ) exists, as n → ∞, then the limit distribution function is one of just three types: H * i (u; γ ) = 1 − H i (−u; γ ), 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.

Corollary 2 If condition
Denote

Theorem 2 The following statements hold for any fixed
then F ∈ D(H * 2 ) and If condition (21) with γ < 2 holds, then u * = u − F > −∞ ( p * = 0) and inf G(u, v) = 0. In other cases (condition (19), or condition (20), or condition (21) with γ > 2 holds), u * > u − F ( p * > 0) and inf G(u, v) > 0. 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.
The risk function is of order 1/n, as n → ∞.
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 · n 1/2 ).
It remains to consider the cases when in a neighborhood of u + F (u + F < ∞) (15) with β < 2 holds and/or in a neighborhood of u − F (u − F > −∞) (21) with γ < 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.
Let q * = 1 (the case p * = 0 can be considered similarly). In general, one can distinguish between three types of sequences {m n } satisfying m n /n → q * = 1, n → ∞: (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 = n − j + 1, j > 1 is fixed; in this case y m n :n = y n− j+1:n , i. e. we deal with other extreme order statistics; (c) m n = n − j + 1, j = j n → ∞, j n /n → 0, n → ∞; in this case y m n :n = y n− j n +1:n , i. e. we deal with intermediate order statistics. The question arises: what type of the sequence one should choose to obtain the better confidence region?
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 ∈ D(H i ), i = 1, 2, 3, iff the limit distribution function of an extreme order statistic y n− j+1:n , as n → ∞, where j is fixed, is of the form [− 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 n− j+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 n− j+1:n , n → ∞, j → ∞, j/n → 0, is standard normal and d n = F −1 (1 − j/n), c n = n f (d n )/j 1/2 . Note that in the case of interest (when in a neighborhood of u + F (15) with β < 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 .
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 (−u) = − f (u), then p * = 1 − q * (see Theorem 10.4 of David andNagaraja 2003 andalso Ogawa 1998 for the proof and discussion).

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.
It is the case III since in a neighborhood of u − F = 0 (21) with γ = 1 holds, while in a neighborhood of u + F = ∞ (16) holds. By simple calculations we obtain The optimal confidence region has the risk of order 1/n 3/2 . Calculations of the risk function for ( p, q) = (0.25, 0.75) (the first table) and for ( p * , q * ) = (0, 0.7968) (the second table): where v * = 1.1106 is the solution of the equation .
The optimal confidence region has the risk of order 1/n. Therefore, the density function of the random vector T n has the form Since a n → ∞, b n → b, d n → d, as n → ∞, statement (i) follows.
Lemma 2 For β > 0 consider two densities and let (a, b) and (a , b ) be such intervals that for given α If β ≤ 1, then for any j > 1 Proof Let F β and G β, j be the distribution functions corresponding to the densities f β and g β, j , respectively, i. e.