Abstract
Conditions are considered under which studentization does not change the limiting distribution of the normalized intermediate order statistics. A similar problem is considered by Berman as applied to a limiting distribution of extreme order statistics.
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INTRODUCTION
Let independent and identically distributed random variables \(X_{1},...,X_{n}\) have a common distribution function (DF) \(F(x)=\mathbf{P}\{X\leqslant x\}\) and distribution density \(f(x)=F^{\prime}(x).\) We denote by \(X_{k}^{(n)}\) the \(k\)th order statistic in variational series \(X_{1}^{(n)}\leqslant...\leqslant X_{n}^{(n)}\) constructed using the variables \(X_{1},...,X_{n}\), \(\bar{X}=\sum\limits_{i=1}^{n}X_{i}/n\), and \(S^{2}=\sum\limits_{i=1}^{n}(X_{i}-\bar{X})^{2}/n.\) For some \(c_{n}>0\) and \(d_{n}\). Below, we study the asymptotic distribution (as \(n\to\infty\)) of the quantity
where
In [1], a similar problem was considered as applied to a limiting distribution of extremal order statistics.
1. MAIN RESULT
Theorem. Let
If, for some \(c_{n}>0\) and \(d_{n}\), the quantity
has a limiting DF \(H(x)\) as \(n\to\infty\) and the limit relation
holds, quantity (1) has the same limiting DF \(H(x).\)
Proof. We write (1) in the form
It follows from conditions (3) that, as \(n\to\infty\),
expression (1) is equivalent to the expression
Let us show that quantity (6) has the same limit as quantity (4). It follows from conditions (3) that quantities \(\sqrt{n}\bar{X}\) and \(\sqrt{n}(1-S)\) as \(n\to\infty\) are asymptotically normal.
We consider the expression
Since both terms on the right-hand side of relation (7) converge in probability to zero as \(n\to\infty\) by the hypotheses of the theorem, we have
It thus follows that the second term in expression (6) converges to zero in probability as \(n\to\infty\). Let us show that as \(n\to\infty\),
Let us consider two cases.
1. Dstribution \(F\) is unbound from the left; i.e., \(F(x)>0\) for all \(x.\) As \(n\to\infty\), we then have \(X_{k}^{(n)}\stackrel{{\scriptstyle P}}{{\longrightarrow}}-\infty,\) and condition (8) implies condition (9).
2. There exists a finite number \(x_{0},\) such that \(d_{n}\to x_{0}\) and \(X_{k}^{(n)}\stackrel{{\scriptstyle P}}{{\longrightarrow}}x_{0}\) as \(n\to\infty,\) and the condition \(\mathbf{E}X_{1}=0\) implies that \(x_{0}<0.\) Again, condition (9) follows from condition (8); consequently, as \(n\to\infty\), the third term in expression (6) converges in probability to zero. The theorem is proved.
2. APPLICATIONS
Suppose that condition (2) holds. A necessary and sufficient condition for asymptotic normality as \(n\to\infty\) of statistics \(T_{n}=(X_{k}^{(n)}-d_{n})/c_{n}\) for some \(c_{n}>0\) and \(d_{n}\) is that the following relation is satisfied for any \(x\) [2, 3]:
For absolutely continuous distributions, quantities \(c_{n}\) and \(d_{n}\) are determined using relations
In [4, 5], it was shown that under condition (2) and as \(n\to\infty\), probable limiting distributions of statistics \(T_{n}\) are normal and lognormal distributions. The joint asymptotic distribution of intermediate order statistics was studied in [6, 7].
We assume below that \(z_{F}=\inf\{x:F(x)>0\}\), \(k=[n^{\alpha}]\), \(0<\alpha<1\), and \([x]\) denotes the integral part of number \(x,\) and we consider typical classes of distributions widely used in statistical applications.
Class \(B_{1}:\)
Class \(B_{2}:\)
Class \(B_{3}:\)
It is easy to show that relation (10) holds for all three classes \(B_{1}\), \(B_{2}\), and \(B_{3}\), and the limiting distribution of the statistics \(T_{n}\) as \(n\to\infty\) is the standard normal distribution. For classes \(B_{1}\) and \(B_{2}\), condition (5) is satisfied,while for class \(B_{3}\) condition (5) is satisfied under the constraint \(\Delta>2.\) Conditions (3) require additional constraints on the parameters for all three classes. For class \(B_{3}\), it follows from (3) that \(z_{F}<0.\)
Examples of the distributions from class \(B_{1}\) that satisfy the hypotheses of the theorem are the standard normal distribution and the Laplace distribution with density \(f(x)=\exp(-\sqrt{2}|x|)/\sqrt{2}\), \(|x|<\infty.\)
3. EXAMPLE OF THE DISTRUBUTION FROM CLASS \(B_{2}\) SATISFYING THE HYPOTHESES OF THE THEOREM
Let us consider a distribution with density \(f(x)=b_{1}/(x^{6}+b_{2}^{6})\),\(|x|<\infty.\) Positive numbers \(b_{1}\) and \(b_{2}\) are determined later by using condition (3). We have
hence, \(b_{2}^{3}=b_{1}\pi/3.\) Further,
Since
it follows that \(b_{1}=6\sqrt{2}/\pi\) and \(b_{2}^{6}=8.\) We have
The conditions of the theorem are met.
4. AN EXAMPLE OF THE DISTRUBUTION FROM CLASS \(B_{3}\) SATISFYING THE HYPOTHESES OF THE THEOREM
Let \(F(x)=a(x-z_{F})^{\Delta}\), \(f(x)=\Delta a(x-z_{F})^{\Delta-1}\), \(x\in(z_{F},b)\), and \(a,\Delta>0.\) In light of condition \(F(b)=1\) and relations (3), we obtain
Under constraint \(\Delta>2\), the conditions of the theorem hold.
5. EXAMPLE OF A DISTRUBUTION FROM CLASS \(B_{3}\) NOT SATISFYING THE HYPOTHESES OF THE THEOREM
Let \(F(x)=1-\exp(-(x+1)),\quad f(x)=\exp(-(x+1))\), \(x>-1.\) We assume that conditions (3) holds,
and condition (5) is not satisfied. We have
but since \(X_{[n^{\alpha}]}^{(n)}\stackrel{{\scriptstyle P}}{{\longrightarrow}}-1\), \(\sqrt{n}\bar{X}\) and \(\sqrt{n}(1-S)\) are asymptotically normal,
the second and third terms in representation (6) grow infinitely in absolute magnitude.
REFERENCES
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N. V. Smirnov, ‘‘On convergence of terms of a variational series to the normal distribution law,’’ Izv. Akad. Nauk Uzb. SSR 3, 24–32 (1966).
D. M. Chibisov, ‘‘On limit distributions for order statistics,’’ Theory Probab. Appl. 9 (1), 142–148 (1964).
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V. A. Ditkin and A. P. Prudnikov, Integral Transforms and Operational Calculus (Fizmatgiz, Moscow, 1961; Pergamon Press, Oxford, 1965).
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Translated by I. Tselishcheva
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Pagurova, V.I. On the Limiting Distribution of Studentized Intermediate Order Statistics. MoscowUniv.Comput.Math.Cybern. 45, 81–84 (2021). https://doi.org/10.3103/S0278641921020047
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DOI: https://doi.org/10.3103/S0278641921020047