1.
Introduction
Order statistic (OS) plays an important role in nonparametric statistics. Under the assumption of large sample size, relative investigations are mainly focused on asymptotic distributions of some functions of these OSs. Among these studies, the elegant one provided by Bahadur in 1966 (see [1]) is the central limit theorem on OSs. As was revealed there, under the situation of an absolute continuous population, the sequence of some normalized OSs usually has an asymptotic standard normal distribution. That is useful in the construction of a confidence interval for estimating some certain quantile of the population. Comparatively, study on some moment convergence of the mentioned sequence is also significant, for instance, if we utilize a sample quantile as an asymptotic unbiased estimator for the corresponding quantile of the population, then the analysis of the second moment convergence of the sequence is significant if we want to make an approximation of the mean square error of the estimate.
However, the analysis of moment convergence of OSs is usually very difficult, the reason, as was interpreted by Thomas and Sreekumar in [2], may lie in the fact that the moment of OS is usually very difficult to obtain.
For a random sequence, although it is well-known that the convergence in distribution does not necessarily guarantee the corresponding moment convergence, usually, that obstacle can be sufficiently overcome by the additional requirement of the uniform integrability of the sequence. For instance, we can see [3] as a reference dealing with some extreme OSs under some populations. In that article Wang et al. discussed uniform integrability of the sequence of some normalized extreme OSs and derived equivalent moment expressions there.
Here in the following theorem we discuss the moment convergence for some common OSs rather than extreme ones.
Theorem 1. For a population X distributed according to a continuous probability density function (pdf) f(x), let p∈(0,1) and xp be the p−quantile of X satisfying f(xp)>0. Let (X1,...,Xn) be a random sample arising from X and Xi:n be the i−th OS. If the cumulative distribution function (cdf) F(x) of X has an inverse function G(x) satisfying
for some constants B>0,q≥0 and all x∈(0,1), then for arbitrary δ>0, we have
provided limn→+∞i/n=p or equivalently rewritten as i/n=p+o(1).
Remark 1. Now we use the symbol ⌈z⌉ for the integer part of a positive number z and mn,p for the p-quantile of a random sample (X1,⋯,Xn), namely, mn,p=(Xpn:n+Xpn+1:n)/2 if pn is an integer and mn,p=X⌈pn+1⌉:n otherwise. As both limiting conclusions limn→∞EXδ⌈pn⌉:n=xδp and limn→∞EXδ⌈pn+1⌉:n=xδp hold under the conditions of Theorem 1 and mδn,p is always squeezed by Xδ⌈pn⌉:n and Xδ⌈pn+1⌉:n, according to the Sandwich Theorem, we have limn→∞Emδn,p=xδp.
Remark 2. For a continuous function H(x) where x∈(0,1), if
then there is a constant C>0 such that the inequality |H(x)|≤C holds for all x∈(0,1). By that reason, the condition (1.1) can be replaced by the statement that there exists some constant V≥0 such that
Remark 3. As the conclusion is on moment convergence of OSs, one may think that the moment of the population X in Theorem 1 should exist. That is a misunderstanding because the existence of the moment of the population is actually unnecessary. We can verify that by a population according to the well-known Cauchy distribution X∼f(x)=1π(1+x2) where x∈(−∞,+∞), in this case, the moment EX of the population does not exist whereas the required conditions in Theorem 1 are satisfied. Even for some population without any moment of positive order, the conclusion of Theorem 1 still holds, for instance, if f(x)=1x(ln(x))2I[e,∞)(x) (where the symbol IA(x) or IA stands for the indicator function of a set A), then we have the conclusion
which leads to
and therefore the condition (1.1) holds, thus we can see that Theorem 1 is workable. That denies the statement in the final part of paper [4] exclaiming that under the situation X∼f(x)=1x(ln(x))2I[e,∞)(x) any OS does not have any moment of positive order.
According to Theorem 1, we known that the OS Xi:n of interest is an asymptotic unbiased estimator of the corresponding population quantile xp. Now we explore the infinitesimal type of the mean error of the estimate and derive
Theorem 2. Let (X1,...,Xn) be a random sample from X who possesses a continuous pdf f(x). Let p∈(0,1) and xp be the p−quantile of X satisfying f(xp)>0 and Xi:n be the i−th OS. If the cdf F(x) of X has an inverse function G(x) with a continuous derivative function G‴(x) in (0,1) and there is a constant U≥0 such that
then under the assumption i/n=p+O(n−1) which indicates the existence of the limit limx→0+i/n−p1/n, the following proposition stands
Remark 4. Obviously we can see that |E(mn,p−xp)|=O(1/n) under the conditions of Theorem 2.
For i.i.d random variables(RVs) X1,...,Xn with an identical expectation μ and a common finite standard deviation σ>0, the famous Levy-Lindeberg central limit theorem reveals that the sequence of normalized sums
converges in distribution to the standard normal distribution N(0,12) which we denote that as
In 1964, Bengt presented his work [5] showing that if it is further assumed that E|X1|k<+∞ for some specific positive k, then the m-th moment convergence conclusion
holds for any positive m satisfying m≤k. Here and throughout our paper, we denote Z a RV of standard normal distribution N(0,12).
Let f(x) be a continuous pdf of a population X and xr be the r−quantile of X satisfying f(xr)>0. Like the Levy-Lindeberg central limit theorem, Bahadur interpreted in [1] (1966) that for the OS Xi:n, following convergence conclusion holds
provided i/n→r as n→∞.
Later in 1967, Peter studied moment convergence on similar topic. He obtained in [6] that for some ε>0, r∈(0,1) and pn=i/n, if the limit condition
holds, then the conclusion
is workable for positive integer k and rn≤i≤(1−r)n as n→+∞.
In addition to the mentioned reference dealing with moment convergence on OSs, we find some more desirable conclusions on similar topic provided by Reiss in reference [7] in 1989, from which we excerpt the one of interest as what follows.
Theorem 3. Respectively let f(x) and F(x) be the pdf and cdf of a population X. Let p∈(0,1) and xp be the p−quantile of X satisfying f(xp)>0. Assume that on a neighborhood of xp the cdf F(x) has m+1 bounded derivatives. If a positive integer i satisfies i/n=p+O(n−1) and E|Xs:j|<∞ holds for some positive integer j and s∈{1,...,j} and a measurable function h(x) meets the requirement |h(x)|≤|x|k for some positive integer k, then
Here the function φ(x) and Φ(x) are respectively the pdf and cdf of a standard normal distribution while Si,n(x), a polynomial of x with degree not more than 3i−1 and coefficients uniformly bounded over n, especially
Remark 5. By putting h(x)=x2 and m=2, we derive under the conditions of Theorem 3 that as n→+∞,
Therefore, we see that the sequence
is uniformly bounded over n≥N0. Here N0 is the positive integer number that the moment EX2i:n exists when n≥N0. In accordance with the inequality |Eξ|≤√Eξ2 if only the moment Eξ2 exists, the sequence
is also uniformly bounded, say, by a number L over n∈{N0,N0+1,...}. Now that
we have
Under the conditions in Theorem 2, when we estimate a population quantile xp by an OS Xi:n, usually the estimate is not likely unbiased, compared with the two conclusions (1.3) and (1.6), the result (1.3) in Theorem 2 is more accurate.
Remark 6. For a random sample (Y1,Y2,...,Yn) from a uniformly distributed population Y∼U[0,1], we write Yi:n the i−th OS. Obviously, conditions in Theorem 3 are fulfilled for any positive integer m≥2. That yields
and
where for each i=1,2,...,5, αi(n) is uniformly bounded over n.
As is above analyzed, we conclude that under the assumption i/n=p+O(n−1),
Based on Theorems 1 and 3, here we give some alternative conditions to those in Theorem 3 to embody its range of applications including situations even when the population X in Theorem 3 has no definite moment of any positive order. We obtain:
Theorem 4. Let (X1,...,Xn) be a random sample derived from a population X who has a continuous pdf f(x). Let p∈(0,1) and xp be the p−quantile of X satisfying f(xp)>0 on a neighborhood of xp and the following three conditions hold,
(i) The cdf F(x) of X has an inverse function G(x) satisfying
for some constants B>0,Q≥0 and all x∈(0,1).
(ii) F(x) has m+1 bounded derivatives where m is a positive integer.
(iii) Let i/n=p+O(n−1) and ai:n=xp+O(n−1) as n→+∞.
Then the following limiting result holds as n→+∞
Remark 7. For the mean ¯Xn of the random sample (X1,...,Xn) of a population X whose moment EXm exists, according to conclusion (1.4), we see
which indicates that the m−th central moment of sample mean E(¯Xn−μ)m is usually of infinitesimal O(n−m/2).
Here under the conditions of Theorem 4, if EXi,n=xp+O(n−1) (we will verify in later section that for almost all continuous populations we may encounter, this assertion holds according to Theorem 2), then by Eq (1.9), we are sure that the central moment E(Xi:n−EXi:m)m is also of an infinitesimal O(n−m/2). Moreover, by putting ai:n=xp, we derive under the assumptions of Theorem 4 that
Similar to Remark 1, we can also show by Sandwich Theorem that
indicating that if we use the sample p−quantile mn,p to estimate xp, the corresponding population p−quantile, then E(mn,p−xp)m=O(n−m/2).
For estimating a parameter of a population without an expectation, estimators based on functions of sample moments are always futile because of uncontrollable fluctuation. Alternatively, estimators obtained by some functions of OSs are usually workable. To find a desirable one of that kind, approximating some moment expressions of OSs is therefore significant. For instance, let a population X be distributed according to a pdf
where constants θ2>0 and θ1 is unknown. Here x0.56=0.19076θ2+θ1 and x0.56+x0.44=2x0.5=2θ1. To estimate x0.5=θ1, we now compare estimators mn,0.5 and (mn,0.56+mn,0.44)/2. Under large sample size, we deduce according to conclusion (1.10) that
whereas
Obviously, both estimators mn,0.5 and (mn,0.56+mn,0.44)/2 are unbiased for θ1. For large n, the main part 0.2554πθ2n−1 of the mean square error (MSE) E[(mn,0.56+mn,0.44)/2−θ1]2 is even less than one-third of 0.785πθ2n−1, the main part of the MSE E(mn,0.5−θ1)2. That is the fundamental reason why Sen obtained in [8] the conclusion that the named optimum mid-range (mn,0.56+mn,0.44)/2 is more effective than the sample median mn,0.5 in estimating θ1.
By statistical comparison of the scores presented in following Table 1 standing for 30 returns of closing prices of German Stock Index(DAX), Mahdizadeh and Zamanzade reasonably applied the previously mentioned Cauchy distribution (1.11) as a stock market return distribution with θ1 and θ2 being respectively estimated as ^θ1=0.0009629174 and ^θ2=0.003635871 (see [9]).
Now we utilize (mn,0.56+mn,0.44)/2 as a quick estimator of θ1 and derive a value 0.00105955 which roughly closes to the estimate value 0.0009629174 in reference [9].
Even now there are many estimate problems (see [10] for a reference) dealing with situations when a population have no expectation, as above analysis, further study on moment convergence for some OSs may be promising.
2.
Preparation of main proof
Lemma 1. (see [11] and [12]) For a random sequence {ξ1,ξ2,⋯} converging in distribution to a RV ξ which we write as ξnD→ξ, if d>0 is a constant and the following uniform integrability holds
then limn→∞E|ξn|d=E|ξ|d and accordingly limn→∞Eξnd=Eξd.
Remark 8. As discarding some definite number of terms from {ξ1,ξ2,⋯} does not affect the conclusion limn→∞E|ξn|d=E|ξ|d, the above condition lims→+∞supnE|ξn|dI|ξn|d≥s=0 can be replaced by lims→+∞supn≥ME|ξdn|I|ξdn|≥s=0 for any positive constant M>0.
Lemma 2. For p∈(0,1) and a random sample (ξ1,ξ2,⋯,ξn) from a population possessing a continuous pdf f(x), if the p-quantile xp of the population satisfies f(xp)>0, then for the i-th OS ξi:n where i/n=p+o(1), we have ξi:nD→xp.
Proof. Obviously, the sequence {f(xp)(ξi:n−xp)√p(1−p)/n,n=1,2,...} has an asymptotic standard normal distribution N(0,12), thus we see that the statistic ξi:n converges to xp in probability. That leads to the conclusion ξi:nD→xp by the reason that, for a sequence of RVs, the convergence to a constant in probability is equivalent to the convergence in distribution.
3.
Proof of theorems
Clarification before presenting the proof:
● Under the assumption i/n=p+o(1) when n→∞, we would better think of i as a function of n and use the symbol an instead of i. Nevertheless, for simplicity concern, we prefer no adjustment.
● Throughout our paper, C1, C2,⋯ are some suitable positive constants.
3.1. Proof of Theorem 1
As in→p∈(0,1) when n→∞, we only need care large numbers n,i and n−i.
Let an integer K>δq be given and M>0 be such a number that if n≥M, then all the following inequalities i−1−δq>0, n−i−δq>0, n−i−K>0 and i+Kn<v=1+p2 hold simultaneously. Here the existence of v in the last inequality is ensured by the fact i+Kn→p as n→∞.
According to Lemmas 1 and 2 as well as Remark 8, to prove Theorem 1 we only need to show that
That is
To show that equation, it suffices for us to prove respectively
and
Equivalently by putting x=F(u), we need to prove respectively
as well as
As both proofs are similar in fashion, we chose to prove the Eq (3.2) only. Actually, according to the given condition |G(x)|≤Bx−q(1−x)−q, we see
Here the positive number C1>0 exists because n/i=1/p+o(1) where p∈(0,1).
Now applying the Stirling's formula n!=√2πn(n/e)neθ12n where θ∈(0,1) (see [13]), we have
Noting that
as n→∞, we see that there exists a positive constant, say Q>0 such that
for all n. Consequently,
Due to the assumptions i+Kn<v=1+p2<1 as n≥M, we derive
Finally, by the fact that if u>0 is given sufficiently small, then the first term of the sequence {un√n,n≥1} is the maximum, thus we can confirm
Combining the five conclusions numbered from (3.3) to (3.7), we obtain Eq (3.2).
3.2. Proof of Theorem 2
Here we would like to assume U>1 (or we may use U+2 instead of U).
By the reason interpreted in Remark 2 and according to condition (1.2), we see that there is a constant A>0 satisfying
Now we define Y=F(X) and Yi:n=F(Xi:n) or equivalently X=G(Y) and Xi:n=G(Yi,n), we have G(p) = x_p . Obviously, the conclusions in Remark 6 are workable here.
By the Taylor expansion formula we have
where
Noting that almost surely 0 < min(Y_{i:n}, p) < \xi < max(Y_{i:n}, p) < 1 , we obtain
by Eq (3.8). Here the last step is in accordance to (1.7).
Now we can draw the conclusion that
That is
provided i/n = p+O(n^{-1}) .
Still according to conclusion (1.7), we have
Finally, as i/n = p+O(n^{-1}) also guarantees i/(n+1)-p = O(n^{-1}) , we can complete the proof of E(X_{i:n}-x_p) = O(n^{-1}) or equivalently
by the assertion of (3.11).
3.3. Proof of Theorem 4
As EZ = 0 , the proposition holds when m = 1 , now we only consider the case of m\geq2 . By Theorem 1, we see EX_{i:n}^2\to x_{p}^{2} , therefore E\vert X_{s:j}\vert exists for some integer j and s\in\{1, ..., j\} and Theorem 3 is workable here when we put h(x) = x^m . We derive
Moreover, for given positive integer m\geq2 , as the coefficients in polynomial S_{i, n}(x) are uniformly bounded over n and \varphi'(x) = -x \varphi(x) , the sequence of the integrals
is also uniformly bounded over n . That indicates that
according to conclusion (3.12).
As a consequence, we can conclude that for explicitly given m\geq 2 the sequence
is uniformly bounded over n . Moreover, due to the inequality
we see that the sequence
is also uniformly bounded over n .
Now that a_{i:n} = x_p+O(n^{-1}) , we complete the proof by the following reasoning
4.
Some verifications and one application
4.1. Verification examples
Now we consider the applicability of our theorems obtained so far. As other conditions can be trivially or similarly verified, here we mainly focus on the verification of condition (1.2).
Example 1: Let the population X have a Cauchy distribution with a pdf f(y) = \frac{1}{\pi(1+y^2)}, -\infty < y < +\infty, correspondingly the inverse function of the cdf of X can be figured out to be
satisfying
Example 2: For X\sim f(x) = \frac{1}{x(\ln(x))^{2}}I_{[e, \infty)}(x) , we have
and
Example 3: For X\sim N(0, 1^2) , on that occasion, f(y) = \frac{1}{\sqrt{2\pi}}e^{-\frac{y^2}{2}} , f'(y) = -yf(y) and y = G(x)\Leftrightarrow x = F(y) = \int_{ - \infty }^y {\frac{1}{{\sqrt {2\pi } }}{e^{ - \frac{{{{t }^2}}}{{2}}}}dt} , therefore, as x \to 0 + , we have
Noting that as x = F(y)\to 0+ or equivalently y\to-\infty ,
we have as x\to 0+ ,
By the same fashion, we can show as x\to 1- that
In conclusion, for x\to 0+ as well as for x\to 1- ,
Accordingly, there exists a positive M > 0 such that for all x\in (0, 1) ,
No matter if x\to 0+ or x\to 1- , we get
Here the last step holds in accordance to Eq (4.2).
For x\to 0+ as well as for x\to 1- ,
Thus we can see the achievement of condition (1.2) by
Remark 9. For a RV X with a cdf F(x) possessing an inverse function G(x) , we can prove that if \sigma > 0 and \mu\in(-\infty, +\infty) are constants, then the cdf of the RV \sigma X+\mu will have an inverse function \sigma G(x)+\mu . Thus for the general case X\sim N(\mu, \sigma^2) , we can still verify the condition (1.2).
Example 4: For a population X\sim U[a, b] , G(x) = (b-a)x+a is the inverse function of the cdf of X . As G^{'''}(x) = 0 , the assumption of condition (1.2) holds.
Generally, for any population distributed over an interval [a, b] according to a continuous pdf f(x) , if G{'''}(0+) and G{'''}(1-) exist, then the condition (1.2) holds.
For length concern, here we only point out without detailed proof that for a population X according to a distribution such as Gamma distribution (including special cases such as the Exponential and the Chi-square distributions) and beta distribution and so on, the requirement of condition (1.2) can be satisfied.
4.2. An application in obtaining a combination formula
For a random sample (X_1, ..., X_n) derived from a population X which is uniformly distributed over the interval [0, 1] , the moment of the i- th OS EX_{i:n} = i/(n+1)\to p if i/n\to p\in(0, 1) as n\to \infty . Let a_{i:n} = i/n . According to conclusion (1.9) where f(x_p) = 1 and x_p = p\in(0, 1) , we have for integer m\geq2 ,
That results in
or equivalently
Consequently we have the following equation
which yields
As i/n\to p\in(0, 1) when n\to+\infty , the above equation indicates that
For convenience sake, now we denote \sum_{k = u}^{v} = 0 and \prod_{k = u}^{v} = 1 if v < u . Noting for given explicit integers m\geq2 and j\in\{0, 1, ..., m\} the expression
is a multinomial of i and n . We see that the nominator of the LHS of Eq (4.10) is also a multinomial which we now denote as
Equivalently, we derive
By Eq (4.10), we see for any given p\in(0, 1) , if i/n\to p\in(0, 1) as n\to+\infty , then
Noting that
we see in accordance to (4.12) that
That indicates that if a non-negative integer k satisfies 2m-k > 3m/2 , or equivalently 0\leq k < m/2 , then the coefficient of n^{2m-k} in the nominator of LHS of Eq (4.13) must be zero for any given p\in(0, 1) , namely
holds for any p\in (0, 1) . Thereby, for the case of non-negative integers s and t satisfying s+t = k < m/2 , we see that the equation a_{s, t}^{(m)} = 0 surely holds.
It is funny to notice that for big m , we immediately have the following three corresponding equations
and
according to the conclusions a_{0, 0}^{(m)} = 0 , a_{1, 0}^{(m)} = 0 and a_{1, 1}^{(m)} = 0 .
As for the structure of a_{s, t}^{(m)} when s\geq2 , t\geq1 and m > 2(s+t) , obviously s < m-t holds on this occasion and the term a_{s, t}^{(m)}i^{m-s}n^{m-t} in the multinomial
is also the term a_{s, t}^{(m)}i^{m-s}n^{m-t} in the multinomial
Noting for given j\in\{s, ..., m-t\} , the monomial
is the term with degree m-s in the polynomial of i
while the monomial
is the term with degree m-t in the polynomial of n
we see for s+t < m/2 ,
Now that a_{s, t}^{m} = 0 holds provided s+t = k < m/2 according to Eq (4.13), we conclude the following Theorem.
Theorem 5. If s , t and m are integers satisfying s\geq2 , t\geq1 and m > 2(s+t) , then
Example 5: For big integer m , according to Theorem 5, we have a_{2, 1}^{(m)} = 0 and a_{2, 2}^{(m)} = 0 . Correspondingly, we obtain equations
and
Both equations can be verified by the aid of Maple software.
5.
Conclusions
Let real \delta > 0 and integer m > 0 be given. For a population satisfying condition (1.1), no matter if the population has an expectation or not, the moment of X_{i:n}^{\delta} exists and the sequence \{EX_{i:n}^{\delta}, n\geq1\} converges for large i and n satisfying i/n\to p\in(0, 1) . Under some further trivial assumptions, for large integer n the m- th moment of the standardized sequence \{X_{i:n}, n\geq1\} can be approximated by the m- th moment of a standard normal distribution EZ^m .
Due to the fact that the existence requirement of some expectation X_{s:j} in Theorem 3 has always been hard to be verified for a population without an expectation, for a long time, real-life world data corresponding to that population of interest has been unavailable in the vast majority of references. Now that the alternative condition (1.8) is presented, maybe things will improve in the future and we still have a long way to go.
Acknowledgments
This work was supported by the Science and Technology Plan Projects of Jiangxi Provincial Education Department, grant number GJJ180891.
Conflict of interest
There exists no conflict of interest between authors.