On asymptotic correlation coefficient for some order statistics

1.   Introduction

Let a random sample $X_1, ..., X_n$ be drawn from a population $X$ according to a distribution $F(x, \theta)$ where $\theta$ is unknown. If a statistic $\widehat{\theta}_{n}$ is unbiased for $\theta$ and there are known positive constants $b_n$'s such that the normalized random sequence $\{({\widehat{\theta}_{n}-\theta})/{b_n}, n\geq 1\}$ converges in both the first and the second moment to a known distribution of a random variable (RV), say $\xi$, with a known variance $\sigma^2$, then the effectiveness of the unbiased estimator $\widehat{\theta}_{n}$ can be assessed by the variance $b_{n}^{2}\sigma^2$, the less the variance, the more effective the estimator. However, the Cramér-Rao inequality indicates that to estimate the unknown parameter $\theta$ of the distribution $F(x, \theta)$, the unbiased estimator usually has a variance not less than $1/(nI(\theta))$ where $I(\theta)$ denotes the Fisher information. That indicates that an unbiased estimator with a variance reaching the lower bound $1/(nI(\theta))$ is sure of minimum variance.

Under a large sample size, the maximum likelihood estimate (m.l.e) method usually (but not always) yields a theoretically desirable estimator, say $\widehat{\theta_n}$, in a sense that $\widehat{\theta_n}$ has an asymptotic normal distribution $N(\theta, 1/(nI(\theta)))$ with a variance reaching the lower bound of the well-known Cramér-Rao inequality (see ^[1] as a reference).

For estimating a parameter of the distribution of a population that has no expectation, the classical moment estimate method is futile. Moreover, the classical method of m.l.e usually becomes invalid too in the sense that it doesn't have a closed solution. Under such a situation, especially in the case of estimating some parameters such as the location of a population, it is worth trying to investigate an unbiased estimator established by a linear function of some sample quantiles. That will be preferable if the efficiency is close to that of the theoretical m.l.e. To approximate the efficiency of the estimator, we need the following conclusions.

Theorem 1.1. For a population $X$ distributed according to a continuous pdf $f(x)$, let $p$ and $r$ be two numbers satisfying $0 < p \leq r < 1$ and ${x_p}$ and ${x_r}$ be respectively the $p$-quantile and r-quantile of $X$ satisfying $f({x_p})f({x_r}) > 0$. Let $({X_1}, ..., {X_n})$ be a random sample derived from $X$. If there are constants $\omega > 0$ and $\nu\in (-\infty, \infty)$ such that the cdf $F(x)$ of $\omega X+\nu$ has an inverse function $Q(u)$ which possesses a continuous third-order derivative function $Q'''(u)$ in the interval $(0, 1)$ satisfying

for some given constants $K > 0$, $A\geq0$ and all $u \in (0, 1)$, then:

(1) we have, as $n\to\infty$,

provided ${i}/{n} = p + o(n^{-1/2})$ and ${j}/{n} = r +o(n^{-1/2})$ as $n\to+\infty$;

(2) the correlation coefficient ${corr(X_{i:n}, X_{j:n})}$ between ${X_{i:n}}$ and ${X_{j:n}}$ satisfies

provided ${i}/{n} = p+ o(1)$ and ${j}/{n} = r+ o(1)$ as $n\to\infty$.

Under the conditions in Theorem 1.1 but without assumption (1.1), it is mentioned (without formal proof) in ^[2] that the same conclusions hold according to some equations given. We have found a gap there, that is, ^[2] uses a partial sum of a Taylor expansion of a function to approximate the function itself without rigorous proof. Here we have to apply the assumption (1.1) to fill that gap.

In exploring the measurement of dependence or independence between two order statistics (OSs), many research works based on the Copula function method are instructive. On that subject, Barakat led the research. For references, we can consult Barakat's ^[3,4,5] and Hürlimann's ^[6] as well.

Item (2) in Theorem 1.1 can be regarded as a corresponding exploration of the relation between two general OSs, where we measure the asymptotic dependence or independence by providing the limiting correlation coefficient for them. That has two main advantages. First, this preferred measurement has its advantage in that the respective normalization of both OSs has no effect. Second, as was discovered by Bahadur (see ^[7]) or as summarized by DasGupta in ^[8], the asymptotic joint distributions of some sample quantiles have a multivariate normal distribution while, for two RVs according to a bivariate normal distribution, being independent is equivalent to being uncorrelated.

Under the conditions of Theorem 1.1, we see that the OSs $X_{i:n}$ and $X_{j:n}$ are asymptotically dependent, which supports Barakat's corresponding conclusion in article ^[3]. Moreover, Theorem 1.1 also supports the exclamation in ^[9] stating that the dependence between $X_{i:n}$ and $X_{j:n}$ decreases as $i$ and $j$ draw apart.

To our studies, conditions in Theorem 1.1 are met for almost all continuous populations including the situation discussed in ^[10] from which we see that the correlation coefficient between a sample maximum and a sample minimum has a limiting value 0 as the sample size $n$ tends to infinity. Here Theorem 1.1 deals with correlation coefficients for common OSs relevant to some general sample quantiles.

Now we use symbol $\lceil z\rceil$ for the integer part of a positive number $z$ and $m_{n, p}$ for the $p$-quantile of the random sample $(X_1, \cdots, X_n)$. Namely $m_{n, p} = (X_{pn:n}+X_{pn+1:n})/2$ if $pn$ is an integer and $m_{n, p} = X_{\lceil pn+1\rceil:n}$ otherwise.

Remark 1.1. Assume the condition (1.1) of Theorem 1.1, then:

(1) Corresponding to the central limit theorem for sample quantiles, the following second moment convergence conclusion holds as $n\to\infty$,

(2) The asymptotic correlation coefficient $corr(m_{n, p}, m_{n, r})$ for $m_{n, p}$ and $m_{n, r}$ satisfies

Remark 1.2. For a given sample $(X_1, ..., X_n)$, it is obvious that the correlation coefficient for two different sample quantiles is relevant to the distribution of the population $X$ from which the sample is drawn. Here Theorem 1 indicates that, as the sample size $n$ tends to infinity, the mentioned correlation coefficient is eventually free from the distribution of $X$.

Corollary 1.2. Under the condition (1.1) of Theorem 1.1, if we use the sample quantile $m_{n, p}$ as an estimator for the corresponding population quantile $x_p$, then we have the following variance equivalence

Moreover, if we further assume that $F(x_r) = r$ and $F'(x_r) = f(x_r) > 0$ where $0 < p\leq r < 1$, then the covariance between $m_{n, p}$ and $m_{n, r}$ is

Generally, for real numbers $u_1, u_2, ..., u_k$ and $p_1, p_2, ..., p_k$ satisfying $0 < p_1 < ... < p_k < 1$ and $f(x_{p_1})f(x_{p_2})...f(x_{p_k}) > 0$, the following analogous expression holds

For a positive integer $r$, we mean that a random sequence $\{\xi_n, n\geq 1\}$ converges in an $r$-th order of moment if the number sequence $\{E\xi_n^r, n\geq 1\}$ converges. In reference ^[11], Wang et al. investigated moment convergence conclusions for some OSs connected to a general continuous population. It is found that not only the sequence of sample quantiles but also the corresponding standardized sequence converges in some positive order of moments even when the population of interest has no expectation. Here Theorem 1.1 is a subsequent exploration.

2.   Preparation for the main proof

Lemma 2.1. (see ^[2]). Let $Y$ be a population uniformly distributed over the interval $[0, 1]$; Let ${Y_{i:n}}$ be the corresponding $i$-th OS of a random sample $({Y_1}, \cdots, {Y_n})$ from $Y$. For nonnegative integers $u$, $v$, $i$ and $j$ satisfying $1\leq i < j\leq n$,

Remark 2.1. Setting $u$, $v$, $i$ and $j$ to be some specified nonnegative integers, we can gain

and

Lemma 2.2. Now we denote $\mu_{k:n} = EY_{k:n} = k/(n+1)$. Under the conditions of Lemma 2.1 and by Eq (2.1), we can conclude that for integer $k$ satisfying $k/n\to \rho\in(0, 1)$, as $n\to\infty$,

and

3.   Main proof

3.1.   The proof of Theorem 1.1

Proof. Denoting $Y = F(X)$ and ${Y_i} = F({X_i})$, we see that Lemmas 2.1 and 2.2 are applicable.

According to the Taylor expansion formula,

where $\Delta \in [\min (t, {t_0}), \max (t, {t_0})]$, there exists some RVs ${\tau_{i:n}}$ satisfying

such that

By the equivalent expressions in Lemma (2.2), we have

and

Moreover, by the assumption $|{Q'''(u)}| \leq K {u^{ - A}}{(1 - u)^{ - A}},$ no matter if $0 < Y_{i:n}\leq \tau_{i:n}\leq \mu_{i:n}\leq 1$ or $0 < \mu_{i:n}\leq \tau_{i:n}\leq Y_{i:n}\leq 1$, we have

Noting that the pdf of $Y_{i:n}$ is $n!/((i-1)! (n-i)!)x^{i-1}(1-x)^{n-i} I_{[0, 1]}(x)$, we see that

Now let $M$ be the nonnegative integer satisfying $u = M-4A\in [0, 1)$. By the formula (see ^[12]) $\Gamma(n+\alpha)\sim n^{\alpha}(n-1)!$ where $\alpha > 0$, we have for $i/n\to p\in (0, 1)$ as $n\to\infty$,

Furthermore, we can utilize the Stirling formula $m!\sim (m/e)^{m}\sqrt{2\pi m}$ to obtain

According to (3.5) and (3.8) and by using Liapunov's inequality in the form $E(\xi^2)\leq [E(\xi^4)]^{1/2}$, we see that there exists a positive constant $R > 0$ such that the inequality

holds uniformly with respect to $n\geq 1$. By the Cauchy-Schwarz inequality $[E(\xi \eta)]^2\leq E\xi^2\cdot E\eta^2$ and Lemma 2.2 as well as the fact $|Y_{i:n}-\mu_{i:n}|\leq 1$, we see that

Similarly

Combining the conclusions (3.1) and (3.11), we get

Similarly to (3.1), there exists some RV $\alpha_{j:n}\in [min({\mu _{j:n}}, {Y_{j:n}}), max({\mu _{j:n}}, {Y_{j:n}})]$ such that

Replacing $i$ with $j$ in (3.2), (3.3), (3.11) and (3.13) yields

and

Moreover, we see that

hence

That results in the following conclusion according to Eqs (3.1) and (3.14):

Now noting the equations numbered (3.2), (3.3), (3.11) and those from (3.15) to (3.17), we derive

from which we conclude that

Substituting the corresponding part in (3.20) by the just obtained above result, we have

Referring to the procedure in obtaining conclusion (3.22), we can also reach the following conclusions

and

(1) Now we notice that as $n\to\infty$,

therefore according to Eq (3.13), we have

provided $i/n = p+o(n^{-1/2})$ which is equivalent to $i/(n+1) = p+o(n^{-1/2})$. Consequently we see that

according to Eq (3.23). Here the reason for the last equation is that the continuous function $Q'(u)$ is positive according to the deduction $Q'(u) = {1}/ {{F'(x)}} = {1}/{{f(x)}} > 0$ at $x = x_p$.

(2) Combining the three conclusions (3.22)–(3.24), we get the asymptotic correlation coefficient ${corr(X_{i:n}, X_{j:n})}$ by the following procedures:

provided ${i}/{n} = p + o(1)$ and ${j}/ {n} = r + o(1)$.

4.   Examples

To continue our discussions, we give the following two propositions beforehand:

Proposition 1. If the inverse function $Q(u)$ of a cdf $F(x)$ has a third-order derivative $Q'''(u)$, then

where $x = Q(u)$ and $f(x) = F'(x)$.

Proposition 2. For a function $-\ln({u}^{A}{(1-u)}^{A})$ (where $A\geq0$ is a constant) and any specified constant $\varepsilon > 0$, there exists a corresponding number $C(\varepsilon) > 0$ such that the inequality $-\ln({u}^{A}{(1-u)}^{A})\leq C(\varepsilon) (u(1-u))^{-\varepsilon}$ holds for all $u\in (0, 1)$.

4.1.   One example of Theorem 1.1

Although almost all commonly applied continuous types of populations satisfy the conditions in Theorem 1.1, due to length concerns in this section, we will present only one example of Theorem 1.1.

Example. For a population $X$ with a gamma distribution (including special cases such as the Exponential as well as the Chi-square distributions), the pdf is

Corresponding to the case $\omega = \beta > 0$ and $\nu = 0$, we now assume $x = Q(u)$ to be the inverse function of the cdf $u = F(x)$ of $\beta X$, the pdf of which can be worked out as

On that occasion, we can easily see that for $x > 0$,

Noting that it is easy to verify that Theorem 1.1 is applicable in the case $\alpha = 1$, we now assume that $\alpha\in (0, 1)\cup (1, +\infty)$.

As a condition like (1.1) is equivalent to verifying the existence of a positive number $q$ such that

and by Proposition 1, it is sufficient to verifying the conditions

and

Now we use the notation $g(x)\asymp h(x)$ to mean that there are positive constants $a < b$ such that $a|g(x)|\leq h(x)\leq b|g(x)|$ as $x\to 0+$ or $x\to\infty$, according to context. Then

● Case 1. $x\to 0+$. We only need consider $x$ confined in a sufficiently small interval $(0, \delta]$. Obviously $F(x)\asymp x^\alpha$, $f(x)\asymp x^{\alpha-1}$, $f'(x)\asymp x^{\alpha-2}$ and $f''(x)\asymp x^{\alpha-3}$. It follows that conditions (4.3) are satisfied if a positive constant $q$ satisfies $q > 3-1/\alpha$.

● Case 2. $x\to\infty$. On that occasion, we see according to (4.2) that the following relations hold simultaneously for a positive number $q > 3$:

and

Consequently we see the realization of (4.3).

As above analyzed, gamma distributions satisfy condition (1.1).

4.2.   One application example

The Cauchy distribution has a wide range of applications in physics, economics as well as in the medical domain. We may perceive its important application in physics by a simple model depicted as what follows: in a coordinate plane, if we place at a point $(\theta_1, \theta_2)$ (where $\theta_2 > 0$) a radioactive material emitting a particle at a random angle $U$ uniformly distributed over an interval $[0, 2\pi]$, then we can show that the particle will reach the abscissa axis at a point $X$ distributed according to a pdf

which is the pdf of a Cauchy distribution. The relevant kinds of literature are huge. For general introduction we recommend ^[13] whereas, for some elegant studies on a similar topic to this article, we consult references ^[14] and ^[15].

There also is a considerable literature on L-estimation, including determining optimal weights. Some of this is in the robustness literature. See ^[2] and ^[16] for more references.

On estimating the location $\theta_1$ in (4.5), Sen verified in ^[17] that the so-called mid-range $(m_{n, 0.56}+m_{n, 0.44})/2$ is more effective than the sample median $m_{n, 0.5}$. By rejecting a fixed number of the largest and the smallest OSs to avoid a large mean squared error of the parameter estimator, Pekasiewicz utilized in ^[18] a method named the truncated quantile least squares method to estimate the location parameter $\theta_1$. Recently, Krykun ^[19] investigated estimating both $\theta_1$ and $\theta_2$, by resorting to an arctangent regression function and rejecting some fraction of the largest and the smallest OSs. Some ideal simulated results are obtained in ^[19]. Comparatively, what we present in the following exploration is a third way using optimal linear combinations of some sample quantiles.

To estimate $\theta_1$ in (4.5), we will, without loss of generality, set $\theta_2 = 1$.

Let $(X_1, ..., X_n)$ be a random sample from a population $X$ according to the pdf

with an unknown $\theta$. As finding the uniformly minimum variance unbiased estimator (UMVUE) for $\theta$ is hopeless, we now think of the estimator

which is named as sample quasi-midrange (see ^[20]). It is trivial to see that $R_{n}(r)$ is unbiased in estimating $\theta$. According to Theorem 1, we see that

As we can easily see that the equivalence for the variance of the sample median

the result of (4.8) seems to indicate that the unbiased estimator $R_{n}(r)$ will be more effective than the sample median $m_{n, 0.5}$ if we can diminish the value $r/sin^4(\pi r)$. As the minimum value of $r/sin^4(\pi r)$ exists but can not be obtained as an explicit expression, here we make an approximation of the minimum value of $r/sin^4(\pi r)$ as 0.4724417292 when $r = 0.4435$. By the equivalence (1.3) in Corollary 1.1, the estimator $R_n(0.4435)$ is preferable for $\theta$ because the equivalent corresponding variance

is a bit smaller than that of the sample median. That is exactly the conclusion drawn in ^[17]. Moreover, for $0 < p < r\leq 0.5$ and $t\in (-\infty, +\infty)$, we see that the estimator $t R_n(p)+(1-t)R_{n}(r)$ is also unbiased for $\theta$ and

According to equivalence (1.4) and by noting that $f(x_p) = f(x_{1-p})$ and $f(x_r) = f(x_{1-r})$, we obtain

and thus for large $n$,

Generally, for two sequences of real numbers $t_1, ..., t_m$ and $p_1, ..., p_m$ respectively satisfying $t_m = 1-\sum_{i = 1}^{m-1}t_i$ and $0\leq p_1 < p_2 < ... < p_m\leq 0.5$, the linear combination $\sum_{i = 1}^{m}t_{i}R_{n}(p_i)$ is an unbiased estimator for $\theta$ and the corresponding asymptotic variance is

For the unknown $\theta$ in the pdf of (4.6), to find an unbiased estimator of the form $\widehat{\theta}_{m, n} = \sum_{i = 1}^{m}t_{i}R_{n}(p_i)$ with minimum variance, what is left is just a matter of some calculations of finding the $t_i$'s and $p_i$'s such that the expression (4.10) attains its minimum value. For instance, by putting $m = 5$ in (4.10) and by some numerical calculations, we obtain such an estimator defined by

With the aid of Matlab software, the asymptotic variance can be shown to be $Var(E_{5, n})\sim 2.0314/n$. The estimator $\widehat{\theta}_{5, n}$ is unbiased and is better than the estimator $R_n(0.4435)$, which was named the optimum mid-range estimator and was admitted in ^[17] as a superior estimator to the sample median in estimating $\theta$. As $p_1 = 0.4435$ can be determined numerically for the case $m = 1$, among unbiased estimators $R_n(p)$ in (4.7), $R_{n}(0.4435)$ is the most efficient one such that (4.10) has a minimum variance when $m = 1$ is specified.

The Fisher information $I(\theta) = 1/2$ for the Cauchy pdf (4.6), so we see that even if the UMVUE, say $\widehat{\theta}^*_n$ for $\theta$ exists, the theoretical variance $Var(\widehat{\theta}^*_n)$ can not be smaller than $\frac{2}{n}$ according to the well-known Cramér-Rao inequality.

Noting that the quotient $\frac{2}{2.0314}\approx0.9845$ is close to 1, we see that the quick unbiased estimator $E_{5, n}$ is close to the theoretical ideal unbiased estimator.

To compare the effectiveness of estimating $\theta_1$ by the three mentioned estimators, namely, the median $m_{n, 0.5}$, the quasi-midrange $R_{n}(0.4435)$ in (4.7) and the just discussed estimator $E_{5, n}$, by the aid of Matlab software, we simulate 30 times a random sample of size $n = 200$ drawn from a specified Cauchy distribution $f(x, \theta_1, \theta_2) = \frac{\theta_2}{\pi [\theta_2^2+(x-\theta_1)^2]}$ with respective true values $\theta_1 = 0.75$ and $\theta_2 = 2$. According to the simulated results, Figure 1 shows the effectiveness of the three estimators in estimating $\theta_1$. The averaged squared errors for the three estimators are respectively, 0.0030, 0.0025 and 0.0019.

As is indicated by the simulated results, among the three estimators $m_{n, 0.5}, E_{5, n}$ and $R_{n}(0.4435)$, the estimator $E_{5, n}$ is the most effective under the assumption of a large sample size.

Acknowledgments

This work was supported by the Science and Technology Plan Project of Jiangxi Province Health Commission (Grant NO. 202311165) and the National Natural Science Foundation of China (Grant NO. 81960618).

Conflict of interest

There exists no conflict of interest between authors.