Mallows’ models for imperfect ranking in ranked set sampling

Abstract

In this paper, we consider some statistical measures of deviation from the perfect ranking in the framework of ranked set sampling. We use nonparametric approach for testing the null hypothesis for perfect ranking. The distance-based Mallows’ models with appropriate distance on permutations are suggested in the case of imperfect ranking. Some asymptotic results for the corresponding error probability matrix are derived for the models based on Spearman’s footrule and Spearman’s rho. We propose an EM algorithm for estimating the unknown parameter in the Mallows’ models in order to compare the power of the presented test statistics.

Appendix

The proofs given below are mainly based on the following combinatorial central limit theorem (CCLT), formulated and proved by Hoeffding (1951).

Theorem 4

(Hoeffding’s CCLT) Let $\pi \sim Uniform(\mathbf {S_{k}})$ and $D(\pi )=\sum \limits _{s=1}^{k}a_{k}\big (\pi (s),s\big )$, where $a_{k}(r,s)\in {\mathbf {R}}$ for $r,s=1,2,\ldots ,k$. Then the mean and variance of D are

$$\begin{aligned} {\mathbf {E}}\left( D\right)&=\frac{1}{k}\sum _{r=1}^{k}\sum _{s=1}^{k}a_{k}(r,s) \end{aligned}$$

(15)

$$\begin{aligned} \mathbf {Var} \left( D\right)&=\frac{1}{k-1}\sum _{r=1}^{k}\sum _{s=1}^{k}b_{k}^{2}(r,s) \, , \end{aligned}$$

(16)

where

$$\begin{aligned} b_{k}(r,s)=a_{k}(r,s)- \frac{1}{k} \sum _{l=1}^{k}a_{k}(l,s)-\frac{1}{k}\sum _{m=1}^ {k}a_{k}(r,m)+\frac{1}{k^{2}}\sum _{l=1}^{k} \sum _{m=1}^{k}a_{k}(l,m) \end{aligned}$$

for $r,s=1,2,\ldots ,k$. Furthermore, the distribution of D is asymptotically normal if

$$\begin{aligned} \lim _{k \rightarrow \infty } \displaystyle \frac{\displaystyle \max \nolimits _{1 \le r,s \le k}b_{k}^{2}(r,s)}{\displaystyle \frac{1}{k}\sum \nolimits _{r=1}^{k}\sum \nolimits _{s=1}^{k}b_{k}^{2}(r,s)} = 0 \,. \end{aligned}$$

(17)

Consider the random variables based on Spearman’s footrule and Spearman’s rho:

$$\begin{aligned} D_{F}\left( \pi \right) =\sum \limits _{s=1}^{k}\mid \pi (s)-s\mid \quad \hbox {and} \quad D_{R}\left( \pi \right) =\sum \limits _{s=1}^{k}\left( \pi (s)-s\right) ^{2}, \end{aligned}$$

where $\pi \sim Uniform(\mathbf {S_{k}})$. By applying Theorem 4 to $D_{F}$ and $D_{R}$ (see, e.g., Marden 1995, p.83) it can be shown that $D_{F}$ and $D_{R}$ are asymptotically normal with means and variances:

$$\begin{aligned} {\mathbf {E}}\left( D_{F}\right)&=\frac{1}{k}\sum _{r=1}^{k}\sum _{s=1}^{k}\mid r-s\mid =\frac{k^2-1}{3},\nonumber \\ \mathbf {Var}\left( D_{F}\right)&=\frac{(k+1)\left( 2k^2+7\right) }{45}, \end{aligned}$$

(18)

$$\begin{aligned} {\mathbf {E}}\left( D_{R}\right)&=\frac{1}{k}\sum _{r=1}^{k}\sum _{s=1}^{k}\left( r-s\right) ^{2}=\frac{k\left( k^2-1\right) }{6},\nonumber \\ \mathbf {Var}\left( D_{R}\right)&=\frac{k^2(k-1)\left( k+1\right) ^{2}}{36}. \end{aligned}$$

(19)

In order to prove Theorem 2, let define the random variables $D_{F}^{(i,j)}=d_{F}\left( \pi ,e_{k}\right)$ for $i,j=1,2,\ldots ,k$, where $d_{F}(\cdot ,\cdot )$ is Spearman’s footrule, and $\pi$ is uniformly and randomly selected from ${\mathbf {S}}_{{\mathbf {k}}}^{(i,j)}=\left\{ \sigma \in \mathbf {S_{k}}: \sigma (j)=i\right\}$, i.e., $\pi \sim Uniform\left( {\mathbf {S}}_{{\mathbf {k}}}^{(i,j)}\right)$. Then, for a fixed pair (i, j),

$$\begin{aligned} D_{F}^{(i,j)}=\sum \limits _{s=1}^{k}\mid \pi (s)-s\mid =\sum \limits _{\begin{array}{c} s=1 \\ s\ne j \end{array}}^{k}\mid \pi (s)-s\mid + \mid i-j\mid =\sum \limits _{s=1}^{k-1}{\tilde{a}}_{k}(\sigma (s),s) + \mid i-j\mid \,, \end{aligned}$$

where

$$\begin{aligned} \sigma (s)= {\left\{ \begin{array}{ll} \pi (s), &{}\quad \hbox {if}\quad s<j \quad \hbox {and}\quad \pi (s)<i,\\ \pi (s)-1, &{}\quad \hbox {if}\quad s<j \quad \hbox { and }\quad \pi (s)>i,\\ \pi (s+1), &{}\quad \hbox {if}\quad s\ge j \quad \hbox { and }\quad \pi (s+1)<i,\\ \pi (s+1)-1, &{}\quad \hbox {if}\quad s\ge j \quad \hbox { and }\quad \pi (s+1)>i, \end{array}\right. } \end{aligned}$$

(20)

and

$$\begin{aligned} {\tilde{a}}_{k}(r,s)= {\left\{ \begin{array}{ll} \mid r-s \mid , &{}\quad \hbox {if }\quad s<j \quad \hbox { and }\quad r<i,\\ \mid r+1-s \mid , &{}\quad \hbox {if }\quad s<j \quad \hbox { and }\quad r\ge i,\\ \mid r-s-1 \mid , &{}\quad \hbox {if }\quad s\ge j \quad \hbox { and }\quad r<i,\\ \mid r+1-s-1 \mid , &{}\quad \hbox {if }\quad s\ge j \quad \hbox { and }\quad r\ge i, \end{array}\right. } \end{aligned}$$

(21)

for $r,s=1,2,\ldots ,k-1$ and $\pi \sim Uniform\left( {\mathbf {S}}_{{\mathbf {k}}}^{(i,j)}\right)$.

Lemma 1

Let $\displaystyle {\tilde{D}}_{F}\left( \sigma \right) =\sum \nolimits _{s=1}^{k-1}{\tilde{a}}_{k}(\sigma (s),s)$, where $\sigma (\cdot )$ and ${\tilde{a}}_{k}(\cdot ,\cdot )$ are given in (20) and (21), respectively. Then the distribution of ${\tilde{D}}_{F}$ is asymptotically normal with mean and variance

$$\begin{aligned} {\mathbf {E}}\left( {\tilde{D}}_{F}\right)&= \displaystyle \frac{k(k+1)}{3}-\frac{f(i)+f(j)-\mid i-j\mid }{k-1}, \end{aligned}$$

(22)

$$\begin{aligned} \mathbf {Var} \left( {\tilde{D}}_{F}\right)&= \displaystyle \frac{1}{k-2}\left\{ \sum _{\begin{array}{c} r=1 \\ r\ne i \end{array}}^{k}\sum _{\begin{array}{c} s=1 \\ s\ne j \end{array}}^{k} \left[ \mid r-s\mid +\frac{k(k+1)}{3(k-1)}\right. \right. \nonumber \\&\qquad \qquad \qquad -\frac{f(r)+f(s)-\mid i-s\mid -\mid r-j\mid }{k-1}\nonumber \\&\qquad \qquad \qquad \left. \left. -\frac{f(i)+f(j)-\mid i-j\mid }{(k-1)^{2}}\right] ^{2}\right\} , \end{aligned}$$

(23)

where

$$\begin{aligned} f(x)=\frac{x(x-1)+(k-x)(k-x+1)}{2}. \end{aligned}$$

Proof

From the definition of $\sigma$ in (20) it is easy to check that $\sigma \sim Uniform(\mathbf {S_{k-1}})$ for $\pi \sim Uniform\left( {\mathbf {S}}_{{\mathbf {k}}}^{(i,j)}\right)$. Therefore, Theorem 4 can be applied to the random variable ${\tilde{D}}_{F}$. By using (15), (21) and the expectation in (18), it follows that

$$\begin{aligned}&{\mathbf {E}}\left( {\tilde{D}}_{F}\right) {\mathop {=}\limits ^{(15)}}\frac{1}{k-1}\sum _{r=1}^{k-1}\sum _{s=1}^{k-1}{\tilde{a}}_{k}(r,s){\mathop {=}\limits ^{(21)}}\frac{1}{k-1}\sum _{\begin{array}{c} r=1 \\ r\ne i \end{array}}^{k}\sum _{\begin{array}{c} s=1 \\ s\ne j \end{array}}^{k}\mid r-s\mid \\&\quad =\frac{1}{k-1}\sum _{r=1}^{k}\sum _{s=1}^{k}\mid r-s\mid -\frac{1}{k-1}\sum _{r=1}^{k}\mid r-j\mid \\&\qquad -\frac{1}{k-1}\sum _{s=1}^{k}\mid i-s\mid +\frac{\mid i-j\mid }{k-1}\\&\quad {\mathop {=}\limits ^{(18)}}\frac{k(k+1)}{3}-\frac{f(i)+f(j)-\mid i-j\mid }{k-1}, \end{aligned}$$

where

$$\begin{aligned} f(x)=\sum _{r=1}^{k}\mid r-x\mid =\frac{x(x-1)+(k-x)(k-x+1)}{2}, \end{aligned}$$

(24)

for $x=1,2,\ldots ,k$. Using (16) of Theorem 4,

$$\begin{aligned} \mathbf {Var} \left( {\tilde{D}}_{F}\right) =\frac{1}{k-2}\sum _{\begin{array}{c} r=1 \\ r\ne i \end{array}}^{k}\sum _{\begin{array}{c} s=1 \\ s\ne j \end{array}}^{k}{\tilde{b}}_{k}^{2}(r,s), \end{aligned}$$

(25)

where

$$\begin{aligned} {\tilde{b}}_{k}(r,s)=\mid r-s\mid - \sum _{\begin{array}{c} l=1 \\ l\ne i \end{array}}^{k}\frac{\mid l-s\mid }{k-1}-\sum _{\begin{array}{c} m=1 \\ m\ne j \end{array}}^{k}\frac{\mid r-m\mid }{k-1}+\frac{1}{\left( k-1\right) ^{2}} \sum _{\begin{array}{c} l=1 \\ l\ne i \end{array}}^{k}\sum _{\begin{array}{c} m=1 \\ m\ne j \end{array}}^{k}\mid l-m\mid , \end{aligned}$$

for $r,s=1,2,\ldots ,k$. Simplifying this expression gives

$$\begin{aligned} {\tilde{b}}_{k}(r,s)= & {} \mid r-s\mid -\frac{f(r)+f(s)-\mid i-s\mid -\mid r-j\mid }{k-1} \nonumber \\&+\frac{k(k+1)}{3(k-1)}-\frac{f(i)+f(j)-\mid i-j\mid }{(k-1)^{2}}. \end{aligned}$$

(26)

The variance of ${\tilde{D}}_{F}$ given in (23) is obtained by substituting (26) in formula (25).

From (24) it is easy to check that

$$\begin{aligned} \frac{k^{2}-1}{4} \le f(x)\le \frac{k(k-1)}{2} \quad \hbox {for } 1\le x \le k. \end{aligned}$$

(27)

Combining

$$\begin{aligned} 1\le \mid x-y\mid \le k-1 \quad \hbox {for } 1\le x,y \le k, \end{aligned}$$

with (26) and (27), it follows that

$$\begin{aligned} \mid r-s \mid - \frac{2k}{3}+\epsilon _{1} \le {\tilde{b}}_{k}(r,s) \le \mid r-s \mid - \frac{k}{6}+\epsilon _{2}, \end{aligned}$$

where $r,s=1,2,\ldots ,k$, $\displaystyle \lim _{k \rightarrow \infty }\frac{\epsilon _{1}}{k}=0$ and $\displaystyle \lim _{k \rightarrow \infty }\frac{\epsilon _{2}}{k}=0$. Therefore, there exists a constant $c_{1}>0$ such that

$$\begin{aligned} \max _{1 \le r,s \le k}{\tilde{b}}_{k}^{2}(r,s) \le c_{1}k^{2}, \end{aligned}$$

(28)

and a number $N>0$ such that for $k\ge N$

$$\begin{aligned} \mid r-s \mid - k \le {\tilde{b}}_{k}(r,s) \le \mid r-s \mid - \frac{k}{7} \; . \end{aligned}$$

Suppose that r is a fixed index from the set $\left\{ 1,2,\ldots ,k\right\}$. Then for $k\ge N$

$$\begin{aligned} \sum _{\begin{array}{c} s=1 \\ s\ne j \end{array}}^{k}{\tilde{b}}_{k}^{2}(r,s)= & {} \sum _{s=1}^{k}{\tilde{b}}_{k}^{2}(r,s)-{\tilde{b}}_{k}^{2}(r,j) \ge \sum _{\mid r-s\mid =0}^{k/7}{\tilde{b}}_{k}^{2}(r,s)-{\tilde{b}}_{k}^{2}(r,j)\\\ge & {} \sum _{v=0}^{k/7}\left( v-\frac{k}{7}\right) ^{2}-{\tilde{b}}_{k}^{2}(r,j), \end{aligned}$$

where $\displaystyle \sum _{\mid r-s\mid =0}^{k/7}$ is a summation over all values of s such that $\displaystyle 0\le \mid r-s\mid \le \frac{k}{7}$. Thus, for $k\ge N$ there exists a constant $c_{2}>0$ such that

$$\begin{aligned} \sum _{\begin{array}{c} s=1 \\ s\ne j \end{array}}^{k}{\tilde{b}}_{k}^{2}(r,s) \ge c_{2}k^{3} \; . \end{aligned}$$

(29)

By using (28) and (29),

$$\begin{aligned} \lim _{k \rightarrow \infty } \displaystyle \frac{\displaystyle \max \nolimits _{1 \le r,s \le k}{\tilde{b}}_{k}^{2}(r,s)}{\displaystyle \frac{1}{k-1}\sum \nolimits _{\begin{array}{c} r=1 \\ r\ne i \end{array}}^{k}\sum \nolimits _{\begin{array}{c} s=1 \\ s\ne j \end{array}}^{k}{\tilde{b}}_{k}^{2}(r,s)} \le \lim _{k \rightarrow \infty } \frac{c_{1}k^2}{\displaystyle \frac{1}{k-1}\sum \nolimits _{\begin{array}{c} r=1 \\ r\ne i \end{array}}^{k} c_{2}k^3} = 0, \end{aligned}$$

i.e., the condition (17) of Theorem 4 is fulfilled and the distribution of ${\tilde{D}}_{F}$ is asymptotically normal. $\square$

Similarly to $\left\{ D_{F}^{(i,j)}\right\} _{i,j=1}^{k}$, consider the random variables $D_{R}^{(i,j)}=d_{R}\big (\pi ,e_{k}\big )$ based on Spearman’s rho. For a fixed pair (i, j),

$$\begin{aligned} D_{R}^{(i,j)}=\sum \limits _{s=1}^{k}\big (\pi (s)-s\big )^{2}=\sum \limits _{\begin{array}{c} s=1 \\ s\ne j \end{array}}^{k}\big (\pi (s)-s\big )^{2} + \left( i-j\right) ^{2}=\sum \limits _{s=1}^{k-1}{\bar{a}}_{k}(\sigma (s),s) + \left( i-j\right) ^{2} \,, \end{aligned}$$

where

$$\begin{aligned} {\bar{a}}_{k}(r,s)= {\left\{ \begin{array}{ll} \left( r-s \right) ^{2}, &{}\quad \hbox {if }\quad s<j \quad \hbox { and }\quad r<i,\\ \left( r+1-s \right) ^{2}, &{}\quad \hbox {if }\quad s<j \quad \hbox { and }\quad r\ge i,\\ \left( r-s-1 \right) ^{2}, &{}\quad \hbox {if }\quad s\ge j \quad \hbox { and }\quad r<i,\\ \left( r+1-s-1 \right) ^{2}, &{}\quad \hbox {if }\quad s\ge j \quad \hbox { and }\quad r\ge i, \end{array}\right. } \end{aligned}$$

(30)

for $r,s=1,2,\ldots ,k-1$, $\pi \sim Uniform\left( {\mathbf {S}}_{{\mathbf {k}}}^{(i,j)}\right)$ and $\sigma (s)$ is defined as in (20).

Lemma 2

Let $\displaystyle {\bar{D}}_{R}\left( \sigma \right) =\sum \nolimits _{s=1}^{k-1}{\bar{a}}_{k}(\sigma (s),s)$, where $\sigma (\cdot )$ and ${\bar{a}}_{k}(\cdot ,\cdot )$ are given in (20) and (30), respectively. Then the distribution of ${\bar{D}}_{R}$ is asymptotically normal with mean and variance

$$\begin{aligned} {\mathbf {E}}\left( {\bar{D}}_{R}\right)&= \displaystyle \frac{k^{2}(k+1)}{6}-\frac{h(i)+h(j)-\left( i-j\right) ^{2} }{k-1},\\ \mathbf {Var} \left( {\bar{D}}_{R}\right)&= \displaystyle \frac{1}{k-2}\left\{ \sum _{\begin{array}{c} r=1 \\ r\ne i \end{array}}^{k}\sum _{\begin{array}{c} s=1 \\ s\ne j \end{array}}^{k} \left[ \left( r-s \right) ^{2} +\frac{k^{2}(k+1)}{6(k-1)}\right. \right. \\&\qquad \qquad \qquad -\frac{h(r)+h(s)-\left( i-s\right) ^{2} -\left( r-j\right) ^{2} }{k-1}\\&\qquad \qquad \qquad \left. \left. -\frac{h(i)+h(j)-\left( i-j\right) ^{2} }{(k-1)^{2}}\right] ^{2}\right\} , \end{aligned}$$

where

$$\begin{aligned} h(x)=\frac{x(x-1)(2x-1)+(k-x)(k-x+1)(2k-2x+1)}{6}. \end{aligned}$$

Proof

By using (19) and the fact that for $x\in \left\{ 1,2,\ldots ,k\right\}$

$$\begin{aligned} h(x)= & {} \sum _{r=1}^{k}\left( r-k\right) ^{2}\nonumber \\= & {} \frac{x(x-1)(2x-1)+(k-x)(k-x+1)(2k-2x+1)}{6}, \end{aligned}$$

(31)

${\mathbf {E}}\left( {\bar{D}}_{R}\right)$ and $\mathbf {Var} \left( {\bar{D}}_{R}\right)$ can be evaluated in a similar way as in the proof of Lemma 1.

Now, consider the quantities

$$\begin{aligned} {\bar{b}}_{k}(r,s)= & {} \left( r-s\right) ^{2} +\frac{k^{2}(k+1)}{6(k-1)}-\frac{h(r)+h(s)-\left( i-s\right) ^{2} -\left( r-j\right) ^{2} }{k-1} \\&-\frac{h(i)+h(j)-\left( i-j\right) ^{2} }{(k-1)^{2}} \end{aligned}$$

for $r,s=1,2,\ldots ,k$. Since (31)

$$\begin{aligned} \frac{k(k^{2}+2)}{12} \le h(x)\le \frac{k(k-1)(2k-1)}{6} \quad \hbox {for } 1\le x \le k, \end{aligned}$$

it follows that

$$\begin{aligned} \left( r-s \right) ^{2} -\frac{k^{2}}{2}+\epsilon _{1} \le {\bar{b}}_{k}(r,s) \le \left( r-s \right) ^{2} +\epsilon _{2}, \end{aligned}$$

where $r,s=1,2,\ldots ,k$, $\displaystyle \lim _{k \rightarrow \infty }\frac{\epsilon _{1}}{k^{2}}=0$ and $\displaystyle \lim _{k \rightarrow \infty }\frac{\epsilon _{2}}{k^{2}}=0$. Hence, there exists a constant $c_{1}>0$ such that

$$\begin{aligned} \max _{1 \le r,s \le k}{\bar{b}}_{k}^{2}(r,s) \le c_{1}k^{4}. \end{aligned}$$

(32)

Further, fix the indexes $\displaystyle 1\le r,s \le \frac{k}{4}$. Then, since

$$\begin{aligned} \frac{k(7k^{2}+12k+8)}{48} \le h(x)\le \frac{k(k-1)(2k-1)}{6} \quad \hbox {for } 1\le x \le \frac{k}{4}, \end{aligned}$$

it follows that

$$\begin{aligned} \left( r-s \right) ^{2} -\frac{k^{2}}{2}+\epsilon _{3} \le {\bar{b}}_{k}(r,s) \le \left( r-s \right) ^{2} -\frac{k^{2}}{8}+\epsilon _{4}, \end{aligned}$$

where $\displaystyle \lim _{k \rightarrow \infty }\frac{\epsilon _{3}}{k^{2}}=0$ and $\displaystyle \lim _{k \rightarrow \infty }\frac{\epsilon _{4}}{k^{2}}=0$. Thus, there exists a number $N>0$ such that for $k\ge N$

$$\begin{aligned} \left( r-s \right) ^{2} - k^{2} \le {\bar{b}}_{k}(r,s) \le \left( r-s \right) ^{2} -\frac{k^{2}}{9}. \end{aligned}$$

Hence, for $k\ge N$

$$\begin{aligned}&\sum _{\begin{array}{c} r=1 \\ r\ne i \end{array}}^{k} \sum _{\begin{array}{c} s=1 \\ s\ne j \end{array}}^{k}{\bar{b}}_{k}^{2}(r,s)\ge \sum _{\begin{array}{c} r=1 \\ r\ne i \end{array}}^{k/4} \sum _{\begin{array}{c} s=1 \\ s\ne j \end{array}}^{k/4}{\bar{b}}_{k}^{2}(r,s)=\sum _{\begin{array}{c} r=1 \\ r\ne i \end{array}}^{k/4} \left\{ \sum _{s=1}^{k/4}{\bar{b}}_{k}^{2}(r,s)-{\bar{b}}_{k}^{2}(r,j)\right\} \\&\quad \ge \sum _{\begin{array}{c} r=1 \\ r\ne i \end{array}}^{k/4} \left\{ \sum _{\left( r-s\right) ^{2}=0}^{k^{2}/9}{\bar{b}}_{k}^{2}(r,s)-{\bar{b}}_{k}^{2}(r,j)\right\} \ge \sum _{\begin{array}{c} r=1 \\ r\ne i \end{array}}^{k/4} \left\{ \sum _{v=0}^{k/3}\left( v^{2}-\frac{k^{2}}{9}\right) ^{2}-{\bar{b}}_{k}^{2}(r,j)\right\} , \end{aligned}$$

where $\sum _{\left( r-s\right) ^{2}=0}^{k^{2}/9}$ is a summation over all values of s, such that $\displaystyle 0\le \left( r-s\right) ^{2} \le \frac{k^{2}}{9}$. Thus, for $k\ge N$ there exists a constant $c_{2}>0$, such that

$$\begin{aligned} \sum _{\begin{array}{c} r=1 \\ r\ne i \end{array}}^{k} \sum _{\begin{array}{c} s=1 \\ s\ne j \end{array}}^{k}{\bar{b}}_{k}^{2}(r,s) \ge c_{2}k^{6} \; . \end{aligned}$$

(33)

From (32) and (33), it is easy to check that the condition (17) of Theorem 4 is fulfilled and the distribution of ${\bar{D}}_{R}$ is asymptotically normal. $\square$

Proof of Theorem 2

From (5), (6) and (7), it follows that

$$\begin{aligned} q(i,j,k,\theta )=\sum _{\pi (j)=i} \exp \left( \theta d(\pi ,e_{k})-\psi _{k}(\theta )\right) =\frac{(k-1)!{\tilde{m}}_{k-1}(\theta )}{k!m_{k}(\theta )}=\frac{1}{k}\frac{{\tilde{m}}_{k-1}(\theta )}{m_{k}(\theta )}, \end{aligned}$$

where $m_{k}(\cdot )$ and ${\tilde{m}}_{k-1}(\cdot )$ are the moment generating functions of $D_{F}(\pi )$ and $D_{F}^{(i,j)}(\sigma )$ for $\pi \sim Uniform(\mathbf {S_{k}})$ and $\sigma \sim Uniform\left( {\mathbf {S}}_{{\mathbf {k}}}^{(i,j)}\right)$. Since $D_{F}^{(i,j)}={\tilde{D}}_{F}+\mid i-j\mid$ and according to Lemma 1${\tilde{D}}_{F}$ is asymptotically normal, it follows that $D_{F}^{(i,j)}$ is asymptotically normal. Therefore, $m_{k}(\cdot )$ and ${\tilde{m}}_{k-1}(\cdot )$ can be approximated with the moment generating function of the normal distribution and

$$\begin{aligned} q(i,j,k,\theta ) \displaystyle \frac{k}{ \exp \left( \theta \mu + \frac{\theta ^{2}\nu ^{2}}{2}\right) } \xrightarrow [k \rightarrow \infty ] \displaystyle 1, \end{aligned}$$

where $\mu ={\mathbf {E}}\left( D_{F}^{(i,j)}\right) -{\mathbf {E}}\left( D_{F}\right)$ and $\nu ^{2}=\mathbf {Var}\left( D_{F}^{(i,j)}\right) -\mathbf {Var}\left( D_{F}\right)$.

The values of $\mu$ and $\nu ^{2}$ given in Theorem 2 are obtained by combining formulas (18), (22) and (23) with

$$\begin{aligned} {\mathbf {E}}\left( D_{F}^{(i,j)}\right) ={\mathbf {E}}\left( {\tilde{D}}_{F}\right) +\mid i-j\mid \quad \hbox {and} \quad \mathbf {Var}\left( D_{F}^{(i,j)}\right) =\mathbf {Var}\left( {\tilde{D}}_{F}\right) . \end{aligned}$$

$\square$

Proof of Theorem 3

The proof is similar to the proof of Theorem 2 using Lemma 2.

$\square$