Selection of statistics for a multinomial goodness-of-fit test and a test of independence for a multi-way contingency table when data are sparse

Abstract

For the goodness-of-fit test for a multinomial distribution and test of some kinds of independence of a multi-way contingency table, we consider test statistics based on the $\phi$-divergence family. Members of the $\phi$-divergence family of statistics all have an equivalent Chi-square limiting distribution under the null hypothesis. We consider a second-order correction term as an index of investigating whether the distributions of statistics are close to the Chi-square limiting distribution. We derive properties for the second-order correction term for selecting a $\phi$-divergence statistic when we consider an asymptotic test in the case of data being sparse. We propose a selection of statistics when we use a power divergence family of statistics and the family of Rukhin’s statistics as special $\phi$-divergence statistics.

Appendices

Appendix: Proof of Theorems 1 and 2

If we assume that statistic $G_{\phi }$ has a continuous distribution, the characteristic function of $G_{\phi }$ is evaluated as

$$\begin{aligned} \psi _{\phi }^G(t)=(1-2it)^{-\lambda /2}+\frac{1}{n} \sum _{j=0}^3(1-2it)^{-(\lambda +2j)/2}r_j^{\phi } + o(n^{-1}), \end{aligned}$$

where

$$\begin{aligned} r_0^{\phi }= & {} \displaystyle {\frac{1}{12}}\biggl [ -(S-1)\biggr ], \\ r_1^{\phi }= & {} \displaystyle {\frac{1}{24}}\biggl [ 3(3S -k^2-2k) +6\phi '''(1)(S -k^2 ) \\{} & {} +\{\phi '''(1)\}^2 (5S -3k^2-6k+4 ) -3\phi ^{(4)}(1)(S - 2k + 1 )\biggr ], \\ r_2^{\phi }= & {} \displaystyle {\frac{1}{24}}\biggl [ -6(2S - k^2-2k+1 ) -4\phi '''(1)(4S - 3k^2 - 3k + 2) \\{} & {} -2\{\phi '''(1)\}^2( 5S - 3k^2 - 6k + 4) +3\phi ^{(4)}(1)(S -2k+1 )\biggr ], \\ r_3^{\phi }= & {} \displaystyle {\frac{1}{24}}\biggl [ \{1+\phi '''(1)\}^2 (5S -3k^2-6k+4 )\biggr ], \end{aligned}$$

and S is given by (7). Here, $r_j^\phi \; (j=0, 1, 2, 3)$ satisfy

$$\begin{aligned} \sum _{j=0}^3r_j^\phi =0. \end{aligned}$$

(A.1)

For an arbitrary natural number s, let $\psi _{\phi }^{G(s)}(t)$ be the sth derivatives of $\psi _{\phi }^G(t)$. Then

$$\begin{aligned} \psi _{\phi }^{G(s)}(t)= & {} i^s\Bigl [\lambda (\lambda +2) \cdots \{\lambda +2(s-1)\} (1-2it)^{-(\lambda +2s)/2} \\{} & {} +\frac{1}{n}\sum _{j=0}^3(\lambda +2j)(\lambda +2j+2) \cdots \{\lambda +2j+2(s-1)\}\\{} & {} \times (1-2it)^{-(\lambda +2j+2s)/2}r_j^{\phi } + o(n^{-1})\Bigr ]. \end{aligned}$$

Therefore, the sth moment about the origin of the statistic $G_{\phi }$ is evaluated as

$$\begin{aligned} E\{(G_{\phi })^s \vert H_0\} = \lambda (\lambda +2) \cdots \{\lambda +2(s-1)\} + \frac{1}{n}m^G_{\phi }(s) +o(n^{-1}) \; (s=1,2,\ldots ), \end{aligned}$$

where

$$\begin{aligned} m_{\phi }^G(s)= \sum _{j=0}^3(\lambda +2j)(\lambda +2j+2) \cdots \{\lambda +2j+2(s-1)\}r_j^{\phi } \; (s=1,2,\ldots ).\nonumber \\ \end{aligned}$$

(A.2)

If we put

$$\begin{aligned} m_{\phi }^G(s)=\sum _{\ell =0}^sb_{s,\ell }\lambda ^\ell \; (s=1,2,\ldots ), \end{aligned}$$

(A.3)

from (A.1) and (A.2), coefficients $b_{s,s},\;b_{s,s-1}$, and $b_{s,s-2}$ are calculated as follows:

$$\begin{aligned} b_{s,s}= & {} \sum _{j=0}^{3}r_j^{\phi }=0 \; (s=1,2,\ldots ), \end{aligned}$$

(A.4)

$$\begin{aligned} b_{s,s-1}= & {} 2s\sum _{j=1}^3jr_j^\phi \nonumber \\= & {} \frac{s}{12}\left\{ 4\phi ^{'''}(1)(S-3k+2) +3\phi ^{(4)}(1)(S-2k+1)\right\} \nonumber \\{} & {} \qquad (s =1,2, \ldots ), \end{aligned}$$

(A.5)

and

$$\begin{aligned} \quad \quad \quad b_{s,s-2}= & {} \frac{s(s-1)}{12} \Big [6(S-k^2-2k+2)\nonumber \\{} & {} +4\phi ^{'''}(1)\left\{ (s+7)S-3k^2-(s+4)(3k-2)\right\} \nonumber \\{} & {} +3\phi ^{(4)}(1)(s+2)(S-2k+1)\nonumber \\{} & {} +2\{\phi ^{'''}(1)\}^2(5S-3k^2-6k+4)\Big ] \; (s=2,3,\ldots ). \qquad \quad \quad \end{aligned}$$

(A.6)

From the assumption that $q_j = O(k^{-1}) \;(j=1, \ldots , k)$ and (7), we find that $S=O(k^2)$. Therefore

$$\begin{aligned} r_j^\phi = O(k^2) \; (j=0,1,2,3). \end{aligned}$$

(A.7)

holds. Furthermore,

$$\begin{aligned} \lambda = k-1 = O(k). \end{aligned}$$

(A.8)

Then, from (A.7) and (A.8),

$$\begin{aligned} \sum _{\ell =0}^{s-3}b_{s,\ell }\lambda ^\ell = O(k^{s-1})\; (s=3,4 \ldots ). \end{aligned}$$

(A.9)

Therefore, from (A.3)–(A.9), we obtain the following expression:

$$\begin{aligned} \begin{array}{lcl} \quad \quad m^G_\phi (s) &{}=&{} \displaystyle {\frac{s}{12}} \left\{ 4\phi ^{'''}(1)(S-3k+2) +3\phi ^{(4)}(1)(S-2k+1)\right\} \lambda ^{s-1} \\ &{} &{} + \frac{\displaystyle {1}}{\displaystyle {12}} s(s-1) \Big [6(S-k^2-2k+2)\\ &{}&{} +4\phi ^{'''}(1)\left\{ (s+7)S-3k^2-(s+4)(3k-2)\right\} \\ &{}&{} +3\phi ^{(4)}(1)(s+2)(S-2k+1) \\ &{}&{} +2\{\phi ^{'''}(1)\}^2(5S-3k^2-6k+4)\Big ] \lambda ^{s-2} \\ &{} &{} + O(k^{s-1}) \\ &{}=&{} \displaystyle {\frac{s}{12}} \Big \{4\phi ^{'''}(1)+3\phi ^{(4)}(1)\Big \}\left( \frac{\displaystyle {S}}{\displaystyle {k^2}}\right) k^{s+1}\\ &{}&{}+ \displaystyle {\frac{s}{12}} \Big [(s-1)\Big \{(s+1)(4\phi ^{'''}(1)+3\phi ^{(4)}(1))\\ &{}&{}+10(\phi ^{'''}(1)+1)^2 -4 \Big \} \left( \frac{\displaystyle {S}}{\displaystyle {k^2}}\right) \\ &{}&{}-2\Big \{(4\phi ^{'''}(1)+3\phi ^{(4)}(1))+3(s-1)(\phi ^{'''}(1)+1)^2 +2\phi ^{'''}(1)\Big \}\Big ]k^s \\ &{}&{} +O(k^{s-1})\; (s=1,2,\ldots ). \qquad \qquad \qquad \qquad \quad \qquad \qquad \quad \end{array} \end{aligned}$$

(A.10)

The second-order correction term $m^G_{\phi }(s)\; (s=1,2,\ldots )$ is evaluated as

$$\begin{aligned} m_{\phi }^G(s) = \{4\phi ^{'''}(1)+3\phi ^{(4)}(1)\}O(k^{s+1})+O(k^s) \; (s=1,2,\ldots ). \end{aligned}$$

(A.11)

From (A.11), if we put $m^G_{\phi }(s) = 0$, then $4\phi ^{'''}(1)+3\phi ^{(4)}(1) = o(1),$ which implies $4\phi ^{'''}(1)+3\phi ^{(4)}(1) \rightarrow 0$ as $k \rightarrow \infty$. We have completed the proof of Theorem 1. On the other hand, by substituting (8) in (A.10), we obtain the result of Theorem 2.

Proof of Corollary 3

Since the former half of Corollary 3 is immediately shown from Theorem 2, we prove the latter half. Proof of $\vert A_1 \vert < \vert B_1 \vert$ is straightforward. The relation $\vert A_s \vert > \vert B_s \vert$ is equivalent to $(A_s - B_s)(A_s + B_s) > 0$. Since

$$\begin{aligned} A_s-B_s = \frac{25}{54}s(s-1)\left\{ \frac{S}{k^2}-1\right\} +\frac{1}{27}s(5s-8), \end{aligned}$$

by noting $S/k^2 \ge 1$, $A_s-B_s > 0$ when $s \ge 2$. Therefore, $\vert A_s \vert > \vert B_s \vert$ is equivalent to $A_s+B_s > 0$, when $s \ge 2$. Since

$$\begin{aligned} A_s+B_s = \frac{25}{54}s(s-1)\left\{ \frac{S}{k^2} - C(s) \right\} , \end{aligned}$$

$S/k^2 > C(s)$ implies $\vert A_s \vert > \vert B_s \vert$, when $s \ge 2$. Similarly, $1 \le S/k^2 < C(s)$ implies $\vert A_s \vert < \vert B_s \vert$, when $s \ge 2$. We obtain the results of the latter half.

Proof of Theorems 3 and 4

Let the characteristic function of $M_\phi ^*$ be $\psi _\phi ^M(t)$. By assuming that the distribution of $M_\phi ^*$ is continuous, $\psi _\phi ^M(t)$ is evaluated as follows:

$$\begin{aligned} \psi _\phi ^M(t) = (1-2it)^{-\mu /2} +\frac{1}{n}\sum _{j=0}^3(1-2it)^{-(\mu +2j)/2}d_j^\phi + o(n^{-1}), \end{aligned}$$

(A.12)

where

$$\begin{aligned} d_0^\phi= & {} \frac{1}{12}\left[ -\prod _{m=1}^MS_m+\sum _{m=1}^MS_m-(M-1)\right] ,\\ d_1^\phi= & {} \displaystyle {\frac{1}{24}}\biggl [ 3(M-1)K^2\{\phi '''(1)+1\}^2 \\{} & {} +\{-3\phi ^{(4)}(1)+5(\phi '''(1))^2+6\phi '''(1)+9\}\displaystyle {\prod _{m=1}^MS_m} \\{} & {} -3\{\phi '''(1)+1\}^2K^2\displaystyle {\sum _{m=1}^M\frac{S_m}{J_m^2}}+6\{\phi ^{(4)}(1)-(\phi '''(1))^2-1\}K \sum _{m=1}^M\displaystyle {\frac{S_m}{J_m}} \\{} & {} +6(M-1)K\{-\phi ^{(4)}(1)+(\phi '''(1))^2+1\} \\{} & {} +\{-3\phi ^{(4)}(1)+4(\phi '''(1))^2\}\displaystyle {\sum _{m=1}^MS_m} \\{} & {} -6\{\phi ^{(4)}(1)-2(\phi '''(1))^2\}\displaystyle {\sum _{m=1}^{M-1}\sum _{\ell =m+1}^MJ_mJ_\ell } \\{} & {} +6\{\phi ^{(4)}(1)-2(\phi '''(1))^2\}(M-1)\displaystyle {\sum _{m=1}^MJ_m} \\{} & {} +\{-3(M-1)^2\phi ^{(4)}(1)+2(M-1)(3M-2)(\phi '''(1))^2\}\biggr ],\\ d_2^\phi= & {} \displaystyle {\frac{1}{24}}\biggl [ -6(M-1)K^2\{\phi '''(1)+1\}^2 \\{} & {} +\{3\phi ^{(4)}(1)-10(\phi '''(1))^2-16\phi '''(1)-12\}\displaystyle {\prod _{m=1}^MS_m} \\{} & {} +6\{\phi '''(1)+1\}^2K^2\displaystyle {\sum _{m=1}^M\frac{S_m}{J_m^2}} \\{} & {} +6\{-\phi ^{(4)}(1)+2(\phi '''(1))^2+2\phi '''(1)+2\} K\displaystyle {\sum _{m=1}^M}\displaystyle {\frac{S_m}{J_m}} \\{} & {} +6\{\phi ^{(4)}(1)-2(\phi '''(1))^2-2\phi '''(1)-2\}(M-1)K\\{} & {} +\{3\phi ^{(4)}(1)-8(\phi '''(1))^2-8\phi '''(1)-6\}\displaystyle {\sum _{m=1}^MS_m} \\{} & {} +6\{\phi ^{(4)}(1)-4(\phi '''(1))^2-4\phi '''(1)-2\}\displaystyle {\sum _{m=1}^{M-1}\sum _{\ell =m+1}^MJ_mJ_\ell } \\{} & {} +6\{-\phi ^{(4)}(1)+4(\phi '''(1))^2+4\phi '''(1)+2\}(M-1)\displaystyle {\sum _{m=1}^MJ_m}\\{} & {} +\{3(M-1)^2(\phi ^{(4)}(1)-2)-4(M-1)(3M-2)((\phi '''(1))^2+\phi '''(1))\}\biggr ],\\ d_3^\phi= & {} \displaystyle {\frac{1}{24}}\{\phi '''(1)+1\}^2\biggl [3(M-1)K^2+5\displaystyle {\prod _{m=1}^M}S_m -3K^2\displaystyle {\sum _{m=1}^M\frac{S_m}{J_m^2}} -6K\displaystyle {\sum _{m=1}^M\frac{S_m}{J_m}} \\{} & {} +6(M-1)K + 4\displaystyle {\sum _{m=1}^MS_m}+12\displaystyle {\sum _{m=1}^{M-1}\sum _{\ell =m+1}^MJ_mJ_\ell } -12(M-1)\displaystyle {\sum _{m=1}^MJ_m} \\{} & {} +2(M-1)(3M-2)\biggr ], \end{aligned}$$

$S_m \; (m=1, \ldots , M)$ is given by (18), and $\mu$ is given by (15). Here, $d_j^\phi \; (j=0, 1, 2, 3)$ satisfy

$$\begin{aligned} \sum _{j=0}^3d_j^\phi =0. \end{aligned}$$

(A.13)

By (A.12), the sth moment about the origin of $M_\phi ^*$ under null hypothesis $H_0^M$ given by (12) is expressed as follows:

$$\begin{aligned} E\{(M_\phi ^*)^s \vert H_0^M\} = \mu (\mu +2) \cdots \{\mu +2(s-1)\} + \frac{1}{n}m_\phi ^M(s)+o(n^{-1}) \; (s=1, 2, \ldots ), \end{aligned}$$

where $m_\phi ^M(s)$ is the second-order correction term of the sth moment about the origin of $M_\phi ^*$ and is given by

$$\begin{aligned} m_\phi ^M(s)=\sum _{j=0}^3(\mu +2j)(\mu +2j+2)\cdots \{\mu +2j+2(s-1)\}d_j^\phi \; (s=1, 2, \ldots ).\nonumber \\ \end{aligned}$$

(A.14)

In (A.14), let the coefficient of $\mu ^\ell$ be $c_{s,\ell }$ for $\ell =0,1,\ldots ,s$, then $m^M_\phi (s)$ can be written as

$$\begin{aligned} m^M_\phi (s)=\sum _{\ell =0}^{s-1}c_{s,\ell }\mu ^{\ell } \; (s=1,2,\ldots ), \end{aligned}$$

(A.15)

since $c_{s,s}=0$, which is shown by (A.13). Here, $m_\phi ^M(s)$ is a polynomial of $\mu$. From (A.14), $c_{s,s-1}$, which is the coefficient of the maximum degree for $\mu$, is given as follows:

$$\begin{aligned} c_{s,s-1} = \displaystyle {2s\sum _{j=1}^3 jd_j^\phi }. \end{aligned}$$

(A.16)

Therefore, by substituting the expression of $d_j^\phi \; (j=1,2,3)$ in (A.16), we obtain

$$\begin{aligned} \begin{array}{lll} c_{s,s-1} &{}=&{}\displaystyle {\frac{s}{12}}\biggl [\{3\phi ^{(4)}(1)+4\phi '''(1)\}\displaystyle {\prod _{m=1}^MS_m} -6\{\phi ^{(4)}(1)+2\phi '''(1)\}K \displaystyle {\sum _{m=1}^M\frac{S_m}{J_m}} \\ &{} &{} + 6\{\phi ^{(4)}(1)+2\phi '''(1)\}(M-1)K + \{3\phi ^{(4)}(1)+8\phi '''(1)\}\displaystyle {\sum _{m=1}^MS_m} \\ &{} &{} +6\{\phi ^{(4)}(1)+4\phi '''(1)+2\}\displaystyle {\sum _{m=1}^{M-1}\sum _{\ell =m+1}^MJ_mJ_\ell } \\ &{} &{} -6\{\phi ^{(4)}(1)+4\phi '''(1)+2\}(M-1)\displaystyle {\sum _{m=1}^MJ_m} \\ &{} &{} +\{3(M-1)^2\phi ^{(4)}(1)+4(M-1)(3M-2)\phi '''(1)+6M(M-1)\}\biggr ] \; \\ &{} &{} \qquad (s=1,2,\ldots ). \end{array} \end{aligned}$$

(A.17)

By the assumption that $p_{j_1 \ldots j_M} =O(K^{-1}) \; (j_m=1, \ldots , J_m; m=1, \ldots , M),$ $p_{\cdot (m, j_m)}=O(J_m^{-1}) \; (j_m=1, \ldots , J_m; m=1, \ldots , M)$ hold. Then, $S_m=O(J_m^2)$ $\; (m =1, \ldots , M)$ hold. Therefore, we consider evaluating $m_\phi ^M(s)$ given by (A.15) as the order of $J_1, \ldots , J_M$. We obtain the following relations:

$$\begin{aligned} \left. \begin{array}{rll} \displaystyle {\prod _{m=1}^M}S_m &{}=&{} O\left( \left( \displaystyle {\prod _{m=1}^M}J_m\right) ^2\right) \\ K\displaystyle {\sum _{m=1}^M}\frac{S_m}{J_m} &{}=&{} O\left( \left( \displaystyle {\prod _{m=1}^M}J_m\right) \left( \displaystyle {\sum _{m=1}^M}J_m\right) \right) \\ K &{}=&{} O\left( \displaystyle {\prod _{m=1}^M}J_m\right) \\ \displaystyle {\sum _{m=1}^M}S_m &{}=&{} O\left( \displaystyle {\sum _{m=1}^M}J_m^2\right) \\ \displaystyle {\sum _{m=1}^{M-1} \sum _{\ell =m+1}^M}J_m J_\ell &{}=&{} O\left( \displaystyle {\sum _{m=1}^{M-1}\sum _{\ell =m+1}^M}J_m J_\ell \right) \\ \displaystyle {\sum _{m=1}^M}J_m &{}=&{} O\left( \displaystyle {\sum _{m=1}^M}J_m \right) \\ M &{}=&{} O(1). \end{array} \right\} \end{aligned}$$

(A.18)

From the above discussion, by assumption $O(J_1) = \cdots = O(J_M)$, we find that $\prod _{m=1}^M S_m$ is the highest order in terms of (A.18). On the other hand, the following evaluation holds:

$$\begin{aligned} d_j^\phi =O\left( \left( \displaystyle {\prod _{m=1}^M J_m} \right) ^2 \right) \; (j=0, 1, 2, 3) \end{aligned}$$

and

$$\begin{aligned} \mu = O\left( \displaystyle {\prod _{m=1}^M J_m} \right) . \end{aligned}$$

Therefore, in the case of statistic for testing complete independence in a multi-way contingency table

$$\begin{aligned} \left. \begin{array}{rll} m_\phi ^M(s) &{}=&{} K^{s-1}c_{s,s-1}-(s-1)K^{s-2}\displaystyle {\left( \sum _{m=1}^MJ_m\right) }c_{s,s-1} \\ &{}&{}+O(K^s)+O\left( K^{s-1}\displaystyle {\left( \sum _{m=1}^M J_m \right) ^2}\right) , \qquad \qquad \qquad \qquad \quad \quad \; \end{array} \right. \end{aligned}$$

(A.19)

since $c_{s,\ell }=O(K^2),$

$$\begin{aligned} \begin{array}{rll} \mu ^{s-1} &{}=&{} \left( K-\displaystyle {\sum _{m=1}^MJ_m+M-1}\right) ^{s-1} \\ &{}=&{} K^{s-1}-(s-1)K^{s-2}\displaystyle {\left( \sum _{m=1}^M J_m\right) } +O\left( K^{s-2}\right) +O\left( K^{s-3}\displaystyle {\left( \sum _{m=1}^M J_m\right) ^2}\right) \end{array} \end{aligned}$$

and

$$\begin{aligned} \mu ^{s-2} = \left( K-\displaystyle {\sum _{m=1}^MJ_m+M-1}\right) ^{s-2} = O\left( K^{s-2}\right) . \end{aligned}$$

From (A.19), we note that it is not necessary to substitute the expression of coefficient $c_{s,s-2}$ in (A.19) to evaluate $m_\phi ^M(s)$, which is different from the case of multinomial goodness-of-fit test statistic. By substituting (A.17) for $c_{s, s-1}$ in (A.19), we obtain the evaluation of $m_\phi ^M(s)$ as follows:

$$\begin{aligned} m_\phi ^M(s)= & {} \displaystyle {\frac{1}{12}}s\left\{ 4\phi '''(1)+3\phi ^{(4)}(1)\right\} K^{s-1}\displaystyle {\prod _{m=1}^M S_m} \nonumber \\{} & {} \displaystyle {-\frac{1}{12}}s \biggl [(s-1)\left\{ 4\phi '''(1)+3\phi ^{(4)}(1) \right\} K^{-2} \left( \displaystyle {\prod _{m=1}^M S_m} \right) \left( \displaystyle {\sum _{m=1}^M J_m} \right) \nonumber \\{} & {} +2\left\{ (4\phi '''(1)+3\phi ^{(4)}(1))+2\phi '''(1)\right\} \displaystyle {\sum _{m=1}^M\frac{S_m}{J_m}}\biggr ]K^s +O\left( K^s \right) \nonumber \\{} & {} +O\left( K^{s-1} \left( \displaystyle {\sum _{m=1}^M J_m}\right) ^2\right) \; \; (s=1, 2, \ldots ) \end{aligned}$$

(A.20)

$$\begin{aligned}= & {} \left\{ 4\phi '''(1)+3\phi ^{(4)}(1)\right\} O\left( K^{s+1} \right) \nonumber \\{} & {} +O\left( K^s \left( \displaystyle {\sum _{m=1}^M J_m} \right) \right) \; \; (s=1, 2, \ldots ). \end{aligned}$$

(A.21)

From (A.21), if we put $m_\phi ^M(s)=0$, then $4\phi '''(1)+3\phi ^{(4)}(1) = o(1)$, which implies $4\phi '''(1)+3\phi ^{(4)}(1) \rightarrow 0$ as $K \rightarrow \infty$. We have completed the proof of Theorem 3. On the other hand, by substituting (8) in (A.20), we obtain the results of Theorem 4.

Proof of Theorems 5 and 6

Let the characteristic function of $C_\phi ^{*}$ be $\psi _\phi ^C(t)$. By assuming that the distribution of $C_\phi ^{*}$ is continuous, $\psi _\phi ^{C}(t)$ is evaluated as follows:

$$\begin{aligned} \psi _\phi ^{C}(t) = (1-2it)^{-\nu /2} +\frac{1}{n}\sum _{j=0}^3(1-2it)^{-(\nu +2j)/2}v_j^\phi + o(n^{-1}), \end{aligned}$$

where

$$\begin{aligned}{} & {} \begin{array}{cll} v_0^{\phi } &{} = &{} \gamma _0, \\ v_1^{\phi } &{} = &{} (\gamma _1+\gamma _2)+2\phi ^{'''}(1)\gamma _2 +\phi ^{(4)}(1)\gamma _3+\{\phi ^{'''}(1)\}^2\gamma _4, \\ v_2^{\phi } &{} = &{} 2(-\gamma _2+\gamma _3)-2\phi ^{'''}(1)(\gamma _2+\gamma _4) -\phi ^{(4)}(1)\gamma _3-2\{\phi ^{'''}(1)\}^2\gamma _4, \\ v_3^{\phi } &{} = &{} \gamma _4\{1+\phi ^{'''}(1)\}^2, \\ \gamma _0 &{} = &{} -\displaystyle {\frac{1}{12}}\Gamma _1-\frac{1}{12}\Gamma _2 + \frac{1}{12}(\Gamma _3+\Gamma _4),\\ \gamma _1 &{} = &{} \displaystyle {\frac{1}{4}}J_1J_2\Gamma _1+\displaystyle {\frac{1}{4}}\Gamma _2- \displaystyle {\frac{1}{4}}(J_2\Gamma _3+J_1\Gamma _4), \\ \gamma _2 &{} = &{} \displaystyle {\frac{1}{8}}J_1^2J_2^2\Gamma _1+\displaystyle {\frac{1}{8}}\Gamma _2-\displaystyle {\frac{1}{8}}(J_2^2\Gamma _3+J_1^2\Gamma _4)\\ \gamma _3 &{}=&{} \left( -\displaystyle {\frac{1}{2}}J_1J_2-\displaystyle {\frac{1}{8}} +\displaystyle {\frac{1}{4}}J_1+\displaystyle {\frac{1}{4}}J_2\right) \Gamma _1 -\displaystyle {\frac{1}{8}}\Gamma _2+\displaystyle {\frac{1}{4}}(J_2\Gamma _3+J_1\Gamma _4) -\displaystyle {\frac{1}{8}}(\Gamma _3+\Gamma _4) \\ \gamma _4 &{}=&{} \left( \displaystyle {\frac{1}{8}}J_1^2J_2^2+\displaystyle {\frac{3}{4}}J_1J_2+ \displaystyle {\frac{1}{3}}-\displaystyle {\frac{1}{2}}J_1-\displaystyle {\frac{1}{2}}J_2\right) \Gamma _1 +\displaystyle {\frac{5}{24}}\Gamma _2-\displaystyle {\frac{1}{8}}(J_2^2\Gamma _3+J_1^2\Gamma _4) \\ &{} &{}-\displaystyle {\frac{1}{4}}(J_2\Gamma _3+J_1\Gamma _4) +\displaystyle {\frac{1}{6}}(\Gamma _3+\Gamma _4), \\ \end{array} \\{} & {} \Gamma _1 = \displaystyle {\sum _{\ell =1}^{J_3}\frac{1}{p_{\cdot \cdot \ell }}}, \quad \Gamma _2 = \displaystyle {\sum _{j=1}^{J_1}\sum _{k=1}^{J_2}\sum _{\ell =1}^{J_3}\frac{1}{q_{j k \ell }}},\quad \Gamma _3 = \displaystyle {\sum _{j=1}^{J_1}\sum _{\ell =1}^{J_3}\frac{1}{p_{j \cdot \ell }}}, \quad \Gamma _4 = \displaystyle {\sum _{k=1}^{J_2}\sum _{\ell =1}^{J_3}\frac{1}{p_{\cdot k \ell }}}, \end{aligned}$$

and $\nu$ is defined in (21). Here, $v_j^{\phi } \; (j=0, 1, 2, 3)$ satisfy

$$\begin{aligned} \sum _{j=0}^3v_j^{\phi }=0. \end{aligned}$$

Similar to the discussion for the proof of Theorems 3 and 4, we calculate $m_\phi ^C(s)$. After that, since $\Gamma _1=O(J_3^2)$, $\Gamma _2=O(K^2)$, $\Gamma _3=O((J_1J_3)^2)$, and $\Gamma _4=O((J_2J_3)^2)$ under the assumption that $p_{jk\ell }=O(K^{-1})$, we can evaluate $m^C_\phi (s)$ as follows:

$$\begin{aligned} m_\phi ^C(s)= & {} \displaystyle {\frac{1}{12}}s\left\{ 4\phi '''(1)+3\phi ^{(4)}(1)\right\} K^{s-1}\Gamma _2 \nonumber \\{} & {} \displaystyle {-\frac{1}{12}}s \biggl [(s-1)\left\{ 4\phi '''(1)+3\phi ^{(4)}(1)\right\} (J_2^{-1}\Gamma _2+J_1^{-1}\Gamma _2) \nonumber \\{} & {} +2\left\{ (4\phi '''(1)+3\phi ^{(4)}(1))+2\phi '''(1)\right\} (J_2 \Gamma _3 + J_1 \Gamma _4) \biggr ] K^{s-1} \nonumber \\{} & {} +O\left( K^{s-1}J_1^2J_3^2 \right) +O\left( K^{s-1} J_2^2 J_3^2 \right) +O\left( K^s J_3 \right) \; (s=1, 2, \ldots ) \nonumber \\ \end{aligned}$$

(A.22)

$$\begin{aligned}= & {} \left\{ 4\phi '''(1)+3\phi ^{(4)}(1)\right\} O\left( K^{s+1} \right) \nonumber \\{} & {} +O\left( K^sJ_1J_3\right) +O\left( K^sJ_2J_3 \right) \qquad (s= 1, 2, \ldots ). \end{aligned}$$

(A.23)

From (A.23), if we put $m_\phi ^C(s) = 0$, then $4\phi '''(1)+3\phi ^{(4)}(1)=o(1)$. On the other hand, by substituting (8) in (A.22), we obtain the result of Theorem 6.