Influence function-based confidence intervals for the Kendall rank correlation coefficient

Abstract

Correlation coefficients measure the association between two random variables. In circumstances in which the typically-used Pearson correlation coefficient does not suffice, the Kendall rank correlation coefficient is routinely used as an alternative measure. In this paper, using the influence function of the Kendall rank correlation coefficient, we develop a normal approximation-based confidence interval and an empirical likelihood-based confidence interval for the Kendall rank correlation coefficient. Simulation studies are conducted to show their good finite sample properties and robustness. We apply the proposed methods to a real dataset on Bitcoin financial data.

Appendix: Proof of Theorem 1

We need the following Lemmas for the proof of Theorem 1.

Lemma 1

Under the conditions in Theorem 1, we have

$$\begin{aligned} n^{-1/2}\sum _{i=1}^{n}(\widehat{V}(W_{i})-R_{K}(H))\xrightarrow {\textit{L}}N(0,\sigma ^{2}), \end{aligned}$$

where $\sigma ^{2} =4P_{H}[(X-\mu _{x})(Y-\mu _{y})>0] \{1-P_{H}[(X-\mu _{x})(Y-\mu _{y})>0]\}$.

Proof

From (14), we have the following decomposition:

$$\begin{aligned} n^{-1/2}\sum _{i=1}^{n}(\widehat{V}(W_{i})-R_{K}(H))&= n^{-1/2}\sum _{i=1}^{n}(\widehat{V}(W_{i}) - V(W_{i}))\nonumber \\&+n^{-1/2}\sum _{i=1}^{n} ({V}(W_{i})-R_{K}(H)) \nonumber \\\equiv & {} I_1 + I_2. \end{aligned}$$

(21)

From $-\infty< E(X)=\mu _x <\infty$, $-\infty< E(Y)=\mu _y <\infty$, $\bar{X}=\mu _x+o(1) \ a.s.$ and $\bar{Y}=\mu _y+o(1) \ a.s.$, it follows that

$$\begin{aligned} \widehat{V}(W_{i}) - V(W_{i})&= 2( I[(X_{i}-\bar{X})(Y_{i}-\bar{Y})>0]- I[(X_{i}-\mu _x)(Y_{i}-\mu _y)>0]) \nonumber \\&= 0 \ a.s. \text { for} \ \forall \ i, as \ n \rightarrow \infty . \end{aligned}$$

(22)

Hence,

$$\begin{aligned} I_1 = 0 \ a.s. \ as \ n \rightarrow \infty . \end{aligned}$$

(23)

From (7), we have

$$\begin{aligned} I_2 \xrightarrow {\textit{L}}N(0,\sigma ^{2}). \end{aligned}$$

(24)

$\square$

Lemma 1 follows from (21), (23) and (24) right away.

Lemma 2

Under the conditions in Theorem 1, we have that

$$\begin{aligned} \dfrac{1}{n}\sum _{i=1}^{n}\left( \widehat{V}(W_{i})-R_{K}(H)\right) ^{2} \xrightarrow {\textit{p}}\sigma ^{2}. \end{aligned}$$

Proof

From $|V(W_{i})-R_{K}(H)| \le 5$ and (22), it follows that

$$\begin{aligned} \dfrac{1}{n}\sum _{i=1}^{n}(\widehat{V}(W_{i})-R_{K}(H))^{2}&= \dfrac{1}{n}\sum _{i=1}^{n}({V}(W_{i})-R_{K}(H))^{2} + \dfrac{1}{n}\sum _{i=1}^{n}(\widehat{V}(W_{i})-V(W_{i}))^{2} \\&+2 \dfrac{1}{n}\sum _{i=1}^{n}(\widehat{V}(W_{i})-V(W_{i}))({V}(W_{i})-R_{K}(H))\\&= E({V}(W_{i})-R_{K}(H))^{2} +o_p(1) \\&= \sigma ^{2} +o_p(1). \end{aligned}$$

$\square$

The Proof of Theorem 1

Using similar arguments in Owen (1990), we can prove that $\lambda =O_p (n^{-1/2})$. Applying Taylor’s expansion to (17), we obtain that

$$\begin{aligned} l(R_{K}(H))&= 2\sum _{i=1}^{n}\log \left[ 1+\lambda \left( \widehat{V}(W_{i})-R_{K}(H)\right) \right] \end{aligned}$$

(25)

$$\begin{aligned}&= 2\sum _{i=1}^{n}\left[ \lambda \left( \widehat{V}(W_{i})-R_{K}(H)\right) -\dfrac{1}{2} \left( \lambda \left( \widehat{V}(W_{i})-R_{K}(H)\right) \right) ^{2}\right] +r_{n} \end{aligned}$$

(26)

where

$$\begin{aligned} |r_{n}|\le C\sum _{i=1}^{n}\left| \lambda \left( \widehat{V}(W_{i})-R_{K}(H)\right) \right| ^{3} \le C|\lambda |^{3}n=O_{p}(n^{-1/2}) \end{aligned}$$

(27)

From (16), if follows that

$$\begin{aligned} \lambda&= \dfrac{\sum _{i=1}^{n}\left( \widehat{V}(W_{i})-R_{K}(H)\right) }{\sum _{i=1}^{n} \left( \widehat{V}(W_{i})-R_{K}(H)\right) ^{2}}+O_{p}(n^{-1/2}), \\ \sum _{i=1}^{n}\lambda \left( \widehat{V}(W_{i})-R_{K}(H)\right)&= \sum _{i=1}^{n} \left( \lambda \left( \widehat{V}(W_{i})-R_{K}(H)\right) \right) ^{2}+o_{p}(1). \end{aligned}$$

Therefore, by Lemmas 1 and 2, we have that

$$\begin{aligned} l(R_{K}(H))&= \sum _{i=1}^{n}\lambda \left( \widehat{V}(W_{i})-R_{K}(H)\right) +o_{p}(1) \end{aligned}$$

(28)

$$\begin{aligned}&= \dfrac{\left[ \sum _{i=1}^{n}\left( \widehat{V}(W_{i})-R_{K}(H)\right) \right] ^{2}}{\sum _{i=1}^{n}\left( \widehat{V}(W_{i})-R_{K}(H)\right) ^{2}}+o_{p}(1) \xrightarrow {\textit{L}}\chi ^{2}_{1} \end{aligned}$$

(29)

The proof of the Theorem 1 is thus completed.$\square$