A Jackknife Empirical Likelihood Approach for Testing the Homogeneity of K Variances

Abstract

A nonparametric test for equality of K variances has been proposed by developing the jackknife empirical likelihood ratio. The standard limiting Chi-squared distribution with degrees freedom of $K-1$ for the test statistic is established, and is used to determine the type I error rate and the power of the test. Simulation studies have been conducted to show that the proposed method is competitive to the current existing methods, Levene’s test and Fligner-Killeen’s test, in terms of power and robustness. The proposed method has been illustrated in an application on a real data set.

Appendix

Define ${\varvec{\lambda }}=(\lambda _{1}, ..., \lambda _{K})$, $n=n_1+n_2+...+n_k$,

$$\begin{aligned}&W_{kn}(\theta , {\varvec{\lambda }})=\dfrac{1}{n} \sum _{l=1}^{n_k} \frac{{\hat{V}}^{k}_l-\theta }{1+\lambda _{k} \left( {\hat{V}}^{k}_l-\theta \right) }, \quad k=1,...,K,\\&W_{{(K+1)n}}(\theta , {\varvec{\lambda }})=\dfrac{1}{n}\sum _{k=1}^K\lambda _{k}\sum _{l=1}^{n_k} \frac{-1}{1+\lambda _{k} \left( {\hat{V}}^{k}_l-\theta \right) }. \end{aligned}$$

Lemma 6.1

(Hoeffding (1948)) Under condition C1,

$$\begin{aligned} \dfrac{\sqrt{n_k}(U_{n_k}-\theta _0)}{2\sigma _{gk}} {\mathop {\rightarrow }\limits ^{d}} N(0, 1) \ \text { as} \ n_k \rightarrow \infty , \end{aligned}$$

Lemma 6.2

Let $S_{k}=\dfrac{1}{n_k} \displaystyle \sum _{l=1}^{n_k} \left( {\hat{V}}^{k}_l- \theta _0\right) ^2, k=1, ..., K,$ Under the conditions of Lemma 6.1,

$$\begin{aligned} S_{k}=4\sigma ^2_{g_k}+o_p(1), \end{aligned}$$

as $n_k \rightarrow \infty , k=1, ..., K$

Lemma 6.3

(Cheng et al. 2018) Under conditions C1 and C2 and $H_0$, with probability tending to one as $\min \{n_1, ..., n_K\} \rightarrow \infty$, there exists a root ${\tilde{\theta }}$ of

$$\begin{aligned}&W_{kn}(\theta , {\varvec{\lambda }})=0, \ k=1,..., K+1, \end{aligned}$$

such that $|{\tilde{\theta }}-\theta _0|< \delta ,$ where $\delta =n^{-1/3}$.

Let $\tilde{{\varvec{\eta }}}=({\tilde{\theta }}, \tilde{{\varvec{\lambda }}})^T$ be the solution to the above equations, and ${\varvec{\eta }}_0=(\theta , 0, ..., 0)^T$. By expanding $W_{kn}(\tilde{{\varvec{\eta }}})$ at ${\varvec{\eta }}_0$, we have, for $k=0, 1,...,K+1$,

$$\begin{aligned} 0&=W_{kn}({\varvec{\eta }}_0)+ \dfrac{\partial W_{kn}}{\partial \theta }({\varvec{\eta }}_0)({\tilde{\theta }}-\theta _0)+\dfrac{\partial W_{kn}}{\partial \lambda }({\varvec{\eta }}_0) {\tilde{\lambda }}\\&\quad +\dfrac{\partial W_{kn}}{\partial \lambda _1}({\varvec{\eta }}_0) {\tilde{\lambda }}_{1}+\cdots +\dfrac{\partial W_{kn}}{\partial \lambda _K}({\varvec{\eta }}_0) {\tilde{\lambda }}_{K}+R_{kn}, \end{aligned}$$

where $R_{kn}=\frac{1}{2} (\tilde{{\varvec{\eta }}}-{\varvec{\eta }}_0)^T \dfrac{\partial ^2 W_{kn}({\varvec{\eta }}^{*})}{\partial {\varvec{\eta }}\partial {\varvec{\eta }}^T} (\tilde{{\varvec{\eta }}}-{\varvec{\eta }}_0)=o_p(n^{-1/2}),$ and ${\varvec{\eta }}^{*}$ lies between ${\varvec{\eta }}_0$ and $\tilde{{\varvec{\eta }}}$.

Lemma 6.4

Under $H_0$, $\text{ Cov }(U_{n_k}, U_{n_l})=0$, $1\le k \ne l \le K$.

Proof of Theorem 2.1

$$\begin{aligned} \begin{pmatrix} W_{1n}({\varvec{\eta }}_0) \\ W_{2n}({\varvec{\eta }}_0) \\ \vdots \\ W_{Kn}({\varvec{\eta }}_0))\\ 0\\ \end{pmatrix} = {\varvec{\mathcal {B}}} \begin{pmatrix} {\tilde{\lambda }}_{1}\\ {\tilde{\lambda }}_{2} \\ \vdots \\ {\tilde{\lambda }}_{K}\\ {\tilde{\theta }}-\theta _0 \end{pmatrix}+o_{p}(n^{-1/2}), \end{aligned}$$

where

$\sigma ^2_k=4\sigma ^2_{gk}, k=1,..,K.$

It is easy to see that ${\varvec{\mathcal {B}}}$ is nonsingular under Conditions C1 and C2. Therefore,

$$\begin{aligned} \begin{pmatrix} {\tilde{\lambda }}_{1}\\ {\tilde{\lambda }}_{2} \\ \vdots \\ {\tilde{\lambda }}_{K}\\ {\tilde{\theta }}-\theta _0 \end{pmatrix}={\varvec{\mathcal {B}}}^{-1}\begin{pmatrix} W_{1n}({\varvec{\eta }}_0) \\ \vdots \\ W_{Kn}({\varvec{\eta }}_0))\\ 0\\ \end{pmatrix}+o_{p}(n^{-1/2}). \end{aligned}$$

Under $H_0$, $\sigma ^2_1=...=\sigma ^2_K= \sigma ^2$.

$$\begin{aligned}&{\tilde{\theta }}-\theta _0=\sum ^K_{k=0} W_{kn}({\varvec{\eta }}_0)+o_p(n^{-1/2}). \end{aligned}$$

By Jing et al. (2009), we have

$$\begin{aligned}&{\tilde{\lambda }}_{k}=\dfrac{U_{n_k}-{\tilde{\theta }}}{{\tilde{S}}_{k}}+o_{p}(n^{-1/2}), \quad k=1,...,K, \end{aligned}$$

where ${\tilde{S}}_{k}=\dfrac{1}{n_k}\sum \nolimits ^{n_k}_{l=1}({\hat{V}}^{k}_{l}-{\tilde{\theta }})^2$. It is easy to check that ${\tilde{S}}_{k}=\sigma ^2_k+o_p(1), k=0,1,..,K$. By the proof of Theorem 1 in Jing et al. (2009),

$$\begin{aligned} -2\log R=\left[ \sum _{k=1}^K \dfrac{n_{k}(U_{n_k}-{\tilde{\theta }})^2}{\sigma ^2_{k}} \right] (1+o_{p}(1)). \end{aligned}$$

With simple algebra, we have

$$\begin{aligned}&\sum _{k=1}^K \dfrac{n_{k}(U_{n_k}-{\tilde{\theta }})^2}{\sigma ^2_{k}}\nonumber \\&=(\sqrt{n}W_{1n}({\varvec{\eta }}_0), ..., \sqrt{n}W_{Kn}({\varvec{\eta }}_0) \times {\varvec{A}}^T {\varvec{\mathcal {W}}} {\varvec{A}} \times (\sqrt{n}W_{1n}({\varvec{\eta }}_0), ..., \sqrt{n}W_{Kn}({\varvec{\eta }}_0))^T +o_p(1), \end{aligned}$$

(8)

where

and

Furthermore, by Shi (1984), we have the central limit theorem for W’s at ${\varvec{\eta }}_0$. This is because each $W({\varvec{\eta }}_0)$ is the average of asymptotically independent psuedo-values. That is,

$$\begin{aligned} \sqrt{n}\begin{pmatrix} W_{1n}({\varvec{\eta }}_0) \\ W_{2n}({\varvec{\eta }}_0) \\ \vdots \\ W_{Kn}({\varvec{\eta }}_0))\\ \end{pmatrix} \overset{D}{\rightarrow } N({\varvec{0}}, {\varvec{\Sigma }}), \end{aligned}$$

where

Therefore, under $H_0$, $-2\log R$ converges to $\sum _{i=1}^{K} \omega _i \chi ^2_i$ in distribution, where $\chi ^2_i, i=1,..., K$ are K independent chi-square random variables with one degree of freedom, and $\omega _i, i=1,.., K$ are eigenvalues of ${\varvec{\Sigma }}_0^{1/2} {\varvec{A}}^T {\varvec{\mathcal {W}_0}} {\varvec{A}} {\varvec{\Sigma }}_0^{1/2}$, where

and

We can show that ${\varvec{A}}^{T} {\varvec{\mathcal {W}_0}} {\varvec{A}}={\varvec{A}}$. Hence, ${\varvec{\Sigma }}_0^{1/2} {\varvec{A}}^T {\varvec{\mathcal {W}_0}} {\varvec{A}} {\varvec{\Sigma }}_0^{1/2}={\varvec{\Sigma }}_0 {\varvec{A}}$ since ${\varvec{A}}$ is symmetric. With algebra calculation, the eigenvalues of ${\varvec{\Sigma }}_0 {\varvec{A}}$ are {0, 1, 1,...,1} with trace(${\varvec{\Sigma }}_0 {\varvec{A}})=K-1.$ By this result, we complete the proof. $\square$

Proof of Theorem 2.2

Under $H_a$, at least one of ${\mathbb {E}}U_k,$ $k=1,...,K$ will be different from the others. Let ${\mathbb {E}}U_k=\theta _k, \ k=1,...,K.$ From (8),

$$\begin{aligned} -2\log R= & {} \sum ^K_{k=1}\dfrac{n_k(U_{n_k}-{\tilde{\theta }})^2}{{\tilde{S}}^2_i}+o(1)\\= & {} \sum ^K_{k=1}\left[ \dfrac{\sqrt{n_k}(U_{n_k}-\theta _k)}{{\tilde{S}}_k}+\dfrac{\sqrt{n_k}(\theta _k-{\tilde{\theta }})}{{\tilde{S}}_k} \right] ^2+o(1), \end{aligned}$$

which is divergent since at least one of $\dfrac{\sqrt{n_k}(\theta _k-{\tilde{\theta }})^2}{{\tilde{S}}_k}, k=1,...,K$ will diverge to $\infty$.$\square$