ned. So we propose to use EL as follows.

Since the blocks are disjoint, ${\widehat}{\theta}_{1m}, \dots, {\widehat}{\theta}_{Km}$ are independent. We can regard them as one sample and apply EL to make inference on $\theta$. For notational convenience, let $Y_{km}={\sqrt{m}}{\widehat}{\theta}_{k m}$ and $\mu=\sqrt{m}\theta$. Hence, the empirical likelihood ratio for $\mu$ is given by $$\label{eq20}
 \mathcal{R}(\mu)=\max\left\{   \prod_{k=1}^{K}K\omega_{k} ~\Big|~~\sum_{k=1}^{K}\omega_{k}Y_{k m}= \mu, \omega_{k}\geq 0,\quad \sum_{k=1}^{K}\omega_{k}=1          \right\}.$$ By the Lagrange multipliers method, we can find the maximum point $$\omega_{k}= \frac{1}{K} \frac{1}{1 + \lambda^\top(Y_{km}- \mu)},$$ where $\lambda=\lambda(\mu)$ satisfies the equation given by $$\label{eq24}
  0=\frac{1}{K}\sum_{k=1}^{K} \frac{Y_{km}- \mu}{1 + \lambda^\top(Y_{km}- \mu)}.$$ As in [@Owen1990], we can get the follow Wilks’ theorem.

\[theorem2\] Under Assumptions \[assumption1\]–\[assumption2\], we have $$-2 \log \mathcal{R}(\mu) \stackrel{d}{\longrightarrow} \chi^{2}_{p}$$ as $K, m\to \infty$.

The accuracy of each block estimator increases as $m$ increases. The power of EL increases as $K$ becomes greater. So there is a trade-off between $K$ and $m$. But we are studying massive data, $K$ and $m$ are large enough to guarantee the accuracy of each step’s inference. In simulations, we set $n=10^5$, $K=\{50, 100, 150\}$. The numerical results show that our proposed method is not sensitive to $K$.

Compared with the BLB and SDB, our method provides a specific asymptotic distribution to make inference on $\theta$. It is unnecessary to apply bootstrap to specify critical values. This reduces the computation burden a lot.

Now, we discuss the computational times of our proposed method, BLB and SDB. Let $t(m)$ be the computational time to estimate ${\widehat}{\theta}_{m}$ based on a sample of size $m$. $c(K)$ denotes the cost time of EL based on $K$ blocks. Table \[table1\] presents the comparison. In Table \[table1\], the column 