and two PM2.5 data sets collected in two major cities in China (i.e. Beijing and Shanghai). Before discussing the real-data analysis, we first use some synthesized data sets containing 500 runs for each case, to show the performance of the proposed methods and compare their results to that of the conventional sequential method (i.e. $M=1$), the latter of which we refer to as ‘SM’. Additionally, we offer the results of multiple parallel sequential estimation procedures ($M > 1$), which we refer to as ‘PSM’.

Simulation studies
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Simple distributed sequential estimation {#simple-distributed-sequential-estimation .unnumbered}
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We first consider $M=2, 5$ with $d=0.5,0.4,0.3$ and $0.2$, without using the ASE feature. We use two different true parameter vectors: ${\bm{{\beta}}_{0}} = (-1,1)$ and $(-1.0,1.0,0.7,0.5,0.2)$, denoted by $S_1$ and $S_2$, and generate data using the following model. $$Y=\mathbf X^T{\bm{{\beta}}_{0}}+\epsilon,$$ where [ $\mathbf X=(X_1,...,X_p)$ with $X_1=1$ and $X_i,i=2,...,p$, generated from normal distributions with mean $1$ and 0.2 for $p=2$ and $5$, and variance $1$; additionally, each component of $\epsilon$ follows the standard normal distribution.]{}

Table \[tab1-2\] summarizes the total number of observations used (i.e. stopping times), empirical coverage probabilities, and computation time (i.e. CPU time). For PSM, $N^*$ is equal to the sum of the numbers 