# On the Subrange and Its Application to the R-Chart
## 2 The Distribution of the Subrange
### The Relative Efficiency of the Subrange  
We proposed the \(R\)-chart based on the subrange. Given this, a natural question is which subrange should be selected? This is essentially the same as how to choose \(k\). We suggest the choice of \(k\) based on the RE value. In the statistics literature, the RE is defined as  
\[\text{RE}\Big{(}\,\widehat{\theta}_{2}\Big{|}\,\widehat{\theta}_{1}\Big{)}\ =\ \frac{\text{Var}\Big{(}\,\widehat{\theta}_{1}\Big{)}}{\text{Var}\Big{(}\, \widehat{\theta}_{2}\Big{)}}\times 100\%, \tag{6}\]  
where \(\widehat{\theta}\) is often a reference or baseline estimator. For more details, see [PERSON] [39] and [PERSON] [40]. We consider the RE of the unbiased scale estimator \(\widehat{\sigma}_{[k]}\) using \(\widehat{\sigma}_{[0]}\) as a baseline estimator, which is given by  
\[\text{RE}\Big{(}\,\widehat{\sigma}_{[k]}\Big{|}\,\widehat{\sigma}_{[0]}\Big{)} \ =\ \frac{\text{Var}\Big{(}\,\widehat{\sigma}_{[0]}\Big{)}}{\text{Var} \Big{(}\,\widehat{\sigma}_{[k]}\Big{)}}\times 100\%\ =\ \Big{(}d_{3}(n,0)^{2}/d_{2}(n,0)^{2}\Big{)}\, \Big{/}\Big{(}d_{3}(n,k)^{2}/d_{2}(n,k)^{2}\Big{)}\times 100\%. \tag{7}\]  
We calculated the values of \(\text{RE}\Big{(}\,\widehat{\sigma}_{[k]}\Big{|}\,\widehat{\sigma}_{[0]}\Big{)}\) in Table 3 for \(n\ =\ 2,3,\cdots 50\) and \(k\ =\ 0,1,\cdots,9\). It is easily seen that the choice of \(k\ =\ 0\) gives the best performance for \(n\ =\ 2,3,\cdots,17\), and the choice of \(k\ =\ 1\) does for \(n\ =\ 18,19,\cdots,31\) etc. In Table 1, we also summarize the best choice of \(k\) which provides the maximum RE value. We also obtain the simple regression line using the least-squares method. The fitted line equation is given by  
\[k\ =\ 0.65806+0.06926\cdot n.\] \begin{table}
\begin{tabular}{c c c c c} \hline \(2\leq n\leq 17\) & \(18\leq n\leq 31\) & \(32\leq n\leq 45\) & \(46\leq n\leq 60\) & \(61\leq n\leq 74\) \\ \hline \(k^{*}\ =\ 0\) & \(k^{*}\ =\ 1\) & \(k^{*}\ =\ 2\) & \(k^{*}\ =\ 3\) & \(k^{*}\ =\ 4\) \\ \hline \(75\leq n\leq 88\) & \(89\leq n\leq 103\) & \(104\leq n\leq 118\) & \(119\leq n\leq 132\) & \(133\leq n\leq 146\) \\ \hline \(k^{*}\ =\ 5\) & \(k^{*}\ =\ 6\) & \(k^{*}\ =\ 7\) & \(k^{*}\ =\ 8\) & \(k^{*}\ =\ 9\) \\ \hline \(147\leq n\leq 162\) & \(163\leq n\leq 175\) & \(176\leq n\leq 188\) & \(189\leq n\leq 203\) & \(204\leq n\leq 217\) \\ \hline \(k^{*}\ =\ 10\) & \(k^{*}\ =\ 11\) & \(k^{*}\ =\ 12\) & \(k^{*}\ =\ 13\) & \(k^{*}\ =\ 14\) \\ \hline \end{tabular}
\end{table} 
Table 1: The number of trimmings, \(k^{*}\), with the best relative efficiency. Thus, when the sample size \(n\) is very large, the value of \(k\) can be approximately selected using the above-fitted value. For example, when \(n~{}=~{}500\), we have \(k\approx 34\). Using the unbiased estimator \(S_{n}/c_{4}(n)\) of \(\sigma\) as a baseline, we also calculated the RE of the unbiased scale estimator \(\widehat{\sigma}_{[k]}\) again. Here, \(c_{4}(n)\) is given by  
\[c_{4}(n)~{}=~{}\sqrt{\frac{2}{n-1}}\cdot\frac{\Gamma\big{(}\frac{n}{2}\big{)}}{ \Gamma\Big{(}\frac{n}{2}-\frac{1}{2}\Big{)}}.\]  
It should be noted that \(E\big{[}S_{n}^{2}\big{]}~{}=~{}\sigma^{2}\) and \(E[S_{n}]~{}=~{}c_{4}(n)\sigma\). Thus, we have \(\text{Var}(S_{n}/c_{4}(n))\)\(=\Big{\{}1/c_{4}(n)^{2}-1\Big{\}}\sigma^{2}\) and  
\[\text{RE}\bigg{(}\,\widehat{\sigma}_{[k]}\big{|}\frac{S_{n}}{c_{4}(n)}\bigg{)} ~{}=~{}\frac{\text{Var}\Big{(}\frac{S_{n}}{c_{4}(n)}\Big{)}}{\text{Var}\Big{(} \widehat{\sigma}_{[k]}\Big{)}}\times 100\%~{}=~{}\frac{1/c_{4}(n)^{2}-1}{d_{3}(n,k)^{2}/d_{2}(n,k)^{2}}\times 100\%.\]  
We also calculated the values of \(\text{RE}\Big{(}\,\widehat{\sigma}_{[k]}\big{|}\frac{S_{n}}{c_{4}(n)}\Big{)}\) in Table A3 for \(n~{}=~{}2,3,\cdots 50\) and \(k~{}=~{}0,1,\cdots,9\). It should be noted that \(\text{RE}\Big{(}\,\widehat{\sigma}_{[k]}\big{|}\frac{S_{n}}{c_{4}(n)}\Big{)} =~{}100\%\) when \(n~{}=~{}2\). We investigated this specific case in more detail as follows. When \(n~{}=~{}2\), \(k\) should be zero.