Implements the bootstrap based finite sample critical value defined in Section 2.6, Bagkavos, Patil and Wood (2021), and calculated as follows:
1. Resample the observations \(\mathcal{X}=\{X_1, \dots, X_n\}\) to obtain \(M\) bootstrap samples, denoted by \(\mathcal{X}_m^\ast=\{ X_{1m}^\ast, \dots, X_{nm}^\ast\}\), where for each \(m=1,\ldots , M\), \(\mathcal{X}_m^\ast\) is sampled randomly, with replacement, from \(\mathcal{X}\). Write \(\hat{\theta}=\theta(\mathcal{X})\) for the estimator of \(\theta\) based on the original sample \(\mathcal{X}\) and, for each \(m\), define the bootstrap estimator of \(\theta\) by \(\hat{\theta}_m^\ast = \theta(\mathcal{X}_m^\ast)\), where \(\theta(\cdot)\) is the relevant functional for the parameter \(\theta\).
2. For \(m=1, \ldots , M\), use \(\mathcal{X}_m^\ast =\{X_{1m}^\ast, \dots, X_{nm}^\ast\}\) and \(\hat \theta_m^\ast\) from the previous step to calculate \(n \Delta^{2d} h^{-d/2} \hat S_{n,m}^\ast(h\rho)\),\(m=1, \dots, M\).
3. Calculate \(\ell_\alpha^\ast\) as the \(1-\alpha\) empirical quantile of the values \(n \Delta^{2d} h^{-d/2} \hat S_{n,m}^\ast(h\rho)\), \(m=1, \dots, M\). Then \(\ell_\alpha^\ast\) approximately satisfies \(P^\ast [ n \Delta^{2d} h^{-d/2}\hat S_{n,m}^\ast(h\rho)> \ell_\alpha^\ast ]=1-\alpha\), where \(P^\ast\) indicates the bootstrap probability measure conditional on \(\mathcal{X}\).