The AstroStat Slog

The following footnotes are from one of Prof. Babu’s slides but I do not recall which occasion he presented the content.

– In the XSPEC packages, the parametric bootstrap is command FAKEIT, which makes Monte Carlo simulation of specified spectral model.
– XSPEC does not provide a nonparametric bootstrap capability.

Parametric Bootstrap: $$X_1^*,…,X_n^* \sim F(\cdot;\theta_n)$$
Both $$\sqrt{n} \sup_x |F_n(x)-F(x;\theta_n)|$$ and $$\sqrt{n} \sup_x |F_n^*(x)-F(x;\theta_n^*)|$$ have the same limiting distribution.^[1]

Nonparametric Bootstrap:$$X_1^*,…,X_n^* \sim F_n.$$
A bias correction $$B_n(x)=F_n(x)-F(x;\theta_n)$$ is needed.
$$\sqrt{n} \sup_x |F_n(x)-F(x;\theta_n)|$$ and $$\sqrt{n} \sup_x |F_n^*(x)-F(x;\theta_n^*)-B_n(x)|$$ have the same limiting distribution.^[2]

1. In the XSPEC packages, the parametric bootstrap is command FAKEIT, which makes Monte Carlo simulation of specified spectral model.[↩]
2. XSPEC does not provide a nonparametric bootstrap capability[↩]