Lymnmeatlypamgfm

2022-05-01

Let $f:\mathrm{\Omega}\Rightarrow \mathbb{R}$ be a Borel-measurable function, X a random variable with values in $\mathrm{\Omega}$ and $X}_{i}\in \mathbb{R},i\in \mathbb{N$ realizations of X.

Literature

In their book on Monte Carlo Methods (Simulation and the Monte Carlo method, Second Edition) Rubinstein and Kroese give in Section 4.2.1 an approximate confidence interval for the mean E[f(X)]:

$(\hat{\mu}\pm {z}_{1-\frac{\alpha}{2}}\frac{\hat{{\sigma}_{f}}}{\sqrt{N}}),$

where we write N for the number of Monte Carlo samples,$z}_{1-\frac{\alpha}{2}$ for the $1-\frac{\alpha}{2}$ quantile of the standard normal distribution, $\hat{\mu}=\frac{1}{N}\sum _{i=1}^{N}f\left({X}_{i}\right)$ for the unbiased estimator of the mean and $\hat{\sigma}}_{f}=\sqrt{\frac{1}{N-1}\sum _{i=1}^{N}{(f\left({X}_{i}\right)-\hat{\mu})}^{2}$ for the empirical standard deviation derived from the unbiased estimator of the variance.

Extension to variance

In my opinion we can derive in a similar way a confidence interval for the variance var[f(X)] if we define$g\left(X\right){\textstyle \phantom{\rule{0.222em}{0ex}}}={(f\left(X\right)-\hat{\mu})}^{2}$ and repeat the construction. This leads to the interval

$({\hat{\sigma}}_{f}^{2}\pm {z}_{1-\frac{\alpha}{2}}\frac{{\hat{\sigma}}_{g}}{\sqrt{N}}),$

where$\hat{\sigma}}_{g}=\frac{1}{N-1}\sum _{i=1}^{N}{(g\left(X\right)-\frac{1}{N}\sum _{i=1}^{N}g\left({X}_{i}\right))}^{2$

How can I transfer this to give a confidence interval for the standard deviation? Surely I can't just take the square root of the boundaries. Any literature, help and thoughts are welcome. Thanks.

Literature

In their book on Monte Carlo Methods (Simulation and the Monte Carlo method, Second Edition) Rubinstein and Kroese give in Section 4.2.1 an approximate confidence interval for the mean E[f(X)]:

where we write N for the number of Monte Carlo samples,

Extension to variance

In my opinion we can derive in a similar way a confidence interval for the variance var[f(X)] if we define

where

How can I transfer this to give a confidence interval for the standard deviation? Surely I can't just take the square root of the boundaries. Any literature, help and thoughts are welcome. Thanks.

Giovanny Howe

Beginner2022-05-02Added 18 answers

The approach is different and yields a confidence interval of the form

$(\sqrt{\frac{(N-1){\hat{\sigma}}_{f}^{2}}{{\chi}_{\left\{\frac{\alpha}{2}\right\}}^{2}}},\sqrt{\frac{(N-1){\hat{\sigma}}_{f}^{2}}{{\chi}_{\{1-\frac{\alpha}{2}\}}^{2}}}),$

where$\chi}_{\alpha}^{2$ denotes the $\alpha$ -quantile of the $\chi}^{2$ distribution with $N-1$ degrees of freedom.

where

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