Let f:Ω⇒R be a Borel-measurable function, X a random variable with values in Ω and Xi∈R,i∈N realizations of X.LiteratureIn 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)]:(μ^±z1−α2σf^N),where we write N for the number of Monte Carlo samples, z1−α2 for the 1−α2 quantile of the standard normal distribution, μ^=1N∑i=1Nf(Xi) for the unbiased estimator of the mean and σ^f=1N−1∑i=1N(f(Xi)−μ^)2 for the empirical standard deviation derived from the unbiased estimator of the variance.Extension to varianceIn my opinion we can derive in a similar way a confidence interval for the variance var[f(X)] if we define g(X)=(f(X)−μ^)2 and repeat the construction. This leads to the interval(σ^f2±z1−α2σ^gN),where σ^g=1N−1∑i=1N(g(X)−1N∑i=1Ng(Xi))2How 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.

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Let f:Ω⇒R be a Borel-measurable function, X a random variable with values in Ω and Xi∈R,i∈N realizations of X.LiteratureIn 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)]:(μ^±z1−α2σf^N),where we write N for the number of Monte Carlo samples, z1−α2 for the 1−α2 quantile of the standard normal distribution, μ^=1N∑i=1Nf(Xi) for the unbiased estimator of the mean and σ^f=1N−1∑i=1N(f(Xi)−μ^)2 for the empirical standard deviation derived from the unbiased estimator of the variance.Extension to varianceIn my opinion we can derive in a similar way a confidence interval for the variance var[f(X)] if we define g(X)=(f(X)−μ^)2 and repeat the construction. This leads to the interval(σ^f2±z1−α2σ^gN),where σ^g=1N−1∑i=1N(g(X)−1N∑i=1Ng(Xi))2How can I transfer this to give a confidence interval for the standard deviation? Surely I can&#039;t just take the square root of the boundaries. Any literature, help and thoughts are welcome. Thanks.

Giovanny Howe · Accepted Answer

The approach is different and yields a confidence interval of the form((N−1)σ^f2χ{α2}2,(N−1)σ^f2χ{1−α2}2),where χα2 denotes the α-quantile of the χ2 distribution with N−1 degrees of freedom.

Let f:\Omega\rightarrow\mathbb{R} be a Borel-measurable function, X a random variable with val

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