Estimating moments with Monte Carlo

Let us denote \vect{Y} = h\left( \vect{X},\vect{d} \right) = \left( Y^1,\ldots,Y^{n_Y} \right), where \vect{X}= \left( X^1,\ldots,X^{n_X} \right) is a random vector, and \vect{d} a deterministic vector. We seek here to evaluate, the characteristics of the central part (central tendency and spread i.e. mean and variance) of the probability distribution of a variable Y^i, using the probability distribution of the random vector \vect{X}.

The Monte Carlo method is a numerical integration method using sampling, which can be used, for example, to determine the mean and standard deviation of a random variable Y^i (if these quantities exist, which is not the case for all probability distributions):

    m_{Y^i} = \int u \, f_{Y^i}(u) \, du,\ \sigma_{Y^i} = \sqrt{\int \left( u-m_{Y^i} \right)^2 \, f_{Y^i}(u) \, du}

where f_{Y^i} represents the probability density function of Y^i.

Suppose now that we have the sample \left\{ y^i_1,\ldots,y^i_N \right\} of N values randomly and independently sampled from the probability distribution f_{Y^i}; this sample can be obtained by drawing a N sample \left\{ \vect{x}_1,\ldots,\vect{x}_N \right\} of the random vector \vect{X} (the distribution of which is known) and by computing \vect{y}_j = h \left( \vect{x}_j,\vect{d} \right) \ \forall 1 \leq j \leq N. Then, the Monte-Carlo estimations for the mean and standard deviation are the empirical mean and standard deviations of the sample:

    \widehat{m}_{Y^i} = \frac{1}{N} \sum_{j=1}^N y^i_j,\ \widehat{\sigma}_{Y^i} = \sqrt{\frac{1}{N} \sum_{j=1}^N \left( y^i_j - \widehat{m}_{Y^i} \right)^2}

These are just estimations, but by the law of large numbers their convergence to the real values m_{Y^i} and \sigma_{Y^i} is assured as the sample size N tends to infinity. The Central Limit Theorem enables the difference between the estimated value and the sought value to be controlled by means of a confidence interval (especially if N is sufficiently large, typically N > a few dozens even if there is now way to say for sure if the asymptotic behavior is reached). For a probability \alpha strictly between 0 and 1 chosen by the user, one can, for example, be sure with a confidence \alpha, that the true value of m_{Y^i} is between \widehat{m}_{i,\inf} and \widehat{m}_{i,\sup} calculated analytically from simple formulae. To illustrate, for \alpha = 0.95:

    \widehat{m}_{i,\inf} = \widehat{m}_{Y^i} - 1.96 \frac{\displaystyle \widehat{\sigma}_{Y^i}}{\displaystyle \sqrt{N}},\ \widehat{m}_{i,\sup} = \widehat{m}_{Y^i} + 1.96 \frac{\widehat{\sigma}_{Y^i}}{\sqrt{N}},\ \textrm{that is to say}\ \textrm{Pr} \left(  \widehat{m}_{i,\inf} \leq m_{Y^i} \leq \widehat{m}_{i,\sup} \right) = 0.95

The size of the confidence interval, which represents the uncertainty of this mean estimation, decreases as N increases but more gradually (the rate is proportional to \sqrt{N}: multiplying N by 100 reduces the length of the confidence interval \left| \widehat{m}_{i,\inf}-\widehat{m}_{i,\sup} \right| by a factor 10).

This method is also referred to as Direct sampling, crude Monte Carlo method, Classical Monte Carlo integration.

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  • Robert C.P., Casella G. (2004). “Monte Carlo Statistical Methods”, Springer, ISBN 0-387-21239-6, 2nd ed.

  • Rubinstein R.Y. (1981). “Simulation and The Monte Carlo methods”, John Wiley & Sons

  • “Guide to the expression of Uncertainty in Measurements (GUM)”, ISO publication