 Computational statistics

Computational statistics develops and studies computational procedures for statistical inference. It includes numerical evaluation of likelihood functions, simulation from probability distributions and assessment of uncertainty when an exact symbolic calculation is unavailable. The statistical question and the numerical approximation both need error control.

For example, if independent simulations $X_1,\dots ,X_M$ follow a model and $g(X)$ has finite variance, the Monte Carlo estimator $\hat{\mu }=M^{-1}{\sum }_{j}g(X_j)$ estimates $\mu =\mathbb{E}[g(X)]$, with variance $\mathrm{Var}(g(X))/M$. Halving its standard deviation therefore requires four times as many independent simulations. An optimization routine finding a maximum of a likelihood function and a simulation routine estimating an integral solve different computational tasks, even when both are steps in the same analysis.

Computational statistics is a field that combines statistical theory and methodologies with computational techniques to analyze complex data sets and solve statistical problems. It involves the use of algorithms, numerical methods, and computer simulations to perform statistical analysis, particularly when traditional analytical methods are impractical or infeasible due to the complexity of the data or the model.