Nonparametric bootstrap, statistical

Porter PS, Rao ST, Ku J-Y, Poirot RL, Dakins M (1997) Small sample properties of nonparametric bootstrap t confidence intervals. J Air Waste Management Assoc 47 1197-1203 Powell R, Hergt J, Woodhead J (2002) Improving isochron calcttlatiorrs with robust statistics and the bootstrap. Chem Geol 185 191- 204... 

The interpercentile interval can be determined by parametric, nonparametric, and bootstrap statistical techniques. ... 

How to Bootstrap. First, the number of subjects in a multistudy data set for the purposes presented needs to be kept constant to maintain the correct statistical interpretations of bootstrap, that is, correctly representing the underlying empirical distribution of the study populations. Second, the nonparametric bootstrap, as opposed to some other more parametric alternatives, was considered more suitable in order to minimize the dependence on having assumed the correct structural model. 

Consider the univariate case where a random variable X is measured n-times and some statistic f(x) is calculated from the sample vector X. In its most basic form, the nonparametric bootstrap is done as follows ... 

The nonparametric bootstrap is useful when distributions cannot be assumed as true or when the sampled statistic is based on few observations. In this setting, an observed data set, for example, Zj,. .., Z , where X could be vector-valued (i.e., concentrations at fixed sampling times) can be summarized in the usual way by a mean, median, and variance. An approximate sampling distribution can be obtained drawing a sample of the same size as the original sample from the original data with replacement, for example, Z/,. .., Z , where i is the index of the bootstrap sample... 

A nonparametric approach can involve the use of synoptic data sets. In a synoptic data set, each unit is represented by a vector of measurements instead of a single measurement. For example, for synoptic data useful for pesticide fate, assessment could take the form of multiple physical-chemical measurements recorded for each of a sample of water bodies. The multivariate empirical distribution assigns equal probability (1/n) to each of n measurement vectors. Bootstrap evaluation of statistical error can involve sampling sets of n measurement vectors (with replacement). Dependencies are accounted for in such an approach because the variable combinations allowed are precisely those observed in the data, and correlations (or other dependency measures) are fixed equal to sample values. 

In a 1988 paper, Lodder and Hieftje used the quantile-BEAST (bootstrap error-adjusted single-sample technique) [77] to assess powder blends. In the study, four benzoic acid derivatives and mixtures were analyzed. The active varied between 0 and 25%. The individual benzoic acid derivatives were classified into clusters using the nonparametric standard deviations (SDs), analogous to SDs in parametric statistics. Ace-tylsalicylic acid was added to the formulations at concentrations of 1 to 20%. All uncontaminated samples were correctly identified. Simulated solid dosage forms containing ratios of the two polymorphs were prepared. They were scanned from 1100 to 2500 nm. The CVs ranged from 0.1 to 0.9%. 

Both nonparametric and parametric bootstrap approaches can be pursued depending on whether we are willing to assume we know the true form of the distribution of the observed sample (parametric case). The parametric bootstrap is particularly useful when the sample statistic of interest is highly complex (as one might expect when trying to bootstrap a pharmacokinetic parameter derived from a nonlinear mixed effect model) or when we happen to know the distribution, since the additional assumption of a known distribution adds power to the estimate. 

Sprent, P. and Smeeton, N. C. 2000. Applied Nonparametric Statistical Methods, 3rd edn, Chapman and Hall/CRC Press, London. (Covers a wide range of significance tests in a practical way, with a good discussion of robust techniques and of bootstrapping and other re-sampling methods.)... 

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