Bootstrapping populations in statistics and mathematics starts with a sample observed from a random variable.

When X has a given distribution law, a parametric inference problem consists of computing suitable values – call them estimates – of these parameters precisely on the basis of the sample. An estimate is suitable if replacing it with the unknown parameter does not cause major damage in next computations. In Algorithmic inference, suitability of an estimate reads in terms of compatibility with the observed sample.

In this framework, resampling methods compatible with an observed sample, where the compatibility of its values has the properties of a probability distribution. By plugging parameters into the expression of the questioned distribution law, we bootstrap entire populations of random variables compatible with the observed sample.

The rationale of the algorithms computing the replicas, which we denote population bootstrap exhibiting specific properties, denoting a well behavior may be expressed as a function of the unknown parameters and a random seed through the sampling mechanism expressions as functions of seeds and parameters – the master equations – that we invert to find values of the latter as a function of: i) the statistics, whose values in turn are fixed at the observed ones; and ii) the seeds, which are random according to their own distribution. Hence from a set of seed samples we obtain a set of parameter replicas.

Method

of a random variable X and a sampling mechanism for X, the realization x. Focusing on well-behaved statistics,

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for their parameters, the master equations read

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fixed to the observed values. Having computed a huge set of compatible vectors, say N, the empirical marginal distribution is obtained by:

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is the indicator function Some indeterminacies remain if X denotes the parameter vector from which the statistics vector is derived.

Algorithm

You may easily see from a table of sufficient statistics that we obtain the curve in the picture on the left by computing the empirical distribution (2) on the population obtained through the above algorithm when: i) X, and

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and the curve in the picture on the right when: i) X, and

1. .

Remark

Note that the accuracy with which a parameter distribution law of populations compatible with a sample is obtained is not a function of the sample size. Instead, it is a function of the number of seeds we draw. In turn, this number is purely a matter of computational time but does not require any extension of the observed data. With other bootstrapping methods focusing on a generation of sample replicas (like those proposed by ) the accuracy of the estimate distributions depends on the sample size.

Example

expected to represent a Pareto distribution and k, we have that the cumulative distribution function reads:

1. .

A sampling mechanism uniform seed U described by:

is constituted by the pair of joint sufficient statistics and K. The master equations read

.

.

Notes

References