For each sample seed you obtain a vector of parameters from the solution of the above system with fixed to the observed values.
Having computed a huge set of compatible vectors, say N, you obtain the empirical marginal distribution of by:
(2)
denoting by the j-th component of the generic solution of (1) and by the indicator function of in the interval .
Some indeterminacies remain with X discrete which we will consider shortly.
The whole procedure may be summed up in the form of the following Algorithm, where the index of denotes the parameters vector which the statics vector refers to.
draw a sample seed of size m from the seed random variable;
get as a solution of (1) in θ with and ;
add to ; population.
You may easily see from the 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 is an Exponential random variable, ii) , and
,
and the curve in the picture on the right when: i) X is a Uniform random variable in , ii) , and
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 (Efron and Tibshirani 1993) harv error: no target: CITEREFEfron_and_Tibshirani1993 (help)) the accuracy of the estimate distributions depends on the sample size.
For expected to represent a Pareto distribution, whose specification requires values for the parameters and k[1], we have that the cumulative distribution function reads:
Efron, B. and Tibshirani, R. (1993). An introduction to the Boostrap. Freeman, New York: Chapman and Hall.{{cite book}}: CS1 maint: multiple names: authors list (link)
Apolloni, B (2006). Algorithmic Inference in Machine Learning. International Series on Advanced Intelligence. Vol. 5 (2nd ed.). Adelaide: Magill. Advanced Knowledge International{{cite book}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)
Apolloni, B., Bassis, S., Gaito. S. and Malchiodi, D. (2007). "Appreciation of medical treatments by learning underlying functions with good confidence". Current Pharmaceutical Design. 13 (15): 1545–1570.{{cite journal}}: CS1 maint: multiple names: authors list (link)