User:Bassis/Bootstrapping populations

sample.

Method

Given a ${\boldsymbol {x}}=\{x_{1},\ldots ,x_{m}\}$ of a random variable X and a sampling mechanism $(g_{\boldsymbol {\theta }},Z)$ for X, we have ${\boldsymbol {x}}=\{g_{\boldsymbol {\theta }}(z_{1}),\ldots ,g_{\boldsymbol {\theta }}(z_{m})\}$, with ${\boldsymbol {\theta }}=(\theta _{1},\ldots ,\theta _{k})$. Focusing on well behaving statistics

$s_{1}=h_{1}(x_{1},\ldots ,x_{m})$

$\vdots \ \ \ \ \ \ \ \ \ \ \ \ \ \ \vdots$

$s_{k}=h_{k}(x_{1},\ldots ,x_{m})$

for their parameters, the master equations read

$s_{1}=h_{1}(g_{\boldsymbol {\theta }}(z_{1}),\ldots ,g_{\boldsymbol {\theta }}(z_{m}))=\rho _{1}({\boldsymbol {\theta }};z_{1},\ldots ,z_{m})$
$\vdots \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \vdots \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \vdots$	(1)
$s_{k}=h_{k}(g_{\boldsymbol {\theta }}(z_{1}),\ldots ,g_{\boldsymbol {\theta }}(z_{m}))=\rho _{k}({\boldsymbol {\theta }};z_{1},\ldots ,z_{m})$

For each sample seed $\{z_{1},\ldots ,z_{m}\}$ you obtain a vector of parameters ${\boldsymbol {\theta }}$ from the solution of the above system with $s_{i}$ fixed to the observed values. Having computed a huge set of compatible vectors, say N, you obtain the empirical marginal distribution of $\Theta _{j}$ by:

${\widehat {F}}_{\Theta _{j}}(\theta )=\sum _{i=1}^{N}{\frac {1}{N}}I_{(-\infty ,\theta ]}({\breve {\theta }}_{j,i})$

(2)

denoting by ${\breve {\theta }}_{j,i}$ the j-th component of the generic solution of (1) and by $I_{(-\infty ,\theta ]}({\breve {\theta }}_{j,i})$ the indicator function of ${\breve {\theta }}_{j,i}$ in the interval $(-\infty ,\theta ]$. 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 ${\boldsymbol {\Theta }}$ of ${\boldsymbol {s}}_{\boldsymbol {\Theta }}$ denotes the parameters vector which the statics vector refers to.

Algorithm

Generating parameter populations through a bootstrap

Given a sample $\{x_{1},\ldots ,x_{m}\}$ from a random variable with parameter vector ${\boldsymbol {\theta }}$ unknown, Identify a vector of well behaving statistics ${\boldsymbol {S}}$ for ${\boldsymbol {\Theta }}$; compute a specification ${\boldsymbol {s}}_{\boldsymbol {\Theta }}$ of ${\boldsymbol {S}}$ from the sample; repeat for a satisfactory number N of iterations: draw a sample seed ${\breve {\boldsymbol {z}}}_{i}$ of size m from the seed random variable; get ${\breve {\boldsymbol {\theta }}}_{i}=\mathrm {Inv} ({\boldsymbol {s}},{\boldsymbol {z}}_{i})$ as a solution of (1) in θ with ${\boldsymbol {s}}={\boldsymbol {s}}_{\boldsymbol {\Theta }}$ and ${\boldsymbol {z}}_{i}=\{{\breve {z}}_{1},\ldots ,{\breve {z}}_{m}\}$; add ${\breve {\boldsymbol {\theta }}}_{i}$ to ${\boldsymbol {\Theta }}$; population.

Cumulative distribution function of the parameter Λ of an Exponential random variable when statistic $s_{\Lambda }=6.36$

Cumulative distribution function of the parameter A of a uniform continuous random variable when statistic $s_{A}=9.91$

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) $s_{\Lambda }=\sum _{j=1}^{m}x_{j}$, and

${\text{ iii) Inv}}(s_{\Lambda },{\boldsymbol {u}}_{i})=\sum _{j=1}^{m}(-\log u_{ij})/s_{\Lambda }$,

and the curve in the picture on the right when: i) X is a Uniform random variable in $[0,a]$, ii) $s_{A}=\max _{j=1,\ldots ,m}x_{j}$, and

${\text{iii) Inv}}(s_{A},{\boldsymbol {u}}_{i})=s_{A}/\max _{j=1,\ldots ,m}\{u_{ij}\}$.

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 (Efron and Tibshirani 1993)) the accuracy of the estimate distributions depends on the sample size.

Example

For ${\boldsymbol {x}}$ expected to represent a Pareto distribution, whose specification requires values for the parameters $a$ and k , we have that the cumulative distribution function reads:

Joint empirical cumulative distribution function of parameters $(A,K)$ of a Pareto random variable when $m=30,s_{1}=83.24$ and $s_{2}=8.37$ based on 5,000 replicas.

$F_{X}(x)=1-\left({\frac {k}{x}}\right)^{a}$.

A sampling mechanism $(g_{(a,k)},U)$ has $[0,1]$ uniform seed U and explaining function $g_{(a,k)}$ described by:

$x=g_{(a,k)}=(1-u)^{-{\frac {1}{a}}}k$

A relevant statistic ${\boldsymbol {s}}_{\boldsymbol {\Theta }}$ is constituted by the pair of joint sufficient statistics for $A$ and K, respectively $s_{1}=\sum _{i=1}^{m}\log x_{i},s_{2}=\min\{x_{i}\}$. The master equations read

$s_{1}=\sum _{i=1}^{m}-{\frac {1}{a}}\log(1-u_{i})+m\log k$

$s_{2}=(1-u_{\min })^{-{\frac {1}{a}}}k$

with $u_{\min }=\min\{u_{i}\}$.

Figure on the right reports the three dimensional plot of the empirical cumulative distribution function (2) of $(A,K)$.

References

• 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)

Category:Computational statistics Category:Data analysis Category:Statistical inference Category:Resampling (statistics)