Bootstraps methods for Tsallis Distributions

Description

Bootstrap functions.

Usage

tsal.bootstrap.errors(dist=NULL, reps=500, confidence=0.95,
    n=if(is.null(dist)) 1 else dist$n,
    shape=if(is.null(dist)) 1 else dist$shape,
    scale=if(is.null(dist)) 1 else dist$scale,
    q = if(is.null(dist)) tsal.q.from.shape(shape) else dist$q,
    kappa = if(is.null(dist)) tsal.kappa.from.ss(shape,scale) else dist$kappa,
    method = if(is.null(dist)) "mle.equation" else dist$method,
    xmin = if(is.null(dist)) 0 else dist$xmin)


tsal.total.magnitude(dist=NULL, n=if(is.null(dist)) 1 else dist$n,
    shape=if(is.null(dist)) 1 else dist$shape,
    scale=if(is.null(dist)) 1 else dist$scale,
    q = if(is.null(dist)) tsal.q.from.shape(shape) else dist$q,
    kappa = if(is.null(dist)) tsal.kappa.from.ss(shape,scale) else dist$kappa,
    xmin = if(is.null(dist)) 0 else dist$xmin,
    mult = 1)

Arguments

Details

tsal.bootstrap.errors finds biases and standard errors for parameter estimates by parametric bootstrapping, and simple confidence intervals Simulate, many times, drawing samples from the estimated distribution, of the same size as the original data; re-estimate the parameters on the simulated data. The distribution of the re-estimates around the estimated parameters is approximately the same as the distribution of the estimate around the true parameters. This function invokes the estimating-equation MLE, but it would be easy to modify to use other methods. Confidence intervals (CI) are calculated for each parameter separately, using a simple pivotal interval (see, e.g., Wasserman, _All of Statistics_, Section 8.3). Confidence regions for combinations of parameters would be a tedious, but straightforward, extension.

tsal.total.magnitude estimates the total magnitude of a tail-sampled population given that we have n samples from the tail of a distribution, i.e., only values >= xmin were retained, provide an estimate of the total magnitude (summed values) of the population. Then it estimates the number of objects, observed and un-observed, as n/pr(X >= xmin) and then multiply by the mean.

Value

tsal.bootstrap.errors returns a structured list, containing the actual parameter settings used, the estimated biases, the estimated standard errors, the lower confidence limits, the upper confidence limits, the sample size, the number of replicates, the confidence level, and the fitting method.

tsal.total.magnitude returns a list, giving estimated total magnitude and estimated total population size.

Author(s)

Cosma Shalizi (original R code), Christophe Dutang (R packaging)

References

Maximum Likelihood Estimation for q-Exponential (Tsallis) Distributions, http://bactra.org/research/tsallis-MLE/ and http://arxiv.org/abs/math.ST/0701854.

Examples

#####
# (1) fit
x <- rtsal(20, 1/2, 1/4)
tsal.loglik(x, 1/2, 1/4)

tsal.fit(x, method="mle.equation")
tsal.fit(x, method="mle.direct")
tsal.fit(x, method="leastsquares")



		

Results