tibber {distillery}  R Documentation 
TestInversion Bootstrap
Description
Calculate (1  alpha) * 100 percent confidence intervals for an estimated parameter using the testinversion bootstrap method.
Usage
tibber(x, statistic, B, rmodel, test.pars, rsize, block.length = 1, v.terms,
shuffle = NULL, replace = TRUE, alpha = 0.05, verbose = FALSE, ...)
tibberRM(x, statistic, B, rmodel, startval, rsize, block.length = 1,
v.terms, shuffle = NULL, replace = TRUE, alpha = 0.05, step.size,
tol = 1e04, max.iter = 1000, keep.iters = TRUE, verbose = FALSE,
...)
Arguments
x 
numeric vector or data frame giving the original data series. 
statistic 
function giving the estimated parameter value. Must minimally contain arguments 
B 
number of replicated bootstrap samples to use. 
rmodel 
function that simulates data based on the nuisance parameter provided by 
test.pars 
single number or vector giving the nuisance parameter value. If a vector of length greater than one, then the interpolation method will be applied to estimate the confidence bounds. 
startval 
one or two numbers giving the starting value for the nuisance parameter in the RobbinsMonro algorithm. If two numbers are given, the first is used as the starting value for the lower bound, and the second for the upper. 
rsize 
(optional) numeric less than the length of the series given by 
block.length 
(optional) length of blocks to use if the circular block bootstrap resampling scheme is to be used (default is iid sampling). 
v.terms 
(optional) gives the positions of the variance estimate in the output from 
shuffle 

replace 
logical stating whether or not to sample with replacement. 
alpha 
significance level for the test. 
step.size 
Step size for the RobbinsMonro algorithm. 
tol 
tolerance giving the value for how close the estimated pvalue needs to be to 
max.iter 
Maximum number of iterations to perform before stopping the RobbinsMonro algorithm. 
keep.iters 
logical, should information from each iteration of the RobbinsMonro algorithm be saved? 
verbose 
logical should progress information be printed to the screen. 
... 
Optional arguments to 
Details
The testinversion bootstrap (Carpenter 1999; Carpenter and Bithell 2000; Kabaila 1993) is a parametric bootstrap procedure that attempts to take advantage of the duality between confidence intervals and hypothesis tests in order to create bootstrap confidence intervals. Let X = X_1,...,X_n be a series of random variables, T, is a parameter of interest, and R(X) is an estimator for T. Further, let x = x_1,...,x_n be an observed realization of X, and r(x) an estimate for R(X), and let x* be a bootstrap resample of x, etc. Suppose that X is distributed according to a distribution, F, with parameter T and nuisance parameter V.
The procedure is carried out by estimating the pvalue, say p*, from r*_1, ..., r*_B estimated from a simulated sample from rmodel
assuming a specific value of V by way of finding the sum of r*_i < r(x) (with an additional correction for the ties r*_i = r(x)). The procedure is repeated for each of k values of V to form a sample of pvalues, p*_1, ..., p*_k. Finally, some form of rootfinding algorithm must be employed to find the values r*_L and r*_U that estimate the lower and upper values, resp., for R(X) associated with (1  alpha) * 100 percent confidence limits. For tibber
, the routine can be executed one time if test.pars
is of length one, which will enable a user to employ their own rootfinding algorithm. If test.pars
is a vector, then an interpolation estimate is found for the confidence end points. tibberRM
makes successive calls to tibber
and uses the RobbinsMonro algorithm (Robbins and Monro 1951) to try to find the appropriate bounds, as suggested by Garthwaite and Buckland (1992).
Value
For tibber, if test.pars is of length one, then a 3 by 1 matrix is returned (or, if v.terms
is supplied, then a 4 by 1 matrix) where the first two rows give estimates for R(X) based on the original simulated series and the median from the bootstrap samples, respectively. the last row gives the estimated pvalue. If v.terms
is supplied, then the fourth row gives the pvalue associated with the Studentized pvalue.
If test.pars is a vector with length k > 1, then a list object of class “tibbed” is returned, which has components:
results 
3 by k matrix (or 4 by k, if 
TIB.interpolated , STIB.interpolated 
numeric vector of length 3 giving the lower bound estimate, the estimate from the original data (i.e., r(x)), and the estimated upper bound as obtained from interpolating over the vector of possible values for V given by test.pars. The Studentized TIB interval, 
Plow , Pup , PstudLow , PstudUp 
Estimated pvalues used for interpolation of pvalue. 
call 
the original function call. 
data 
the original data passed by the x argument. 
statistic , B , rmodel , test.pars , rsize , block.length , alpha , replace 
arguments passed into the orignal function call. 
n 
original sample size. 
total.time 
Total time it took for the function to run. 
For tibberRM, a list of class “tibRMed” is returned with components:
call 
the original function call. 
x , statistic , B , rmodel , rsize , block.length , alpha , replace 
arguments passed into the orignal function call. 
result 
vector of length 3 giving the estimated confidence interval with the original parameter estimate in the second component. 
lower.p.value , upper.p.value 
Estimated achieved pvalues for the lower and upper bounds. 
lower.nuisance.par , upper.nuisance.par 
nuisance parameter values associated with the lower and upper bounds. 
lower.iterations , upper.iterations 
number of iterations of the RobbinsMonro algorithm it took to find the lower and upper bounds. 
total.time 
Total time it took for the function to run. 
Author(s)
Eric Gilleland
References
Carpenter, James (1999) Test inversion bootstrap confidence intervals. J. R. Statist. Soc. B, 61 (1), 159–172.
Carpenter, James and Bithell, John (2000) Bootstrap confidence intervals: when, which, what? A practical guide for medical statisticians. Statist. Med., 19, 1141–1164.
Garthwaite, P. H. and Buckland, S. T. (1992) Generating Monte Carlo confidence intervals by the RobbinsMonro process. Appl. Statist., 41, 159–171.
Kabaila, Paul (1993) Some properties of profile bootstrap confidence intervals. Austral. J. Statist., 35 (2), 205–214.
Robbins, Herbert and Monro, Sutton (1951) A stochastic approximation method. Ann. Math Statist., 22 (3), 400–407.
See Also
Examples
# The following example follows the example provided at:
#
# http://influentialpoints.com/Training/bootstrap_confidence_intervals.htm
#
# which is provided with a creative commons license:
#
# https://creativecommons.org/licenses/by/3.0/
#
y < c( 7, 7, 6, 9, 8, 7, 8, 7, 7, 7, 6, 6, 6, 8, 7, 7, 7, 7, 6, 7,
8, 7, 7, 6, 8, 7, 8, 7, 8, 7, 7, 7, 5, 7, 7, 7, 6, 7, 8, 7, 7,
8, 6, 9, 7, 14, 12, 10, 13, 15 )
trm < function( data, ... ) {
res < try( mean( data, trim = 0.1, ... ) )
if( class( res ) == "tryerror" ) return( NA )
else return( res )
} # end of 'trm' function.
genf < function( data, par, n, ... ) {
y < data * par
h < 1.06 * sd( y ) / ( n^( 1 / 5 ) )
y < y + rnorm( rnorm( n, 0, h ) )
y < round( y * ( y > 0 ) )
return( y )
} # end of 'genf' function.
look < tibber( x = y, statistic = trm, B = 500, rmodel = genf,
test.pars = seq( 0.85, 1.15, length.out = 100 ) )
look
plot( look )
# outer vertical blue lines should cross horizontal blue lines
# near where an estimated pvalue is located.
tibber( x = y, statistic = trm, B = 500, rmodel = genf, test.pars = 1 )
## Not run:
look2 < tibberRM(x = y, statistic = trm, B = 500, rmodel = genf, startval = 1,
step.size = 0.03, verbose = TRUE )
look2
# lower achieved est. pvalue should be close to 0.025
# upper should be close to 0.975.
plot( look2 )
trm2 < function( data, par, n, ... ) {
a < list( ... )
res < try( mean( data, trim = a$trim ) )
if( class( res ) == "tryerror" ) return( NA )
else return( res )
} # end of 'trm2' function.
tibber( x = y, statistic = trm2, B = 500, rmodel = genf,
test.pars = seq( 0.85, 1.15, length.out = 100 ), trim = 0.1 )
# Try getting the STIB interval. v.terms = 2 below because mfun
# returns the variance of the estimated parameter in the 2nd position.
#
# Note: the STIB interval can be a bit unstable.
mfun < function( data, ... ) return( c( mean( data ), var( data ) ) )
gennorm < function( data, par, n, ... ) {
return( rnorm( n = n, mean = mean( data ), sd = sqrt( par ) ) )
} # end of 'gennorm' function.
set.seed( 1544 )
z < rnorm( 50 )
mean( z )
var( z )
# Trialanderror is necessary to get a good result with interpolation method.
res < tibber( x = z, statistic = mfun, B = 500, rmodel = gennorm,
test.pars = seq( 0.95, 1.10, length.out = 100 ), v.terms = 2 )
res
plot( res )
# Much trialanderror is necessary to get a good result with RM method.
# If it fails to converge, try increasing the tolerance.
res2 < tibberRM( x = z, statistic = mfun, B = 500, rmodel = gennorm,
startval = c( 0.95, 1.1 ), step.size = 0.003, tol = 0.001, v.terms = 2,
verbose = TRUE )
# Note that it only gives the STIB interval.
res2
plot( res2 )
## End(Not run)