Bootstrap confidence interval

`ci = bootci(nboot,bootfun,...)`

ci = bootci(nboot,{bootfun,...},'alpha',alpha)

ci = bootci(nboot,{bootfun,...},...,'type',* type*)

ci = bootci(nboot,{bootfun,...},...,'type','student','nbootstd',nbootstd)

ci = bootci(nboot,{bootfun,...},...,'type','student','stderr',stderr)

ci = bootci(nboot,{bootfun,...},...,'Weights',weights)

ci = bootci(nboot,{bootfun,...},...,'Options',options)

[ci,bootstat] = bootci(...)

`ci = bootci(nboot,bootfun,...)`

computes
the 95% bootstrap confidence interval of the statistic computed by
the function `bootfun`

. `nboot`

is
a positive integer indicating the number of bootstrap samples used
in the computation. `bootfun`

is a function handle
specified with `@`

. The third and later input arguments
to `bootci`

are data (scalars, column vectors, or
matrices) that are used to create inputs to `bootfun`

. `bootci`

creates
each bootstrap sample by sampling with replacement from the rows of
the non-scalar data arguments (these must have the same number of
rows). Scalar data are passed to `bootfun`

unchanged.

If `bootfun`

returns a scalar, `ci`

is
a vector containing the lower and upper bounds of the confidence interval.
If `bootfun`

returns a vector of length *m*, `ci`

is
an array of size 2-by-*m*, where `ci(1,:)`

are
lower bounds and `ci(2,:)`

are upper bounds. If `bootfun`

returns
an array of size *m*-by-*n*-by-*p*-by-..., `ci`

is
an array of size 2-by-*m*-by-*n*-by-*p*-by-...,
where `ci(1,:,:,:,...)`

is an array of lower bounds
and `ci(2,:,:,:,...)`

is an array of upper bounds.

`ci = bootci(nboot,{bootfun,...},'alpha',alpha)`

computes
the `100*(1-alpha)`

bootstrap confidence interval
of the statistic defined by the function `bootfun`

. `bootfun`

and
the data that `bootci`

passes to it are contained
in a single cell array. `alpha`

is a scalar between `0`

and `1`

.
The default value of `alpha`

is `0.05`

.

`ci = bootci(nboot,{bootfun,...},...,'type',`

computes
the bootstrap confidence interval of the statistic defined by the
function * type*)

`bootfun`

. `type`

is
the confidence interval type, chosen from among the following:`'norm'`

or`'normal'`

— Normal approximated interval with bootstrapped bias and standard error.`'per'`

or`'percentile'`

— Basic percentile method.`'cper'`

or`'corrected percentile'`

— Bias corrected percentile method.`'bca'`

— Bias corrected and accelerated percentile method. This is the default.`'stud'`

or`'student'`

— Studentized confidence interval.

`ci = bootci(nboot,{bootfun,...},...,'type','student','nbootstd',nbootstd)`

computes
the studentized bootstrap confidence interval of the statistic defined
by the function `bootfun`

. The standard error of
the bootstrap statistics is estimated using bootstrap, with `nbootstd`

bootstrap
data samples. `nbootstd`

is a positive integer value.
The default value of `nbootstd`

is `100`

.

`ci = bootci(nboot,{bootfun,...},...,'type','student','stderr',stderr)`

computes
the studentized bootstrap confidence interval of statistics defined
by the function `bootfun`

. The standard error of
the bootstrap statistics is evaluated by the function `stderr`

. `stderr`

is
a function handle. `stderr`

takes the same arguments
as `bootfun`

and returns the standard error of the
statistic computed by `bootfun`

.

`ci = bootci(nboot,{bootfun,...},...,'Weights',weights)`

specifies
observation weights. `weights`

must be a vector of
non-negative numbers with at least one positive element. The number
of elements in `weights`

must be equal to the number
of rows in non-scalar input arguments to `bootfun`

.
To obtain one bootstrap replicate, `bootstrp`

samples *N* out
of *N* with replacement using these weights as multinomial
sampling probabilities.

`ci = bootci(nboot,{bootfun,...},...,'Options',options)`

specifies
options that govern the computation of bootstrap iterations. One option
requests that `bootci`

perform bootstrap iterations
using multiple processors, if the Parallel Computing Toolbox™ is
available. Two options specify the random number streams to be used
in bootstrap resampling. This argument is a struct that you can create
with a call to `statset`

. You can
retrieve values of the individual fields with a call to `statget`

. Applicable `statset`

parameters
are:

`'UseParallel'`

— If`true`

and if a`parpool`

of the Parallel Computing Toolbox is open, compute bootstrap iterations in parallel. If the Parallel Computing Toolbox is not installed, or a`parpool`

is not open, computation occurs in serial mode. Default is`false`

, or serial computation.`UseSubstreams`

— Set to`true`

to compute in parallel in a reproducible fashion. Default is`false`

. To compute reproducibly, set`Streams`

to a type allowing substreams:`'mlfg6331_64'`

or`'mrg32k3a'`

.`Streams`

— A`RandStream`

object or cell array of such objects. If you do not specify`Streams`

,`bootci`

uses the default stream or streams. If you choose to specify`Streams`

, use a single object except in the caseYou have an open Parallel pool

`UseParallel`

is`true`

`UseSubstreams`

is`false`

In that case, use a cell array the same size as the Parallel pool.

`[ci,bootstat] = bootci(...)`

also returns
the bootstrapped statistic computed for each of the `nboot`

bootstrap
replicate samples. Each row of `bootstat`

contains
the results of applying `bootfun`

to one bootstrap
sample. If `bootfun`

returns a matrix or array, then
this output is converted to a row vector for storage in `bootstat`

.

Compute the confidence interval for the capability index in statistical process control:

y = normrnd(1,1,30,1); % Simulated process data LSL = -3; USL = 3; % Process specifications capable = @(x)(USL-LSL)./(6* std(x)); % Process capability ci = bootci(2000,capable,y) % BCa confidence interval ci = 0.8122 1.2657 sci = bootci(2000,{capable,y},'type','student') % Studentized ci sci = 0.7739 1.2707

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