Documentation

### This is machine translation

Mouseover text to see original. Click the button below to return to the English version of the page.

Note: This page has been translated by MathWorks. Click here to see
To view all translated materials including this page, select Country from the country navigator on the bottom of this page.

# bootci

Bootstrap confidence interval

## Syntax

```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(...) ```

## Description

`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',type)` computes the bootstrap confidence interval of the statistic defined by the function `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 case

• `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`.

## Examples

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```

Download ebook