paramci
Confidence intervals for probability distribution parameters
Description
returns
confidence intervals with additional options specified by one or more
name-value pair arguments. For example, you can specify a different
percentage for the confidence interval, or compute confidence intervals
only for selected parameters.ci
= paramci(pd
,Name,Value
)
Examples
Parameter Confidence Intervals
Load the sample data. Create a vector containing the first column of students’ exam grade data.
load examgrades
x = grades(:,1);
Fit a normal distribution object to the data.
pd = fitdist(x,'Normal')
pd = NormalDistribution Normal distribution mu = 75.0083 [73.4321, 76.5846] sigma = 8.7202 [7.7391, 9.98843]
The intervals next to the parameter estimates are the 95% confidence intervals for the distribution parameters.
You can also obtain these intervals by using the function paramci
.
ci = paramci(pd)
ci = 2×2
73.4321 7.7391
76.5846 9.9884
Column 1 of ci
contains the lower and upper 95% confidence interval boundaries for the mu parameter, and column 2 contains the boundaries for the sigma parameter.
Change Parameter Confidence Intervals
Load the sample data. Create a vector containing the first column of students’ exam grade data.
load examgrades
x = grades(:,1);
Fit a normal distribution object to the data.
pd = fitdist(x,'Normal')
pd = NormalDistribution Normal distribution mu = 75.0083 [73.4321, 76.5846] sigma = 8.7202 [7.7391, 9.98843]
Compute the 99% confidence interval for the distribution parameters.
ci = paramci(pd,'Alpha',.01)
ci = 2×2
72.9245 7.4627
77.0922 10.4403
Column 1 of ci
contains the lower and upper 99% confidence interval boundaries for the mu parameter, and column 2 contains the boundaries for the sigma parameter.
Input Arguments
pd
— Probability distribution
probability distribution object
Probability distribution, specified as one of the probability distribution objects in the following table.
Distribution Object | Function or App Used to Create Probability Distribution Object |
---|---|
BetaDistribution | makedist , fitdist , Distribution Fitter |
BinomialDistribution | makedist , fitdist ,
Distribution Fitter |
BirnbaumSaundersDistribution | makedist , fitdist ,
Distribution Fitter |
BurrDistribution | makedist , fitdist ,
Distribution Fitter |
ExponentialDistribution | makedist , fitdist ,
Distribution Fitter |
ExtremeValueDistribution | makedist , fitdist ,
Distribution Fitter |
GammaDistribution | makedist , fitdist ,
Distribution Fitter |
GeneralizedExtremeValueDistribution | makedist , fitdist ,
Distribution Fitter |
GeneralizedParetoDistribution | makedist , fitdist ,
Distribution Fitter |
HalfNormalDistribution | makedist , fitdist ,
Distribution Fitter |
InverseGaussianDistribution | makedist , fitdist ,
Distribution Fitter |
KernelDistribution | fitdist , Distribution Fitter |
LogisticDistribution | makedist , fitdist ,
Distribution Fitter |
LoglogisticDistribution | makedist , fitdist ,
Distribution Fitter |
LognormalDistribution | makedist , fitdist ,
Distribution Fitter |
LoguniformDistribution | makedist |
MultinomialDistribution | makedist |
NakagamiDistribution | makedist , fitdist ,
Distribution Fitter |
NegativeBinomialDistribution | makedist , fitdist ,
Distribution Fitter |
NormalDistribution | makedist , fitdist ,
Distribution Fitter |
PiecewiseLinearDistribution | makedist |
PoissonDistribution | makedist , fitdist ,
Distribution Fitter |
RayleighDistribution | makedist , fitdist ,
Distribution Fitter |
RicianDistribution | makedist , fitdist ,
Distribution Fitter |
StableDistribution | makedist , fitdist ,
Distribution Fitter |
tLocationScaleDistribution | makedist , fitdist ,
Distribution Fitter |
TriangularDistribution | makedist |
UniformDistribution | makedist |
WeibullDistribution | makedist , fitdist ,
Distribution Fitter |
Name-Value Arguments
Specify optional pairs of arguments as
Name1=Value1,...,NameN=ValueN
, where Name
is
the argument name and Value
is the corresponding value.
Name-value arguments must appear after other arguments, but the order of the
pairs does not matter.
Before R2021a, use commas to separate each name and value, and enclose
Name
in quotes.
Example: 'Alpha',0.01
specifies a 99% confidence
interval.
Alpha
— Significance level
0.05
(default) | scalar value in the range (0,1)
Significance level for the confidence interval, specified as the
comma-separated pair consisting of 'Alpha'
and a
scalar value in the range (0,1). The confidence level of
ci
is 100(1–Alpha)
%. The
default value 0.05
corresponds to a 95% confidence
interval.
Example: 'Alpha',0.01
Data Types: single
| double
Parameter
— Parameter list
character vector | string array | cell array of character vectors
Parameter list for which to compute confidence intervals, specified as the comma-separated
pair consisting of 'Parameter'
and a character
vector, string array, or cell array of character vectors containing the
parameter names. By default, paramci
computes
confidence intervals for all distribution parameters.
Example: 'Parameter','mu'
Data Types: char
| string
| cell
Type
— Computation method
'exact'
| 'Wald'
| 'lr'
Computation method for the confidence intervals, specified as
the comma-separated pair consisting of 'Type'
and 'exact'
, 'Wald'
,
or 'lr'
.
'exact'
computes the confidence intervals
using an exact method, and is available for the following distributions.
Distribution | Computation Method |
---|---|
Binomial | Compute using the Clopper-Pearson method based on exact probability calculations. This method does not provide exact coverage probabilities. |
Exponential | Compute using a method based on a chi-square distribution. This method provides exact coverage for complete and Type 2 censored samples. |
Normal | Computation method based on t and chi-square
distributions for uncensored samples provides exact coverage for uncensored
samples. For censored samples, paramci uses the
Wald method if Type is exact . |
Lognormal | Computation method based on t and chi-square
distributions for uncensored samples provides exact coverage. For
censored samples, paramci uses the Wald method
if Type is exact . |
Poisson | Computation method based on a chi-square distribution provides exact coverage. For large degrees of freedom, the chi-square is approximated by a normal distribution for numerical efficiency. |
Rayleigh | Computation method based on a chi-square distribution provides exact coverage probabilities. |
Alternatively, you can specify 'Wald'
to compute
the confidence intervals using the Wald method, or
'lr'
to compute the confidence intervals using
the likelihood ratio method.
'exact'
is the default when it is available. Otherwise, the default is
'Wald'
.
Example: 'Type','Wald'
LogFlag
— Boolean flag for log scale
vector
Boolean flag for the log scale, specified as the comma-separated
pair consisting of 'LogFlag'
and a vector containing
Boolean values corresponding to each distribution parameter. The flag
specifies which Wald intervals to compute on a log scale. The default
values depend on the distribution.
Example: 'LogFlag',[0,1]
Data Types: logical
Output Arguments
ci
— Confidence interval
array
Confidence interval, returned as a p-by-2 array containing the lower and
upper bounds of the 100(1–Alpha)
% confidence interval for
each distribution parameter. p is the number of
distribution parameters.
If you create pd
by using makedist
and specifying the
distribution parameters, the lower and upper bounds are equal to the
specified parameters.
Extended Capabilities
GPU Arrays
Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.
This function fully supports GPU arrays. For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).
Version History
Introduced in R2013a
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