paramci

R2026b

Confidence intervals for probability distribution parameters

collapse all in page

Syntax

ci = paramci(pd)
ci = paramci(pd,Name=Value)

Description

ci = paramci(pd) returns the array ci containing the lower and upper boundaries of the 95% confidence interval for each parameter in probability distribution pd.

example

ci = paramci(pd,Name=Value) returns confidence intervals with additional options specified by one or more name-value arguments. For example, you can specify a different percentage for the confidence interval, or compute confidence intervals only for selected parameters.

Examples

collapse all

Open Live Script

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.

Open Live Script

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=0.01)
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

collapse all

Probability distribution, specified as one of the probability distribution objects in the following table.

Distribution ObjectFunction or App Used to Create Probability Distribution Object
BetaDistributionmakedist, fitdist, Distribution Fitter
BinomialDistributionmakedist, fitdist, Distribution Fitter
BirnbaumSaundersDistributionmakedist, fitdist, Distribution Fitter
BurrDistributionmakedist, fitdist, Distribution Fitter
EmpiricalDistributionfitdist
ExponentialDistributionmakedist, fitdist, Distribution Fitter
ExtremeValueDistributionmakedist, fitdist, Distribution Fitter
GammaDistributionmakedist, fitdist, Distribution Fitter
GeneralizedExtremeValueDistributionmakedist, fitdist, Distribution Fitter
GeneralizedParetoDistributionmakedist, fitdist, Distribution Fitter
HalfNormalDistributionmakedist, fitdist, Distribution Fitter
InverseGaussianDistributionmakedist, fitdist, Distribution Fitter
KernelDistributionfitdist, Distribution Fitter
LogisticDistributionmakedist, fitdist, Distribution Fitter
LoglogisticDistributionmakedist, fitdist, Distribution Fitter
LognormalDistributionmakedist, fitdist, Distribution Fitter
LoguniformDistributionmakedist
MultinomialDistributionmakedist
NakagamiDistributionmakedist, fitdist, Distribution Fitter
NegativeBinomialDistributionmakedist, fitdist, Distribution Fitter
NormalDistributionmakedist, fitdist, Distribution Fitter
PearsonDistributionmakedist
PiecewiseLinearDistributionmakedist
PoissonDistributionmakedist, fitdist, Distribution Fitter
RayleighDistributionmakedist, fitdist, Distribution Fitter
RicianDistributionmakedist, fitdist, Distribution Fitter
ShiftedExponentialDistributionmakedist, fitdist, Distribution Fitter
ShiftedGammaDistributionmakedist, fitdist, Distribution Fitter
ShiftedWeibullDistributionmakedist, fitdist, Distribution Fitter
StableDistributionmakedist, fitdist, Distribution Fitter
tLocationScaleDistributionmakedist, fitdist, Distribution Fitter
TriangularDistributionmakedist
UniformDistributionmakedist
WeibullDistributionmakedist, fitdist, Distribution Fitter

Name-Value Arguments

collapse all

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.

Example: paramci(pd,Alpha=0.01) specifies a 99% confidence interval.

Significance level for the confidence interval, specified as 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 list for which to compute confidence intervals, specified as 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

Computation method for the confidence intervals, specified as "exact", "wald", or "lr".

"exact" computes the confidence intervals using an exact method, and is available for the following distributions.

DistributionComputation Method
BinomialCompute using the Clopper-Pearson method based on exact probability calculations. This method does not provide exact coverage probabilities.
ExponentialCompute using a method based on a chi-square distribution. This method provides exact coverage for complete and Type 2 censored samples.
NormalComputation 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.
LognormalComputation 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.
PoissonComputation 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.
RayleighComputation method based on a chi-square distribution provides exact coverage probabilities.
Shifted exponentialFor uncensored samples, paramci uses a computation method based on the F-distribution to provide exact coverage. For right-censored samples, paramci uses the likelihood ratio method if Type is "exact".

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. The Wald method is not supported for shifted exponential distributions. It is supported for shifted gamma and shifted Weibull distributions, but the software issues a warning if the fitted shape parameter is less than 1. In this case, the likelihood surface can be irregular, leading the Wald interval to be unreliable.

"exact" is the default when it is available. "lr" is the default for shifted gamma and shifted Weibull distributions. Otherwise, the default is "wald".

Example: Type="wald"

Data Types: char | string

Boolean flag for the log scale, specified as 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

collapse all

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

expand all

Version History

Introduced in R2013a

expand all

See Also

| | | | |

Topics