paramci

Confidence intervals for probability distribution parameters

Description

example

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.

ci = paramci(pd,Name,Value) 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.

Examples

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Load the sample data. Create a vector containing the first column of students’ exam grade data.

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.

Load the sample data. Create a vector containing the first column of students’ exam grade data.

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

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

Distribution ObjectFunction or App to Create Probability Distribution Object
BinomialDistributionmakedist, fitdist, Distribution Fitter
BirnbaumSaundersDistributionmakedist, fitdist, Distribution Fitter
BurrDistributionmakedist, fitdist, Distribution Fitter
ExponentialDistributionmakedist, fitdist, Distribution Fitter
ExtremeValueDistributionmakedist, 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
PiecewiseLinearDistributionmakedist
PoissonDistributionmakedist, fitdist, Distribution Fitter
RayleighDistributionmakedist, fitdist, Distribution Fitter
RicianDistributionmakedist, fitdist, Distribution Fitter
StableDistributionmakedist, fitdist, Distribution Fitter
tLocationScaleDistributionmakedist, fitdist, Distribution Fitter
TriangularDistributionmakedist
UniformDistributionmakedist
WeibullDistributionmakedist, fitdist, Distribution Fitter

Name-Value Arguments

Specify optional comma-separated pairs of Name,Value arguments. Name is the argument name and Value is the corresponding value. Name must appear inside quotes. You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

Example: 'Alpha',0.01 specifies a 99% confidence interval.

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 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

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.

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.

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'

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

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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.