paramci

Confidence intervals for probability distribution parameters

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.

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

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 a probability distribution object created using one of the following.

Function or App	Description
makedist	Create a probability distribution object using specified parameter values.
fitdist	Fit a probability distribution object to sample data.
Distribution Fitter	Fit a probability distribution to sample data using the interactive Distribution Fitter app and export the fitted object to the workspace.

Name-Value Pair 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.

'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.

'exact' is the default when it is available. 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.

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.