dfba_beta_descriptive {DFBA} R Documentation

## Descriptive Statistics for a Beta Distribution

### Description

Given the two shape parameters for a beta distribution, the function provides central tendency statistics, interval limits, and density and cumulative probabilities.

### Usage

dfba_beta_descriptive(a, b, prob_interval = 0.95)

### Arguments

 a The first shape parameter for the beta distribution. Must be positive and finite. b The second shape parameter for the beta distribution. Must be positive and finite. prob_interval Desired probability within interval limits (default is .95)

### Details

The density function for a beta variate is

f(x) = \begin{cases} Kx^{a-1}(1-x)^{b-1} & \quad \textrm{if } 0 \le x \le 1, \\0 & \quad \textrm{otherwise} \end{cases}

where

K = \frac{\Gamma(a + b)}{\Gamma(a)\Gamma(b)}.

(Johnson, Kotz, & Balakrishnan, 1995). The two shape parameters a and b must be positive values.

The dfba_beta_descriptive() function provides features to complement the beta distribution functions available in the stats package. The function provides the mean, median, mode, and variance for a beta variate in terms of its two shape parameters.

While the mean, variance, and median are straightforward, there are several conditions that result in an undefined mode. When either (1) a = b = 1, (2) a < 1, or (3) b < 1, the mode is undefined. For example, when a = b = 1, the function is the uniform distribution, which does not have a modal value. The other cases above result in the density function diverging at either x = 0 or x = 1. The function returns a value of NA for the mode for all the cases where a unique mode does not exist.

For interval estimation, the function finds an equal-tail interval limits in all cases, and it also provides the highest-density limits when there is a well-defined mode. When the mode does not exist, the function returns NA for the limits for the highest-density interval (HDI). For interval estimation, the probability between the lower and upper limit is the probability specified in the prob_interval input. The dfba_beta_descriptive() output object includes a dataframe that has density and cumulative probability information that can be used for plotting.

### Value

A list containing the following components:

 a The first beta shape parameter b The second beta shape parameter prob_interval The probability for interval estimates x_mean The mean of the distribution x_median The median of the distribution x_mode The mode for the distribution x_variance The variance for the distribution eti_lower The equal-tail lower interval limit eti_upper The equal-tail upper interval limit hdi_lower The lower limit for the highest-density interval hdi_upper The upper limit for the highest-density interval outputdf A dataframe of x, density, and cumulative probability for x from 0 to 1 in steps of .005

### References

Johnson, N. L., Kotz S., and Balakrishnan, N. (1995). Continuous Univariate Distributions, Vol. 1, New York: Wiley.

Distributions for additional details on functions for the beta distribution in the stats package.

### Examples

dfba_beta_descriptive(a = 38,
b = 55)

dfba_beta_descriptive(38,
55,
prob_interval=.99)

[Package DFBA version 0.1.0 Index]