gold_cdf: Estimate of CDF from GOLD
In reykp/biRd: What the Package Does in One 'Title Case' Line

Description Usage Arguments Details Value Examples

Calculates a posterior estimate of the CDF based on the output from gold for an individual or vector of values.

1	gold_cdf(x, gold_output, burnin = NA)

`x`	A single value or vector of values at which to calculate the CDF. These values are to be entered on the scale of the data (i.e. values can fall outside of 0 and 1).
`gold_output`	The list returned by `gold` containing the density estimate results.
`burnin`	The desired burnin to discard from the results. If no values is entered, the default is half the number of iterations.

The function gold_cdf returns the posterior mean CDF. The CDF is calculated based on the following equation at each iteration:

F(x) =

\int_{0}^{x} exp(g(y)) / (\int_{0}^{1} exp(g(u)) du)

where g(x) is an unknown log density.

Given that g(x) is unknown, the normalizing constant is estimated using a weighted average and the set of unknown paramters that recieve a prior is g(x) at a finite set of points. These weights are also used in the estimate of the integral in the numerator.

gold_cdf returns a list of the CDF results from gold.

`cdf`	A vector of the posterior mean CDF values at each value in `x`.
`cri`	A matrix with 2 columns and rows equaling the length of `x` containing the 95% credible interval for the CDF at each of the points in `x`.
`mat`	A matrix containing the CDF values for each `x` at EACH itertation after the burnin is discarded. The number of columns is the length of `x`.
`x`	A vector containing the values at which to estimate the CDF.

## --------------------------------------------------------------------------------
## Beta Distribution
## --------------------------------------------------------------------------------

# First run 'duos' on data sampled from a Beta(2,5) distribution wiht 100 data points.
y <- rbeta(100, 2, 5)
gold_beta <- gold(y, s1=1, c1=1, s2=0.8, c2=.8, MH_N=20000)
#Calculate cdf at a variety of values
cdf_beta <- gold_cdf(x = c(.01, .25, .6, .9), gold_beta)

#Examine the CDF at 'x'
cdf_beta$cdf

#Examine the credibal intervals of the CDF at 'x'
cdf_beta$cri

## --------------------------------------------------------------------------------
## Normal Distribution
## --------------------------------------------------------------------------------

# First run 'gold' on data sampled from a Normal(0,1) distribution with 50 data points.
y <- rnorm(50, 0, 1)
gold_norm <- gold(y, s1 = 1, c1 = 0.8, s2 = 1, c2 = 0.5, MH_N=20000)
cdf_norm <- gold_cdf(x=c(-2, -1, 0, .8, 1.8), gold_norm)

#Examine the CDF at 'x'
cdf_norm$cdf

#Examine the credibal intervals of the CDF at 'x'
cdf_norm$cri

Histogram of distribution of the CDF density estimate at 0.8
hist(cdf_norm$mat[, 4])