duos_cdf: Estimate of the CDF from 'duos'
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 duos for an individual or vector of values.

1	duos_cdf(x, duos_output, burnin = NA, scale = FALSE)

`x`	A single value or vector of values at which to calculate the CDF. These values should be entered on the scale of the data (i.e. values can fall outside of 0 and 1 if data does).
`duos_output`	The list returned by `duos` containing the density estimate results.
`burnin`	The desired burnin to discard from the results. If no value is entered, the default is half the number of iterations.
`scale`	This value TRUE/FALSE indicates whether to return scaled or unscaled results IF the original data does not fall between 0 and 1. The default is FALSE (i.e. returns results on the original data scale).

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

F(x) =

(π_1) / (γ_1) * x , 0 ≤ x < γ_1

π_1 + (π_2) / (γ_2-γ_1) * (x-γ_1) , γ_1 ≤ x < γ_2

π_1 + π_2 + (π_3) / (γ_3-γ_2) * (x-γ_2) , γ_2 ≤ x < γ_3

... , ... ≤ x < ...

∑_{i=1}^{k-1} π_i + (π_k) / (γ_k-γ_{k-1}) * (x-γ_{k-1}) , γ_{k-1} ≤ x < γ_k

∑_{i=1}^{k}π_i + (π_{k+1}) / (1-γ_k) * (x-γ_k) , γ_k ≤ x < 1

where γ_1 < γ_2 < ... < γ_k is in (0,1) and π_1 + π_2 + ... + π_{k+1} = 1

duos_cdf returns a list of the CDF results from duos.

`cdf`	A vector of the posterior mean CDF at each value in `x`.
`cri`	A matrix with 2 columns and a row for each value in `x`. It contains the 95% credible interval for the CDF at each point in `x`.
`mat`	A matrix containing the CDF values for `x` at EACH iteration after the burnin is discarded. There is a column for each value in `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 with 100 data points.
y <- rbeta(100, 2, 5)
duos_beta <- duos(y = y, k = 5, MH_N = 20000)
# Estimate the CDF at a vector of values
cdf_beta <- duos_cdf(x = c(.01, .25, .6, .9), duos_beta)

# Examine the CDF at each value in 'x'
cdf_beta$cdf

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

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

# First run 'duos' on data sampled from a Normal(0,1) distribution with 50 data points.
y <- rnorm(50, 0, 1)
duos_norm <- duos(y = y, k = 4, MH_N = 20000, scale_l = sd(y), scale_u = sd(y))
# Estimate the CDF at a vector of values
cdf_norm <- duos_cdf(x = c(-2, -1, 0, .8, 1.8), duos_norm)

# Examine the CDF at 'x'
cdf_norm$cdf

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

# Plot a histogram of distribution of the posterior draws for the CDF estimate at 0.8
hist(cdf_norm$mat[, 4])

# Find probability of being greater than 2
1 - duos_cdf(2, duos_norm)$cdf