Step (piecewise-constant) CDF

Returns the CDF of a discrete distribution whose mass is concentrated at the right boundary of each bin. The CDF is right-continuous and piecewise constant with jumps at boundaries[2], ..., boundaries[K+1].

Usage

pdiscretestep(q, boundaries = NULL, pmf)

Arguments

q: Numeric vector of quantiles.
boundaries: Numeric vector of length \(K+1\) defining the bin edges. Must be strictly increasing. Defaults to 0:K (unit-width daily bins) where K is inferred from length(pmf).
pmf: Numeric vector of length \(K\) giving the probability mass for each bin. Must be non-negative and sum to approximately 1; if either condition is violated the function returns a vector of zeros (a soft simplex penalty for use inside optimisation).

Value

Numeric vector of CDF values, the same length as q.

Details

Below boundaries[1] the function returns 0. At boundaries[i+1] (the right edge of bin i), F jumps by pmf[i], so \(F(boundaries[i+1]) = \sum_{j=1}^{i} pmf_j\). For q in \([boundaries[i], boundaries[i+1])\), F equals \(\sum_{j=1}^{i-1} pmf_j\). At or above boundaries[K+1] the function returns 1.

Use with `fitdistdoublecens()`

This function carries the attribute vector_param = "pmf" so that fitdistdoublecens can drive it from a flat list of scalar parameters p1, ..., p_{K-1}. The free parameters are the first \(K-1\) bin probabilities; the last is set to 1 - sum(p1, ..., p_{K-1}). When the implied probabilities violate the simplex (any negative entry, or sum departing from 1 by more than \(10^{-8}\)), the function returns 0 (or near-zero density in ddiscretestep) rather than erroring; this drives the optimiser back to the feasible region.

Examples

# Two-bin PMF: mass 0.3 at x=1, mass 0.7 at x=2
pdiscretestep(c(0.5, 1, 1.5, 2), boundaries = 0:2, pmf = c(0.3, 0.7))
#> [1] 0.0 0.3 0.3 1.0