Handle bounds for discrete and zero-inflated densities

docs/overview-continuous.dox

@@ -37,12 +37,25 @@ An explicitly supplied `bandwidth` is interpreted on the transformed scale.
 `get_bandwidth()` reports that bulk bandwidth before `multiplier` is applied;
 it does not report an endpoint-expert bandwidth.
 
+@section continuous-zero-inflated Zero-inflated fits
+
+For a zero-inflated fit, exact zeros form a separate point mass and the
+remaining observations form a continuous hurdle component. Finite `xmin` and
+`xmax` constrain only this continuous component: zeros need not lie in the
+interval, while every nonzero observation must. After zeros are removed, the
+continuous observations use the same support transform, normalization, and
+adaptive boundary repair as an ordinary continuous fit. The bounds are used
+as supplied, without the half-unit adjustment used for discrete jitter cells.
+If all observations are zero, the point mass is one and no continuous
+component is fitted.
+
 @section continuous-repair Boundary repair
 
 `boundary_repair=true` makes each known finite endpoint eligible for repair;
 it does not force an expert to be used. Repair is considered only for
-continuous samples with at least 16 cleaned observations and effective sample
-size
+continuous samples, bounded discrete jitter samples, or nonzero components of
+zero-inflated samples with at least 16 cleaned observations and effective
+sample size
 
 @f[
 n_e=\frac{(\sum_i w_i)^2}{\sum_i w_i^2}

docs/overview-discrete.dox

@@ -1,4 +1,62 @@
 /** @page overview-discrete Discrete Distributions
 
+Discrete fits model integer-valued observations, including negative integers.
+Any finite `xmin` or `xmax` must also be an integer. A finite bound describes
+the support of the discrete variable itself; probabilities outside those
+integer bounds are zero.
+
+@section discrete-jitter Jittering and support transforms
+
+Before fitting, tied observations are spread deterministically within their
+unit cells by adding conditionally equidistant noise in @f$(-1/2,1/2)@f$.
+Consequently, if the discrete support has a finite lower endpoint @f$a@f$ or
+upper endpoint @f$b@f$, the continuous jitter support has the corresponding
+endpoint
+
+@f[
+a_J=a-1/2,
+\qquad
+b_J=b+1/2.
+@f]
+
+The local-likelihood bulk estimator applies the same support transforms
+described in @ref continuous-bulk to these jitter-adjusted endpoints. Thus
+`xmin=0`, for example, invokes a one-sided transformation at @f$-1/2@f$;
+finite `xmin` and `xmax` invoke a two-sided transformation over the full
+jitter cells. The public bounds remain unchanged and integer-valued.
+
+@section discrete-pmf From jitter density to probability mass
+
+The fitted interpolation grid represents a continuous jitter density, but the
+discrete probability mass function is formed only from its values at integer
+levels:
+
+@f[
+\widehat p(k)=
+\frac{\widehat f_J(k)}{\sum_{j\in\mathcal S}\widehat f_J(j)},
+\qquad k\in\mathcal S.
+@f]
+
+Here @f$\mathcal S@f$ is the integer support inferred from the interpolation
+grid and restricted by any finite public bounds. Noninteger arguments and
+integer values outside this support have probability zero. The CDF and
+quantile function use the same normalized masses.
+
+This distinction also explains why an unconstrained jitter density can give a
+good discrete estimate despite having a nonzero continuous integral outside
+the integer support: that integral is not assigned to any discrete level.
+Only the density ordinates at integer centers enter the PMF. Supplying bounds
+uses the known support during fitting and prevents the transformed bulk from
+extrapolating through the ends of the boundary cells.
+
+@section discrete-repair Adaptive boundary repair
+
+When `boundary_repair=true`, each supplied discrete endpoint is eligible for
+the adaptive boundary expert described in @ref continuous-repair and
+@ref continuous-fusion. Classification and repair operate on the jittered
+observations and the half-unit endpoint. The expert is not forced: an endpoint
+whose local behavior appears vanishing, exploding, or ambiguous retains the
+transformed bulk estimate. This guards finite-looking boundary cells without
+replacing a stable bulk tail based only on the existence of a support bound.
 
 */