DHILLON: OBSERVED-DATA POSTERIOR CONDITIONS
fitdistrBayes 0.6.0

Source for the complete-data priors and results:
Ramos, P. L., Achire Quispe, E., de Oliveira, R. P., and Achcar, J. A.
(2026), Objective Bayesian inference for the Dhillon distribution,
unpublished manuscript supplied by the authors.

This derivation extends the conditions to independent right censoring.
It does not claim that the source manuscript already proves these censored
results. Priors remain those of the COMPLETE-DATA sampling model.

1. SETUP

S(t)=1/(1+theta*t^beta), beta>0, theta>0. Write l_i=log(x_i),
r=sum(delta_i), N=r+c after discarding zero censoring limits, and
y=log(theta). Exact observations must be positive. Define
h(v)=exp(v)/(1+exp(v))^2 and s(v)=1/(1+exp(v)).

Under Jeffreys/reference-beta, the posterior measure in (beta,y) is
proportional, apart from positive data-only factors, to

  beta^(r-1) prod_events h(y+beta*l_i) prod_censored s(y+beta*l_i) d beta d y.

Under reference-theta it is multiplied by
  w(y)=(pi^2+3+3*y^2)^(-1/2).

Uniformly, (1/4)*exp(-abs(v)) <= h(v) <= exp(-abs(v)), and
(1/2)*exp(-max(v,0)) <= s(v) <= exp(-max(v,0)).

2. PROPRIETY

For r=0, fix y in a bounded interval and let beta approach zero. The
integrand has order beta^(-1) for both priors; the normalizer diverges.

For r>=1 set y=beta*z and
  A_0(z)=sum_events abs(z+l_i)+sum_censored max(z+l_i,0).
The Jeffreys normalizer is comparable, up to fixed positive constants, to
the integral of beta^r exp(-beta*A_0(z)) over beta>0 and real z.
A_0 has slopes -r and N at the left and right tails. Its minimum is
strictly positive iff either exact event times differ, or a censoring limit
exceeds the common event time. If the minimum is positive, integration in
beta gives Gamma(r+1)*A_0(z)^(-r-1), integrable on both z tails and compact
sets. Reference-theta follows by w(y)<=1/sqrt(pi^2+3).

If the minimum is zero, all events share l and all censoring logs are <=l.
In the tube y=-beta*l+v, abs(v)<=1, beta>=1, all likelihood factors are
bounded below by positive constants. The Jeffreys beta integral dominates
beta^(r-1). Since w(y)>=C/beta on this tube, reference-theta dominates
beta^(r-2). Both diverge for r>=1, proving necessity as well.

3. POSITIVE BETA MOMENTS

In proper cases, multiplying by beta^q, q>0, gives the finite upper bound
Gamma(r+q+1)*integral A_0(z)^(-r-q-1) dz. Reference-theta follows by bounded
domination. Thus all positive beta moments are finite for both priors.

4. POSITIVE THETA MOMENTS

For beta in a fixed compact positive interval, the y->infinity integrand
has order exp((q-N)*y), with an additional factor of order 1/y for
reference-theta. Both theta moments diverge for q>=N, including logarithmic
divergence for reference-theta at q=N.

For 0<q<N define
  A_q(z)=sum_events abs(z+l_i)+sum_censored max(z+l_i,0)-q*z.
Its slopes at the two tails are -(r+q) and N-q. If min A_q>0, integration
in beta gives the finite bound Gamma(r+1)*integral A_q(z)^(-r-1) dz.
Both priors then have finite moments. If A_q is negative on an interval,
exponential growth causes divergence even after the factor w(beta*z).

If the minimum is zero at z0, A_q(z)<=C*abs(z-z0) locally. Restrict beta>=1
and abs(z-z0)<=1/beta. The exponential is bounded below on this tube,
whose z-width is 2/beta. Its Jeffreys contribution dominates
integral beta^(r-1) d beta. For reference-theta, w(beta*z)>=C'/beta,
so it dominates integral beta^(r-2) d beta. Both diverge since r>=1.
This argument includes a flat zero interval. Thus the exact condition for
both priors is q<N and min A_q>0.

The minimum occurs at a knot z=-l_i. Give l_i weight 1+delta_i, sort in
ascending order, and choose the first l_i whose cumulative weight reaches
N-q. This selects a minimizing knot in O(N log N) operations. The code
withholds certification for signs unresolved by floating-point arithmetic.
When all delta_i=1 this reduces to the complete-data criterion.

5. MDI

The MDI kernel is beta*theta^(1/beta). For fixed beta>0 the observed
likelihood has theta-tail order theta^(-N). Choose an interval of positive
beta sufficiently small that 1/beta-N>=-1. The theta integral diverges.
Tonelli gives an improper posterior for every finite sample. Zero censoring
limits are discarded before defining N because they contribute survival one.

6. PREDICTIVE MOMENTS

For every q>0 the conditional lifetime moment is infinite whenever beta<=q.
Each registered proper posterior is positive throughout the positive
parameter quadrant, so this set has positive posterior probability.
Tonelli gives an infinite q-th predictive moment. The predictive distribution
is nevertheless proper and has finite interior quantiles. The predictive
mean is infinite; its variance and standard deviation are undefined.

7. SAMPLING AND MODEL COMPARISON

Coordinates are a=log(beta), eta=log(theta)+beta*l0, for fixed l0. The
Jacobian to the original parameters is beta*theta and is included in the
target. Geometry is fixed during retained simulation. Augmentation samples
log S(T)=log S(censoring limit)-Exp(1) by exact inverse-survival transformation,
without clipping. It targets the same posterior as the observed likelihood.
Latent log times are stored and their logistic log density is used for parameter
updates. This removes the need to exponentiate possibly enormous imputed times.
The log-time Jacobian is parameter-independent and does not affect those updates.
Public natural-time predictions still reject nonrepresentable draws explicitly.
The transformed posterior target also avoids exponentiating log(theta), so
parameter representability does not impose a hidden truncation of that target.
Conversion of an unrepresentable parameter state to the public natural scale
stops explicitly, rather than returning a bounded approximation.

WAIC and LOO use observed-data contributions, not imputed complete-data
densities. Each LOO training posterior is checked by section 2. DIC requires
finite posterior parameter means. Marginal-likelihood Bayes factors are not
provided under these improper priors. Raw predictive sample averages should
not be interpreted as estimates of a finite predictive mean.
