I have spent a little time reading "Generalized fiducial inference for normal mixed models" by Cisewski and Hannig. First of all I am interested in understanding how to simulate the fiducial distribution of the parameters. But I don't understand this point.
Consider for instance the one-way random effects model $y_{ij}=\mu + \alpha_i + \epsilon_{ij}$ with $\alpha_i \sim {\cal N}(0,\sigma_a)$ independent of $\epsilon_{ij} \sim {\cal N}(0,\sigma)$.
First of all the authors write $\alpha_i= \sigma_a u_{i}$ and $\epsilon_{ij}=\sigma v_{ij}$ with $u_{i}$ and $v_{ij}$ independent standard normal random variables.
In fact they consider the data are rounded. For instance when $y_{ij}=1.32$ then actually the $(i,j)$-datum is the interval $[a_{ij}, b_{ij}[=[1.315, 1.325[$ because the "real" observation is rounded with two decimals. Thus the model is $$a_{ij} \leq \mu + \sigma_a u_i+\sigma v_{ij} < b_{ij}.$$.
Denote by $Z$ the random vector made of all the random components of the model: all the $u_i$'s and all the $v_{ij}'s$.
Define the set $Q\bigl(a,b, u, v\bigr) = \bigl\{ (\mu,\sigma_a,\sigma) \mid a \leq \mu + \sigma_a u + \sigma v < b\bigr\}$ and then define the random set $$Q(\text{data}, Z) = \bigcap_{i,j}Q\bigl(a_{ij},b_{ij},u_i,v_{ij}\bigr)$$ (with randomness contained in $Z$, and the data are fixed). When this set is nonempty we denote by $V\left(Z\right)$ an element of this set selected according to a specified rule (at least in a measurable way). The (generalized) fiducial distribution of $(\mu,\sigma_a,\sigma_b)$ is the conditional distribution of $V(Z)$ given $Q(\text{data}, Z) \neq \varnothing$.
Then the authors propose a sequential Monte-Carlo (SMC) generator of the fiducial distribution, and this is the point that confuses me. It starts on page 9 of the paper.
The authors claim that generating the fiducial distribution is equivalent to generating $Z$ conditional on $Q(\text{data}, Z) \neq \varnothing$. I agree with this part but they do not propose a measurable rule $V$, or I have missed this point while reading the paper (or is it an obvious point ?).
For $t=1, \ldots, n$ where $n$ is the sample size, denote by $Z_{1:t}$ the random components corresponding to the first $t$ data values (my notation is not appropriate here). Then the authors propose a sequential algorithm whose first step at time $t$ requires them to simulate the distribution $\pi_t$ of $Z_{1:t}$.
There are some undefined notations in the preprinted version whose link is given above: $Q_t$ and $Q_t^{(J)}$ and the definition of the set $C_t$ is not well-written. It is easy to guess the definition of $Q_t$: for $t=1,\ldots,n$ the random set $Q_t$ is constructed similarly to $Q(\text{data}, Z)$ but only with the first $t$ available data points and their associated random components in the model (my notation is not appropriate for defining $Q_t$), and $C_t$ are the possible values of the random components for which $Q_t$ is not empty.
Then $\pi_t$ is the independent product of the standard normal distributions truncated to $C_t$. I am totally lost here. In particular, does the algorithm require one to derive $C_t$ explicitly ?
I would be glad if you someone could clear up the points that I find confusing, or give me a link to a similar algorithm which is more detailed.