Bounded ML and REML degrees for one-way variance component models with general mean structure

Consider the one-way layout

$$Y_{ij}=_{ij}+_i+_{ij},$$

where $i=1,,,q$, $j=1,,,n_i$, the random effects satisfy $_i\boldsymbol{\sim}(0,)$, the errors satisfy $_{ij}\boldsymbol{\sim}(0,)$, and all these variables are mutually independent. Let $N=n_1+++n_q$, let $X\in^{Np}$ be a full-rank design matrix whose column span contains $(1,,,1)^T$, and suppose that $ec(_{ij})=X$ for a fixed mean parameter vector $^p$. There are $q$ random group effects. Bounded-degree conjecture. The ML degree for this model with mean space $(X)$ is at most $3q-3$, and the REML degree is at most $2q-3$. The conjecture is motivated by numerical experiments with smaller models and randomly chosen design matrices; it asserts that general mean structures do not exceed the largest respective degrees for the common-mean model, whose largest degrees occur in the entirely unbalanced case with pairwise distinct group sizes.