We know that when the error $\epsilon$ is Gaussian in a linear regression model $$Y=X\beta +\epsilon$$ the BLUP(Best linear unbiased predictor, which minimizes MSPE at any new location $x_0$, i.e. the quantity $\mathrm{E}\{[\hat{Y}(x_0)-Y(x_0)]^2\}$ subject to $\mathrm{E}\{\hat{Y}(x_0)\}=Y(x_0)$) is also LSP (Least square predictor, by plugging in the Least square estimator, which minimizes $[\hat{Y}(x_0)-Y(x_0)]^2$ into the mean function), which is also conditional expectation of $\mathrm{E}\{Y(x_0)\mid X\}$.
Similar claim holds for random fields and multidimensional processes.
Question.
BLUP($X$),LSP($X$),$E\{\bullet \mid X\}$ are the corresponding statistical functionals based on $X$.
(1) Conversely, if BLUP($X$)=LSP($X$), does it necessarily mean the error belong to Gaussian family? If not, what other families on error terms satisfies this property such that BLUP=LSP?
(2) if BLUP($X$)=$E\{\bullet \mid X\}$, does it necessarily mean the error belong to Gaussian family? If not, what other families on error terms satisfies this property such that BLUP=conditional expectation?
