In what family of distributions of error in a linear regression model the BLUP is LSP?

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?

• Related, maybe duplicate: stats.stackexchange.com/questions/173621/… – kjetil b halvorsen Sep 12 '17 at 21:29
• @kjetilbhalvorsen It does not seem related nor relevant that MLE=LSE. We are discussing prediction performance and I already did my part of work of searching related topic... – Henry.L Sep 13 '17 at 12:00