Linear Prediction Sufficiency for New Observations in the General Gauss–Markov Model

Best linear unbiased prediction for linear combinations in general mixed linear models

“…As pointed out by a referee, we can see that Z = 0 in conjunction with K = 0 results in an interesting topic. It is actually a combination of ordinary linear sufficiency and the so-called linear prediction sufficiency proposed by Isotalo and Puntanen [20] w.r.t. a special prediction model in some sense.…”

“…[1] and Groß [12]), and linear prediction sufficiency (cf. Isotalo and Puntanen [20]). Furthermore, Ibarrola and Pérez-Palomares [17,18] applied linear sufficiency and linear completeness to a continuous time linear model and corresponding characterizations were offered.…”

“…It is known that, when estimating Xβ, linear completeness together with (ordinary) linear sufficiency is equivalent to (ordinary) linear minimal sufficiency. Isotalo and Puntanen [20] argued that the corresponding relation (among their linear prediction sufficiency, linear minimal prediction sufficiency, and linear completeness) does not seem to hold. However, this is not necessarily the case as regards the BLUP problem, when predicting f (l, m).…”

“…Such linear statistics will be called linearly (combination) sufficient statistics, combining the notion of linear sufficiency introduced by Drygas [8] when Xβ was estimated and the notion of linear error-sufficiency investigated by Groß [12] while the random error term was predicted in a general Gauss-Markov model. This is as argued by Isotalo and Puntanen [20] that since the uniformly minimum variance unbiased estimators (UMVUEs) are based on the concepts of sufficiency and completeness and the well known Rao-Blackwell and Lehmann-Scheffé Theorems, linear versions of above can be defined correspondingly and then used as an alternative to obtaining the BLUE or BLUP.…”

Some notes on linear sufficiency

“…11.1). For the corresponding concept, sometimes called linear error-sufficiency or linear prediction sufficiency, defined for the best linear unbiased predictor, see Groß (1998) and Isotalo and Puntanen (2006b …”