Abstract
The availability of some prior information, along with the current, may help us to improve the properties of statistical techniques. In this study, Bayesian best linear predictor is derived for simple and multivariate calibration situations. A comparative study of the mean squared errors of the Bayesian best linear predictor and the best linear predictor (classical) shows that Bayesian best linear predictor performs equally well. Interval estimates, both for known and unknown parameters, of the best linear predictor have been considered using different pivotal functions and different distributions for The outcomes have shown that the error probabilities depend upon
and to some extent on
the same invariants upon which the mean squared error of the estimator depends.
Acknowledgements
The authors are grateful to the anonymous reviewers for their valuable suggestions that helped in improving the initial version of the manuscript.
Appendix. Mean squared error
There are two situations. i: and
known; it is just of theoretical nature. ii:
and
unknown; it is the most practical situation.
and
known
The expression is quadratic in
and is minimized by
and
and its minimum is
Thus
Here is a
symmetric matrix i.e.,
Now trace
here
Finally
For it reduces to simple linear calibration and
Using Taylor series is
It depends only on four invariants i.e., and
Here
is size of the experiment;
relative concentration of the experiment;
relative bias of the experiment and
is squared correlation coefficient.
According to Muhammad and Riaz (Citation2016) in an invariant quantity and depends only on
and
as mean squared error in univariate case is
Where =
For the multivariate case mean squared error is
The mean squared error for the Bayesian linear predictor is
As is a function of the four invariants, N,
and
so would be the
So comparing and
is equivalent to comparing MSEπ/σ² and M
E/σ².
where f is
Note that when N is large; MLE or unbiased estimators of Γ are approximately equal to posterior expectationE (Γ ∣ expt.). Also when N is large; remaining fixed and
then f ≈ 1 Thus
One may see more details of this appendix in Muhammad (Citation1987) and Muhammad and Riaz (Citation2016)