On best linear and Bayesian linear predictor in calibration

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 $p(t).$ The outcomes have shown that the error probabilities depend upon ${\text{N,B}}_{\text{N}},{\text{C}}_{\text{N}}$ and to some extent on $\rho ,$ the same invariants upon which the mean squared error of the estimator depends.

Keywords:

• Bayesian approach
• best linear predictor
• conditional and unconditional intervals
• mean square error
• pivotal functions

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:$\alpha ,\mathrm{ }\mathrm{\beta }$ and$\Gamma$ known; it is just of theoretical nature. ii: $\alpha ,\mathrm{ }\beta$ and$\Gamma$ unknown; it is the most practical situation.

1. $\alpha ,\mathrm{ }\beta$ and $\Gamma$ known $${\text{MSE = E[T -(C + D}}^{\text{T}}{\text{X)]}}^{\text{2}}{\sigma }^{\text{2}}{\text{ - D}}^{\text{T}}\text{COV(T,X) = }{\sigma }^{\text{2}}\text{[1 - }{\sigma }^{\text{2}}{\beta }^{\text{T}}{\text{(}\Gamma \text{ + }{\sigma }^{\text{2}}\beta {\beta }^{\text{T}}\text{)}}^{\text{-1}}\beta \text{]}$$ $$\text{= }{\sigma }^{\text{2}}\text{(1 - }{\rho }^{\text{2}})$$

2. $\text{MSE}={\text{E(T }-\widehat{\mathrm{C}}\mathrm{ }-{\widehat{\mathrm{D}}}^{\mathrm{T}}\text{ X}\text{)}}^2={\text{EE}[\text{(T }-\widehat{\mathrm{C}}\mathrm{ }-{\widehat{\mathrm{D}}}^{\mathrm{T}}\text{ X}\text{)}}^2\mathrm{|}\widehat{\text{C}},\widehat{\mathrm{D}}\text{]}$

The expression $\mathrm{E}[{\text{(T }-\widehat{\mathrm{C}}\mathrm{ }-{\widehat{\mathrm{D}}}^{\mathrm{T}}\text{ X)}}^2\mathrm{|}\widehat{\text{C}}\text{, }\widehat{\mathrm{D}}\text{] }$ is quadratic in $\widehat{\mathrm{C}},\mathrm{ }\widehat{\mathrm{D}}$ and is minimized by $\widehat{\mathrm{C}}=\mathrm{C}$ and $\widehat{\mathrm{D}}=\mathrm{D }$ and its minimum is$\mathrm{ }{\mathrm{\sigma }}^2\mathrm{ }(1\mathrm{\hspace{0.17em}}-\mathrm{\hspace{0.17em}}{\rho }^2).$ Thus $$\mathrm{E}[{\text{(T }-\widehat{\mathrm{C}}\mathrm{ }-{\widehat{\mathrm{D}}}^{\mathrm{T}}\text{ X)}}^2\mathrm{|}\widehat{\mathrm{C}}\text{, }\widehat{\mathrm{D}}\text{]}={(\begin{array}{cc}\widehat{\mathbf{C}}-\mathbf{C}& \widehat{\mathbf{D}}-\mathbf{D}\end{array})}^{\mathrm{T}}\mathbf{M}(\begin{array}{cc}\widehat{\mathbf{C}}-\mathbf{C}& \widehat{\mathbf{D}}-\mathbf{D}\end{array})+\mathrm{ }{\sigma }^2\mathrm{ }(1\mathrm{\hspace{0.17em}}-\mathrm{\hspace{0.17em}}{\rho }^2).$$

Here $\mathrm{M}$ is a $\mathrm{(q}+1\text{)×(q}+1\mathrm{) }$ symmetric matrix i.e., $\mathrm{M}=\left[\begin{array}{ccccc}1& \mathit{E}{\mathit{X}}_1& .& .& \mathit{E}{\mathit{X}}_{\mathit{q}}\\ & \mathit{E}{\mathit{x}}_1{}^2& .& .& \mathit{E}{\mathit{X}}_1{\mathit{X}}_{\mathit{q}}\\ & & .& .& .\\ & & & .& .\\ & & & & \mathit{E}{\mathit{X}}_{\mathit{q}}{}^2\end{array}\right]$

Now$\mathrm{ }\text{MSE}={\sigma }^2\mathrm{ }(1\mathrm{\hspace{0.17em}}-\mathrm{\hspace{0.17em}}{\rho }^2)+E$ trace $\text{MN},$ here $\mathrm{N}=$ $\left({}_{\widehat{\mathbf{D}}-\mathbf{D}}^{\widehat{\mathbf{C}}-\mathbf{C}}\right)(\begin{array}{cc}\widehat{\mathbf{C}}-\mathbf{C}& {\widehat{\mathbf{D}}}^{\mathbf{T}}\mathbf{-}{\mathbf{D}}^{\mathbf{T}}\end{array}).$ Finally $$\begin{array}{l}\text{MSE=E}{\text{(}\widehat{\text{C}}\text{−C)}}^2+2({\text{EX}}_1)\text{E(}\widehat{\text{C}}\text{-C)}\text{(}{\widehat{\text{D}}}_1\text{-}{\text{D}}_1\text{)}+\dots +2({\text{EX}}_{\mathrm{q}})\text{E(}\widehat{\text{C}}\text{-C)}\text{(}{\widehat{\text{D}}}_{\mathrm{q}}\text{-}{\text{D}}_{\mathrm{q}}\text{)}\\ \text{+E(}{\text{X}}_1^2)\text{E}{\text{(}{\widehat{\text{D}}}_1\text{-}{\text{D}}_1\text{)}}^2+\dots +2({\text{EX}}_1{\text{X}}_{\mathrm{q}})\text{E(}{\widehat{\text{D}}}_1\text{-}{\text{D}}_1\text{)}\text{(}{\widehat{\text{D}}}_{\mathrm{q}}\text{-}{\text{D}}_{\mathrm{q}}\text{)}\\ \text{+}\dots \text{+}\dots \text{+(E}{\text{X}}_{\mathrm{q}}^2){\text{E(}{\widehat{\text{D}}}_{\mathrm{q}}\text{-}\text{Dq)}}^2+{\mathrm{\sigma }}^2(1-{\rho }^2)\end{array}$$

For $\mathrm{q}=\mathrm{p}=1,$ it reduces to simple linear calibration and $$\mathrm{ }\text{MSE}=(1\mathrm{\hspace{0.17em}}-\mathrm{\hspace{0.17em}}{\rho }^2){\sigma }^2(1+{\text{Q}}_{\text{s}}).$$

Using Taylor series ${\text{Q}}_{\text{s}}$ is $${\mathrm{Q}}_{\mathrm{A }}=\mathrm{ }\frac{{\rho }^2}{\mathrm{N}}+\frac{1}{\mathrm{N}-2}[2{\rho }^2\mathrm{(}1-{\rho }^2\mathrm{)}+\mathrm{(}1-2{\rho }^2{\text{)C}}_{\mathrm{N}}+{\rho }^2{\text{B}}_{\text{N}}]\mathrm{ }$$

It depends only on four invariants i.e., ${\mathrm{N},\mathrm{ B}}_{\mathrm{N}}{,\mathrm{ C}}_{\text{N }}$ and ${\rho }^2.$ Here $\mathrm{N}$ is size of the experiment;${\mathrm{C}}_{\mathrm{N }}=(N-2){\sigma }^2/{\mathrm{S}}_{\mathit{\text{TT}}},$ relative concentration of the experiment; ${\mathrm{B}}_{\mathrm{N }}=(\mathrm{N}-2){(t-\mu )}^2/{\mathrm{S}}_{\mathit{\text{TT}}},$ relative bias of the experiment and ${\rho }^2$ is squared correlation coefficient.

According to Muhammad and Riaz (2016)$\mathrm{ }\text{MSE}/{\sigma }^2$ in an invariant quantity and depends only on ${\mathrm{N},\mathrm{ B}}_{\mathrm{N}}{,\mathrm{ C}}_{\text{N}}{}_{,\rho }$ and $\mathrm{q}$ as mean squared error in univariate case is $$\text{MSE}=\mathrm{E}{(\widehat{\mathrm{C}}-\mathrm{C})}^2+2(\mathrm{EX})\mathrm{E}(\widehat{\mathrm{C}}-\mathrm{C})(\widehat{\mathrm{D}}-\mathrm{D})+(\mathrm{E}{\mathrm{X}}^2)\mathrm{E}{(\widehat{\mathrm{D}}-\mathrm{D})}^2+(1-{\mathrm{\rho }}^2){\mathrm{\sigma }}^2$$ $$\frac{\mathrm{M}\widehat{\mathrm{S}}\mathrm{E}}{{\text{σ}}^2}= (1-{\widehat{\rho }}^2)(1+{\widehat{\mathrm{Q}}}_{\text{s}})$$

Where ${\widehat{\mathrm{Q}}}_{\mathrm{s}}$=$\frac{\mathrm{E}{(\widehat{\mathrm{C}}-\mathrm{C})}^2+2(\mathrm{EX})\mathrm{E}(\widehat{\mathrm{C}}-\mathrm{C})(\widehat{\mathrm{D}}-\mathrm{D})+(\mathrm{E}{\mathrm{X}}^2)\mathrm{E}{(\widehat{\mathrm{D}}-\mathrm{D})}^2}{(1-{\rho }^2){\sigma }^2}$ $$\text{So}{\widehat{\mathrm{Q}}}_{\mathrm{s}}=\frac{\mathrm{M}\widehat{\text{S}}\mathrm{E}}{{\sigma }^2\mathrm{(}1\mathrm{\hspace{0.17em}}-\mathrm{\hspace{0.17em}}{\widehat{\rho }}^2\mathrm{ )}}-1$$

For the multivariate case mean squared error is $$\begin{array}{l}\mathrm{MSE}=\mathrm{E}{(\widehat{\mathrm{C}}-\mathrm{C})}^2+2(\mathrm{E}{\mathrm{X}}_1)\mathrm{E}(\widehat{\mathrm{C}}-\mathrm{C})({\widehat{\mathrm{D}}}_1-{\mathrm{D}}_1)+\mathrm{...}+2(\mathrm{E}{\mathrm{X}}_{\mathrm{q}})\mathrm{E}(\widehat{\mathrm{C}}-\mathrm{C})({\widehat{\mathrm{D}}}_{\mathrm{q}}-{\mathrm{D}}_{\mathrm{q}})+\\ +(\mathrm{E}{\mathrm{X}}_1{}^2)\mathrm{E}{({\widehat{\mathrm{D}}}_1-{\mathrm{D}}_1)}^2+\mathrm{...}+2(\mathrm{E}{\mathrm{X}}_1{\mathrm{X}}_{\mathrm{q}})\mathrm{E}({\widehat{\mathrm{D}}}_1-{\mathrm{D}}_1)({\widehat{\mathrm{D}}}_{\mathrm{q}}-{\mathrm{D}}_{\mathrm{q}})+\mathrm{...}\\ +\mathrm{E}{({\mathrm{X}}_{\mathrm{q}})}^2\mathrm{E}{({\widehat{\mathrm{D}}}_{\mathrm{q}}-{\mathrm{D}}_{\mathrm{q}})}^2+(1-{\mathrm{\rho }}^2){\mathrm{\sigma }}^2\end{array}$$ $${\widehat{\mathrm{Q}}}_{\mathrm{s}}=\frac{\mathrm{M}\widehat{\text{S}}\mathrm{E}}{{\sigma }^2\mathrm{(}1\mathrm{\hspace{0.17em}}-\mathrm{\hspace{0.17em}}{\widehat{\rho }}^2\mathrm{ )}}-1$$

The mean squared error for the Bayesian linear predictor is $$\begin{array}{l}{\text{MSE}}_{\text{π}}={\text{σ}}^{\text{2}}\left({\text{1-ρ}}_{\text{π}}^{\text{2}}\right)\\ ={\sigma }^2\left(1-{\widehat{\rho }}^2\right)({\text{1+Q}}_{\text{π}}\text{)}\\ \text{So}{\mathrm{Q}}_{\pi }=\frac{(1-{\rho }_{\pi }^2)}{1-{\widehat{\rho }}^2}-\text{1}\end{array}$$

As ${\rho }_{\pi }^2$ is a function of the four invariants, N,${\text{B}}_{\text{N}},$${\text{C}}_{\text{N}}$ and ${\widehat{\rho }}^2$ so would be the ${\mathrm{Q}}_{\pi }.$

So comparing ${\mathrm{Q}}_{\pi }$ and ${\widehat{\mathrm{Q}}}_{\mathrm{s}}$ is equivalent to comparing MSE_π/σ² and M$\widehat{\mathrm{S}}$ E/σ². $$\begin{array}{c}{\mathrm{Q}}_{\pi }=\frac{(1-{\rho }_{\pi }^2)}{(1-{\widehat{\rho }}^2)}-1\\ =\frac{\mathrm{f}+\mathrm{(}1-\mathrm{f}){\widehat{\rho }}^2-{\widehat{\rho }}^2}{\mathrm{(}1-{\widehat{\rho }}^2\left)\right[\mathrm{f}+\mathrm{(}1-\mathrm{f}\left){\widehat{\rho }}^2\right]}-1\\ =\frac{-(1-\mathrm{f}){\widehat{\rho }}^2}{\mathrm{f}+(1-\mathrm{f}){\widehat{\rho }}^2}\\ =\frac{(\mathrm{f}-1){\widehat{\rho }}^2}{\mathrm{f}-(\mathrm{f}-1){\widehat{\rho }}^2}\end{array}$$ where f is $$\begin{array}{c}\mathrm{f }=\mathrm{ }(\mathrm{N}-2)/(\mathrm{N}-\mathrm{q}-3)[1+1/\mathrm{N }+\mathrm{ }\{\mathrm{\sigma }²+(\mathrm{\mu }-\overline{t}\overline{t})²\}/{\text{S}}_{\text{TT}}]\\ \mathrm{\hspace{0.17em}}=\mathrm{\hspace{0.17em}}(\mathrm{N}-2)/(\mathrm{N}-\mathrm{q}-3)[1\mathrm{ }+\mathrm{ }1/\mathrm{N }+\mathrm{ }{\text{B}}_{\text{N}}/(\mathrm{N}-2)\mathrm{ }+\mathrm{ }{\text{C}}_{\text{N}}/(\mathrm{N}-2\left)\right]\end{array}$$

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;$\overline{t}$ remaining fixed and $\text{STT → ∞},$ then f ≈ 1 Thus $${\text{C}}_{\text{π}}+\mathrm{ }{\mathbf{\text{D}}}_{\pi }{}^{\mathbf{\text{T}}}\mathbf{\text{X}}\mathrm{\hspace{0.17em}}=\mathrm{\hspace{0.17em}}\widehat{\mathrm{C}}+\mathrm{ }{\widehat{\mathrm{D}}}^{\mathrm{T}}\mathrm{X}\text{ and MSE }=\text{ MSEπ}$$

One may see more details of this appendix in Muhammad (1987) and Muhammad and Riaz (2016)