Variable selection in data envelopment analysis via Akaike’s information criteria

Abstract

The decision makers always suffer from predicament in choosing appropriate variable set to evaluate/improve production efficiencies in many applications of data envelopment analysis (DEA). The selected data set may exist information redundancy. On that account, this study proposes an alternative approach to screen out proper input and output variables set for evaluation via Akaike’s information criteria (AIC) rule. This method mainly focuses on assessing the importance of subset of original variables rather than testing the marginal role of variables one by one in many other methods. In terms of the proposed approach, the most optimized variable set contains the least redundant information, which provides decision support to the decision makers. Besides, we also define redundant/cross redundant variables with the form of theorems and give the proofs subsequently. In addition, the AIC approach is firstly extended to stochastic data set to select an appropriate set of stochastic variables as well. Finally, the proposed approach has been applied to some data sets from given data and prior DEA literatures.

Appendices

Appendix 1

Based on Wagner and Shimshak (2007), ACE is shown as follows:

$$\begin{aligned} { ACE}_j \{X_i \}=\frac{\mathop {\sum }\nolimits _{j=1}^n {[E_j (M,S)-E_j (M/{\{X_i \}},S)]} }{n} \end{aligned}$$

where $ACE_j \{X_i \}$ is the ith input’s average change efficiency of $DMU_{\mathrm{j}}$, and $E_j(M,S)$ and $E_j(M/{\{X_i\}},S)$ are the efficiencies of $DMU_{\mathrm{j}}$ based upon CCR model with considering output set S and input set M and $M/{\{X_i\}}$, respectively. n is the number of DMUs.

ECM can be described below according to Pastor et al. (2002):

$$\begin{aligned} { ECM}_j \{X_{\mathrm{j}}\}=\frac{E_j(M,S)}{E_j (M/\{X_i\},S)} \end{aligned}$$

where $ECM_j \{i\}$ denotes the efficiency contribution measure of the ith input to $DMU_{\mathrm{j}}$, and $E_j (M,S)$ and $E_j(M/{\{X_i\}},S)$ have the same meanings as before.

Appendix 2

$$\begin{aligned} \theta _j =\mathop {\sum }\limits _{r\in {Q}'} {\beta _r Y_{rj}} -\mathop {\sum }\limits _{i\in {P}'} {\alpha _i X_{ij}} +\varepsilon _j ,j\in N \nonumber \\ \alpha _i ,\beta _r \ge 0,\forall i\in {P}',r\in {Q}' \end{aligned}$$

(2)

We show how to calculate the maximum likelihood estimate (${{ MLE}}$) in model (4). From model (2), we obtain a probability density function

$$\begin{aligned}&f\left( \theta _j+\mathop {\sum }\limits _{i\in {P}'} {\alpha _i X_{ij} } -\mathop {\sum }\limits _{r\in {Q}'} {\beta _r Y_{rj} } \left. \right| \alpha _i ,\beta _r \right) \nonumber \\&\quad =\frac{1}{\sqrt{2\pi }\sigma _\varepsilon }\exp \left\{ -\frac{\left( \theta _j +\mathop {\sum }\nolimits _{i\in {P}'} {\alpha _i X_{ij} } -\mathop {\sum }\nolimits _{r\in {Q}'} {\beta _r Y_{rj} } \right) ^2 }{2\sigma _\varepsilon ^2 }\right\} \end{aligned}$$

(14)

So, the MLE was obtained through following system of equations

$$\begin{aligned} { MLE}(P,Q)= & {} \prod _{j=1}^n f\left( \theta _j +\mathop {\sum }\limits _{i\in {P}'} {\alpha _i X_{ij}} -\mathop {\mathop {\sum }\limits }\limits _{r\in {Q}'} {\beta _r Y_{rj} } \bigg |\alpha _i ,\beta _r \right) \nonumber \\= & {} \prod _{j=1}^n \frac{1}{\sqrt{2\pi }\sigma _\varepsilon }\exp \left\{ -\frac{(\theta _j +\mathop {\sum }\nolimits _{i\in {P}'} {\alpha _i X_{ij} } -\mathop {\sum }\nolimits _{r\in {Q}'} {\beta _r Y_{rj} } )^2 }{2\sigma _\varepsilon ^2 }\right\} \nonumber \\= & {} \left( \frac{1}{\sqrt{2\pi }}\right) ^{n}\sigma _\varepsilon ^{-n} \exp \left\{ -\frac{\mathop {\sum }\nolimits _{j=1}^n {(\theta _j +\mathop {\sum }\nolimits _{i\in {P}'} {\alpha _i X_{ij} } -\mathop {\sum }\nolimits _{r\in {Q}'} {\beta _r Y_{rj}} )^2 } }{2\sigma _\varepsilon ^2 }\right\} \nonumber \\ \end{aligned}$$

(15)

Then, the natural logarithmic transformation of MLE is as follows

$$\begin{aligned} \ln [{ MLE}(P,Q)]= & {} -n\ln \left( \sqrt{2\pi }\right) -n\ln \sigma _\varepsilon -\frac{1}{2\sigma _\varepsilon ^2 }\mathop {\sum }\limits _{j=1}^n \nonumber \\&\times \,{\left( \theta _j +\mathop {\sum }\limits _{i\in {P}'} {\alpha _i X_{ij}} -\mathop {\sum }\limits _{r\in {Q}'} {\beta _r Y_{rj} } \right) ^2 } \end{aligned}$$

(16)

$\ln [{ MLE}(P,Q)]$ can be maximized by setting the first derivative with respect to $\sigma _\varepsilon$, equal to zero and solving the resulting equation for $\sigma _\varepsilon$. So we have

$$\begin{aligned} \frac{\partial \ln [{ MLE}(P,Q)]}{\partial \sigma _\varepsilon }= & {} -n\sigma _\varepsilon ^{-1} +\sigma _\varepsilon ^{-3} \mathop {\sum }\limits _{j=1}^n {\left( \theta _j +\mathop {\sum }\limits _{i\in {P}'} {\alpha _i X_{ij}} -\mathop {\sum }\limits _{r\in {Q}'} {\beta _r Y_{rj} } \right) ^2 } =0 \end{aligned}$$

(17)

and

$$\begin{aligned} \mathop {\sigma _\varepsilon ^2 }\limits ^\wedge =\frac{1}{n}\mathop {\sum }\limits _{j=1}^n {\left( \theta _j +\mathop {\sum }\limits _{i\in {P}'} {\alpha _i X_{ij}} -\mathop {\sum }\limits _{r\in {Q}'} {\beta _r Y_{rj} } \right) ^2 } \end{aligned}$$

(18)

Then, the remaining question is how to solve following model

$$\begin{aligned} \mathop {\min }\limits _{\alpha _i ,\beta _r } \mathop {\sum }\limits _{j=1}^n {\left( \theta _j +\mathop {\sum }\limits _{i\in {P}'} {\alpha _i X_{ij}} -\mathop {\sum }\limits _{r\in {Q}'} {\beta _r Y_{rj}} \right) ^2 }\nonumber \\ s.t. \alpha _i ,\beta _r \ge 0,\forall i\in {P}',r\in {Q}' \end{aligned}$$

(19)

where the constraints of model (19) are from model (2). The objective function of model (19) is nonlinear, and its constraints are linear. It can be easily solved by using the function “Isqlin” in Matlab software. Suppose the optimal solution to model (3) is ($\alpha _i^*,\beta _r^*,\forall i\in P,r\in Q)$. So, based upon (18), we have

$$\begin{aligned} \mathop {\sigma _\varepsilon ^2 }\limits ^\wedge =\frac{1}{n}\mathop {\sum }\limits _{j=1}^n {\left( \theta _j +\mathop {\sum }\limits _{i\in {P}'} {\alpha _i^*X_{ij}} -\mathop {\sum }\limits _{r\in {Q}'} {\beta _r^*Y_{rj}} \right) ^2 } \end{aligned}$$

(20)

and the minimum of AIC estimator is

$$\begin{aligned} \mathop {\min }\limits _{P,Q} { AIC}(P,Q)= & {} n[\ln (2\pi )+\ln \mathop {\sigma _\varepsilon ^2}\limits ^\wedge +1]+2(| P|+| Q |) \end{aligned}$$

(21)

$$\begin{aligned} { AIC}(P,Q)= & {} -2\ln [{ MLE}(P,Q)]+2(|P|+|Q|) \end{aligned}$$

(3)

Appendix 3

Theorem 1

If an input variable is the exact nonnegative linear combination of other variables in input variable subset P, such as $X_i ={ ENLC}(P/{(X_i )}),\quad \forall X_i\in P, i.e.\, X_i= \sum \nolimits _{\begin{array}{c} k=1 \\ k\ne i \end{array}}^p {\lambda _k} X_k, \forall X_k\in P/{\{X_i\}}$, it’s noteworthy that there are at least one $\lambda _k$ greater than zero. Then the following equation will be achieved: ${ AIC}(P,Q)-{ AIC}(P/{(X_i^ )},Q)= 2$, here, Q is a nonvoid subset of output variables. Then, it comes to a conclusion that $X_i$ is a redundant variable.

Proof of Theorem 1

Firstly, we prove that $E_0^*(P/\{X_i\},Q)=E_0^*(P,Q),$ Denote the optimal solution to the model (1) based upon input subset P and output subset Q as ($\mu _r^*,\upsilon _i^*,\forall r,i;)$, and its corresponding optimal efficiency of DMU$_{0 }$ as $E_0^*(P,Q)$.

The proof of this theorem can be divided into two parts:

Part 1: The efficiency based upon the model including a variable (input or output) is not less than the value based upon the model excluding the variable (Hughes and Yaisawarng 2004; Li et al. 2008). So we obtain

$$\begin{aligned} E_0^*(P/\{X_i \},Q)\le E_0^*(P,Q), \forall i\in {P}'\subset {M}'. \end{aligned}$$

(24)

Part 2: The second part of the proof can operate via two cases as follows:

Case 1: $\upsilon _i^*=0$

Let $\mu _r^{\prime } =\mu _r^*,\upsilon _k^{\prime } =\upsilon _k^*,\forall r\in {Q}',k\in {P}'/\{i\};$, then $(\mu _r^{\prime } ,\upsilon _i^{\prime } ,\forall r\in {Q}',k\in {P}'/\{i\};)$ is a feasible solution to model (1) corresponding to $M=P/\{X_i\},S=Q$, because it satisfies all the constraints of model (1), such as

$$\begin{aligned} \mathop {\sum }\limits _{r\in {Q}'} {\mu _r^{\prime } y_{rj} } -\mathop {\sum }\limits _{k\in {P}'/\{i\}} {\upsilon _k^{\prime } x_{kj} } =\mathop {\sum }\limits _{r\in {Q}'}{\mu _r^*y_{rj} } -\mathop {\sum }\limits _{k\in {P}'/\{i\}} {\upsilon _k^*x_{kj} } \le 0,\forall j, \end{aligned}$$

and

$$\begin{aligned} \mathop {\sum }\limits _{k\in {P}'/\{i\}} {\upsilon _k^{\prime } x_{k0} } =\mathop {\sum }\limits _{k\in {P}'/\{i\}} {\upsilon _k^*x_{k0} } =\mathop {\sum }\limits _{k\in {P}'} {\upsilon _k^*x_{k0} } -\upsilon _i^*x_{i0} =1-0=1. \end{aligned}$$

Thus, when input and output subset is $P/\{X_i\}$ and Q, the optimal efficiency of $\hbox {DMU}_{0}$ in model (1) is

$$\begin{aligned} E_0^*(P/\{X_i\},Q)\ge \mathop {\sum }\limits _{r\in {Q}'} {\mu _r^{\prime } y_{rj} } =\mathop {\sum }\limits _{r\in {Q}'} {\mu _r^*y_{rj} } =E_0^*(P,Q),\forall i\in {P}'\subset {M}' \end{aligned}$$

(25)

Case 2: $\upsilon _i^*>0$

From Definition 1, we obtain $X_i =\mathop {\sum }\limits _{k\in {P}'/\{i\}} {\lambda _k X_k}$, for $X_i ={ ENLC}(X_1 ,\ldots ,X_{i-1} ,X_{i+1} ,\ldots ,X_p )$.

Let $\mu _r^{\prime } =\mu _r^*,\upsilon _k^{\prime } =\upsilon _k^*+\lambda _k *\upsilon _i^*,\forall r\in {Q}',k\in {P}'/\{i\};$ naturally, $\mu _r'\ge 0,\;\upsilon _k' \ge 0,$then $(\mu _r' ,\upsilon _k^{\prime } ,\forall r,k;)$ is a feasible solution to model (1) corresponding to $P/\{X_i\}$ and Q, because it also satisfies all the constraints of model (1), such as

$$\begin{aligned} \mathop {\sum }\limits _{r\in {Q}'} {\mu _r' y_{rj} } -\mathop {\sum }\limits _{k\in P{\prime }/\{i\}} {\upsilon _k' x_{kj} }= & {} \mathop {\sum }\limits _{r\in Q'} {\mu _r^*y_{rj} } -\mathop {\sum }\limits _{k\in P'/\{i\}} {(\upsilon _k^*+\lambda _k *\upsilon _i^*)x_{kj} }\nonumber \\= & {} \mathop {\sum }\limits _{r\in Q{\prime }} {\mu _r^*y_{rj} } -\mathop {\sum }\limits _{k\in P{\prime }/\{i\}} {\upsilon _k^*x_{kj} } -\upsilon _i^**\mathop {\sum }\limits _{k\in P'/\{i\}} {\lambda _k x_{kj} } , \nonumber \\= & {} \mathop {\sum }\limits _{r\in Q'} {\mu _r^*y_{rj} } -\mathop {\sum }\limits _{k\in P'/\{i\}} {\upsilon _k^*x_{kj} } -\upsilon _i^**x_{ij} \le 0,\forall j \end{aligned}$$

(26)

and $\mathop {\sum }\limits _{k\in P'/\{i\}} {\upsilon _k' x_{k0} } =\mathop {\sum }\limits _{k\in P'/\{i\}} {(\upsilon _k^*+\lambda _k *\upsilon _i^*)x_{kj} } =\mathop {\sum }\limits _{k\in P'} {\upsilon _k^*x_{k0} } =1$.

Thus, the optimal efficiency of $\hbox {DMU}_{0}$ in case of $P/\{X_i\}$ and Q in model (1) is

$$\begin{aligned} E_0^*(P/\{X_i\},Q)\ge \mathop {\sum }\limits _{r\in Q'} {\mu _r' y_{rj} } =\mathop {\sum }\limits _{r\in Q'} {\mu _r^*y_{rj} } =E_0^*(P,Q),\forall i\in P{\prime }\subset M' \end{aligned}$$

(27)

From (25) and (27), we obtain

$$\begin{aligned} E_0^*(P/\{X_i\},Q)\ge E_0^*(P,Q),\forall i\in P{\prime }\subset M' \end{aligned}$$

(28)

From (24) and (28), we get

$$\begin{aligned} E_0^*(P/\{X_i\},Q)=E_0^*(P,Q),\forall i\in P'\subset M';\forall Q\subset S,Q\ne \emptyset \end{aligned}$$

By arbitrariness, we get $E_j^*(P/\{X_i\},Q)=E_j^*(P,Q),\forall j\in N$.

Secondly, we prove that $\mathop {\sum }\nolimits _{r\in Q'} {{\beta }_r^{'*} y_{rj} } -\mathop {\sum }\nolimits _{k\in P'/\{i\}} {{\alpha }_k^{'*} x_{kj} } =\mathop {\sum }\nolimits _{r\in Q'} {\beta _r^*y_{rj} } -\mathop {\sum }\nolimits _{k\in P'} {\alpha _k^*x_{kj} }$, where ($\beta _r^*,\alpha _k^*)$ (or (${\beta }_r^{'*} ,{\alpha }_k^{'*} ))$ is the optimal solution of model (19) with considering input set P (or $P/\{X_i\})$ and output Q.

From (26), we have $X_i =\mathop {\sum }\nolimits _{k\in P'/\{i\}} {\lambda _k X_k }$, set ${\alpha }'_k =\alpha _k^*+\lambda _k *\alpha _i^*\;, \beta _r' =\beta _r^*$. Then, $\alpha _k' \ge 0,\;\beta _r' \ge 0.$ To each $DMU_j$,

$$\begin{aligned} \mathop {\sum }\limits _{r\in Q'} {\beta _r' y_{rj} } -\mathop {\sum }\limits _{k\in P'/\{i\}} {\alpha _k' x_{kj} }= & {} \mathop {\sum }\limits _{r\in Q'} {\beta _r^*y_{rj} } -\mathop {\sum }\limits _{k\in P'/\{i\}} {(\alpha _k^*+\lambda _k*\alpha _i^*)x_{kj} } \\= & {} \mathop {\sum }\limits _{r\in Q'} {\beta _r^*y_{rj} } -\mathop {\sum }\limits _{k\in P'/\{i\}} {\alpha _k^*x_{kj} } +\alpha _i^**\mathop {\sum }\limits _{k\in P'/\{i\}} {\lambda _k x_{kj} } , \\= & {} \mathop {\sum }\limits _{r\in Q'} {\beta _r^*y_{rj} } -\mathop {\sum }\limits _{k\in P'/\{i\}} {\alpha _k^*x_{kj} } +\alpha _i^**x_{ij} , \\= & {} \mathop {\sum }\limits _{r\in Q'} {\beta _r^*y_{rj} } -\mathop {\sum }\limits _{k\in P'} {\alpha _k^*x_{kj} } . \end{aligned}$$

Then, (${\beta }'_r ,{\alpha }'_k)$ is the optimal solution of model (19) with considering input data set $P/\{X_i\}$ and output data set Q. Thus, $\beta _r' ={\beta }_r^{'*} ,{\alpha }'_i ={\alpha }_i^{'*}$. Therefore,

$$\begin{aligned} \mathop {\sigma _\varepsilon ^2 }\limits ^\wedge= & {} \frac{1}{n}\mathop {\sum }\limits _{j=1}^n {\left( E_j^*(P,Q)+\mathop {\sum }\limits _{k\in P'} {\alpha _k^*X_{kj}} -\mathop {\sum }\limits _{r\in Q'} {\beta _r^*Y_{rj} } \right) ^2 } \\= & {} \mathop {{\sigma }_\varepsilon ^{'2} }\limits ^\wedge =\frac{1}{n}\mathop {\sum }\limits _{j=1}^n {\left( E_j^*(P/{\{X_i\},Q})+\mathop {\sum }\limits _{k\in {P'}/\{i\}} {{\alpha }_k^{'*} X_{kj}} -\mathop {\sum }\limits _{r\in Q'} {{\beta }_r^{'*} Y_{rj} } \right) ^2 } \end{aligned}$$

Then, $\ln [{ MLE}(P/{\{X_i \}},Q)]=\ln [{ MLE}(P,Q)]$, according to the definition of AIC value,

$$\begin{aligned} { AIC}(P,Q)-{ AIC}(P/{\{X_i \}},Q)= & {} \{-2\ln [{ MLE}(P,Q)]+2(| P |+| Q |)\}\\&\quad -\{-2\ln [{ MLE}(P/{\{X_i\}},Q)]+2(\left| {P/{\{X_i \}}} \right| +\left| Q \right| )\} \\= & {} 2\left| P \right| -2\left| {P/{\{X_i \}}} \right| =2. \end{aligned}$$

$\square$

Appendix 4

Proof of Theorem 3

Firstly, we denote the optimal solution to the model (1) based upon $M=P,S=Q$ as ($\mu _r^*,\upsilon _i^*,\forall r,i;)$. According to Corollary 1 of Theorem 1 in Lee and Choi (2009), we obtain that $E_j^*(P,Q/{\{Y_i\}})=E_j^*(P,Q),\forall j\in N$. Similar to proof in theorem 1, since $Y_i =\mathop {\sum }\nolimits _{r\in {Q'}/{\{i\}}} {\lambda _r} Y_r -\mathop {\sum }\nolimits _{k\in P'} {\lambda _k} X_k ,\forall j$ and we set ${\beta }'_r =\beta _r^*+\lambda _r *\beta _i^*, \quad {\alpha }'_k =\alpha _k^*+\lambda _k *\beta _i^*$. So, to each $DMU_j$,

$$\begin{aligned} \mathop {\sum }\limits _{r\in Q'/\{i\}} {\beta _r' y_{rj} } -\mathop {\sum }\limits _{k\in P'} {\alpha _k' x_{kj} }= & {} \mathop {\sum }\limits _{r\in Q'/\{i\}} {(\beta _r^*+\lambda _r *\beta _i^*)y_{rj} } -\mathop {\sum }\limits _{k\in P'} {(\alpha _k^*+\lambda _k *\beta _i^*)x_{kj} } \\= & {} \mathop {\sum }\limits _{r\in Q'/\{i\}} {\beta _r^*y_{rj} } +\beta _i^**\mathop {\sum }\limits _{r\in Q'/\{i\}} {\lambda _r y_{rj} } -\beta _i^**\mathop {\sum }\limits _{k\in P'} {\lambda _k x_{kj} } -\mathop {\sum }\limits _{k\in P'} {\alpha _k^*x_{kj} } \\= & {} \mathop {\sum }\limits _{r\in Q'/\{i\}} {\beta _r^*y_{rj} } +\beta _i^**\left( \mathop {\sum }\limits _{r\in Q'/\{i\}} {\lambda _r y_{rj} } -\mathop {\sum }\limits _{k\in P'} {\lambda _k x_{kj} } \right) -\mathop {\sum }\limits _{k\in P'} {\alpha _k^*x_{kj} } \\= & {} \mathop {\sum }\limits _{r\in Q'/\{i\}} {\beta _r^*y_{rj} } +\beta _i^**y_{ij} -\mathop {\sum }\limits _{k\in P'} {\alpha _k^*x_{kj} } \\= & {} \mathop {\sum }\limits _{r\in Q'} {\beta _r^*y_{rj} } -\mathop {\sum }\limits _{k\in P'} {\alpha _k^*x_{kj} } \end{aligned}$$

Then, (${\beta }'_r ,{\alpha }'_k)$ is the optimal solution of model (19) with considering input data set P and output data set $Q/\{Y_i\}$. So $\beta _r' ={\beta }_r^{'*} ,{\alpha }'_i ={\alpha }_i^{'*}$, and we get $\mathop {\sum }\nolimits _{r\in Q'/\{i\}} {{\beta }_r^{'*} y_{rj} } -\mathop {\sum }\nolimits _{k\in P'} {{\alpha }_k^{'*} x_{kj} } =\mathop {\sum }\nolimits _{r\in Q'} {\beta _r^*y_{rj} } -\mathop {\sum }\nolimits _{k\in P'} {\alpha _k^*x_{kj} }$. Therefore,

$$\begin{aligned} \mathop {\sigma _\varepsilon ^2 }\limits ^\wedge= & {} \frac{1}{n}\mathop {\sum }\limits _{j=1}^n {\left( E_j^*(P,Q)+\mathop {\sum }\limits _{k\in P'} {\alpha _k^*X_{kj}} -\mathop {\sum }\limits _{r\in Q'} {\beta _r^*Y_{rj}}\right) ^2 } \nonumber \\= & {} \mathop {{\sigma }_\varepsilon ^{'2} }\limits ^\wedge =\frac{1}{n}\mathop {\sum }\limits _{j=1}^n {\left( E_j^*(P,Q/{\{Y_i\}})+\mathop {\sum }\limits _{k\in P'} {{\alpha }_k^{'*} X_{kj}} -\mathop {\sum }\limits _{r\in {Q'}/{\{i\}}} {{\beta }_r^{'*} Y_{rj} } \right) ^2 } \end{aligned}$$

(29)

Then, $\ln [{ MLE}(P,Q/{\{Y_i\}})]=\ln [{ MLE}(P,Q)]$, according to the definition of AIC value,

$$\begin{aligned}&{ AIC}(P,Q)-{ AIC}({P,Q}/{\{Y_i\}}) \\&\quad =\{-2\ln [{ MLE}(P,Q)]+2(\left| P \right| +\left| Q \right| )\}\\&\quad -\{-2\ln [{ MLE}(P,Q/{\{Y_i\}})]+2(\left| P \right| +\left| {Q/{\{Y_i\}}} \right| )\} \\&\quad =2\left| Q \right| -2\left| {Q/{\{Y_i\}}} \right| =2. \end{aligned}$$

$\square$

Appendix 5

$$\begin{aligned} {\tilde{\theta }}_j^d =\mathop {\sum }\limits _{r\in {Q}'} {\beta _r^d {\tilde{Y}}_{rj}^d } -\mathop {\sum }\limits _{i\in {P}'} {\alpha _i^d {\tilde{X}}_{ij}^d } +\varepsilon _j^d ,j\in n \\ \alpha _i^d ,\beta _r^d \ge 0,\forall i\in {P}',r\in {Q}' \end{aligned}$$

We show how to calculate the maximum likelihood estimate ($\mathop {{ MLE}}\nolimits _d )$ in model (12). From model (2), we obtain a probability density function

$$\begin{aligned}&f\left( \theta _j^d +\mathop {\sum }\limits _{i\in {P}'} {\alpha _i X_{ij}^d } -\mathop {\sum }\limits _{r\in {Q}'} {\beta _r Y_{rj}^d } \left. \right| \alpha _i ,\beta _r \right) \nonumber \\&\quad =\frac{1}{\sqrt{2\pi }\sigma _\varepsilon }\exp \left\{ -\frac{\left( \theta _j^d +\mathop {\sum }\nolimits _{i\in {P}'} {\alpha _i X_{ij}^d } -\mathop {\sum }\nolimits _{r\in {Q}'} {\beta _r Y_{rj}^d } \right) ^2 }{2\sigma _\varepsilon ^2 }\right\} \end{aligned}$$

(30)

So, the ($\mathop {{ MLE}}\limits _d )$ was obtained through following system of equations

$$\begin{aligned} \mathop {{ MLE}}\limits _d (P,Q)= & {} \prod _{j=1}^n f\left( \theta _j^d +\mathop {\sum }\limits _{i\in {P}'} {\alpha _i X_{ij}^d } -\mathop {\sum }\limits _{r\in {Q}'} {\beta _r Y_{rj}^d } \left. \right| \alpha _i ,\beta _r\right) \nonumber \\= & {} \prod _{j=1}^n \frac{1}{\sqrt{2\pi }\sigma _\varepsilon }\exp \left\{ -\frac{\left( \theta _j^d +\mathop {\sum }\nolimits _{i\in {P}'} {\alpha _i X_{ij}^d } -\mathop {\sum }\nolimits _{r\in {Q}'} {\beta _r Y_{rj}^d } \right) ^2 }{2\sigma _\varepsilon ^2 }\right\} \nonumber \\= & {} \left( \frac{1}{\sqrt{2\pi }}\right) ^{n} (\sigma _\varepsilon ^d )^{-n} \exp \left\{ -\frac{\mathop {\sum }\nolimits _{j=1}^n {\left( \theta _j^d +\mathop {\sum }\nolimits _{i\in {P}'} {\alpha _i X_{ij}^d } -\mathop {\sum }\nolimits _{r\in {Q}'}{\beta _r Y_{rj}^d } \right) ^2 } }{2\sigma _\varepsilon ^2 }\right\} \nonumber \\ \end{aligned}$$

(31)

Then, the natural logarithmic transformation of ($\mathop {{ MLE}}\limits _d )$ is as follows

$$\begin{aligned} \ln [\mathop {{ MLE}}\limits _d (P,Q)]=-n\ln \left( \sqrt{2\pi }\right) -n\ln \sigma _\varepsilon ^d -\frac{1}{2\sigma _\varepsilon ^2 }\mathop {\sum }\limits _{j=1}^n {\left( \theta _j^d +\mathop {\sum }\limits _{i\in {P}'} {\alpha _i X_{ij}^d } -\mathop {\sum }\limits _{r\in {Q}'} {\beta _r Y_{rj}^d } \right) ^2 }\nonumber \\ \end{aligned}$$

(32)

$\ln [\mathop {{ MLE}}\limits _d (P,Q)]$ can be maximized by setting the first derivative with respect to $\sigma _\varepsilon ^d$, equal to zero and solving the resulting equation for $\sigma _\varepsilon ^d$. So we have

$$\begin{aligned} \frac{\partial \ln [\mathop {{ MLE}}\limits _d (P,Q)]}{\partial \sigma _\varepsilon ^d }= & {} -n(\sigma _\varepsilon ^d )^{-1} +(\sigma _\varepsilon ^d )^{-3} \nonumber \\&\quad \mathop {\sum }\limits _{j=1}^n{\left( \theta _j^d +\mathop {\sum }\limits _{i\in {P}'} {\alpha _i X_{ij}^d } -\mathop {\sum }\limits _{r\in {Q}'}{\beta _r Y_{rj}^d } \right) ^2 } =0 \end{aligned}$$

(33)

and

$$\begin{aligned} \mathop {(\sigma _\varepsilon ^d }\limits ^\wedge )^2 =\frac{1}{n}\mathop {\sum }\limits _{j=1}^n {\left( \theta _j^d +\mathop {\sum }\limits _{i\in {P}'} {\alpha _i X_{ij}^d } -\mathop {\sum }\limits _{r\in {Q}'}{\beta _r Y_{rj}^d } \right) ^2 } \end{aligned}$$

(34)

Then, the remaining question is how to solve following model

$$\begin{aligned}&\mathop {\min }\limits _{\alpha _i ,\beta _r } \mathop {\sum }\limits _{j=1}^n {\left( \theta _j^d +\mathop {\sum }\limits _{i\in {P}'}{\alpha _i X_{ij}^d } -\mathop {\sum }\limits _{r\in {Q}'} {\beta _r Y_{rj}^d } \right) ^2 } \text {s.t.}\nonumber \\&\quad \alpha _i ,\beta _r \ge 0,\forall i\in {P}',r\in {Q}' \end{aligned}$$

(35)

where the constraints of model (35) are from model (2). Suppose the optimal solution to model (11) is ($\alpha _i^*,\beta _r^*,\forall i\in P,r\in Q)$. So, based upon (34), we have

$$\begin{aligned} \mathop {(\sigma _\varepsilon ^d }\limits ^\wedge )^2= & {} \frac{1}{n}\mathop {\sum }\limits _{j=1}^n {\left( \theta _j^d +\mathop {\sum }\limits _{i\in {P}'} {\alpha _i^*X_{ij}^d } -\mathop {\sum }\limits _{r\in {Q}'}{\beta _r^*Y_{rj}^d } \right) ^2 } \end{aligned}$$

(36)

and the minimum of AIC estimator is

$$\begin{aligned} \mathop {Min}\limits _ {\begin{array}{c} \tilde{X}_i ,i \in P\\ {\tilde{Y}}_r ,r \in Q \end{array}} \mathop {Min}\limits _{P,Q} { AIC}(P,Q)= & {} \mathop {Min}\limits _{P,Q} \frac{1}{N}\mathop {\sum }\limits _{d=1}^N {\mathop {Min}\limits _{\begin{array}{c} X_i^d ,i \in P\\ Y_r^d ,r\in Q \end{array}}\mathop {{ AIC}}\limits _d (P,Q)}\nonumber \\= & {} \frac{1}{N}\mathop {\sum }\limits _{d=1}^N {n\left[ \ln (2\pi )+\ln (\mathop {\sigma _\varepsilon ^d }\limits ^\wedge )^2 +1\right] +2(|P|+| Q |)} \end{aligned}$$

(37)