A General Inferential Framework for Singly-Truncated Bivariate Normal Models with Applications in Economics

Abstract

To analyze the singly-truncated bivariate economic data, we establish a class of singly-truncated bivariate normal distributions via stochastically representing the original bivariate normal random vector as a mixture of the singly-truncated part and its complementary components. Aided with the stochastic representaion, we creatively construct two novel unified and simple algorithms—the expectation–maximization algorithm as well as the minorization–maximization algorithm—to calculate the maximum likelihood estimates of the means and covariance matrix for the model of interest. In addition, we also develop a DA algorithm for posterior sampling in Bayesian analysis. Both simulation results and two real data applications in economics, collaborated by comparisons with existing methods, demonstrate the effectiveness and stability of proposed methodologies.

Appendices

Appendix A: Proof of Theorem 2.1

To obtain the conditional distribution of $\textbf{x}|\textbf{y}$, we first derive the conditional distribution of $(\textbf{x},\textbf{u})|\textbf{y}$. Given both $\textbf{y}={\varvec{y}}$ and $U_4=1$, i.e., $\textbf{u}=(0,0,0,1)$, the complete-data random vector $\textbf{x}$ and the observed-data random vector $\textbf{y}$ share the same distribution, i.e., the conditional distribution of $\textbf{x}|(\textbf{y}={\varvec{y}},\textbf{u}=(0,0,0,1))$ degenerates to the point ${\varvec{y}}$. Therefore,

$$\begin{aligned}{} & {} \Pr (\textbf{x}={\varvec{x}},\textbf{u}=(0,0,0,1)|\textbf{y}={\varvec{y}}) \nonumber \\= & {} \Pr (\textbf{x}={\varvec{x}}|\textbf{y}={\varvec{y}},\textbf{u}=(0,0,0,1))\cdot \Pr (\textbf{u}=(0,0,0,1)) = p_{11}({\varvec{\mu }},{\varvec{\Sigma }})I({\varvec{x}}={\varvec{y}}). \end{aligned}$$

(A.1)

Note that when $U_4=0$, then $\textbf{u}$ could be (1,0,0,0) or (0,1,0,0) or (0,0,1,0), thus, given $\textbf{y}={\varvec{y}}$ and $U_4=0$, i.e., $\textbf{u}\ne (0,0,0,1)$, the conditional distribution of $\textbf{x}|(\textbf{y}={\varvec{y}},\textbf{u}\ne (0,0,0,1))$ may have the same distribution of the unobserved random vector $\textbf{y}_1$ or $\textbf{y}_2$ or $\textbf{y}_3$ depending on the value of $\textbf{u}$. Therefore,

$$\begin{aligned}{} & {} f(\textbf{x}={\varvec{x}},\textbf{u}\ne (0,0,0,1)|\textbf{y}={\varvec{y}}) \nonumber \\= & {} f(\textbf{x}={\varvec{x}}|\textbf{y}_1^{\small \rm C}={\varvec{y}}_1^{\small \rm C},\textbf{u}=(1,0,0,0))\cdot \Pr (\textbf{u}=(1,0,0,0)) \nonumber \\{} & {} + f(\textbf{x}={\varvec{x}}|\textbf{y}_2^{\small \rm C}={\varvec{y}}_2^{\small \rm C},\textbf{u}=(0,1,0,0))\cdot \Pr (\textbf{u}=(0,1,0,0)) \nonumber \\{} & {} + f(\textbf{x}={\varvec{x}}|\textbf{y}_3^{\small \rm C}={\varvec{y}}_3^{\small \rm C},\textbf{u}=(0,0,1,0))\cdot \Pr (\textbf{u}=(0,0,1,0)) \nonumber \\= & {} p_{00}({\varvec{\mu }},{\varvec{\Sigma }})f_1(\textbf{y}_1^{\small \rm C}={\varvec{x}}; {\varvec{\mu }}, {\varvec{\Sigma }}, {{\mathbb {D}}}_1)\cdot I({\varvec{x}}\ne {\varvec{y}}) \nonumber \\{} & {} + p_{01}({\varvec{\mu }},{\varvec{\Sigma }})f_2(\textbf{y}_2^{\small \rm C}={\varvec{x}}; {\varvec{\mu }}, {\varvec{\Sigma }}, {{\mathbb {D}}}_2)\cdot I({\varvec{x}}\ne {\varvec{y}}) \nonumber \\{} & {} + p_{10}({\varvec{\mu }},{\varvec{\Sigma }})f_3(\textbf{y}_3^{\small \rm C}={\varvec{x}}; {\varvec{\mu }}, {\varvec{\Sigma }}, {{\mathbb {D}}}_3)\cdot I({\varvec{x}}\ne {\varvec{y}}), \end{aligned}$$

(A.2)

where $f_1(\cdot )$, $f_2(\cdot )$ and $f_3(\cdot )$ denote the density functions of the complementary singly-truncated bivariate normal vector $\textbf{y}_1$, $\textbf{y}_2$ and $\textbf{y}_3$, respectively.

By combining Eq. (A.1) with Eq. (A.2), the conditional distribution of $\textbf{x}|\textbf{y}$ is determined by the following density function:

$$\begin{aligned}{} & {} f_{\textbf{x}|\textbf{y}}(\textbf{x}={\varvec{x}}|\textbf{y}={\varvec{y}}) = \Pr (\textbf{x}={\varvec{x}},\textbf{u}=(0,0,0,1)|\textbf{y}={\varvec{y}}) + f(\textbf{x}={\varvec{x}},\textbf{u}\ne (0,0,0,1)|\textbf{y}={\varvec{y}}) \\= & {} p_{11}({\varvec{\mu }},{\varvec{\Sigma }})I({\varvec{x}}={\varvec{y}}) + p_{00}({\varvec{\mu }},{\varvec{\Sigma }})f_1(\textbf{y}_1^{\small \rm C}={\varvec{x}}; {\varvec{\mu }}, {\varvec{\Sigma }}, {{\mathbb {D}}}_1)\cdot I({\varvec{x}}\ne {\varvec{y}}) \\{} & {} + p_{01}({\varvec{\mu }},{\varvec{\Sigma }})f_2(\textbf{y}_2^{\small \rm C}={\varvec{x}}; {\varvec{\mu }}, {\varvec{\Sigma }}, {{\mathbb {D}}}_2)\cdot I({\varvec{x}}\ne {\varvec{y}}) + p_{10}({\varvec{\mu }},{\varvec{\Sigma }})f_3(\textbf{y}_3^{\small \rm C}={\varvec{x}}; {\varvec{\mu }}, {\varvec{\Sigma }}, {{\mathbb {D}}}_3)\cdot I({\varvec{x}}\ne {\varvec{y}}). \end{aligned}$$

Appendix B: Details for Deriving $Q({\varvec{\mu }},{\varvec{\Sigma }}|{\varvec{\mu }}^{(t)},{\varvec{\Sigma }}^{(t)})$ specified by Eq. (3.9) in the proposed MM algorithm

The difficulty in deriving the explicit expressions of the MLEs for $({\varvec{\mu }},{\varvec{\Sigma }})$ from the observed-data log-likelihood function $\ell ({\varvec{\mu }},{\varvec{\Sigma }}|Y_\textrm{obs})$ defined by (3.1) lies in the integration part, i.e., $-n\log [p_{11}({\varvec{\mu }},{\varvec{\Sigma }})]$. Thus, the key to find the surrogate Q-function for $\ell ({\varvec{\mu }},{\varvec{\Sigma }}|Y_\textrm{obs})$ is to make an appropriate amplification and minification on this term.

Note that $p_{00}({\varvec{\mu }},{\varvec{\Sigma }})+p_{01}({\varvec{\mu }},{\varvec{\Sigma }})+p_{10}({\varvec{\mu }},{\varvec{\Sigma }})+p_{11}({\varvec{\mu }},{\varvec{\Sigma }})=1$ and $\log (\cdot )$ is a concave function, thus,

$$\begin{aligned} 0&=\; \log \left[ p_{00}({\varvec{\mu }},{\varvec{\Sigma }})+p_{01}({\varvec{\mu }},{\varvec{\Sigma }})+p_{10}({\varvec{\mu }},{\varvec{\Sigma }})+p_{11}({\varvec{\mu }},{\varvec{\Sigma }})\right] \nonumber \\&=\; \log \left[ \frac{p_{11}({\varvec{\mu }}^{(t)},{\varvec{\Sigma }}^{(t)})}{p_{11}({\varvec{\mu }}^{(t)},{\varvec{\Sigma }}^{(t)})}\cdot p_{11}({\varvec{\mu }},{\varvec{\Sigma }}) + \frac{1-p_{11}({\varvec{\mu }}^{(t)},{\varvec{\Sigma }}^{(t)})}{1-p_{11}({\varvec{\mu }}^{(t)},{\varvec{\Sigma }}^{(t)})}\cdot (1-p_{11}({\varvec{\mu }},{\varvec{\Sigma }})) \right] \nonumber \\&\geqslant \; p_{11}({\varvec{\mu }}^{(t)},{\varvec{\Sigma }}^{(t)})\log \left[ \frac{p_{11}({\varvec{\mu }},{\varvec{\Sigma }})}{p_{11}({\varvec{\mu }}^{(t)},{\varvec{\Sigma }}^{(t)})} \right] \nonumber \\&\quad+ (1-p_{11}({\varvec{\mu }}^{(t)},{\varvec{\Sigma }}^{(t)}))\log \left[ \frac{1-p_{11}({\varvec{\mu }},{\varvec{\Sigma }})}{1-p_{11}({\varvec{\mu }}^{(t)},{\varvec{\Sigma }}^{(t)})} \right] , \end{aligned}$$

(B.1)

where $({\varvec{\mu }}^{(t)},{\varvec{\Sigma }}^{(t)})$ is the t-th approximation of the MLEs $({\varvec{{\hat{\mu }}}}, {\varvec{{\hat{\Sigma }}}})$. Based on (B.1), we have

$$\begin{aligned} {}- \log [p_{11}({\varvec{\mu }},{\varvec{\Sigma }})] \geqslant&-\log [p_{11}({\varvec{\mu }}^{(t)},{\varvec{\Sigma }}^{(t)})] - \frac{1-p_{11}({\varvec{\mu }}^{(t)},{\varvec{\Sigma }}^{(t)})}{p_{11}({\varvec{\mu }}^{(t)},{\varvec{\Sigma }}^{(t)})}\log [1-p_{11}({\varvec{\mu }}^{(t)},{\varvec{\Sigma }}^{(t)})] \\&+ \frac{1-p_{11}({\varvec{\mu }}^{(t)},{\varvec{\Sigma }}^{(t)})}{p_{11}({\varvec{\mu }}^{(t)},{\varvec{\Sigma }}^{(t)})}\log [1-p_{11}({\varvec{\mu }},{\varvec{\Sigma }})]. \end{aligned}$$

Then,

$$\begin{aligned} \ell ({\varvec{\mu }},{\varvec{\Sigma }}|Y_\textrm{obs}) \geqslant&\; c_1^{(t)} - \frac{n}{2}\log |{\varvec{\Sigma }}| - \frac{1}{2}\sum _{j=1}^n ({\varvec{y}}_j-{\varvec{\mu }})^{\top}{\varvec{\Sigma }}^{-1}({\varvec{y}}_j-{\varvec{\mu }}) \nonumber \\&+ \frac{n[1-p_{11}({\varvec{\mu }}^{(t)},{\varvec{\Sigma }}^{(t)})]}{p_{11}({\varvec{\mu }}^{(t)},{\varvec{\Sigma }}^{(t)})}\log \left[ p_{00}({\varvec{\mu }},{\varvec{\Sigma }})+p_{01}({\varvec{\mu }},{\varvec{\Sigma }})+p_{10}({\varvec{\mu }},{\varvec{\Sigma }})\right] \nonumber \\ \,\hat{=}\,&\; Q_1({\varvec{\mu }},{\varvec{\Sigma }}|{\varvec{\mu }}^{(t)},{\varvec{\Sigma }}^{(t)}), \end{aligned}$$

(B.2)

where $c_1^{(t)}$ is a constant that does not depend on $({\varvec{\mu }},{\varvec{\Sigma }})$.

However, from $Q_1({\varvec{\mu }},{\varvec{\Sigma }}|{\varvec{\mu }}^{(t)},{\varvec{\Sigma }}^{(t)})$, we still can not resolve the MLEs due to the complex integral form of $\log [ p_{00}({\varvec{\mu }},{\varvec{\Sigma }})+p_{01}({\varvec{\mu }},{\varvec{\Sigma }})+p_{10}({\varvec{\mu }},{\varvec{\Sigma }}) ]$. Next, we try to construct a surrogate function for it, again, we need to use the concavity of the log function. Since

$$\begin{aligned}&\; \log \left[ p_{00}({\varvec{\mu }},{\varvec{\Sigma }})+p_{01}({\varvec{\mu }},{\varvec{\Sigma }})+p_{10}({\varvec{\mu }},{\varvec{\Sigma }})\right] \\&=\; \log \bigg [ \frac{p_{00}({\varvec{\mu }}^{(t)},{\varvec{\Sigma }}^{(t)})}{1-p_{11}({\varvec{\mu }}^{(t)},{\varvec{\Sigma }}^{(t)})}\cdot \frac{1-p_{11}({\varvec{\mu }}^{(t)},{\varvec{\Sigma }}^{(t)})}{p_{00}({\varvec{\mu }}^{(t)},{\varvec{\Sigma }}^{(t)})}\cdot p_{00}({\varvec{\mu }},{\varvec{\Sigma }}) \\&\quad+ \frac{p_{01}({\varvec{\mu }}^{(t)},{\varvec{\Sigma }}^{(t)})}{1-p_{11}({\varvec{\mu }}^{(t)},{\varvec{\Sigma }}^{(t)})}\cdot \frac{1-p_{11}({\varvec{\mu }}^{(t)},{\varvec{\Sigma }}^{(t)})}{p_{01}({\varvec{\mu }}^{(t)},{\varvec{\Sigma }}^{(t)})}\cdot p_{01}({\varvec{\mu }},{\varvec{\Sigma }}) \\&\quad+ \frac{p_{10}({\varvec{\mu }}^{(t)},{\varvec{\Sigma }}^{(t)})}{1-p_{11}({\varvec{\mu }}^{(t)},{\varvec{\Sigma }}^{(t)})}\cdot \frac{1-p_{11}({\varvec{\mu }}^{(t)},{\varvec{\Sigma }}^{(t)})}{p_{10}({\varvec{\mu }}^{(t)},{\varvec{\Sigma }}^{(t)})}\cdot p_{10}({\varvec{\mu }},{\varvec{\Sigma }})\bigg ] \\&\geqslant \; \frac{p_{00}({\varvec{\mu }}^{(t)},{\varvec{\Sigma }}^{(t)})}{1-p_{11}({\varvec{\mu }}^{(t)},{\varvec{\Sigma }}^{(t)})}\log \left[ \frac{1-p_{11}({\varvec{\mu }}^{(t)},{\varvec{\Sigma }}^{(t)})}{p_{00}({\varvec{\mu }}^{(t)},{\varvec{\Sigma }}^{(t)})}\cdot p_{00}({\varvec{\mu }},{\varvec{\Sigma }}) \right] \\&\quad+ \frac{p_{01}({\varvec{\mu }}^{(t)},{\varvec{\Sigma }}^{(t)})}{1-p_{11}({\varvec{\mu }}^{(t)},{\varvec{\Sigma }}^{(t)})}\log \left[ \frac{1-p_{11}({\varvec{\mu }}^{(t)},{\varvec{\Sigma }}^{(t)})}{p_{01}({\varvec{\mu }}^{(t)},{\varvec{\Sigma }}^{(t)})}\cdot p_{01}({\varvec{\mu }},{\varvec{\Sigma }}) \right] \\&\quad+ \frac{p_{10}({\varvec{\mu }}^{(t)},{\varvec{\Sigma }}^{(t)})}{1-p_{11}({\varvec{\mu }}^{(t)},{\varvec{\Sigma }}^{(t)})}\log \left[ \frac{1-p_{11}({\varvec{\mu }}^{(t)},{\varvec{\Sigma }}^{(t)})}{p_{10}({\varvec{\mu }}^{(t)},{\varvec{\Sigma }}^{(t)})}\cdot p_{10}({\varvec{\mu }},{\varvec{\Sigma }}) \right] , \end{aligned}$$

therefore,

$$\begin{aligned} Q_1({\varvec{\mu }},{\varvec{\Sigma }}|{\varvec{\mu }}^{(t)},{\varvec{\Sigma }}^{(t)}) \geqslant&\; c_2^{(t)} - \frac{n}{2}\log |{\varvec{\Sigma }}| - \frac{1}{2}\sum _{j=1}^n ({\varvec{y}}_j-{\varvec{\mu }})^{\top}{\varvec{\Sigma }}^{-1}({\varvec{y}}_j-{\varvec{\mu }}) ] \nonumber \\&+ \frac{np_{00}({\varvec{\mu }}^{(t)},{\varvec{\Sigma }}^{(t)})}{p_{11}({\varvec{\mu }}^{(t)},{\varvec{\Sigma }}^{(t)})}\log [p_{00}({\varvec{\mu }},{\varvec{\Sigma }})] \nonumber \\&+ \frac{np_{01}({\varvec{\mu }}^{(t)},{\varvec{\Sigma }}^{(t)})}{p_{11}({\varvec{\mu }}^{(t)},{\varvec{\Sigma }}^{(t)})}\log [p_{01}({\varvec{\mu }},{\varvec{\Sigma }})] \nonumber \\&+ \frac{np_{10}({\varvec{\mu }}^{(t)},{\varvec{\Sigma }}^{(t)})}{p_{11}({\varvec{\mu }}^{(t)},{\varvec{\Sigma }}^{(t)})}\log [p_{10}({\varvec{\mu }},{\varvec{\Sigma }})] \nonumber \\ \,\hat{=}\,&\; Q_2({\varvec{\mu }},{\varvec{\Sigma }}|{\varvec{\mu }}^{(t)},{\varvec{\Sigma }}^{(t)}), \end{aligned}$$

(B.3)

where $c_2^{(t)}$ is a constant that does not depend on $({\varvec{\mu }},{\varvec{\Sigma }})$.

Unfortunately, the Q-function given by Eq. (B.3) is still not the right one because the complex integral forms included in $\log [p_{00}({\varvec{\mu }},{\varvec{\Sigma }})]$, $\log [p_{01}({\varvec{\mu }},{\varvec{\Sigma }})]$ and $\log [p_{10}({\varvec{\mu }},{\varvec{\Sigma }})]$. To solve this problem, we need to use the following integration version of the Jensen’s inequality:

$$\begin{aligned} \log \left[ \int _{{\varvec{z}}\in {{\mathbb {Z}}}} f({\varvec{z}})\cdot g({\varvec{z}}) \ \text{ d }{\varvec{z}}\right] \geqslant \int _{{\varvec{z}}\in {{\mathbb {Z}}}} \log [f({\varvec{z}})]\cdot g({\varvec{z}}) \ \text{ d }{\varvec{z}}, \end{aligned}$$

(B.4)

where ${\varvec{z}}$ is a real number vector defined on the domain ${{\mathbb {Z}}}$, $f(\cdot )$ is a positive multivariate real function defined on ${{\mathbb {Z}}}$ and $g(\cdot )$ is a multivariate pdf defined on ${{\mathbb {Z}}}$. By employing Eq. (B.4), we have

$$\begin{aligned} \log [p_{00}({\varvec{\mu }},{\varvec{\Sigma }})] =&\; \log \left[ \int _{{\varvec{y}}_1^{\small \rm C}\in {{\mathbb {D}}}_1} \frac{\phi ({\varvec{y}}_1^{\small \rm C}| {\varvec{\mu }},{\varvec{\Sigma }})}{f_1({\varvec{y}}_1^{\small \rm C}; {\varvec{\mu }}^{(t)},{\varvec{\Sigma }}^{(t)}, {{\mathbb {D}}}_1)} \cdot f_1({\varvec{y}}_1^{\small \rm C}; {\varvec{\mu }}^{(t)},{\varvec{\Sigma }}^{(t)}, {{\mathbb {D}}}_1) \ \text{ d }{\varvec{y}}_1^{\small \rm C} \right] \nonumber \\ \quad \geqslant&\; \int _{{\varvec{y}}_1^{\small \rm C}\in {{\mathbb {D}}}_1} \log \left[ \frac{\phi ({\varvec{y}}_1^{\small \rm C}| {\varvec{\mu }},{\varvec{\Sigma }})}{f_1({\varvec{y}}_1^{\small \rm C}; {\varvec{\mu }}^{(t)},{\varvec{\Sigma }}^{(t)}, {{\mathbb {D}}}_1)} \right] \cdot f_1({\varvec{y}}_1^{\small \rm C}; {\varvec{\mu }}^{(t)},{\varvec{\Sigma }}^{(t)}, {{\mathbb {D}}}_1) \ \text{ d }{\varvec{y}}_1^{\small \rm C} \\&\!\!\!\!\!\!\!= c_{31}^{(t)} - \frac{1}{2}\log |{\varvec{\Sigma }}| \nonumber \\&\quad - \frac{1}{2} \int _{{\varvec{y}}_1^{\small \rm C}\in {{\mathbb {D}}}_1} \left[ ({\varvec{y}}_1^{\small \rm C}-{\varvec{\mu }})^{\top}{\varvec{\Sigma }}^{-1}({\varvec{y}}_1^{\small \rm C}-{\varvec{\mu }})\right] f_1({\varvec{y}}_1^{\small \rm C}; {\varvec{\mu }}^{(t)},{\varvec{\Sigma }}^{(t)}, {{\mathbb {D}}}_1) \ \text{ d }{\varvec{y}}_1^{\small \rm C}, \end{aligned}$$

(B.5)

where $c_{31}^{(t)}$ is a constant that free of $({\varvec{\mu }},{\varvec{\Sigma }})$. Similarly,

$$\begin{aligned} \log [p_{01}({\varvec{\mu }},{\varvec{\Sigma }})] \geqslant&\; c_{32}^{(t)} - \frac{1}{2}\log |{\varvec{\Sigma }}| \nonumber \\&- \frac{1}{2} \int _{{\varvec{y}}_2^{\small \rm C}\in {{\mathbb {D}}}_2} \left[ ({\varvec{y}}_2^{\small \rm C}-{\varvec{\mu }})^{\top}{\varvec{\Sigma }}^{-1}({\varvec{y}}_2^{\small \rm C}-{\varvec{\mu }})\right] f_2({\varvec{y}}_2^{\small \rm C}; {\varvec{\mu }}^{(t)},{\varvec{\Sigma }}^{(t)}, {{\mathbb {D}}}_2) \ \text{ d }{\varvec{y}}_2^{\small \rm C}, \hspace{0.5cm} \end{aligned}$$

(B.6)

$$\begin{aligned} \log [p_{10}({\varvec{\mu }},{\varvec{\Sigma }})] \geqslant&\; c_{33}^{(t)} - \frac{1}{2}\log |{\varvec{\Sigma }}| \nonumber \\&- \frac{1}{2} \int _{{\varvec{y}}_3^{\small \rm C}\in {{\mathbb {D}}}_3} \left[ ({\varvec{y}}_3^{\small \rm C}-{\varvec{\mu }})^{\top}{\varvec{\Sigma }}^{-1}({\varvec{y}}_3^{\small \rm C}-{\varvec{\mu }})\right] f_3({\varvec{y}}_3^{\small \rm C}; {\varvec{\mu }}^{(t)},{\varvec{\Sigma }}^{(t)}, {{\mathbb {D}}}_3) \ \text{ d }{\varvec{y}}_3^{\small \rm C}, \hspace{0.5cm} \end{aligned}$$

(B.7)

where $c_{32}^{(t)}$ and $c_{33}^{(t)}$ are normalizing constants. By combining Eqs. (B.5)–(B.7), the result shown in Eq. (3.9) can be obtained immediately.

Appendix C: Numerical Simulation Results

See Tables 10, 11, 12, 13, 14, 15, 16

Table 10 Estimation performances under four types of singly-truncation with $\sigma _{12}=\sigma _{21}=0.1$, i.e., $\rho =0.1155$

Table 11 Estimation performances under four types of singly-truncation with $\sigma _{12}=\sigma _{21}=0.5$, i.e., $\rho =0.5774$

Table 12 Estimation performances under four types of singly-truncation with $\sigma _{12}=\sigma _{21}=0.8$, i.e., $\rho =0.9238$

Table 13 Comparison of estimation performances among three methods with $n=100$

Table 14 Comparison of estimation performances among three methods with $n=1000$

Table 15 Posterior estimation comparison with Griffiths’ method when ${\varvec{r}}=(2.6000,3.3147)$

Table 16 Posterior estimation comparison with Griffiths’ method when ${\varvec{r}}=(0,0)$