Sample selection models for discrete and other non-Gaussian response variables

Abstract

Consider observation of a phenomenon of interest subject to selective sampling due to a censoring mechanism regulated by some other variable. In this context, an extensive literature exists linked to the so-called Heckman selection model. A great deal of this work has been developed under Gaussian assumption of the underlying probability distributions; considerably less work has dealt with other distributions. We examine a general construction which encompasses a variety of distributions and allows various options of the selection mechanism, focusing especially on the case of discrete response. Inferential methods based on the pertaining likelihood function are developed.

Appendix: Score function and Hessian matrix

In cases of interest in applications, the density function f is a member of the exponential family which enter the formulation of generalized linear models; hence we focus on this situation. Following essentially the notation of McCullagh and Nelder (1989), we write the baseline density (or probability function, in the discrete case) as

$$\begin{aligned} f(y;\vartheta ,\psi ) = \exp \left\{ \frac{y\vartheta -b(\vartheta )}{a(\psi )} + d(y,\psi )\right\} \end{aligned}$$

(27)

where $a(\cdot ), b(\cdot )$ and $d(\cdot )$ are known functions. In some cases, the dispersion parameters $\psi$ is known; important instances of this type are the Poisson and the binomial distribution.

On inserting expression (27) in (14), the log-likelihood function becomes

$$\begin{aligned} \log L (\alpha , \theta , \psi ) = \sum _{d_i=1} \left[ \frac{y_i\vartheta _i-b(\vartheta _i)}{a_i (\psi )} + d(y_i,\psi ) + \log G_0\{h(y_i)\}\right] + \sum _{d_i=0} \log (1-\pi _i) \end{aligned}$$

(28)

whose derivatives with respect to the parameters $\beta , \gamma , \psi$ are as follows:

$$\begin{aligned} s(\beta _j) = \displaystyle {\frac{\partial \log L (\alpha , \theta , \psi )}{\partial \beta _j}}= & {} \sum _{d_i=1} \left[ \frac{y_i-\mu _i}{V_i} + \frac{g_0\{h(y_i)\}}{G_0\{h(y_i)\}} \frac{\partial h(y_i)}{\partial \mu _i}\right] \frac{1}{g'(\mu _i)}x_{ij} \\&- \sum _{d_i=0} \left[ \frac{\partial \pi _i/\partial \mu _i}{1-\pi _i} \right] \frac{1}{g'(\mu _i)}x_{ij}, \quad \hbox {for}~j=1, \ldots , p,\\ s(\gamma _h) = \displaystyle {\frac{\partial \log L (\alpha , \theta , \psi )}{\partial \gamma _h}}= & {} \sum _{d_i=1} \left[ \frac{g_0\{h(y_i)\}}{G_0\{h(y_i)\}} \frac{\partial h(y_i)}{\partial \tau _i}\right] w_{ih} \\&- \sum _{d_i=0} \left[ \frac{\partial \pi _i/\partial \tau _i}{1-\pi _i} \right] w_{ih}, \quad \hbox {for}~ h=1, \ldots , q,\\ s(\psi ) = \displaystyle {\frac{\partial \log L (\alpha , \theta , \psi )}{\partial \psi }}= & {} \sum _{d_i=1} \left[ \frac{b(\vartheta _i) - y_i\vartheta _i}{a^2_i(\psi )} a'_i(\psi ) + \frac{\partial d(y_i, \psi )}{\partial \psi } \right] \\&- \sum _{d_i=0} \frac{\partial \pi _i/\partial \psi }{1-\pi _i} \end{aligned}$$

where $V_i=a_i(\psi )b''(\vartheta _i)= {\text {var}}_{}\!\left\{ \displaystyle {Y_i}\right\}$, ${\mathbb {E}}_{}\!\left\{ \displaystyle {Y_i}\right\} =\mu _i=b'(\vartheta _i)$, $g_0=G_0'$ and $g(\mu _i)=x_i^{\top }\beta$ is called the link function.

The second order derivatives of (28) are given by the following expressions:

$$\begin{aligned} H(\beta _j, \beta _h)= & {} \sum _{d_i=1} \left[ - \frac{1}{a(\psi )} + \left( \frac{g_0'\{h(y_i)\}}{G_0\{h(y_i)\}} - \left( \frac{g_0\{h(y_i)\}}{G_0\{h(y_i)\}} \right) ^2 \right) \left( \frac{\partial h(y_i)}{\partial \mu _i} \right) ^2 b''(\vartheta _i) \right. \\&+ \frac{g_0\{h(y_i)\}}{G_0\{h(y_i)\}} \left( \frac{\partial ^2 h(y_i)}{\partial \mu _i^2} b''(\vartheta _i) +\frac{\partial h(y_i)}{\partial \mu _i} \frac{b'''(\vartheta _i)}{b''(\vartheta _i)} \right) \\&\left. - \left\{ \frac{y_i -b'(\vartheta _i)}{V_i} + \frac{g_0\{h(y_i)\}}{G_0\{h(y_i)\}} \frac{\partial h(y_i)}{\partial \mu _i} \right\} \cdot \left\{ \frac{b'''(\vartheta _i)}{b''(\vartheta _i)} + \frac{b''(\vartheta _i) g''(\mu _i)}{g'(\mu _i)} \right\} \right] \\&\times \frac{x_{ih}x_{ij}}{b''(\vartheta _i)(g'(\mu _i))^2}+ \sum _{d_i=0} \left[ \frac{1}{\pi _i-1} \left( \frac{\partial ^2 \pi _i}{\partial \mu _i^2} b''(\vartheta _i) + \frac{\partial \pi _i}{\partial \mu _i} \frac{b'''(\vartheta _i)}{b''(\vartheta _i)} \right) \right. \\&\left. -\frac{1}{(1-\pi _i)^2} \left( \frac{\partial \pi _i}{\partial \mu _i} \right) ^2 b''(\vartheta _i) \right. \\&\left. + \frac{\partial \pi _i/\partial \mu _i}{1-\pi _i} \cdot \left\{ \frac{b'''(\vartheta _i)}{b''(\vartheta _i)} + \frac{b''(\vartheta _i) g''(\mu _i)}{g'(\mu _i)} \right\} \right] \frac{x_{ih}x_{ij}}{b''(\vartheta _i)(g'(\mu _i))^2},\\ H(\beta _j, \gamma _h)= & {} \sum _{d_i=1} \left[ \left\{ \frac{g_0' \{h(y_i)\}}{G_0 \{h(y_i)\}} -\left( \frac{g_0 \{h(y_i)\}}{G_0\{h(y_i)\}} \right) ^2 \right\} \frac{\partial h(y_i)}{\partial \tau _i} \frac{\partial h(y_i)}{\partial \mu _i} \frac{w_{ih} x_{ij}}{ g'(\mu _i)} \right. \\&- \left. \sum _{d_i=0} \left( \frac{\frac{\partial ^2 \pi _i}{\partial \tau _i \partial \mu _i} }{1-\pi _i} +\frac{\partial \pi _i}{\partial \tau _i} \frac{\partial \pi _i}{\partial \mu _i} \frac{1}{(1-\pi _i)^2} \right) \right] \frac{w_{ih} x_{ij}}{ g'(\mu _i)}, \end{aligned}$$

$$\begin{aligned} H(\beta _j, \psi )= & {} \sum _{d_i=1} \frac{a' (\psi ) (\mu _i - y_i)}{a^2_i (\psi ) b'' (\vartheta _i)}\frac{x_{ij}}{g' (\mu _i)} \\&- \sum _{d_i=0} \frac{1}{(1-\pi _i)^2} \left\{ \frac{\partial ^2 \pi _i}{\partial \psi \partial \mu _i} (1-\pi _i) + \frac{\partial \pi _i}{\partial \psi } \frac{\partial \pi _i}{\partial \mu _i} \right\} \frac{x_{ij}}{g' (\mu _i)},\\ H(\gamma _j, \gamma _h)= & {} \sum _{d_i=1} \left[ \left\{ \frac{g_0' \{h(y_i)\}}{G_0 \{h(y_i)\}} -\left( \frac{g_0 \{h(y_i)\}}{G_0\{h(y_i)\}} \right) ^2 \right\} \left( \frac{\partial h(y_i)}{\partial \tau _i}\right) ^2\right. \\&\left. + \frac{g_0 \{h(y_i)\}}{G_0 \{h(y_i)\}} \frac{\partial ^2 h(y_i)}{\partial \tau _i^2} \right] w_{ij} w_{ih}\\&- \sum _{d_i=0} \frac{1}{(1-\pi _i)^2} \left( \frac{\partial ^2 \pi _i}{\partial \tau _i^2} (1-\pi _i) + \left( \frac{\partial \pi _i}{\partial \tau _i} \right) ^2\right) w_{ij} w_{ih},\\ H(\gamma _j, \psi )= & {} -\sum _{d_i=0} \frac{1}{(1-\pi _i)^2} \left\{ \frac{\partial ^2 \pi _i}{\partial \psi \partial \tau _i} (1-\pi _i) + \frac{\partial \pi _i}{\partial \psi } \frac{\partial \pi _i}{\partial \tau _i} \right\} w_{ij},\\ H(\psi , \psi )= & {} \sum _{d_i=1} \left\{ \frac{2(y_i\vartheta _i -b(\vartheta _i))}{a^3_i(\psi )} (a'_i(\psi ))^2 - \frac{y_i\vartheta _i -b(\vartheta _i)}{a^2_i(\psi )}a''_i(\psi ) + \frac{\partial ^2 d(y_i, \psi )}{\partial \psi ^2}\right\} \\&- \sum _{d_i=0} \frac{1}{(1-\pi _i)^2} \left\{ \frac{\partial ^2 \pi _i}{\partial \psi ^2} (1-\pi _i) + \left( \frac{\partial \pi _i}{\partial \psi } \right) ^2 \right\} . \end{aligned}$$