Regression analysis of biased case–control data

Abstract

The data obtained from case–control sampling may suffer from selection or reporting bias, resulting in biased estimation of the parameter(s) of interest by standard analysis of case–control data. In this work, the problem of this bias is dealt with by introducing the concept of reporting probability. Then, considering a reference sample from the source population, we obtain asymptotically unbiased estimate of the population parameters by fitting a pseudo-likelihood, assuming the exposure distribution in the population to be unknown and arbitrary. The proposed estimates of the model parameters follow asymptotically a normal distribution and are semiparametrically fully efficient. We motivate the need for such methodology by considering the analysis of spontaneous adverse drug reaction (ADR) reports in presence of reporting bias.

Appendices

Appendix A: Pseudo-log-likelihood

The ‘pseudo-log-likelihood’ (6) is obtained from the log-likelihood (5). In order to obtain the profile likelihood, as discussed in Sect. 3, the log-likelihood (5) is maximized over $\varvec{\delta }$ for fixed $\varvec{\phi }$. Introducing the Lagrange multiplier $\lambda$ to take care of the constraint $\sum _{i=1}^{K}\delta _{i}=1$ and equating the derivative of the log-likelihood (5) with respect to $\delta _{i}$ to zero, we get

$$\begin{aligned} \sum _{j=0}^{J} \left\{ \frac{n_{ji}}{\delta _{i}} - \frac{n_{j}\mu _{ji}p_{ji}}{\sum _{k=1}^{K}\mu _{jk}p_{jk}\delta _{k}} \right\} + \frac{n_{J+1,i}}{\delta _{i}} + \lambda = 0. \end{aligned}$$

(10)

Multiplying (10) by $\delta _{i}$ and summing over $i$, we have $\lambda = -n_{J+1}$. Using this value of $\lambda$ in (10), the expression for $\delta _{i}$ can be written as

$$\begin{aligned} \delta _{i} = \frac{n_{J+1,i} + \sum _{j=0}^{J} n_{ji}}{n_{J+1}\left[ 1 + \sum _{j=0}^{J} \frac{n_{j}}{n_{J+1}}\frac{\mu _{ji}p_{ji}}{\sum _{k=1}^{K}\mu _{jk}p_{jk}\delta _{k}} \right] } \cdot \end{aligned}$$

(11)

From (11), after setting an offset parameter $\rho$ as

$$\begin{aligned} \mathrm{e}^{\rho } = n_{j}/\left( n_{J+1}\sum _{i=1}^{K}\mu _{ji}p_{ji}\delta _{i}\right) , \quad \text{ for } j=0,1,\ldots ,J, \end{aligned}$$

(12)

we have $\delta _{i} = (n_{J+1,i} + \sum _{j=0}^{J} n_{ji})/(n_{J+1} (1 + \sum _{j=0}^{J}\mathrm{e}^{\rho }\mu _{ji}{p_{ji}}))$, which is substituted in (5) to get the pseudo-log-likelihood (6). Note that the $\rho$ in (12) satisfies $\partial l^{*}(\psi )/\partial \rho = 0$, where $l^{*}(\psi )$ is given by (6).

To justify this offset parameter $\rho$ being independent of $j$, consider $n_{0}/n_{j}$ as a consistent estimator of $P(R=1, Y=0)/P(R=1, Y=j)$ (Scott and Wild 1997) so that

$$\begin{aligned} \frac{n_{0}}{n_{j}} = \frac{P(R=1, Y=0)}{P(R=1, Y=j)} + o_{p}(1), \quad \text{ for } j=1,\ldots ,J. \end{aligned}$$

(13)

Note that $n_{j}/n$ tends to $\omega _{j}$ in probability and $n_{0}/n_{j}$ tends to $\omega _{0}/\omega _{j}$ in probability as $n \rightarrow \infty$ with $n = \sum _{l=0}^{J+1}n_{l}$, resulting in

$$\begin{aligned} \frac{\omega _{0}}{\omega _{j}} = \frac{P(R=1, Y=0)}{P(R=1, Y=j)}, \end{aligned}$$

which leads to

$$\begin{aligned} \frac{\omega _{0}}{\omega _{J+1}P(R=1, Y=0)} = \frac{\omega _{j}}{\omega _{J+1}P(R=1, Y=j)}, \quad \text{ for } j=1,\ldots ,J, \end{aligned}$$

(14)

the population counterpart of (12). The implicit dependence of $\rho$ on $\varvec{\phi }$, written as $\rho = \rho (\varvec{\phi })$, is clear from the above description.

Appendix B: Asymptotics

The asymptotic properties of the estimator $\varvec{\hat{\psi }} = (\varvec{\hat{\phi }}, \hat{\rho })$ obtained by maximizing the pseudo log-likelihood (6) are established by considering the multi-sample representation of Hirose (2005) and Lee et al. (2006). Let $E_{j}$ denote the expectation with respect to the conditional distribution of exposure $X$, given $Y=j$, having density $f_{j}(x,\varvec{\phi },g)=\mu _{j}(x,\varvec{\gamma })p_{j}(x,\varvec{\beta })g(x)/\pi _{j}$ with $\pi _{j} = \int \mu _{j}(x,\varvec{\gamma })p_{j}(x,\varvec{\beta })g(x) \mathrm{d}x$, for $j=0,\ldots ,J,$ and $E_{J+1}$ denote the expectation with respect to the unconditional distribution of $X$ having density $f_{J+1}(x,\varvec{\phi },g)=g(x)$.

As in Lee et al. (2006), the estimating equation from (6), the pseudo log-likelihood equation, can be written as

$$\begin{aligned} \frac{\partial l^{*}(\varvec{\psi })}{\partial \varvec{\psi }} = \sum _{j=0}^{J+1} \sum _{i=1}^{n_{j}} \frac{\partial \log Z_{j}(x_{ji}, \varvec{\psi })}{\partial \varvec{\psi }} = 0, \end{aligned}$$

(15)

where

$$\begin{aligned} Z_{j}(x, \varvec{\psi })= & {} \left( \frac{\mathrm{e}^{\rho }\mu _{j}(x,\varvec{\gamma })p_{j}(x,\varvec{\beta })}{1 + \sum _{l=0}^{J}\mathrm{e}^{\rho }\mu _{l}(x,\varvec{\gamma })p_{l}(x,\varvec{\beta })} \right) , \quad \text{ for } j=0,\ldots , J, \text{ and } \\ Z_{J+1}(x, \varvec{\psi })= & {} \left( \frac{1}{1 + \sum _{l=0}^{J}\mathrm{e}^{\rho }\mu _{l}(x,\varvec{\gamma })p_{l}(x,\varvec{\beta })} \right) . \end{aligned}$$

Note that $x_{ji}$, for $i=1,\ldots ,n_{j}$, are independent random variables with common density $f_{j}(x,\varvec{\phi }, g)$, for $j=0,1,\ldots ,J+1$, as mentioned above. Then, we have

$$\begin{aligned} E_{j}\left( \frac{\partial \mathrm{logZ}_{j} }{\partial \varvec{\psi }}\right)= & {} \int \frac{1}{Z_{j}} \frac{\partial Z_{j} }{\partial \varvec{\psi }} \frac{\mu _{j}(x,\varvec{\gamma })p_{j}(x,\varvec{\beta })g(x)}{\pi _{j}}\,\mathrm{d}x \nonumber \\= & {} \int \left( 1 + \sum _{l=0}^{J}\mathrm{e}^{\rho }\mu _{l}(x,\varvec{\gamma })p_{l}(x,\varvec{\beta })\right) \frac{\partial Z_{j} }{\partial \varvec{\psi }} \frac{g(x)}{\mathrm{e}^{\rho }\pi _{j}}\,\mathrm{d}x \nonumber \\= & {} \frac{1}{\omega _{j}} E_{X}\left[ \omega _{J+1} \left( 1 + \sum _{l=0}^{J}\mathrm{e}^{\rho }\mu _{l}(X,\varvec{\gamma })p_{l}(X,\varvec{\beta })\right) \frac{\partial Z_{j} }{\partial \varvec{\psi }} \right] , \end{aligned}$$

(16)

for $j=0,1,\ldots ,J$, using (12) and (14). Similarly,

$$\begin{aligned} E_{J+1}\left( \frac{\partial \mathrm{logZ}_{J+1} }{\partial \varvec{\psi }}\right) = \frac{1}{\omega _{J+1}} E_{X}\left[ \omega _{J+1} \left( 1 + \sum _{l=0}^{J}\mathrm{e}^{\rho }\mu _{l}(X,\varvec{\gamma })p_{l}(X,\varvec{\beta })\right) \frac{\partial Z_{J+1} }{\partial \varvec{\psi }}\right] .\nonumber \\ \end{aligned}$$

(17)

Hence,

$$\begin{aligned} \sum _{j=0}^{J+1} \omega _{j} E_{j}\left[ \frac{\partial \mathrm{logZ}_{j}}{\partial \varvec{\psi }}\right] = 0, \quad \text{ since } \sum _{j=0}^{J+1}Z_{j} =1. \end{aligned}$$

(18)

Now, we use the results related to estimating function and asymptotic linear estimator (see (Hirose 2005, p 72–79)). Here, the estimating function is $\partial Z_{j}(x, \varvec{\psi })/\partial \varvec{\psi }$ with the corresponding asymptotic linear estimator $\varvec{\hat{\psi }}$. Then, the asymptotic distribution of $\sqrt{n}(\varvec{\hat{\psi }} - \varvec{\psi })$ is multivariate normal (see Hirose 2005, p 67–79) with mean zero and variance–covariance matrix given by

$$\begin{aligned} I(\varvec{\psi })^{-1}\Sigma I(\varvec{\psi })^{-1}, \end{aligned}$$

(19)

where

$$\begin{aligned} I(\varvec{\psi })= & {} \sum _{j=0}^{J+1} \omega _{j} E_{j}\left[ -\frac{\partial ^{2} \mathrm{logZ}_{j} }{\partial \varvec{\psi } \partial \varvec{\psi }^{T}}\right] \text{ and } \nonumber \\ \Sigma= & {} \sum _{j=0}^{J+1} \omega _{j} E_{j}\left[ \left\{ \frac{\partial \mathrm{logZ}_{j} }{\partial \varvec{\psi }} - E_{j}\left( \frac{\partial \mathrm{logZ}_{j} }{\partial \varvec{\psi }}\right) \right\} \left\{ \frac{\partial \mathrm{logZ}_{j} }{\partial \varvec{\psi }} - E_{j}\left( \frac{\partial \mathrm{logZ}_{j} }{\partial \varvec{\psi }}\right) \right\} ^{T} \right] .\nonumber \\ \end{aligned}$$

(20)

In our context, it can be shown that the variance–covariance matrix (19) has the form

$$\begin{aligned} I(\varvec{\psi })^{-1} - \begin{bmatrix} 0&\quad 0 \\ 0^{T}&\quad H \end{bmatrix}, \end{aligned}$$

(21)

where $H$ is a scalar element. The resulting variance–covariance matrix of $\sqrt{n}(\varvec{\hat{\phi }} - \varvec{\phi })$ is $[I_{\varvec{\phi } \varvec{\phi }} - I_{\varvec{\phi } \rho }I_{\rho \rho }^{-1}I_{\rho \varvec{\phi }}]^{-1}$, where $I(\varvec{\psi })$ is partitioned as

$$\begin{aligned} I(\varvec{\psi }) = \begin{bmatrix} I_{\varvec{\phi \phi }}&\quad I_{\varvec{\phi }\rho } \\ I_{\rho \varvec{\phi }}^{T}&\quad I_{\rho \rho } \end{bmatrix}. \end{aligned}$$

(22)

Note that $nI(\varvec{\psi })$ can be consistently estimated by $-\partial ^{2} l^{*}(\varvec{\psi })/\partial \varvec{\psi } \partial \varvec{\psi }^{T}$ evaluated at $\varvec{\psi } = \varvec{\hat{\psi }}$. From (20), $\Sigma$ can be written as

$$\begin{aligned}&\sum _{j=0}^{J+1} \omega _{j} E_{j}\left[ \left( \frac{\partial \mathrm{logZ}_{j} }{\partial \varvec{\psi }}\right) \left( \frac{\partial \mathrm{logZ}_{j} }{\partial \varvec{\psi }}\right) ^{T}\right] - \sum _{j=0}^{J+1} \omega _{j}E_{j}\left( \frac{\partial \mathrm{logZ}_{j} }{\partial \varvec{\psi }}\right) E_{j}\left( \frac{\partial \mathrm{logZ}_{j} }{\partial \varvec{\psi }}\right) ^{T} \nonumber \\&\quad = I(\varvec{\psi }) - \sum _{j=0}^{J+1} \omega _{j}E_{j}\left( \frac{\partial \mathrm{logZ}_{j} }{\partial \varvec{\psi }}\right) E_{j}\left( \frac{\partial \mathrm{logZ}_{j} }{\partial \varvec{\psi }}\right) ^{T}. \end{aligned}$$

(23)

To establish (21), we need to show that the second term of (23) is (see Neuhaus et al. 2002)

$$\begin{aligned} I(\varvec{\psi }) \begin{bmatrix} 0&\quad 0 \\ 0^{T}&\quad H \end{bmatrix} I(\varvec{\psi }). \end{aligned}$$

(24)

Note that,

$$\begin{aligned}&E_{j}\left( \frac{\partial \mathrm{logZ}_{j} }{\partial \varvec{\psi }}\right) \left( \frac{\partial \mathrm{logZ}_{j} }{\partial \varvec{\psi }}\right) ^{T} \nonumber \\&\quad = \frac{1}{\omega _{j}} E_{X}\left[ \omega _{J+1} \left( 1 + \sum _{l=0}^{J}\mathrm{e}^{\rho }\mu _{l}(X,\varvec{\gamma })p_{l}(X,\varvec{\beta })\right) \frac{1}{Z_{j}} \frac{\partial Z_{j} }{\partial \varvec{\psi }}\frac{\partial Z_{j} }{\partial \varvec{\psi }^{T}} \right] , \end{aligned}$$

(25)

for $j=0,\ldots ,J+1$. Using (16) and (17), the second term of (23) becomes

$$\begin{aligned} \sum _{j=0}^{J+1} \frac{1}{\omega _{j}} E_{X}\left[ \omega _{J+1} T(X) \frac{\partial Z_{j} }{\partial \varvec{\psi }} \right] E_{X}\left[ \omega _{J+1} T(X) \frac{\partial Z_{j} }{\partial \varvec{\psi }}\right] . \end{aligned}$$

(26)

Using (25), the information matrix can be written as

$$\begin{aligned} I(\varvec{\psi })= & {} \sum _{j=0}^{J+1} E_{X}\left[ \omega _{J+1} T(X) \frac{1}{Z_{j}} \frac{\partial Z_{j} }{\partial \varvec{\psi }}\frac{\partial Z_{j} }{\partial \varvec{\psi }^{T}}\right] \nonumber \\= & {} E_{X}\left[ \omega _{J+1} T(X) \sum _{j=0}^{J+1} \frac{1}{Z_{j}} \frac{\partial Z_{j} }{\partial \varvec{\psi }}\frac{\partial Z_{j} }{\partial \varvec{\psi }^{T}} \right] . \end{aligned}$$

(27)

Now,

$$\begin{aligned} I_{\varvec{\phi \phi }}= & {} E_{X}\left[ \omega _{J+1} T(X) \sum _{j=0}^{J+1} \frac{1}{Z_{j}} \frac{\partial Z_{j} }{\partial \varvec{\phi }}\frac{\partial Z_{j} }{\partial \varvec{\phi }^{T}} \right] \end{aligned}$$

$$\begin{aligned} I_{\rho \varvec{\phi }}= & {} E_{X}\left[ \omega _{J+1} T(X) \sum _{j=0}^{J+1} \frac{1}{Z_{j}} \frac{\partial Z_{j} }{\partial \rho }\frac{\partial Z_{j} }{\partial \varvec{\phi }^{T}} \right] \\= & {} E_{X}\left[ \omega _{J+1} T(X) \left\{ \sum _{j=0}^{J} Z_{J+1} \frac{\partial Z_{j} }{\partial \varvec{\phi }^{T}} - (1 - Z_{J+1}) \frac{\partial Z_{J+1} }{\partial \varvec{\phi }^{T}} \right\} \right] \\= & {} E_{X}\left[ \omega _{J+1} T(X) \left\{ - Z_{J+1} \frac{\partial Z_{J+1} }{\partial \varvec{\phi }^{T}} - (1 - Z_{J+1}) \frac{\partial Z_{J+1} }{\partial \varvec{\phi }^{T}} \right\} \right] \\= & {} - E_{X}\left[ \omega _{J+1} T(X) \frac{\partial Z_{J+1} }{\partial \varvec{\phi }^{T}} \right] \end{aligned}$$

$$\begin{aligned} I_{\rho \rho }= & {} E_{X}\left[ \omega _{J+1} T(X) \left\{ \sum _{j=0}^{J} Z_{j}Z_{J+1}^{2} + Z_{J+1}(1 - Z_{J+1})^{2} \right\} \right] \\= & {} E_{X}\left[ \omega _{J+1} T(X) \left\{ (1- Z_{J+1})Z_{J+1}^{2} + Z_{J+1}(1 - Z_{J+1})^{2} \right\} \right] \\= & {} - E_{X}\left[ \omega _{J+1} T(X) \frac{\partial Z_{J+1} }{\partial \rho }\right] , \end{aligned}$$

using the results,

$$\begin{aligned} \frac{\partial Z_{j}}{\partial \rho } = Z_{j}Z_{J+1} , \quad \text{ for } j=0,1,\ldots ,J, \quad \text{ and } \quad \frac{\partial Z_{J+1}}{\partial \rho } = -Z_{J+1}(1 - Z_{J+1}).\nonumber \\ \end{aligned}$$

(28)

Since the last column of $I(\varvec{\psi })$ is $-E_{X}[\omega _{J+1} T(X) \frac{\partial Z_{J+1} }{\partial \varvec{\psi }}]$, it can be checked that (see Neuhaus et al. 2002)

$$\begin{aligned} -I(\varvec{\psi })^{-1}E_{X}\left[ \omega _{J+1} T(X) \frac{\partial Z_{J+1} }{\partial \varvec{\psi }} \right] = \begin{bmatrix} \mathbf{0} \\ 1 \end{bmatrix}. \end{aligned}$$

(29)

Note that, $\sum _{j=0}^{J+1}Z_{j} =1$ implies

$$\begin{aligned} \sum _{j=0}^{J} E_{X}\left[ \omega _{J+1} T(X) \frac{\partial Z_{j} }{\partial \varvec{\psi }}\right] = - E_{X}\left[ \omega _{J+1} T(X) \frac{\partial Z_{J+1} }{\partial \varvec{\psi }}\right] , \end{aligned}$$

where $T(X) = (1 + \sum _{l=0}^{J}\mathrm{e}^{\rho }\mu _{l}(X,\varvec{\gamma })p_{l}(X,\varvec{\beta }))$. Now, we claim that

$$\begin{aligned} E_{X}\left[ \omega _{J+1} T(X) \frac{\partial Z_{j} }{\partial \varvec{\psi }}\right] = - \tau _{j}(\varvec{\psi }) E_{X}\left[ \omega _{J+1} T(X) \frac{\partial Z_{J+1} }{\partial \varvec{\psi }}\right] , \end{aligned}$$

(30)

where $\sum _{j=0}^{J} \tau _{j}(\varvec{\psi }) = 1$. Using (29) and (30), we have

$$\begin{aligned} I(\varvec{\psi })^{-1}E_{X}\left[ \omega _{J+1} T(X) \frac{\partial Z_{j} }{\partial \varvec{\psi }}\right] = \tau _{j}(\varvec{\psi }) \begin{bmatrix} \mathbf{0} \\ 1 \end{bmatrix}. \end{aligned}$$

(31)

From (29) and (31),

$$\begin{aligned}&\sum _{j=0}^{J+1} \frac{1}{\omega _{j}} I(\varvec{\psi })^{-1}E_{X}\left[ \omega _{J+1} T(X) \frac{\partial Z_{j} }{\partial \varvec{\psi }}\right] E_{X}\left[ \omega _{J+1} T(X) \frac{\partial Z_{j} }{\partial \varvec{\psi }^{T}}\right] I(\varvec{\psi })^{-1} \\&\quad = \left( \frac{1}{\omega _{J+1}} + \sum _{j=0}^{J} \tau _{j}(\varvec{\psi }) \right) \begin{bmatrix} \mathbf{0}&\mathbf{0} \\ \mathbf{0}&1 \end{bmatrix}. \end{aligned}$$

Hence, using (23), (21) is established.

Appendix C: Semiparametric efficiency

Following Bickel et al. (1993), the asymptotic variance matrix for a regular asymptotically linear (RAL) estimate $\varvec{\hat{\phi }}$ of $\varvec{\phi }$ satisfies $V(\varvec{\hat{\phi }}) \ge B$, where $B$ is the semiparametric efficiency bound. Lee and Hirose (2010) have obtained this bound $B$ for the semiparametric maximum likelihood estimate of parameters in general regression model when data are collected under response-selective sampling scheme. In order to apply their results in our context, let us consider the “population expected likelihood” (see also Newey 1990; Lee et al. 2006) as given by

$$\begin{aligned} \sum _{j=0}^{J+1}\omega _{j}E_{j}[\mathrm{logf}_{j}(X,\varvec{\phi },g)], \end{aligned}$$

(32)

where $E_{j}$ is the expectation with respect to the conditional distribution of exposure $X$, given $Y=j$, having density $f_{j}(x,\varvec{\phi },g)=\mu _{j}(x,\varvec{\gamma })p_{j}(x,\varvec{\beta })g(x)/\pi _{j}$ with $\pi _{j} = \int \mu _{j}(x,\varvec{\gamma })p_{j}(x,\varvec{\beta })g(x) \mathrm{d}x$, for $j=0,\ldots ,J$ and $E_{J+1}$ is the expectation with respect to the unconditional distribution of $X$ having density $f_{J+1}(x,\varvec{\phi },g)=g(x)$, where $g(x)$ is the density corresponding to the exposure distribution $G(x)$ of $X$. Then, the efficient scores are given by

$$\begin{aligned} S_{j} = \frac{\partial \mathrm{logf}_{j}(x,\varvec{\phi },\hat{g}(\varvec{\phi }))}{\partial \varvec{\phi }}\biggl |_{\varvec{\phi } = \varvec{\phi }_{0}}, \quad j = 0,1,\ldots ,J+1, \end{aligned}$$

(33)

where $\hat{g}(\varvec{\phi })$ is the maximizer of (32), for fixed $\varvec{\phi }$. Then, the corresponding efficiency bound $B$ is given by

$$\begin{aligned} B^{-1} = \sum _{j=0}^{J+1} \omega _{j}E_{j}(S_{j}S_{j}^{T}). \end{aligned}$$

(34)

To show that the asymptotic variance matrix of $\varvec{\hat{\phi }}$ is equal to the semiparametric efficiency bound, we need to show that

$$\begin{aligned} B^{-1} = I_{\varvec{\phi } \varvec{\phi }} - I_{\varvec{\phi } \rho }I_{\rho \rho }^{-1}I_{\rho \varvec{\phi }}. \end{aligned}$$

(35)

From (32), the expected log-likelihood

$$\begin{aligned}&\sum _{j=0}^{J}\omega _{j} \int \{ \log (\mu _{j}(x,\varvec{\gamma })p_{j}(x,\varvec{\beta })) + \log (g(x)) \} \frac{\mu _{j}(x,\varvec{\gamma _{0}})p_{j}(x,\varvec{\beta _{0}})g_{0}(x)\mathrm{d}x}{\pi _{j}^{0}} \nonumber \\&\quad + \omega _{J+1}\int \{\log (g(x)) \}g_{0}(x)\mathrm{d}x - \sum _{j=0}^{J}\omega _{j}\log \pi _{j}, \end{aligned}$$

(36)

where $\pi ^{0}_{j}= \int \mu _{j}(x,\varvec{\gamma _{0}})p_{j}(x,\varvec{\beta _{0}})g_{0}(x)\mathrm{d}x$. Considering the terms which involve $g(x)$, (36) can be written as

$$\begin{aligned} \omega _{J+1} \int \log (g(x)) \tilde{p}(x) g_{0}(x) \mathrm{d}x - \sum _{j=0}^{J}\omega _{j}\log \pi _{j}, \nonumber \\ \text{ where } \tilde{p}(x) = 1 + \sum _{j=0}^{J} \frac{\omega _{j}}{\omega _{J+1}} \frac{\mu _{j}(x,\varvec{\gamma _{0}})p_{j}(x,\varvec{\beta _{0}})}{\pi _{j}^{0}}. \end{aligned}$$

(37)

Now, we need to find $\hat{g}$ which maximizes (37). Consider the class of distribution of $X$ to be discrete with finite support $\{x_{1},\ldots ,x_{M} \}$. Suppose a general member $g(\cdot )$ of this class has mass $g_{i}$ at $x_{i}$. Note that the true distribution $g_{0}(\cdot )$ is a member of this class having mass $g_{0}(x_{i})$, say, at $x_{i}$. Then, (37) can be written along with Lagrange multiplier $\lambda$ to take care of the constraint $\sum _{i=1}^{M}g_{i} =1$, as

$$\begin{aligned} \omega _{J+1} \sum _{i=1}^{M} \log (g_{i}) \tilde{p}(x_{i}) g_{0}(x_{i}) - \sum _{j=0}^{J}\omega _{j}\log \pi _{j}(\varvec{g}) + \lambda \left( \sum _{i=1}^{M}g_{i} - 1\right) , \end{aligned}$$

(38)

where $\pi _{j}(\varvec{g}) = \sum _{i=1}^{M} \mu _{ji}p_{ji} g_{i}$. Differentiating (38) with respect to $g_{i}$, we have

$$\begin{aligned} \omega _{J+1} \frac{\tilde{p}(x_{i})g_{0}(x_{i})}{g_{i}} - \sum _{j=0}^{J}\omega _{j}\frac{\mu _{j}(x_{i},\varvec{\gamma })p_{j}(x_{i},\varvec{\beta }))}{\pi _{j}(\varvec{g})} + \lambda = 0. \end{aligned}$$

(39)

Multiplying (39) by $g_{i}$ and summing over $i$ give $\lambda = -\omega _{J+1}$. Putting the value of $\lambda$ in (39), we get the estimate of $g_{i}$ as

$$\begin{aligned} \hat{g}_{i} = \frac{\tilde{p}(x_{i})g_{0}(x_{i})}{1 + \sum _{j=0}^{J}\frac{\omega _{j}}{\pi _{j}(\varvec{g})\omega _{J+1}}\mu _{j}(x_{i},\varvec{\gamma })p_{j}(x_{i},\varvec{\beta }))}. \end{aligned}$$

(40)

In case of general $g$, not having finite support, the maximizer of (37) is of the form

$$\begin{aligned} \hat{g}(x; \varvec{\phi }, \rho ) = \frac{\tilde{p}(x)g_{0}(x)}{1 + \sum _{j=0}^{J}\frac{\omega _{j}}{\pi _{j}(\rho )\omega _{J+1}}\mu _{j}(x,\varvec{\gamma })p_{j}(x,\varvec{\beta })} \end{aligned}$$

(41)

(see Lee and Hirose 2010; Lee et al. 2006), where $\pi _{j}(\rho )$ satisfies $\mathrm{e}^{\rho } \!=\! \omega _{j}/(\pi _{j}(\rho )\omega _{J+1})$ (see (14)) for $j=0,1\ldots ,J$ and $\rho = \rho (\varvec{\phi })$ is the solution of the equations

$$\begin{aligned}&\int \mu _{j}(x,\varvec{\gamma }) p_{j}(x,\varvec{\beta }) \hat{g}(x; \varvec{\phi }, \rho )\mathrm{d}x = \pi _{j}(\rho ), \text{ or, } \text{ equivalently, } \nonumber \\&\quad \int \left\{ \frac{\mathrm{e}^{\rho }\mu _{j}(x,\varvec{\gamma }) p_{j}(x,\varvec{\beta })}{1 + \sum _{j=0}^{J}\mathrm{e}^{\rho }\mu _{j}(x,\varvec{\gamma }) p_{j}(x,\varvec{\beta })}\right\} \tilde{p}(x) g_{0}(x) \mathrm{d}x = \frac{\omega _{j}}{\omega _{J+1}}, \end{aligned}$$

(42)

for $j=0,1,\ldots ,J$. Putting the value of $\hat{g}$ in the densities $f_{j}(x,\varvec{\phi },g)$, for $j=0,1,\ldots ,J$, we have

$$\begin{aligned} \mathrm{logf}_{j}(x,\varvec{\phi },\hat{g})= & {} \log \left\{ \frac{\mathrm{e}^{\rho (\varvec{\phi })}\mu _{j}(x,\varvec{\gamma }) p_{j}(x,\varvec{\beta })}{1 + \sum _{j=0}^{J}\mathrm{e}^{\rho (\varvec{\phi })}\mu _{j}(x,\varvec{\gamma }) p_{j}(x,\varvec{\beta })} \right\} + c_{j} \nonumber \\= & {} \log \{q_{j}(x,\varvec{\phi },\rho (\varvec{\phi }))\} + c_{j} \end{aligned}$$

(43)

and

$$\begin{aligned} \mathrm{logf}_{J+1}(x,\varvec{\phi },\hat{g})= & {} \log \left\{ \frac{1}{1 + \sum _{j=0}^{J}\mathrm{e}^{\rho (\varvec{\phi })}\mu _{j}(x,\varvec{\gamma }) p_{j}(x,\varvec{\beta })} \right\} + c_{J+1} \nonumber \\= & {} \log \left\{ 1 - \sum _{j=0}^{J}q_{j}(x,\varvec{\phi },\rho (\varvec{\phi }))\right\} + c_{J+1}, \end{aligned}$$

(44)

where $c_{j}$’s are constants with respect to $\varvec{\psi }=(\rho ,\varvec{\phi })$. Using (33), the efficient scores are given as

$$\begin{aligned} S_{j} = \frac{\partial }{\partial \varvec{\phi }}\log \{q_{j}(x,\varvec{\phi },\rho (\varvec{\phi })) \}, \quad \text{ for } j=0,1,\ldots ,J+1, \end{aligned}$$

(45)

where $q_{J+1}(x,\varvec{\phi },\rho (\varvec{\phi })) = 1 - \sum _{j=0}^{J}q_{j}(x,\varvec{\phi },\rho (\varvec{\phi }))$ and all the derivatives are evaluated at $\varvec{\phi }=\varvec{\phi }_{0}$. Applying chain rule,

$$\begin{aligned} S_{j} = \frac{\partial }{\partial \varvec{\phi }}\log \{q_{j}(x,\varvec{\phi },\rho (\varvec{\phi })) \} + \left\{ \frac{\partial \rho (\varvec{\phi })}{\partial \varvec{\phi }} \right\} ^{T}\frac{\partial }{\partial \rho }\log \{q_{j}(x,\varvec{\phi },\rho (\varvec{\phi })) \}. \end{aligned}$$

(46)

Note that the information matrix (See Lee et al. 2006) is given by

$$\begin{aligned} I(\varvec{\psi }) = I(\rho ,\varvec{\phi }) = \sum _{j=0}^{J+1} \omega _{j}E_{j}\left\{ \left( \frac{\partial }{\partial \varvec{\psi }} \mathrm{logq}_{j}\right) \left( \frac{\partial }{\partial \varvec{\psi }} \mathrm{logq}_{j}\right) ^{T} \right\} . \end{aligned}$$

(47)

From (34) and (46), we get

$$\begin{aligned} B^{-1} = I_{\varvec{\phi }\varvec{\phi }} + \left\{ \frac{\partial \rho (\varvec{\phi })}{\partial \varvec{\phi }} \right\} ^{T}I_{\rho \varvec{\phi }} + I_{\varvec{\phi }\rho }\left\{ \frac{\partial \rho (\varvec{\phi })}{\partial \varvec{\phi }} \right\} + \left\{ \frac{\partial \rho (\varvec{\phi })}{\partial \varvec{\phi }} \right\} ^{T}I_{\rho \rho }\left\{ \frac{\partial \rho (\varvec{\phi })}{\partial \varvec{\phi }} \right\} . \end{aligned}$$

(48)

Differentiating (42) under the integral sign

$$\begin{aligned} \int \frac{\partial q_{j}}{\partial \varvec{\phi }}\tilde{p} g\, \mathrm{d}x + \left\{ \frac{\partial \rho (\varvec{\phi })}{\partial \varvec{\phi }} \right\} ^{T}\int \frac{\partial q_{j}}{\partial \rho }\tilde{p} g\, \mathrm{d}x = 0, \quad j = 0,1,\ldots ,J. \end{aligned}$$

(49)

It can be easily checked that

$$\begin{aligned}&\frac{\partial q_{j}}{\partial \rho } = q_{j}\left( 1-\sum _{j=0}^{J}q_{j}\right) ; -\frac{\partial ^{2}\log q_{j}}{\partial \rho ^{2}} = \sum _{j=0}^{J}\frac{\partial q_{j}}{\partial \rho } \nonumber \\&\qquad \text{ and } -\frac{\partial }{\partial \varvec{\phi }}\left\{ \frac{\partial \log q_{j}}{\partial \rho } \right\} = \sum _{j=0}^{J} \frac{\partial q_{j}}{\partial \varvec{\phi }}. \end{aligned}$$

(50)

Now, using (47) and (50),

$$\begin{aligned} I_{\rho \rho } = \omega _{J+1} \int \sum _{j=0}^{J}\frac{\partial q_{j}}{\partial \rho } \tilde{p} g\, \mathrm{d}x \text{ and } I_{\rho \varvec{\phi }} = \omega _{J+1} \int \sum _{j=0}^{J}\frac{\partial q_{j}}{\partial \varvec{\phi }} \tilde{p} g\, \mathrm{d}x. \end{aligned}$$

(51)

Summing over $j=0,1,\ldots ,J$ in (49) and using (50)

$$\begin{aligned} \frac{\partial \rho (\varvec{\phi })}{\partial \varvec{\phi }} = - I_{\rho \rho }^{-1}I_{\rho \varvec{\phi }}. \end{aligned}$$

(52)

From (34), (48) and (52)

$$\begin{aligned} B^{-1} = I_{\varvec{\phi }\varvec{\phi }} - I_{\varvec{\phi }\rho }I_{\rho \rho }^{-1}I_{\rho \varvec{\phi }}. \end{aligned}$$

(53)

This establish the efficiency bound of the semiparametric procedure.