An EM algorithm for the estimation of parameters of a flexible cure rate model with generalized gamma lifetime and model discrimination using likelihood- and information-based methods

Abstract

In this paper, we consider the Conway–Maxwell Poisson (COM-Poisson) cure rate model based on a competing risks scenario. This model includes, as special cases, some of the well-known cure rate models discussed in the literature. By assuming the time-to-event to follow the generalized gamma distribution, which contains some of the commonly used lifetime distributions as special cases, we develop exact likelihood inference based on the expectation maximization algorithm. The standard errors of the maximum likelihood estimates are obtained by inverting the observed information matrix. An extensive Monte Carlo simulation study is performed to examine the method of inference developed here. Model discrimination within the generalized gamma family is also carried out by means of likelihood- and information-based methods to select the particular lifetime distribution that provides an adequate fit to the data. Finally, a data on cancer recurrence is analyzed to illustrate the flexibility of the COM-Poisson family and the generalized gamma family so as to select a parsimonious competing cause distribution and a lifetime distribution that jointly provide an adequate fit to the data.

Appendices

Appendix 1: First- and second-order derivatives of the Q-function

Bernoulli cure rate model The expressions of the first- and second-order derivatives of the $Q_1(\varvec{\beta },\varvec{\pi }^{(k)})$ function with respect to $\varvec{\beta }$ are as follows:

$$\begin{aligned} \frac{\partial Q_1}{\partial \beta _l}&= \sum _{I_1}x_{il}+\sum _{I_0}\pi _i^{(k)}x_{il}-\sum _{I^*}x_{il}\frac{\exp (\varvec{x}_i^{\prime }\varvec{\beta })}{1+\exp (\varvec{x}_i^{\prime }\varvec{\beta })},\\ \frac{\partial ^2Q_1}{\partial \beta _l\partial \beta _{l^{\prime }}}&= -\sum _{I^*}x_{il}x_{il^{\prime }}\frac{\exp (\varvec{x}_i^{\prime }\varvec{\beta })}{\left( 1+\exp (\varvec{x}_i^{\prime }\varvec{\beta })\right) ^2} \end{aligned}$$

for $l,l^{\prime }=0,1,\ldots ,k$, and $x_{i0}=1 \forall i=1,2,\ldots ,n$. The required first- and second-order derivatives of the $Q_2(\varvec{\gamma }^*,\varvec{\pi }^{(k)})$ function with respect to $\varvec{\gamma }^*$, for a fixed value of $q$, are as follows:

$$\begin{aligned} \frac{\partial Q_2}{\partial \sigma }&= -\frac{n_1}{\sigma }\bigg \{1+\frac{\log \lambda }{q\sigma }\bigg \}+\frac{1}{q\sigma ^2}\sum _{I_1}\{(\lambda t_i)^{q/\sigma }\log (\lambda t_i)-\log t_i\}\\&\quad +\,\sum _{I_0}\pi _i^{(k)}\frac{\partial }{\partial \sigma }\log (S(t_i;\varvec{\gamma })),\\ \frac{\partial Q_2}{\partial \lambda }&= \frac{n_1}{q\sigma \lambda }-\frac{1}{q\sigma \lambda }\sum _{I_1}(\lambda t_i)^{q/\sigma }+\sum _{I_0}\pi _i^{(k)}\frac{\partial }{\partial \lambda }\log (S(t_i;\varvec{\gamma })),\\ \frac{\partial ^2 Q_2}{\partial {\sigma }^2}&= \frac{n_1}{\sigma ^2}\bigg \{1+\frac{2\log \lambda }{q\sigma }\bigg \}+\frac{2}{q\sigma ^3}\sum _{I_1}\log t_i\\&\quad -\,\frac{1}{\sigma ^3}\sum _{I_1}(\lambda t_i)^{q/\sigma }\log (\lambda t_i)\bigg \{\frac{2}{q}+\frac{\log (\lambda t_i)}{\sigma }\bigg \}+\sum _{I_0}\pi _i^{(k)}\frac{\partial ^2}{\partial \sigma ^2}\log (S(t_i;\varvec{\gamma })),\\ \frac{\partial ^2 Q_2}{\partial {\lambda }^2}&= -\frac{n_1}{q\sigma \lambda ^2}-\bigg (\frac{1}{\sigma }-\frac{1}{q}\bigg )\frac{1}{\sigma \lambda ^2}\sum _{I_1}(\lambda t_i)^{q/\sigma }+\sum _{I_0}\pi _i^{(k)}\frac{\partial ^2}{\partial \lambda ^2}\log (S(t_i;\varvec{\gamma })),\\ \frac{\partial ^2 Q_2}{\partial {\sigma }\partial {\lambda }}&= -\frac{n_1}{q\sigma ^2\lambda }+\frac{1}{q\sigma ^2\lambda }\sum _{I_1}(\lambda t_i)^{q/\sigma }\bigg \{1+\frac{q}{\sigma }\log (\lambda t_i)\bigg \}\\&\quad +\,\sum _{I_0}\pi _i^{(k)}\frac{\partial ^2}{\partial \sigma \partial \lambda }\log (S(t_i;\varvec{\gamma })), \end{aligned}$$

where $\pi _i^{(k)}$ is as defined in (12).

Poisson cure rate model The required first- and second-order derivatives of the $Q(\varvec{\theta }^*,\varvec{\pi }^{(k)})$ function with respect to $\varvec{\beta }$ and $\varvec{\gamma }^*$, for a fixed value of $q$, are as follows:

$$\begin{aligned} \frac{\partial Q}{\partial \beta _l}&= \sum _{I_1}x_{il}\frac{\exp (\varvec{x}_i^{'}\varvec{\beta })S(t_i;\varvec{\gamma })}{1+\exp (\varvec{x}_i^{'}\varvec{\beta })}+\sum _{I_1}x_{il}\frac{\exp (\varvec{x}_i^{'}\varvec{\beta })}{(1+\exp (\varvec{x}_i^{'}\varvec{\beta }))\log (1+\exp (\varvec{x}_i^{'}\varvec{\beta }))}\\&\quad +\,\sum _{I_0}\pi _i^{(k)}x_{il}\frac{\exp (\varvec{x}_i^{'}\varvec{\beta })P(t_i,\varvec{x}_i;\varvec{\beta },\varvec{\gamma })S(t_i;\varvec{\gamma })}{1+\exp (\varvec{x}_i^{'}\varvec{\beta })}-\sum _{I^*}x_{il}\frac{\exp (\varvec{x}_i^{'}\varvec{\beta })}{1+\exp (\varvec{x}_i^{'}\varvec{\beta })},\\ \frac{\partial Q}{\partial \sigma }&= -\frac{n_1}{\sigma }\bigg \{1+\frac{\log \lambda }{q\sigma }\bigg \}+\frac{1}{q\sigma ^2}\sum _{I_1}\{(\lambda t_i)^{q/\sigma }\log (\lambda t_i)-\log t_i\}\\&\quad +\,\sum _{I_1}\log (1+\exp (\varvec{x}_i^{'}\varvec{\beta }))\frac{\partial }{\partial \sigma }S(t_i;\varvec{\gamma })\\&\quad +\,\sum _{I_0}\pi _i^{(k)}\log (1+\exp (\varvec{x}_i^{'}\varvec{\beta }))P(t_i,\varvec{x}_i;\varvec{\beta },\varvec{\gamma })\frac{\partial }{\partial \sigma }S(t_i;\varvec{\gamma }),\\ \frac{\partial Q}{\partial \lambda }&= \frac{n_1}{q\sigma \lambda }-\frac{1}{q\sigma \lambda }\sum _{I_1}(\lambda t_i)^{q/\sigma }+\sum _{I_1}\log (1+\exp (\varvec{x}_i^{'}\varvec{\beta }))\frac{\partial }{\partial \lambda }S(t_i;\varvec{\gamma })\\&\quad +\,\sum _{I_0}\pi _i^{(k)}\log (1+\exp (\varvec{x}_i^{'}\varvec{\beta }))P(t_i,\varvec{x}_i;\varvec{\beta },\varvec{\gamma })\frac{\partial }{\partial \lambda }S(t_i;\varvec{\gamma }),\\ \frac{\partial ^2Q}{\partial \beta _l\partial \beta _{l^{\prime }}}&= \sum _{I_1}x_{il}x_{il^{\prime }}\frac{\exp (\varvec{x}_i^{'}\varvec{\beta })}{(1+\exp (\varvec{x}_i^{'}\varvec{\beta }))^2}\bigg \{\frac{\log (1+\exp (\varvec{x}_i^{'}\varvec{\beta }))-\exp (\varvec{x}_i^{'}\varvec{\beta })}{(\log (1+\exp (\varvec{x}_i^{'}\varvec{\beta })))^2}\\&\quad +\,S(t_i;\varvec{\gamma })\bigg \}+\sum _{I_0}\pi _i^{(k)}x_{il}x_{il^{\prime }}\frac{\exp (\varvec{x}_i^{'}\varvec{\beta })S(t_i;\varvec{\gamma })P(t_i,\varvec{x}_i;\varvec{\beta },\varvec{\gamma })}{(1+\exp (\varvec{x}_i^{'}\varvec{\beta }))^2}\\&\quad \bigg \{1\!-\!\exp (\varvec{x}_i^{'}\varvec{\beta })S(t_i;\varvec{\gamma })(P(t_i,\varvec{x}_i;\varvec{\beta },\varvec{\gamma })\!-\!1)\bigg \}\!-\!\sum _{I^*}x_{il}x_{il^{\prime }}\frac{\exp (\varvec{x}_i^{'}\varvec{\beta })}{(1\!+\!\exp (\varvec{x}_i^{'}\varvec{\beta }))^2},\\ \frac{\partial ^2Q}{\partial \beta _l\partial \sigma }&= \sum _{I_1}x_{il}\bigg \{\frac{\exp (\varvec{x}_i^{'}\varvec{\beta })}{1+\exp (\varvec{x}_i^{'}\varvec{\beta })}\bigg \}\frac{\partial }{\partial \sigma }S(t_i;\varvec{\gamma }) +\sum _{I_0}\pi _i^{(k)}x_{il}\\&\quad \times \,\bigg \{\frac{\exp (\varvec{x}_i^{'}\varvec{\beta })P(t_i,\varvec{x}_i;\varvec{\beta },\varvec{\gamma })}{1+\exp (\varvec{x}_i^{'}\varvec{\beta })}\bigg \}\bigg \{1-A(t_i,\varvec{x}_i;\varvec{\beta },\varvec{\gamma })(P(t_i,\varvec{x}_i;\varvec{\beta },\varvec{\gamma })-1)\bigg \}\\&\quad \times \,\frac{\partial }{\partial \sigma }S(t_i;\varvec{\gamma }),\\ \end{aligned}$$

$$\begin{aligned} \frac{\partial ^2Q}{\partial \beta _l\partial \lambda }&= \sum _{I_1}x_{il}\bigg \{\frac{\exp (\varvec{x}_i^{'}\varvec{\beta })}{1+\exp (\varvec{x}_i^{'}\varvec{\beta })}\bigg \}\frac{\partial }{\partial \lambda }S(t_i;\varvec{\gamma }) +\sum _{I_0}\pi _i^{(k)}x_{il}\\&\quad \times \,\bigg \{\frac{\exp (\varvec{x}_i^{'}\varvec{\beta })P(t_i,\varvec{x}_i;\varvec{\beta },\varvec{\gamma })}{1+\exp (\varvec{x}_i^{'}\varvec{\beta })}\bigg \}\bigg \{1-A(t_i,\varvec{x}_i;\varvec{\beta },\varvec{\gamma })(P(t_i,\varvec{x}_i;\varvec{\beta },\varvec{\gamma })-1)\bigg \}\\&\quad \times \,\frac{\partial }{\partial \lambda }S(t_i;\varvec{\gamma }),\\ \frac{\partial ^2 Q}{\partial {\sigma }^2}&= \frac{n_1}{\sigma ^2}\bigg \{1+\frac{2\log \lambda }{q\sigma }\bigg \}+\frac{2}{q\sigma ^3}\sum _{I_1}\log t_i-\frac{1}{\sigma ^3}\sum _{I_1}(\lambda t_i)^{q/\sigma }\log (\lambda t_i)\\&\quad \times \,\bigg \{\frac{2}{q}+\frac{\log (\lambda t_i)}{\sigma }\bigg \} +\sum _{I_1}\log (1+\exp (\varvec{x}_i^{'}\varvec{\beta }))\frac{\partial ^2}{\partial \sigma ^2}S(t_i;\varvec{\gamma })\\&\quad +\,\sum _{I_0}\pi _i^{(k)}\log (1+\exp (\varvec{x}_i^{'}\varvec{\beta }))\bigg [P(t_i,\varvec{x}_i;\varvec{\beta },\varvec{\gamma })\frac{\partial ^2}{\partial \sigma ^2}S(t_i;\varvec{\gamma })\\&\quad +\,\frac{\partial }{\partial \sigma }S(t_i;\varvec{\gamma })\frac{\partial }{\partial \sigma }P(t_i,\varvec{x}_i;\varvec{\beta },\varvec{\gamma })\bigg ],\\ \frac{\partial ^2 Q}{\partial {\lambda }^2}&= -\frac{n_1}{q\sigma \lambda ^2}-\bigg (\frac{1}{\sigma }-\frac{1}{q}\bigg )\frac{1}{\sigma \lambda ^2}\sum _{I_1}(\lambda t_i)^{q/\sigma }\\&\quad +\,\sum _{I_1}\log (1+\exp (\varvec{x}_i^{'}\varvec{\beta }))\frac{\partial ^2}{\partial \lambda ^2}S(t_i;\varvec{\gamma }) +\sum _{I_0}\pi _i^{(k)}\log (1+\exp (\varvec{x}_i^{'}\varvec{\beta }))\\&\quad \times \,\bigg [P(t_i,\varvec{x}_i;\varvec{\beta },\varvec{\gamma })\frac{\partial ^2}{\partial \lambda ^2}S(t_i;\varvec{\gamma })+\frac{\partial }{\partial \lambda }S(t_i;\varvec{\gamma })\frac{\partial }{\partial \lambda }P(t_i,\varvec{x}_i;\varvec{\beta },\varvec{\gamma })\bigg ],\\ \frac{\partial ^2Q}{\partial \sigma \partial \lambda }&= -\frac{n_1}{q\sigma ^2\lambda }+\frac{1}{q\sigma ^2\lambda }\sum _{I_1}(\lambda t_i)^{q/\sigma }\bigg \{1+\frac{q}{\sigma }\log (\lambda t_i)\bigg \}\\&\quad +\,\sum _{I_1}\log (1+\exp (\varvec{x}_i^{'}\varvec{\beta }))\frac{\partial ^2}{\partial \sigma \partial \lambda }S(t_i;\varvec{\gamma }) +\sum _{I_0}\pi _i^{(k)}\log (1+\exp (\varvec{x}_i^{'}\varvec{\beta }))\\&\quad \times \,\bigg [P(t_i,\varvec{x}_i;\varvec{\beta },\varvec{\gamma })\frac{\partial ^2}{\partial \sigma \partial \lambda }S(t_i;\varvec{\gamma })+\frac{\partial }{\partial \sigma }S(t_i;\varvec{\gamma })\frac{\partial }{\partial \lambda }P(t_i,\varvec{x}_i;\varvec{\beta },\varvec{\gamma })\bigg ] \end{aligned}$$

for $l,l^{\prime }=0,1,\ldots ,k$, and $x_{i0}=1 \forall i=1,2,\ldots ,n,$ where

$$\begin{aligned} P(t_i,\varvec{x}_i;\varvec{\beta },\varvec{\gamma })=\frac{\exp (A(t_i,\varvec{x}_i;\varvec{\beta },\varvec{\gamma }))}{\exp (A(t_i,\varvec{x}_i;\varvec{\beta },\varvec{\gamma }))-1} \end{aligned}$$

for $i\in I_0,$ and $A(t_i,\varvec{x}_i;\varvec{\beta },\varvec{\gamma })$ and $\pi _i^{(k)}$ are as defined in (13) and (14), respectively.

Geometric cure rate model The required first- and second-order derivatives of the $Q(\varvec{\theta }^*,\varvec{\pi }^{(k)})$ function with respect to $\varvec{\beta }$ and $\varvec{\gamma }^*$, for a fixed value of $q$, are as follows:

$$\begin{aligned} \frac{\partial Q}{\partial \beta _l}&= \sum _{I_1}x_{il}\{1-2B(t_i,\varvec{x}_i;\varvec{\beta },\varvec{\gamma })(1-S(t_i;\varvec{\gamma }))\}-\sum _{I_0}x_{il}\frac{\exp (\varvec{x}_i^{'}\varvec{\beta })}{1+\exp (\varvec{x}_i^{'}\varvec{\beta })}\\&\quad +\,\sum _{I_0}\pi _i^{(k)}x_{il}\{1-B(t_i,\varvec{x}_i;\varvec{\beta },\varvec{\gamma })(1-S(t_i;\varvec{\gamma }))\},\\ \frac{\partial Q}{\partial \sigma }&= -\frac{n_1}{\sigma }\bigg \{1+\frac{\log \lambda }{q\sigma }\bigg \}+\frac{1}{q\sigma ^2}\sum _{I_1}\{(\lambda t_i)^{q/\sigma }\log (\lambda t_i)-\log t_i\}\\&\quad +\,2\sum _{I_1}B(t_i,\varvec{x}_i;\varvec{\beta },\varvec{\gamma })\frac{\partial }{\partial \sigma }S(t_i;\varvec{\gamma })\\&\quad +\,\sum _{I_0}\pi _i^{(k)}\bigg [\frac{\partial }{\partial \sigma }\log (S(t_i;\varvec{\gamma }))+B(t_i,\varvec{x}_i;\varvec{\beta },\varvec{\gamma })\frac{\partial }{\partial \sigma }S(t_i;\varvec{\gamma })\bigg ],\\ \frac{\partial Q}{\partial \lambda }&= \frac{n_1}{q\sigma \lambda }-\frac{1}{q\sigma \lambda }\sum _{I_1}(\lambda t_i)^{q/\sigma }+2\sum _{I_1}B(t_i,\varvec{x}_i;\varvec{\beta },\varvec{\gamma })\frac{\partial }{\partial \lambda }S(t_i;\varvec{\gamma })\\&\quad +\,\sum _{I_0}\pi _i^{(k)}\bigg [\frac{\partial }{\partial \lambda }\log (S(t_i;\varvec{\gamma }))+B(t_i,\varvec{x}_i;\varvec{\beta },\varvec{\gamma })\frac{\partial }{\partial \lambda }S(t_i;\varvec{\gamma })\bigg ],\\ \frac{\partial ^2Q}{\partial \beta _l\partial \beta _{l^{\prime }}}&= -2\sum _{I_1}x_{il}x_{il^{\prime }}\frac{B(t_i,\varvec{x}_i;\varvec{\beta },\varvec{\gamma })(1-S(t_i;\varvec{\gamma }))}{1+\exp (\varvec{x}_i^{'}\varvec{\beta })(1-S(t_i;\varvec{\gamma }))}\\&\quad -\,\sum _{I_0}x_{il}x_{il^{\prime }}\frac{\exp (\varvec{x}_i^{'}\varvec{\beta })}{(1+\exp (\varvec{x}_i^{'}\varvec{\beta }))^2}\\&\quad -\,\sum _{I_0}\pi _i^{(k)}x_{il}x_{il^{\prime }}\frac{B(t_i,\varvec{x}_i;\varvec{\beta },\varvec{\gamma })(1-S(t_i;\varvec{\gamma }))}{1+\exp (\varvec{x}_i^{'}\varvec{\beta })(1-S(t_i;\varvec{\gamma }))},\\ \frac{\partial ^2Q}{\partial \beta _l\partial \sigma }&= 2\sum _{I_1}x_{il}\bigg \{\frac{B(t_i,\varvec{x}_i;\varvec{\beta },\varvec{\gamma })}{1+\exp (\varvec{x}_i^{'}\varvec{\beta })(1-S(t_i;\varvec{\gamma }))}\bigg \}\frac{\partial }{\partial \sigma }S(t_i;\varvec{\gamma })\\&\quad +\,\sum _{I_0}\pi _i^{(k)}x_{il}\bigg \{\frac{B(t_i,\varvec{x}_i;\varvec{\beta },\varvec{\gamma })}{1+\exp (\varvec{x}_i^{'}\varvec{\beta })(1-S(t_i;\varvec{\gamma }))}\bigg \}\frac{\partial }{\partial \sigma }S(t_i;\varvec{\gamma }),\\ \frac{\partial ^2Q}{\partial \beta _l\partial \lambda }&= 2\sum _{I_1}x_{il}\bigg \{\frac{B(t_i,\varvec{x}_i;\varvec{\beta },\varvec{\gamma })}{1+\exp (\varvec{x}_i^{'}\varvec{\beta })(1-S(t_i;\varvec{\gamma }))}\bigg \}\frac{\partial }{\partial \lambda }S(t_i;\varvec{\gamma })\\&\quad +\,\sum _{I_0}\pi _i^{(k)}x_{il}\bigg \{\frac{B(t_i,\varvec{x}_i;\varvec{\beta },\varvec{\gamma })}{1+\exp (\varvec{x}_i^{'}\varvec{\beta })(1-S(t_i;\varvec{\gamma }))}\bigg \}\frac{\partial }{\partial \lambda }S(t_i;\varvec{\gamma }),\\ \end{aligned}$$

$$\begin{aligned} \frac{\partial ^2 Q}{\partial {\sigma }^2}&= \frac{n_1}{\sigma ^2}\bigg \{1+\frac{2\log \lambda }{q\sigma }\bigg \}+\frac{2}{q\sigma ^3}\sum _{I_1}\log t_i\\&\quad -\,\frac{1}{\sigma ^3}\sum _{I_1}(\lambda t_i)^{q/\sigma }\log (\lambda t_i)\bigg \{\frac{2}{q}+\frac{\log (\lambda t_i)}{\sigma }\bigg \}\\&\quad +\,2\sum _{I_1}\bigg [B(t_i,\varvec{x}_i;\varvec{\beta },\varvec{\gamma })\frac{\partial ^2}{\partial \sigma ^2}S(t_i;\varvec{\gamma })+\frac{\partial }{\partial \sigma }S(t_i;\varvec{\gamma })\frac{\partial }{\partial \sigma }B(t_i,\varvec{x}_i;\varvec{\beta },\varvec{\gamma })\bigg ]\\&\quad +\,\sum _{I_0}\pi _i^{(k)}\bigg [B(t_i,\varvec{x}_i;\varvec{\beta },\varvec{\gamma })\frac{\partial ^2}{\partial \sigma ^2}S(t_i;\varvec{\gamma })+\frac{\partial }{\partial \sigma }S(t_i;\varvec{\gamma })\frac{\partial }{\partial \sigma }B(t_i,\varvec{x}_i;\varvec{\beta },\varvec{\gamma })\bigg ]\\&\quad +\,\sum _{I_0}\pi _i^{(k)}\frac{\partial ^2}{\partial \sigma ^2}\log (S(t_i;\varvec{\gamma })),\\ \frac{\partial ^2 Q}{\partial {\lambda }^2}&= -\frac{n_1}{q\sigma \lambda ^2}-\bigg (\frac{1}{\sigma }-\frac{1}{q}\bigg )\frac{1}{\sigma \lambda ^2}\sum _{I_1}(\lambda t_i)^{q/\sigma }\\&\quad +\,2\sum _{I_1}\bigg [B(t_i,\varvec{x}_i;\varvec{\beta },\varvec{\gamma })\frac{\partial ^2}{\partial \lambda ^2}S(t_i;\varvec{\gamma })+\frac{\partial }{\partial \lambda }S(t_i;\varvec{\gamma })\frac{\partial }{\partial \lambda }B(t_i,\varvec{x}_i;\varvec{\beta },\varvec{\gamma })\bigg ]\\&\quad +\,\sum _{I_0}\pi _i^{(k)}\bigg [B(t_i,\varvec{x}_i;\varvec{\beta },\varvec{\gamma })\frac{\partial ^2}{\partial \lambda ^2}S(t_i;\varvec{\gamma })+\frac{\partial }{\partial \lambda }S(t_i;\varvec{\gamma })\frac{\partial }{\partial \lambda }B(t_i,\varvec{x}_i;\varvec{\beta },\varvec{\gamma })\bigg ]\\&\quad +\,\sum _{I_0}\pi _i^{(k)}\frac{\partial ^2}{\partial \lambda ^2}\log (S(t_i;\varvec{\gamma })),\\ \frac{\partial ^2Q}{\partial \sigma \partial \lambda }&= -\frac{n_1}{q\sigma ^2\lambda }+\frac{1}{q\sigma ^2\lambda }\sum _{I_1}(\lambda t_i)^{q/\sigma }\bigg \{1+\frac{q}{\sigma }\log (\lambda t_i)\bigg \}\\&\quad +\,2\sum _{I_1}\bigg [B(t_i,\varvec{x}_i;\varvec{\beta },\varvec{\gamma })\frac{\partial ^2}{\partial \sigma \partial \lambda }S(t_i;\varvec{\gamma })+\frac{\partial }{\partial \sigma }S(t_i;\varvec{\gamma })\frac{\partial }{\partial \lambda }B(t_i,\varvec{x}_i;\varvec{\beta },\varvec{\gamma })\bigg ]\\&\quad +\,\sum _{I_0}\pi _i^{(k)}\bigg [B(t_i,\varvec{x}_i;\varvec{\beta },\varvec{\gamma })\frac{\partial ^2}{\partial \sigma \partial \lambda }S(t_i;\varvec{\gamma })+\frac{\partial }{\partial \sigma }S(t_i;\varvec{\gamma })\frac{\partial }{\partial \lambda }B(t_i,\varvec{x}_i;\varvec{\beta },\varvec{\gamma })\bigg ]\\&\quad +\,\sum _{I_0}\pi _i^{(k)}\frac{\partial ^2}{\partial \sigma \partial \lambda }\log (S(t_i;\varvec{\gamma })) \end{aligned}$$

for $l,l^{\prime }=0,1,\ldots ,k$, and $x_{i0}=1 \forall i=1,2,\ldots ,n,$ where

$$\begin{aligned} B(t_i,\varvec{x}_i;\varvec{\beta },\varvec{\gamma })=\frac{\exp (\varvec{x}_i^{'}\varvec{\beta })}{1+\exp (\varvec{x}_i^{'}\varvec{\beta })(1-S(t_i;\varvec{\gamma }))} \end{aligned}$$

(17)

for $i=1,2,\ldots ,n,$ and $\pi _i^{(k)}$ is as defined in (15).

COM-Poisson cure rate model The required first- and second-order derivatives of the $Q(\varvec{\theta }^*,\varvec{\pi }^{(k)})$ function with respect to $\varvec{\beta }$ and $\varvec{\gamma }^*$, for fixed values of $q$ and $\phi$, are as follows:

$$\begin{aligned} \frac{\partial Q}{\partial \beta _l}&= \sum _{I_1}x_{il}\frac{\exp (\varvec{x}_i^{'}\varvec{\beta })z_{21i}}{z_{2i}z_{01i}} +\sum _{I_0}\pi _i^{(k)}x_{il}\frac{\exp (\varvec{x}_i^{'}\varvec{\beta })z_{2i}}{z_{1i}z_{01i}}-\sum _{I^*}x_{il}\frac{\exp (\varvec{x}_i^{'}\varvec{\beta })}{1+\exp (\varvec{x}_i^{'}\varvec{\beta })},\\ \frac{\partial Q}{\partial \sigma }&= -\frac{n_1}{\sigma }\bigg \{1+\frac{\log \lambda }{q\sigma }\bigg \}+\frac{1}{q\sigma ^2}\sum _{I_1}\{(\lambda t_i)^{q/\sigma }\log (\lambda t_i)-\log t_i\}\\&\quad +\,\sum _{I_1}\bigg (\frac{z_{21i}}{z_{2i}}-1\bigg )\frac{\partial }{\partial \sigma }\log (S(t_i;\varvec{\gamma }))+\sum _{I_0}\pi _i^{(k)}\frac{z_{2i}}{z_{1i}}\frac{\partial }{\partial \sigma }\log (S(t_i;\varvec{\gamma })),\\ \frac{\partial Q}{\partial \lambda }&= \frac{n_1}{q\sigma \lambda }-\frac{1}{q\sigma \lambda }\sum _{I_1}(\lambda t_i)^{q/\sigma }+\sum _{I_1}\bigg (\frac{z_{21i}}{z_{2i}}-1\bigg )\frac{\partial }{\partial \lambda }\log (S(t_i;\varvec{\gamma }))\\&\quad +\,\sum _{I_0}\pi _i^{(k)}\frac{z_{2i}}{z_{1i}}\frac{\partial }{\partial \lambda }\log (S(t_i;\varvec{\gamma })),\\ \frac{\partial ^2Q}{\partial \beta _l\partial \beta _{l^{\prime }}}&= \sum _{I_1}x_{il}x_{il^{\prime }}\frac{\exp (\varvec{x}_i^{'}\varvec{\beta })}{(z_{2i}z_{01i})^2}\bigg [\exp (\varvec{x}_i^{'}\varvec{\beta })\bigg \{z_{2i}z_{22i}-z_{21i}^2\bigg \}+z_{2i}z_{21i}\bigg \{z_{01i}\\&\quad -\,\frac{\exp (\varvec{x}_i^{'}\varvec{\beta })z_{02i}}{z_{01i}}\bigg \}\bigg ]+\sum _{I_0}\pi _i^{(k)}x_{il}x_{il^{\prime }}\frac{\exp (\varvec{x}_i^{'}\varvec{\beta })}{(z_{1i}z_{01i})^2}\bigg [\exp (\varvec{x}_i^{'}\varvec{\beta })\bigg \{z_{1i}z_{21i}-z_{2i}^2\bigg \}\\&\quad +\,z_{1i}z_{2i}\bigg \{z_{01i}-\frac{\exp (\varvec{x}_i^{'}\varvec{\beta })z_{02i}}{z_{01i}}\bigg \}\bigg ]-\sum _{I^*}x_{il}x_{il^{\prime }}\frac{\exp (\varvec{x}_i^{'}\varvec{\beta })}{(1+\exp (\varvec{x}_i^{'}\varvec{\beta }))^2},\\ \frac{\partial ^2Q}{\partial \beta _l\partial \sigma }&= \sum _{I_1}x_{il}\frac{\exp (\varvec{x}_i^{'}\varvec{\beta })}{z_{01i}}\bigg \{\frac{z_{2i}z_{22i}-z_{21i}^2}{z_{2i}^2}\bigg \}\frac{\partial }{\partial \sigma }\log (S(t_i;\varvec{\gamma }))\\&\quad +\,\sum _{I_0}\pi _i^{(k)}x_{il}\frac{\exp (\varvec{x}_i^{'}\varvec{\beta })}{z_{01i}}\bigg \{\frac{z_{1i}z_{21i}-z_{2i}^2}{z_{1i}^2}\bigg \}\frac{\partial }{\partial \sigma }\log (S(t_i;\varvec{\gamma })),\\ \frac{\partial ^2Q}{\partial \beta _l\partial \lambda }&= \sum _{I_1}x_{il}\frac{\exp (\varvec{x}_i^{'}\varvec{\beta })}{z_{01i}}\bigg \{\frac{z_{2i}z_{22i}-z_{21i}^2}{z_{2i}^2}\bigg \}\frac{\partial }{\partial \lambda }\log (S(t_i;\varvec{\gamma }))\\&\quad +\,\sum _{I_0}\pi _i^{(k)}x_{il}\frac{\exp (\varvec{x}_i^{'}\varvec{\beta })}{z_{01i}}\bigg \{\frac{z_{1i}z_{21i}-z_{2i}^2}{z_{1i}^2}\bigg \}\frac{\partial }{\partial \lambda }\log (S(t_i;\varvec{\gamma })),\\ \end{aligned}$$

$$\begin{aligned} \frac{\partial ^2 Q}{\partial {\sigma }^2}&= \frac{n_1}{\sigma ^2}\bigg \{1+\frac{2\log \lambda }{q\sigma }\bigg \}+\frac{2}{q\sigma ^3}\sum _{I_1}\log t_i-\frac{1}{\sigma ^3}\sum _{I_1}(\lambda t_i)^{q/\sigma }\log (\lambda t_i)\\&\quad \times \,\bigg \{\frac{2}{q}+\frac{\log (\lambda t_i)}{\sigma }\bigg \} +\sum _{I_1}\bigg [\bigg (\frac{z_{21i}}{z_{2i}}-1\bigg )\frac{\partial ^2}{\partial \sigma ^2}\log (S(t_i;\varvec{\gamma }))\\&\quad +\,\bigg \{\frac{z_{2i}z_{22i}-z_{21i}^2}{z_{2i}^2}\bigg \}\bigg \{\frac{\partial }{\partial \sigma }\log (S(t_i;\varvec{\gamma }))\bigg \}^2\bigg ]\\&\quad +\,\sum _{I_0}\pi _i^{(k)}\bigg [\frac{z_{2i}}{z_{1i}}\frac{\partial ^2}{\partial \sigma ^2}\log (S(t_i;\varvec{\gamma }))\\&\quad +\,\bigg \{\frac{z_{1i}z_{21i}-z_{2i}^2}{z_{1i}^2}\bigg \}\bigg \{\frac{\partial }{\partial \sigma }\log (S(t_i;\varvec{\gamma }))\bigg \}^2\bigg ],\\ \frac{\partial ^2 Q}{\partial {\lambda }^2}&= -\frac{n_1}{q\sigma \lambda ^2}-\bigg (\frac{1}{\sigma }-\frac{1}{q}\bigg )\frac{1}{\sigma \lambda ^2}\sum _{I_1}(\lambda t_i)^{q/\sigma }\\&\quad +\,\sum _{I_1}\bigg [\bigg (\frac{z_{21i}}{z_{2i}}-1\bigg )\frac{\partial ^2}{\partial \lambda ^2}\log (S(t_i;\varvec{\gamma }))+\bigg \{\frac{z_{2i}z_{22i}-z_{21i}^2}{z_{2i}^2}\bigg \}\\&\quad \times \,\bigg \{\frac{\partial }{\partial \lambda }\log (S(t_i;\varvec{\gamma }))\bigg \}^2\bigg ] +\sum _{I_0}\pi _i^{(k)}\bigg [\frac{z_{2i}}{z_{1i}}\frac{\partial ^2}{\partial \lambda ^2}\log (S(t_i;\varvec{\gamma }))\\&\quad +\,\bigg \{\frac{z_{1i}z_{21i}-z_{2i}^2}{z_{1i}^2}\bigg \}\bigg \{\frac{\partial }{\partial \lambda }\log (S(t_i;\varvec{\gamma }))\bigg \}^2\bigg ],\\ \frac{\partial ^2Q}{\partial \sigma \partial \lambda }&= -\frac{n_1}{q\sigma ^2\lambda }+\frac{1}{q\sigma ^2\lambda }\sum _{I_1}(\lambda t_i)^{q/\sigma }\bigg \{1+\frac{q}{\sigma }\log (\lambda t_i)\bigg \}\\&\quad +\,\sum _{I_1}\bigg [\bigg (\frac{z_{21i}}{z_{2i}}-1\bigg )\frac{\partial ^2}{\partial \sigma \partial \lambda }\log (S(t_i;\varvec{\gamma }))+\bigg \{\frac{z_{2i}z_{22i}-z_{21i}^2}{z_{2i}^2}\bigg \}\\&\quad \times \,\frac{\partial }{\partial \sigma }\log (S(t_i;\varvec{\gamma }))\frac{\partial }{\partial \lambda }\log (S(t_i;\varvec{\gamma }))\bigg ] +\sum _{I_0}\pi _i^{(k)}\bigg [\frac{z_{2i}}{z_{1i}}\frac{\partial ^2}{\partial \sigma \partial \lambda }\log (S(t_i;\varvec{\gamma }))\\&\quad +\,\bigg \{\frac{z_{1i}z_{21i}-z_{2i}^2}{z_{1i}^2}\bigg \}\frac{\partial }{\partial \sigma }\log (S(t_i;\varvec{\gamma }))\frac{\partial }{\partial \lambda }\log (S(t_i;\varvec{\gamma }))\bigg ] \end{aligned}$$

for $l,l^{\prime }=0,1,\ldots ,k$, and $x_{i0}=1 \forall i=1,2,\ldots ,n,$ where

$$\begin{aligned} z_{21}&= z_{21}(\varvec{\theta };\varvec{x},t)=\sum _{j=1}^{\infty }\frac{j^2\{H_{\phi }^{-1}(1+\exp (\varvec{x}^{'}\varvec{\beta }))S(t;\varvec{\gamma })\}^j}{(j!)^{\phi }},\\ z_{22}&= z_{22}(\varvec{\theta };\varvec{x},t)=\sum _{j=1}^{\infty }\frac{j^3\{H_{\phi }^{-1}(1+\exp (\varvec{x}^{'}\varvec{\beta }))S(t;\varvec{\gamma })\}^j}{(j!)^{\phi }},\\ z_{01}&= z_{01}(\varvec{\theta }_1;\varvec{x})=\sum _{j=1}^{\infty }\frac{j\{H_{\phi }^{-1}(1+\exp (\varvec{x}^{'}\varvec{\beta }))\}^j}{(j!)^{\phi }},\\ z_{02}&= z_{02}(\varvec{\theta }_1;\varvec{x})=\sum _{j=1}^{\infty }\frac{j^2\{H_{\phi }^{-1}(1+\exp (\varvec{x}^{'}\varvec{\beta }))\}^j}{(j!)^{\phi }} \end{aligned}$$

and $\pi _i^{(k)}$ is as defined in (16).

The quantities $z_{1i}, z_{2i}, z_{21i}, z_{22i}, z_{01i}$, and $z_{02i}$ are all computed by truncating the numerical series for each $i$. Let $S_n$ denote the $n$-th order partial sum of the infinite series. Then, the infinite series can be approximated with the first $n$ terms of the series if $S_{n+1}-S_n \le \epsilon$, where $\epsilon$ is a pre-fixed small tolerance value. In our study, we chose $\epsilon =0.01$ for computational ease and efficiency.

Appendix 2: Observed information matrix

Bernoulli cure rate model The components of the score function, for a fixed value of $q$, are

$$\begin{aligned} \frac{\partial l}{\partial \beta _l}&= \sum _{I_1}x_{il}+\sum _{I_0}x_{il}w_i-\sum _{I^*}x_{il}\frac{\exp (\varvec{x}_i^{'}\varvec{\beta })}{1+\exp (\varvec{x}_i^{'}\varvec{\beta })},\\ \frac{\partial l}{\partial \sigma }&= -\frac{n_1}{\sigma }\bigg \{1+\frac{\log \lambda }{q\sigma }\bigg \}+\frac{1}{q\sigma ^2}\sum _{I_1}\{(\lambda t_i)^{q/\sigma }\log (\lambda t_i)-\log t_i\}\\&\quad +\,\sum _{I_0}w_i\frac{\partial }{\partial \sigma }\log (S(t_i;\varvec{\gamma })),\\ \frac{\partial l}{\partial \lambda }&= \frac{n_1}{q\sigma \lambda }-\frac{1}{q\sigma \lambda }\sum _{I_1}(\lambda t_i)^{q/\sigma }+\sum _{I_0}w_i\frac{\partial }{\partial \lambda }\log (S(t_i;\varvec{\gamma })). \end{aligned}$$

The observed information matrix has its components as

$$\begin{aligned} -\frac{\partial ^2l}{\partial \beta _l\partial \beta _{l^{'}}}&= -\sum _{I_0}x_{il}x_{il^{'}}w_i(1-w_i)+\sum _{I^*}x_{il}x_{il^{'}}\frac{\exp (\varvec{x}_i^{'}\varvec{\beta })}{(1+\exp (\varvec{x}_i^{'}\varvec{\beta }))^2},\\ -\frac{\partial ^2l}{\partial \beta _l\partial \sigma }&= -\sum _{I_0}x_{il}\frac{\partial }{\partial \sigma }w_i,\\ -\frac{\partial ^2l}{\partial \beta _l\partial \lambda }&= -\sum _{I_0}x_{il}\frac{\partial }{\partial \lambda }w_i,\\ -\frac{\partial ^2l}{\partial \sigma ^2}&= -\frac{n_1}{\sigma ^2}\bigg \{1+\frac{2\log \lambda }{q\sigma }\bigg \}-\frac{2}{q\sigma ^3}\sum _{I_1}\log t_i\\&\quad +\,\frac{1}{\sigma ^3}\sum _{I_1}(\lambda t_i)^{q/\sigma }\log (\lambda t_i)\bigg \{\frac{2}{q}+\frac{\log (\lambda t_i)}{\sigma }\bigg \}\\&\quad -\,\sum _{I_0}\bigg \{w_i\frac{\partial ^2}{\partial \sigma ^2}\log (S(t_i;\varvec{\gamma }))+\frac{\partial }{\partial \sigma }\log (S(t_i;\varvec{\gamma }))\frac{\partial }{\partial \sigma }w_i\bigg \},\\ -\frac{\partial ^2l}{\partial \lambda ^2}&= \frac{n_1}{q\sigma \lambda ^2}+\bigg (\frac{1}{\sigma }-\frac{1}{q}\bigg )\frac{1}{\sigma \lambda ^2}\sum _{I_1}(\lambda t_i)^{q/\sigma }\\&\quad -\,\sum _{I_0}\bigg \{w_i\frac{\partial ^2}{\partial \lambda ^2}\log (S(t_i;\varvec{\gamma }))+\frac{\partial }{\partial \lambda }\log (S(t_i;\varvec{\gamma }))\frac{\partial }{\partial \lambda }w_i\bigg \},\\ -\frac{\partial ^2l}{\partial \sigma \partial \lambda }&= \frac{n_1}{q\sigma ^2\lambda }-\frac{1}{q\sigma ^2\lambda }\sum _{I_1}(\lambda t_i)^{q/\sigma }\bigg \{1+\frac{q}{\sigma }\log (\lambda t_i)\bigg \}\\&\quad -\,\sum _{I_0}\bigg \{w_i\frac{\partial ^2}{\partial \sigma \partial \lambda }\log (S(t_i;\varvec{\gamma }))+\frac{\partial }{\partial \sigma }\log (S(t_i;\varvec{\gamma }))\frac{\partial }{\partial \lambda }w_i\bigg \}. \end{aligned}$$

The above are defined for $l,l^{\prime }=0,1,\ldots ,k,\,x_{i0}=1 \forall i=1,2,\ldots ,n$, where

$$\begin{aligned} w_i=\frac{\exp (\varvec{x}_i^{'}\varvec{\beta })S(t_i;\varvec{\gamma })}{1+\exp (\varvec{x}_i^{'}\varvec{\beta })S(t_i;\varvec{\beta })} \end{aligned}$$

for $i\in I_0$.

Poisson cure rate model The components of the score function, for a fixed value of $q$, are

$$\begin{aligned} \frac{\partial l}{\partial \beta _l}&= \sum _{I_1}x_{il}\frac{\exp (\varvec{x}_i^{'}\varvec{\beta })}{(1+\exp (\varvec{x}_i^{'}\varvec{\beta }))\log (1+\exp (\varvec{x}_i^{'}\varvec{\beta }))}+\sum _{I^*}x_{il}\frac{\exp (\varvec{x}_i^{'}\varvec{\beta })\{S(t_i;\varvec{\gamma })-1\}}{1+\exp (\varvec{x}_i^{'}\varvec{\beta })},\\ \frac{\partial l}{\partial \sigma }&= -\frac{n_1}{\sigma }\bigg \{1+\frac{\log \lambda }{q\sigma }\bigg \}+\frac{1}{q\sigma ^2}\sum _{I_1}\{(\lambda t_i)^{q/\sigma }\log (\lambda t_i)-\log t_i\}\\&\quad +\,\sum _{I^*}\log (1+\exp (\varvec{x}_i^{'}\varvec{\beta }))\frac{\partial }{\partial \sigma }S(t_i;\varvec{\gamma }),\\ \frac{\partial l}{\partial \lambda }&= \frac{n_1}{q\sigma \lambda }-\frac{1}{q\sigma \lambda }\sum _{I_1}(\lambda t_i)^{q/\sigma }+\sum _{I^*}\log (1+\exp (\varvec{x}_i^{'}\varvec{\beta }))\frac{\partial }{\partial \lambda }S(t_i;\varvec{\gamma }). \end{aligned}$$

The observed information matrix has its components as

$$\begin{aligned} -\frac{\partial ^2l}{\partial \beta _l\partial \beta _{l^{'}}}&= -\sum _{I_1}x_{il}x_{il^{'}}\frac{\exp (\varvec{x}_i^{'}\varvec{\beta })}{(1+\exp (\varvec{x}_i^{'}\varvec{\beta }))^2}\bigg \{\frac{\log (1+\exp (\varvec{x}_i^{'}\varvec{\beta }))-\exp (\varvec{x}_i^{'}\varvec{\beta })}{(\log (1+\exp (\varvec{x}_i^{'}\varvec{\beta })))^2}\bigg \}\\&\quad -\,\sum _{I^*}x_{il}x_{il^{'}}\frac{\exp (\varvec{x}_i^{'}\varvec{\beta })\{S(t_i;\varvec{\gamma })-1\}}{(1+\exp (\varvec{x}_i^{'}\varvec{\beta }))^2},\\ -\frac{\partial ^2l}{\partial \beta _l\partial \sigma }&= -\sum _{I^*}x_{il}\bigg \{\frac{\exp (\varvec{x}_i^{'}\varvec{\beta })}{1+\exp (\varvec{x}_i^{'}\varvec{\beta })}\bigg \}\frac{\partial }{\partial \sigma }S(t_i;\varvec{\gamma }),\\ -\frac{\partial ^2l}{\partial \beta _l\partial \lambda }&= -\sum _{I^*}x_{il}\bigg \{\frac{\exp (\varvec{x}_i^{'}\varvec{\beta })}{1+\exp (\varvec{x}_i^{'}\varvec{\beta })}\bigg \}\frac{\partial }{\partial \lambda }S(t_i;\varvec{\gamma }),\\ -\frac{\partial ^2 l}{\partial {\sigma }^2}&= -\frac{n_1}{\sigma ^2}\bigg \{1+\frac{2\log \lambda }{q\sigma }\bigg \}-\frac{2}{q\sigma ^3}\sum _{I_1}\log t_i\\&\quad +\,\frac{1}{\sigma ^3}\sum _{I_1}(\lambda t_i)^{q/\sigma }\log (\lambda t_i)\bigg \{\frac{2}{q}+\frac{\log (\lambda t_i)}{\sigma }\bigg \}\\&\quad -\,\sum _{I^*}\log (1+\exp (\varvec{x}_i^{'}\varvec{\beta }))\frac{\partial ^2}{\partial \sigma ^2}S(t_i;\varvec{\gamma }),\\ -\frac{\partial ^2 l}{\partial {\lambda }^2}&= \frac{n_1}{q\sigma \lambda ^2}+\bigg (\frac{1}{\sigma }-\frac{1}{q}\bigg )\frac{1}{\sigma \lambda ^2}\sum _{I_1}(\lambda t_i)^{q/\sigma }\\&\quad -\,\sum _{I^*}\log (1+\exp (\varvec{x}_i^{'}\varvec{\beta }))\frac{\partial ^2}{\partial \lambda ^2}S(t_i;\varvec{\gamma }),\\ -\frac{\partial ^2l}{\partial \sigma \partial \lambda }&= \frac{n_1}{q\sigma ^2\lambda }-\frac{1}{q\sigma ^2\lambda }\sum _{I_1}(\lambda t_i)^{q/\sigma }\bigg \{1+\frac{q}{\sigma }\log (\lambda t_i)\bigg \}\\&\quad -\,\sum _{I^*}\log (1+\exp (\varvec{x}_i^{'}\varvec{\beta }))\frac{\partial ^2}{\partial \sigma \partial \lambda }S(t_i;\varvec{\gamma }). \end{aligned}$$

The above are defined for $l,l^{\prime }=0,1,\ldots ,k,\,x_{i0}=1 \forall i=1,2,\ldots ,n$.

Geometric cure rate model The components of the score function, for a fixed value of $q$, are as follows

$$\begin{aligned} \frac{\partial l}{\partial \beta _l}&= \sum _{I_1}x_{il}\{1-2B(t_i,\varvec{x}_i;\varvec{\beta },\varvec{\gamma })(1-S(t_i;\varvec{\gamma }))\}-\sum _{I_0}x_{il}\frac{\exp (\varvec{x}_i^{'}\varvec{\beta })}{1+\exp (\varvec{x}_i^{'}\varvec{\beta })}\\&\quad +\,\sum _{I_0}x_{il}w_i\{1-B(t_i,\varvec{x}_i;\varvec{\beta },\varvec{\gamma })(1-S(t_i;\varvec{\gamma }))\},\\ \frac{\partial l}{\partial \sigma }&= -\frac{n_1}{\sigma }\bigg \{1+\frac{\log \lambda }{q\sigma }\bigg \}+\frac{1}{q\sigma ^2}\sum _{I_1}\{(\lambda t_i)^{q/\sigma }\log (\lambda t_i)-\log t_i\}\\&\quad +\,2\sum _{I_1}B(t_i,\varvec{x}_i;\varvec{\beta },\varvec{\gamma })\frac{\partial }{\partial \sigma }S(t_i;\varvec{\gamma }) +\sum _{I_0}\bigg \{\frac{\exp (\varvec{x}_i^{'}\varvec{\beta })}{1+\exp (\varvec{x}_i^{'}\varvec{\beta })}\bigg \}\frac{\partial }{\partial \sigma }S(t_i;\varvec{\gamma })\\&\quad +\,\sum _{I_0}w_iB(t_i,\varvec{x}_i;\varvec{\beta },\varvec{\gamma })\frac{\partial }{\partial \sigma }S(t_i;\varvec{\gamma }),\\ \frac{\partial l}{\partial \lambda }&= \frac{n_1}{q\sigma \lambda }-\frac{1}{q\sigma \lambda }\sum _{I_1}(\lambda t_i)^{q/\sigma }+2\sum _{I_1}B(t_i,\varvec{x}_i;\varvec{\beta },\varvec{\gamma })\frac{\partial }{\partial \lambda }S(t_i;\varvec{\gamma })\\&\quad +\,\sum _{I_0}\bigg \{\frac{\exp (\varvec{x}_i^{'}\varvec{\beta })}{1+\exp (\varvec{x}_i^{'}\varvec{\beta })}\bigg \}\frac{\partial }{\partial \lambda }S(t_i;\varvec{\gamma }) +\sum _{I_0}w_iB(t_i,\varvec{x}_i;\varvec{\beta },\varvec{\gamma })\frac{\partial }{\partial \lambda }S(t_i;\varvec{\gamma }). \end{aligned}$$

The observed information matrix has its components as

$$\begin{aligned} -\frac{\partial ^2l}{\partial \beta _l\partial \beta _{l^{\prime }}}&= 2\sum _{I_1}x_{il}x_{il^{\prime }}\frac{B(t_i,\varvec{x}_i;\varvec{\beta },\varvec{\gamma })(1-S(t_i;\varvec{\gamma }))}{1+\exp (\varvec{x}_i^{'}\varvec{\beta })(1-S(t_i;\varvec{\gamma }))}\\&\quad +\,\sum _{I_0}x_{il}x_{il^{\prime }}\frac{\exp (\varvec{x}_i^{'}\varvec{\beta })(1-S(t_i;\varvec{\gamma }))}{(1+\exp (\varvec{x}_i^{'}\varvec{\beta }))^2}\\&\quad +\,\sum _{I_0}x_{il}x_{il^{\prime }}w_iB(t_i,\varvec{x}_i;\varvec{\beta },\varvec{\gamma })(1-S(t_i;\varvec{\gamma }))\\&\quad \times \,\bigg \{\frac{1}{1+\exp (\varvec{x}_i^{'}\varvec{\beta })(1-S(t_i;\varvec{\gamma }))}+\frac{1}{1+\exp (\varvec{x}_i^{'}\varvec{\beta })}\bigg \},\\ -\frac{\partial ^2l}{\partial \beta _l\partial \sigma }&= -2\sum _{I_1}x_{il}\bigg \{\frac{B(t_i,\varvec{x}_i;\varvec{\beta },\varvec{\gamma })}{1+\exp (\varvec{x}_i^{'}\varvec{\beta })(1-S(t_i;\varvec{\gamma }))}\bigg \}\frac{\partial }{\partial \sigma }S(t_i;\varvec{\gamma })\\&\quad -\,\sum _{I_0}x_{il}\bigg \{\frac{\exp (\varvec{x}_i^{'}\varvec{\beta })}{(1+\exp (\varvec{x}_i^{'}\varvec{\beta }))^2}\bigg \}\frac{\partial }{\partial \sigma }S(t_i;\varvec{\gamma })\\&\quad -\,\sum _{I_0}x_{il}w_iB(t_i,\varvec{x}_i;\varvec{\beta },\varvec{\gamma })\frac{\partial }{\partial \sigma }S(t_i;\varvec{\gamma })\\&\quad \times \,\bigg \{\frac{1}{1+\exp (\varvec{x}_i^{'}\varvec{\beta })(1-S(t_i;\varvec{\gamma }))}+\frac{1}{1+\exp (\varvec{x}_i^{'}\varvec{\beta })}\bigg \},\\ -\frac{\partial ^2l}{\partial \beta _l\partial \lambda }&= -2\sum _{I_1}x_{il}\bigg \{\frac{B(t_i,\varvec{x}_i;\varvec{\beta },\varvec{\gamma })}{1+\exp (\varvec{x}_i^{'}\varvec{\beta })(1-S(t_i;\varvec{\gamma }))}\bigg \}\frac{\partial }{\partial \lambda }S(t_i;\varvec{\gamma })\\&\quad -\,\sum _{I_0}x_{il}\bigg \{\frac{\exp (\varvec{x}_i^{'}\varvec{\beta })}{(1+\exp (\varvec{x}_i^{'}\varvec{\beta }))^2}\bigg \}\frac{\partial }{\partial \lambda }S(t_i;\varvec{\gamma })\\&\quad -\,\sum _{I_0}x_{il}w_iB(t_i,\varvec{x}_i;\varvec{\beta },\varvec{\gamma })\frac{\partial }{\partial \lambda }S(t_i;\varvec{\gamma })\\&\quad \times \,\bigg \{\frac{1}{1+\exp (\varvec{x}_i^{'}\varvec{\beta })(1-S(t_i;\varvec{\gamma }))}+\frac{1}{1+\exp (\varvec{x}_i^{'}\varvec{\beta })}\bigg \},\\ -\frac{\partial ^2 l}{\partial {\sigma }^2}&= -\frac{n_1}{\sigma ^2}\bigg \{1+\frac{2\log \lambda }{q\sigma }\bigg \}-\frac{2}{q\sigma ^3}\sum _{I_1}\log t_i\\&\quad +\,\frac{1}{\sigma ^3}\sum _{I_1}(\lambda t_i)^{q/\sigma }\log (\lambda t_i)\bigg \{\frac{2}{q}+\frac{\log (\lambda t_i)}{\sigma }\bigg \}\\&\quad -\,2\sum _{I_1}\bigg \{B(t_i,\varvec{x}_i;\varvec{\beta },\varvec{\gamma })\frac{\partial ^2}{\partial \sigma ^2}S(t_i;\varvec{\gamma })+\frac{\partial }{\partial \sigma }S(t_i;\varvec{\gamma })\frac{\partial }{\partial \sigma }B(t_i,\varvec{x}_i;\varvec{\beta },\varvec{\gamma })\bigg \}\\&\quad -\,\sum _{I_0}\bigg \{\frac{\exp (\varvec{x}_i^{\prime }\varvec{\beta })}{1+\exp (\varvec{x}_i^{\prime }\varvec{\beta })}\bigg \}\frac{\partial ^2}{\partial \sigma ^2}S(t_i;\varvec{\gamma })\\&\quad -\,\sum _{I_0}\bigg [w_iB(t_i,\varvec{x}_i;\varvec{\beta },\varvec{\gamma })\frac{\partial ^2}{\partial \sigma ^2}S(t_i;\varvec{\gamma })+\frac{\partial }{\partial \sigma }S(t_i;\varvec{\gamma })\\&\quad \times \,\bigg \{w_i\frac{\partial }{\partial \sigma }B(t_i,\varvec{x}_i;\varvec{\beta },\varvec{\gamma })+\bigg \{\frac{\exp (\varvec{x}_i^{\prime }\varvec{\beta })B(t_i,\varvec{x}_i;\varvec{\beta },\varvec{\gamma })}{1+\exp (\varvec{x}_i^{\prime }\varvec{\beta })}\bigg \}\frac{\partial }{\partial \sigma }S(t_i;\varvec{\gamma })\bigg \}\bigg ],\\ \end{aligned}$$

$$\begin{aligned} -\frac{\partial ^2 l}{\partial {\lambda }^2}&= \frac{n_1}{q\sigma \lambda ^2}+\bigg (\frac{1}{\sigma }-\frac{1}{q}\bigg )\frac{1}{\sigma \lambda ^2}\sum _{I_1}(\lambda t_i)^{q/\sigma }\\&\quad -\,2\sum _{I_1}\bigg \{B(t_i,\varvec{x}_i;\varvec{\beta },\varvec{\gamma })\frac{\partial ^2}{\partial \lambda ^2}S(t_i;\varvec{\gamma })+\frac{\partial }{\partial \lambda }S(t_i;\varvec{\gamma })\frac{\partial }{\partial \lambda }B(t_i,\varvec{x}_i;\varvec{\beta },\varvec{\gamma })\bigg \}\\&\quad -\,\sum _{I_0}\bigg \{\frac{\exp (\varvec{x}_i^{\prime }\varvec{\beta })}{1+\exp (\varvec{x}_i^{\prime }\varvec{\beta })}\bigg \}\frac{\partial ^2}{\partial \lambda ^2}S(t_i;\varvec{\gamma })\\&\quad -\,\sum _{I_0}\bigg [w_iB(t_i,\varvec{x}_i;\varvec{\beta },\varvec{\gamma })\frac{\partial ^2}{\partial \lambda ^2}S(t_i;\varvec{\gamma })+\frac{\partial }{\partial \lambda }S(t_i;\varvec{\gamma })\bigg \{w_i\frac{\partial }{\partial \lambda }B(t_i,\varvec{x}_i;\varvec{\beta },\varvec{\gamma })\\&\quad +\,\bigg \{\frac{\exp (\varvec{x}_i^{\prime }\varvec{\beta })B(t_i,\varvec{x}_i;\varvec{\beta },\varvec{\gamma })}{1+\exp (\varvec{x}_i^{\prime }\varvec{\beta })}\bigg \}\frac{\partial }{\partial \lambda }S(t_i;\varvec{\gamma })\bigg \}\bigg ],\\ \end{aligned}$$

$$\begin{aligned} -\frac{\partial ^2l}{\partial \sigma \partial \lambda }&= \frac{n_1}{q\sigma ^2\lambda }-\frac{1}{q\sigma ^2\lambda }\sum _{I_1}(\lambda t_i)^{q/\sigma }\bigg \{1+\frac{q}{\sigma }\log (\lambda t_i)\bigg \}\\&\quad -\,2\sum _{I_1}\bigg \{B(t_i,\varvec{x}_i;\varvec{\beta },\varvec{\gamma })\frac{\partial ^2}{\partial \sigma \partial \lambda }S(t_i;\varvec{\gamma })+\frac{\partial }{\partial \sigma }S(t_i;\varvec{\gamma })\frac{\partial }{\partial \lambda }B(t_i,\varvec{x}_i;\varvec{\beta },\varvec{\gamma })\bigg \}\\&\quad -\,\sum _{I_0}\bigg \{\frac{\exp (\varvec{x}_i^{\prime }\varvec{\beta })}{1+\exp (\varvec{x}_i^{\prime }\varvec{\beta })}\bigg \}\frac{\partial ^2}{\partial \sigma \partial \lambda }S(t_i;\varvec{\gamma })\\&\quad \!-\!\sum _{I_0}\bigg [\!w_iB(t_i,\varvec{x}_i;\varvec{\beta },\!\varvec{\gamma })\frac{\partial ^2}{\partial \sigma \partial \lambda }S(t_i;\varvec{\gamma })\!+\!\frac{\partial }{\partial \sigma }S(t_i;\varvec{\gamma })\bigg \{\!w_i\frac{\partial }{\partial \lambda }B(t_i,\varvec{x}_i;\varvec{\beta },\varvec{\gamma })\\&\quad +\,\bigg \{\frac{\exp (\varvec{x}_i^{\prime }\varvec{\beta })B(t_i,\varvec{x}_i;\varvec{\beta },\varvec{\gamma })}{1+\exp (\varvec{x}_i^{\prime }\varvec{\beta })}\bigg \}\frac{\partial }{\partial \lambda }S(t_i;\varvec{\gamma })\bigg \}\bigg ]. \end{aligned}$$

The above are defined for $l,l^{\prime }=0,1,\ldots ,k,\,x_{i0}=1 \forall i=1,2,\ldots ,n$, where

$$\begin{aligned} w_i=\frac{\exp (\varvec{x}_i^{\prime }\varvec{\beta })S(t_i;\varvec{\gamma })}{1+\exp (\varvec{x}_i^{\prime }\varvec{\beta })} \end{aligned}$$

for $i\in I_0,$ and $B(t_i,\varvec{x}_i;\varvec{\beta },\varvec{\gamma })$ is as defined in (17).

COM-Poisson cure rate model The components of the score function, for a fixed value of $q$ and $\phi$, are

$$\begin{aligned} \frac{\partial l}{\partial \beta _l}&= \sum _{I_1}x_{il}\frac{\exp (\varvec{x}_i^{'}\varvec{\beta })z_{21i}}{z_{2i}z_{01i}} +\sum _{I_0}x_{il}\frac{\exp (\varvec{x}_i^{'}\varvec{\beta })z_{2i}}{(1+z_{1i})z_{01i}}-\sum _{I^*}x_{il}\frac{\exp (\varvec{x}_i^{'}\varvec{\beta })}{1+\exp (\varvec{x}_i^{'}\varvec{\beta })},\\ \frac{\partial l}{\partial \sigma }&= -\frac{n_1}{\sigma }\bigg \{1+\frac{\log \lambda }{q\sigma }\bigg \}+\frac{1}{q\sigma ^2}\sum _{I_1}\{(\lambda t_i)^{q/\sigma }\log (\lambda t_i)-\log t_i\}\\&\quad +\,\sum _{I_1}\bigg \{\frac{z_{21i}}{z_{2i}}-1\bigg \}\frac{\partial }{\partial \sigma }\log (S(t_i;\varvec{\gamma }))+\sum _{I_0}\bigg \{\frac{z_{2i}}{1+z_{1i}}\bigg \}\frac{\partial }{\partial \sigma }\log (S(t_i;\varvec{\gamma })),\\ \frac{\partial l}{\partial \lambda }&= \frac{n_1}{q\sigma \lambda }-\frac{1}{q\sigma \lambda }\sum _{I_1}(\lambda t_i)^{q/\sigma }+\sum _{I_1}\bigg \{\frac{z_{21i}}{z_{2i}}-1\bigg \}\frac{\partial }{\partial \lambda }\log (S(t_i;\varvec{\gamma }))\\&\quad +\,\sum _{I_0}\bigg \{\frac{z_{2i}}{1+z_{1i}}\bigg \}\frac{\partial }{\partial \lambda }\log (S(t_i;\varvec{\gamma })). \end{aligned}$$

The observed information matrix has its components as

$$\begin{aligned} -\frac{\partial ^2l}{\partial \beta _l\partial \beta _{l^{\prime }}}&= -\sum _{I_1}x_{il}x_{il^{\prime }}\frac{\exp (\varvec{x}_i^{\prime }\varvec{\beta })}{(z_{01i}z_{2i})^2}\bigg [\exp (\varvec{x}_i^{\prime }\varvec{\beta })\bigg \{z_{2i}z_{22i}-z_{21i}^2\bigg \}\\&\quad +\,z_{2i}z_{21i}\bigg \{z_{01i}-\frac{\exp (\varvec{x}_i^{\prime }\varvec{\beta })z_{02i}}{z_{01i}}\bigg \}\bigg ] -\sum _{I_0}x_{il}x_{il^{\prime }}\frac{\exp (\varvec{x}_i^{\prime }\varvec{\beta })}{(z_{01i}(1+z_{1i}))^2}\\&\quad \times \,\bigg [\exp (\varvec{x}_i^{\prime }\varvec{\beta })\bigg \{z_{21i}(1+z_{1i})-z_{2i}^2\bigg \}+z_{2i}(1+z_{1i})\bigg \{z_{01i}\\&-\frac{\exp (\varvec{x}_i^{\prime }\varvec{\beta })z_{02i}}{z_{01i}}\bigg \}\bigg ] +\sum _{I^*}x_{il}x_{il^{\prime }}\frac{\exp (\varvec{x}_i^{\prime }\varvec{\beta })}{(1+\exp (\varvec{x}_i^{\prime }\varvec{\beta }))^2}, \end{aligned}$$

$$\begin{aligned} -\frac{\partial ^2l}{\partial \beta _l\partial \sigma }&= -\sum _{I_1}x_{il}\frac{\exp (\varvec{x}_i^{\prime }\varvec{\beta })}{z_{01i}}\bigg \{\frac{z_{2i}z_{22i}-z_{21i}^2}{z_{2i}^2}\bigg \}\frac{\partial }{\partial \sigma }\log (S(t_i;\varvec{\gamma }))\\&\quad -\,\sum _{I_0}x_{il}\frac{\exp (\varvec{x}_i^{\prime }\varvec{\beta })}{z_{01i}}\bigg \{\frac{z_{21i}(1+z_{1i})-z_{2i}^2}{(1+z_{1i})^2}\bigg \}\frac{\partial }{\partial \sigma }\log (S(t_i;\varvec{\gamma })),\\ -\frac{\partial ^2l}{\partial \beta _l\partial \lambda }&= -\sum _{I_1}x_{il}\frac{\exp (\varvec{x}_i^{\prime }\varvec{\beta })}{z_{01i}}\bigg \{\frac{z_{2i}z_{22i}-z_{21i}^2}{z_{2i}^2}\bigg \}\frac{\partial }{\partial \lambda }\log (S(t_i;\varvec{\gamma }))\\&\quad -\,\sum _{I_0}x_{il}\frac{\exp (\varvec{x}_i^{\prime }\varvec{\beta })}{z_{01i}}\bigg \{\frac{z_{21i}(1+z_{1i})-z_{2i}^2}{(1+z_{1i})^2}\bigg \}\frac{\partial }{\partial \lambda }\log (S(t_i;\varvec{\gamma })),\\ -\frac{\partial ^2 l}{\partial {\sigma }^2}&= -\frac{n_1}{\sigma ^2}\bigg \{1+\frac{2\log \lambda }{q\sigma }\bigg \}-\frac{2}{q\sigma ^3}\sum _{I_1}\log t_i\\&\quad +\,\frac{1}{\sigma ^3}\sum _{I_1}(\lambda t_i)^{q/\sigma }\log (\lambda t_i)\bigg \{\frac{2}{q}+\frac{\log (\lambda t_i)}{\sigma }\bigg \}\\&\quad -\,\sum _{I_1}\bigg [\bigg \{\frac{z_{21i}}{z_{2i}}-1\bigg \}\frac{\partial ^2}{\partial \sigma ^2}\log (S(t_i;\varvec{\gamma }))\\&\quad +\,\bigg \{\frac{z_{2i}z_{22i}-z_{21i}^2}{z_{2i}^2}\bigg \}\bigg \{\frac{\partial }{\partial \sigma }\log (S(t_i;\varvec{\gamma }))\bigg \}^2\bigg ]\\&\quad -\,\sum _{I_0}\bigg [\bigg \{\frac{z_{2i}}{1+z_{1i}}\bigg \}\frac{\partial ^2}{\partial \sigma ^2}\log (S(t_i;\varvec{\gamma }))\\&\quad +\,\bigg \{\frac{z_{21i}(1+z_{1i})-z_{2i}^2}{(1+z_{1i})^2}\bigg \}\bigg \{\frac{\partial }{\partial \sigma }\log (S(t_i;\varvec{\gamma }))\bigg \}^2\bigg ], \end{aligned}$$

$$\begin{aligned} -\frac{\partial ^2 l}{\partial {\lambda }^2}&= \frac{n_1}{q\sigma \lambda ^2}+\bigg (\frac{1}{\sigma }-\frac{1}{q}\bigg )\frac{1}{\sigma \lambda ^2}\sum _{I_1}(\lambda t_i)^{q/\sigma }\\&\quad -\,\sum _{I_1}\bigg [\bigg \{\frac{z_{21i}}{z_{2i}}-1\bigg \}\frac{\partial ^2}{\partial \lambda ^2}\log (S(t_i;\varvec{\gamma }))\\&\quad +\,\bigg \{\frac{z_{2i}z_{22i}-z_{21i}^2}{z_{2i}^2}\bigg \}\bigg \{\frac{\partial }{\partial \lambda }\log (S(t_i;\varvec{\gamma }))\bigg \}^2\bigg ]\\&\quad -\,\sum _{I_0}\bigg [\bigg \{\frac{z_{2i}}{1+z_{1i}}\bigg \}\frac{\partial ^2}{\partial \lambda ^2}\log (S(t_i;\varvec{\gamma }))\\&\quad +\,\bigg \{\frac{z_{21i}(1+z_{1i})-z_{2i}^2}{(1+z_{1i})^2}\bigg \}\bigg \{\frac{\partial }{\partial \lambda }\log (S(t_i;\varvec{\gamma }))\bigg \}^2\bigg ],\\ -\frac{\partial ^2l}{\partial \sigma \partial \lambda }&= \frac{n_1}{q\sigma ^2\lambda }-\frac{1}{q\sigma ^2\lambda }\sum _{I_1}(\lambda t_i)^{q/\sigma }\bigg \{1+\frac{q}{\sigma }\log (\lambda t_i)\bigg \}\\&\quad -\,\sum _{I_1}\bigg [\bigg \{\frac{z_{21i}}{z_{2i}}-1\bigg \}\frac{\partial ^2}{\partial \sigma \partial \lambda }\log (S(t_i;\varvec{\gamma }))\\&\quad +\,\bigg \{\frac{z_{2i}z_{22i}-z_{21i}^2}{z_{2i}^2}\bigg \}\frac{\partial }{\partial \sigma }\log (S(t_i;\varvec{\gamma }))\frac{\partial }{\partial \lambda }\log (S(t_i;\varvec{\gamma }))\bigg ]\\&\quad -\,\sum _{I_0}\bigg [\bigg \{\frac{z_{2i}}{1+z_{1i}}\bigg \}\frac{\partial ^2}{\partial \sigma \partial \lambda }\log (S(t_i;\varvec{\gamma }))\\&\quad +\,\bigg \{\frac{z_{21i}(1+z_{1i})-z_{2i}^2}{(1+z_{1i})^2}\bigg \}\frac{\partial }{\partial \sigma }\log (S(t_i;\varvec{\gamma }))\frac{\partial }{\partial \lambda }\log (S(t_i;\varvec{\gamma }))\bigg ]. \end{aligned}$$

The above are defined for $l,l^{\prime }=0,1,\ldots ,k,\,x_{i0}=1 \forall i=1,2,\ldots ,n$.