Inference for A Generalized Family of Distributions Under Partially Observed Left Truncated and Right Censored Competing Risks Data

Abstract

We make inference for a competing risks model under the assumption that observations are left-truncated and right-censored and failure causes are partially observed. When the latent failure times follow a generalized family of distributions, inference for unknown parameters is provided using classical and Bayesian approaches. Particularly existence-uniqueness properties of maximum likelihood estimators are established. Subsequently interval estimators are constructed based on observed Fisher information matrix. Bayes estimates and associated highest posterior density intervals are developed using gamma-beta prior distributions by considering squared error loss function. We also study estimation problem when parameters are order restricted. The performance of all estimators is evaluated based on an extensive simulation study and comments are obtained. A real data set is also analyzed for illustration purposes.

Appendices

Appendix

A Proof of Theorem 1

For $(t_{i}, \delta _{i}, \mu _{i}) = (t_{i}, 1, 1)$ indicate that ith unit is not truncated and failure of unit occurred due to cause 1, then $t_{i} = T_{i1}$ and $t_{i} < T_{i2}$. Therefore, the likelihood contribution is

$$\begin{aligned} \begin{aligned} P(t_{i}< T_{i1} < t_{i} + dt, T_{i2} > t_{i}) =&f(t_{i}, \beta _{1}, \alpha )\cdot S(t_{i}, \beta _{2}, \alpha )dt_{i} \\&\alpha \beta _{1}h^{'}(x_{i})[h(x_{i})]^{\alpha -1}e^{-\beta _{12}[h(x_{i})]^{\alpha }} \end{aligned} \end{aligned}$$

For $(t_{i}, \delta _{i}, \mu _{i}) = (t_{i}, 1, 0)$ indicate that ith unit has been left truncated implying min$(T_{i1}, T_{i2}) > t_{i}$. Moreover, from $\delta _{i}=1$, $t_{i}=T_{i1}$ and $t_{i} < T_{i2}$. Therefore,

$$\begin{aligned} \begin{aligned} P(t_{i}< T_{i1} < t_{i} + dt, T_{i2}> t_{i}|T_{i1}> \tau _{iL}, T_{i2} > \tau _{iL}) =&\frac{f(t_{i}, \beta _{1}, \alpha )\cdot S(t_{i}, \beta _{2}, \alpha )dt_{i}}{S(\tau _{iL}, \beta _{12}, \alpha )dt_{i}} \\&\frac{\alpha \beta _{1}h^{'}(x_{i})[h(x_{i})]^{\beta -1}e^{-\beta _{12}[h(x_{i})]^{\alpha }}}{e^{-\beta _{12}[h(\tau _{iL})]^{\alpha }}} \end{aligned} \end{aligned}$$

For $(t_{i}, \delta _{i}, \mu _{i}) = (t_{i}, 0, 1)$ indicate that the ith unit is not truncated but right censored at $t_{i}$ which implies that min$(T_{i1}, T_{i2}) > t_{i}$. Therefore,

$$\begin{aligned} P(t_{i} < \text {min}(T_{i1}, T_{i2})) = S(t_{i}, \beta _{12}, \alpha ) = e^{-\beta _{12}[h(x_{i})]^{\alpha }} \end{aligned}$$

For $(t_{i}, \delta _{i}, \mu _{i}) = (t_{i}, 3, 1)$ indicate that the ith unit is not truncated, but the failure of unit is due to an unknown cause, which implies that min$(T_{i1}, T_{i2}) = t_{i}$. Since min$(T_{i1}, T_{i2}) \sim \text {GLD}(\beta _{12}, \alpha )$. Therefore,

$$\begin{aligned} \begin{aligned} P(t_{i}< min(T_{i1}, T_{i2}) < t_{i} + dt) =&f(t_{i}, \beta _{12}, \alpha )dt_{i}= \alpha \beta _{12}h^{'}(x_{i})[h(x_{i})]^{\alpha -1}e^{-\beta _{12}[h(x_{i})]^{\alpha }} \end{aligned} \end{aligned}$$

For $(t_{i}, \delta _{i}, \mu _{i}) = (t_{i}, 0, 0)$, the ith unit is truncated and right censored at $t_{i}$. Therefore,

$$\begin{aligned} P(t_{i} < \text {min}(T_{i1}, T_{i2}) | T_{i1}> \tau _{iL}, T_{i2} > \tau _{iL}) = \frac{S(t_{i}, \beta _{12}, \alpha )}{S(\tau _{iL}, \beta _{12}, \alpha )} = \frac{e^{-\beta _{12}[h(x_{i})]^{\alpha }} }{e^{-\beta _{12}[h(\tau _{L}^{i})]^{\alpha }} } \end{aligned}$$

Likelihood contribution for $(t_{i}, \delta _{i}, \mu _{i}) = (t_{i}, 2, 1)$, $(t_{i}, \delta _{i}, \mu _{i}) = (t_{i}, 2, 0)$, $(t_{i}, \delta _{i}, \mu _{i}) = (t_{i}, 3, 0)$ can be similarly obtained when $(t_{i}, \delta _{i}, \mu _{i}) = (t_{i}, 1, 1)$, $(t_{i}, \delta _{i}, \mu _{i}) = (t_{i}, 1, 0)$, $(t_{i}, \delta _{i}, \mu _{i}) = (t_{i}, 3, 1)$ respectively.

B Proof of Theorem 2

From (9)

$$\begin{aligned} \begin{aligned} l(\beta _{1}, \beta _{2}, \alpha ) = n_{123}\ln \alpha + n_{1}\ln \beta _{1} + n_{2}\ln \beta _{2} +&n_{3}\ln \beta _{12} + \sum _{i\in I_{1}\cup I_{2}\cup I_{3}}\ln h^{'}(x_{i}) + \\&(\alpha -1)\sum _{i\in I_{1}\cup I_{2}\cup I_{3}}\ln h(x_{i}) - \beta _{12}\psi (\alpha ) \end{aligned} \end{aligned}$$

Taking partial derivative with respect to $\beta _{1}$,$\beta _{2}$ and equating to 0, we get

$$\begin{aligned} \frac{\partial l}{\partial \beta _{1}}=\frac{n_{1}}{\beta _{1}}+\frac{n_{3}}{(\beta _{1}+\beta _{2})}-\psi (\alpha )=0 \end{aligned}$$

and

$$\begin{aligned} \frac{\partial l}{\partial \beta _{2}}=\frac{n_{2}}{\beta _{2}}+\frac{n_{3}}{(\beta _{1}+\beta _{2})}-\psi (\alpha )=0 \end{aligned}$$

From the above two equations, we get

$$\begin{aligned} \hat{\beta _{j}}=\frac{n_{j}n_{123}}{n_{12}\psi (\alpha )} , j=1,2 \end{aligned}$$

Further, to verify that these values maximize the likelihood function, we find

$$\begin{aligned} \frac{\partial ^2 l}{\partial \beta _{1}^2}=-\frac{n_{1}}{\beta ^2}-\frac{n_{3}}{(\beta _{1}+\beta _{2})^2} \end{aligned}$$

$$\begin{aligned} \frac{\partial ^2 l}{\partial \beta _{2}^2}=-\frac{n_{2}}{\beta ^2}-\frac{n_{3}}{(\beta _{1}+\beta _{2})^2} \end{aligned}$$

$$\begin{aligned} \frac{\partial ^2 l}{\partial \beta _{2}\partial \beta _{1}}=-\frac{n_{3}}{(\beta _{1}+\beta _{2})^2} \end{aligned}$$

Thus, the Hessian matrix of second-order partial derivatives of $ln (\beta _{1},\beta _{2},\alpha )$ with respect to $\beta _{1}$,$\beta _{2}$ is

$$\begin{aligned} \begin{bmatrix} \frac{\partial ^2l}{\partial \beta _{1}^2} &{} \frac{\partial ^2l}{\partial \beta _{2}\partial \beta _{1}} \\ \frac{\partial ^2l}{\partial \beta _{1}\partial \beta _{2}} &{} \frac{\partial ^2l}{\partial \beta _{2}^2} \end{bmatrix} = \begin{bmatrix} -\frac{n_{1}}{\beta ^2}-\frac{n_{3}}{(\beta _{1}+\beta _{2})^2} &{} -\frac{n_{3}}{(\beta _{1}+\beta _{2})^2}\\ -\frac{n_{3}}{(\beta _{1}+\beta _{2})^2} &{} -\frac{n_{2}}{\beta ^2}-\frac{n_{3}}{(\beta _{1}+\beta _{2})^2} \end{bmatrix} \end{aligned}$$

Here the matrix is negative definite as it is symmetric and $(-1)^kD_{k}>0$.where $D_{k}$ is the leading principal minor of the Hessian matrix of order k, $1\le k \le 2$.This ensures that the likelihood function is maximized at $\hat{\beta _{j}}$.

C Proof of Theorem 3

From using (10), we can obtain

$$\begin{aligned} l^{'}_{1}(\alpha ) = \frac{n_{123}}{\alpha } + \sum _{i\in I_{1}\cup I_{2}\cup I_{3}}\ln h(x_{i}) - n_{123}\frac{\psi ^{'}(\alpha )}{\psi (\alpha )} = 0 \end{aligned}$$

and

$$\begin{aligned} l^{''}_{1}(\alpha ) = -\frac{n_{123}}{\alpha ^2} - n_{123}\bigg [\frac{\psi ^{''}(\alpha )\psi (\alpha ) - (\psi ^{'}(\alpha ))^2}{(\psi (\alpha ))^2}\bigg ] = 0 \end{aligned}$$

Next, one can obtain

$$\begin{aligned} \begin{aligned}&\psi ^{''}(\alpha )\psi (\alpha ) - (\psi ^{'}(\alpha ))^2 = \bigg \{\sum _{i=1}^{n}[h(x_{i})]^{\alpha } \big (\ln [h(x_{i})]\big )^2 - \sum _{i=1}^{n}(1-\mu _{i})[h(\tau _{iL})]^{\alpha }\big (\ln [h(\tau _{iL})]\big )^2\bigg \}\cdot \\ `&\bigg \{\sum _{i=1}^{n}[h(x_{i})]^{\alpha } - \sum _{i=1}^{n}(1-\mu _{i})[h(\tau _{iL})]^{\alpha } \bigg \} - \bigg \{\sum _{i=1}^{n}[h(x_{i})]^{\alpha } \ln [h(x_{i})] - \sum _{i=1}^{n}(1-\mu _{i})[h(\tau _{iL})]^{\alpha }\ln [h(\tau _{iL})] \bigg \}^{2} \\&\implies -\sum _{i=1}^{n}[h(\tau _{iL})]^{\alpha }[h(x_{i})]^{\alpha }\bigg (\ln [h(\tau _{iL})] - \ln [h(x_{i})]\bigg )^2 \le 0 \end{aligned} \end{aligned}$$

This implies that $l^{''}_{1}(\alpha ) < 0$. Thus, $l_{1}(\alpha )$ is a log-concave function. Further

$$\begin{aligned} \lim _{\alpha \rightarrow 0}l_{1}(\alpha ) = -\infty \quad \lim _{\alpha \rightarrow \infty }l_{1}(\alpha ) = -\infty \end{aligned}$$

This proves the $l_{1}(\alpha )$ is unimodal, and so unique MLE of $\alpha$ can be obtained by taking the derivates of $l_{1}(\alpha )$, which is given as

$$\begin{aligned} \frac{n_{123}}{\alpha } + \sum _{i\in I_{1}\cup I_{2}\cup I_{3}}\ln h(x_{i}) - n_{123}\frac{\psi ^{'}(\alpha )}{\psi (\alpha )} = 0 \end{aligned}$$

D Proof of Theorem 4

If $n_{1} \le n_{2}$, the MLEs of $\beta _{j},j=1,2$ obtained in Theorem 1 satisfies the order restriction condition, $\hat{\beta }_{1} \le \hat{\beta }_{2}$ for given $\alpha$. So for the given $\alpha$, MLEs of $\beta _{j}$ under the order restriction is given by

$$\begin{aligned} \tilde{\beta _{j}}= \frac{n_{j}n_{123}}{n_{12}\psi (\alpha )}; j=1,2,\quad \text {if}~ n_{1}< n_{2} \end{aligned}$$

Further, if $n_{1} \ge n_{2}$, MLEs obtained in Theorem 2 do not hold order restriction conditions. In this case, the log-likelihood function $l(\beta _{1},\beta _{2},\alpha )$ in (9) is maximized on $\beta _{1} = \beta _{2} = \beta$ under the restriction $\beta _{1} \le \beta _{2}$. Therefore, associated MLEs $\tilde{\beta }_{1}$ and $\tilde{\beta }_{2}$ can be obtained from

$$\begin{aligned} \max _{\beta } n_{123}\ln \alpha + n_{123}\ln \beta + \sum _{i\in I_{1}\cup I_{2}\cup I_{3}}\ln h^{'}(x_{i}) - 2\beta \psi (\alpha ). \end{aligned}$$

Take the partial derivative of the above equation w.r.to $\beta$ and equate it to 0; one gets

$$\begin{aligned} \tilde{\beta }_{1} = \tilde{\beta }_{2} = \frac{n_{123}}{2\psi (\alpha )}, ~\text {if}~ n_{1} \ge n_{2}. \end{aligned}$$

Therefore, the assertion is completed.