A Cure Rate Model for Exponentially Distributed Lifetimes with Competing Risks

Abstract

In real life, more often experimental units are susceptible to more than one risk factor. Moreover, some experimental units may not fail even if they are observed over a long period of time. In statistical analysis, competing risks models handle the first scenario while cure rate models have been introduced to analyze the long-term survivors in the population. In this paper, we consider a cure rate model when the failure of a unit can be due to either of the two competing causes. To analyze the competing risk data in presence of long-term survivors, we consider the latent failure times approach introduced by Cox (J R Stat Soc Ser B (Methodol) 21(2):411–421, 1959). The latent failure times are assumed to follow exponential distributions, and they are independently distributed. Under this setup, a random censoring scheme is applied and the observed data consist of either censored times or actual failure times along with the cause of failures. We derive the maximum likelihood estimators (MLEs) using the expectation-maximization (EM) algorithm based on the missing value principle. As the overall survival function is not a proper survival function, the asymptotic behavior of the MLEs is not immediate. We provide the sufficient conditions for the existence, uniqueness, consistency and the asymptotic normality of the MLEs. Monte Carlo simulations are performed to support the theoretical validation numerically. For illustrative purposes, we have analyzed one real dataset, and the results are quite satisfactory.

This is a preview of subscription content, access via your institution.

References

  1. 1.

    Balakrishnan N, Pal S (2013) COM Poisson cure rate models and associated likelihood based inference with exponential and Weibull lifetimes. In: Frenkel IB, Karagrigoriou A, Lisnianski A et al (eds) Probabilistic models and statistical inference, applied reliability engineering and risk analysis. Wiley, Chichester, pp 308–348

    Google Scholar 

  2. 2.

    Berkson J, Gage RP (1952) Survival curve for cancer patients following treatment. J Am Stat Assoc 47(259):501–515

    Article  Google Scholar 

  3. 3.

    Boag JW (1949) Maximum likelihood estimates of the proportion of patients cured by cancer therapy. J R Stat Soc Ser B (Methodol) 11(1):15–53

    MATH  Google Scholar 

  4. 4.

    Crowder M (2001) Classical competing risks model. Chapman & Hall/CRC, New York

    Google Scholar 

  5. 5.

    Cox DR (1959) The analysis of exponentially distributed lifetimes with two types of failures. J R Stat Soc Ser B (Methodol) 21(2):411–421

    MATH  Google Scholar 

  6. 6.

    Dunsmuir W, Tweedie R, Flack L, Mengersen K (1989) Modelling of transitions between employment states for young Australians. Aust N Z J Stat 31(1):165–196

    Article  Google Scholar 

  7. 7.

    Farewell VT (1982) The use of mixture models for the analysis of survival data with long-term survivors. Biometrics 38:1041–1046

    Article  Google Scholar 

  8. 8.

    Ghitany ME, Maller RA (1992) Asymptotic results for exponential mixture models with long-term survivors. Stat A J Theor Appl Stat 23(4):321–336

    MathSciNet  MATH  Google Scholar 

  9. 9.

    Ghitany ME, Maller RA, Zhou S (1994) Exponential mixture models with long-term survivors and covariates. J Multivar Anal 49(2):218–241

    MathSciNet  Article  Google Scholar 

  10. 10.

    Gallardo DI, Gómez YM, de Castro M (2018) A flexible cure rate model based on the polylogarithm distribution. J Stat Comput Simul 88(11):2137–2149

  11. 11.

    Goldman AI (1984) Survivorship analysis when cure is a possibility: a Monte Carlo study. Stat Med 3(2):153–163

    MathSciNet  Article  Google Scholar 

  12. 12.

    Greenhouse JB, Wolfe RA (1984) A competing risks derivation of a mixture model for the analysis of survival data. Commun Stat Theory Methods 13(25):3133–3154

    Article  Google Scholar 

  13. 13.

    Gulland JA (1955) On the estimation of population parameters from marked members. Biometrika 42(1–2):269–270

    Article  Google Scholar 

  14. 14.

    Kannan N, Kundu D, Nair P, Tripathi RC (2010) The generalized exponential cure rate model with covariates. J Appl Stat 37(10):1625–1636

    MathSciNet  Article  Google Scholar 

  15. 15.

    Kundu D (2004) Parameter estimation for partially complete time and type of failure data. Biom J J Math Methods Biosci 46(2):165–179

    MathSciNet  MATH  Google Scholar 

  16. 16.

    Maltz DM, McCleary R (1977) The mathematics of behavioral change: recidivism and construct validity. Eval Q 1:421–38

    Article  Google Scholar 

  17. 17.

    Maltz DM (1984) Recidivism. Academic Press, Florida

    Google Scholar 

  18. 18.

    Prentice RL, Kalbfleish JD, Peterson JAV, Flurnoy N, Farewell VT, Breslow NE (1978) The analysis of failure times in presence of competing risks. Biometrics 34:541–554

    Article  Google Scholar 

  19. 19.

    Self SG, Liang KL (1987) Asymptotic properties of maximum likelihood estimators and likelihood ratio tests under nonstandard conditions. J Am Stat Assoc 82:605–610

    MathSciNet  Article  Google Scholar 

  20. 20.

    Wiangnak P, Pal S (2018) Gamma lifetimes and associated inference for interval-censored cure rate model with COM Poisson competing cause. Commun Stat Theory Methods 47(6):1491–1509

    MathSciNet  Article  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the unknown reviewers for their constructive comments which have helped to improve the manuscript significantly.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Debasis Kundu.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix

Appendix

In the Appendix, we provide the proofs of Theorems 4.14.2 and 4.3. To prove these Theorems, we need a series of lemmas and some preliminaries. We provide a flowchart to show how these Lemmas have been used to prove the Theorems.

Result Prerequisite results
Fisher information matrix Lemmas 6.1 and 6.2.
Lemma 6.3 Fisher information matrix
Lemma 6.6 Lemmas 6.4 and 6.5
Lemma 6.7 Lemma 6.6
Lemma 6.8 Lemma 6.7
Lemma 6.10 Lemmas 6.6 and 6.9
Theorem 4.1 Lemmas 6.3 and 6.7
Theorem 4.2 Lemmas 6.3 and 6.8
Theorem 4.3 Lemmas 6.3 and 6.10

The following notations have been used throughout this section.

Let \(\{ X_n \}\) be a sequence of random variables.

  • \(X_n=o_p(1)\) implies \(\lim \limits _{n \rightarrow \infty } P(|X_n|>\epsilon )=0\) \(\forall \epsilon >0.\)

  • \(X_n=O_p(1)\) implies for any \(\epsilon > 0,\) there exists a finite \(M > 0\) and a finite \(N > 0\) such that \(P ( | X_n |> M ) < \epsilon , \ \forall n > N\).

  • \(\lambda _{\mathrm{{min}}}\{\mathbf {A}\}\) is the minimum eigen value of the matrix \(\mathbf {A}.\)

  • The observed Fisher information matrix is

    $$\begin{aligned} {\mathbf {F}}_n(\varvec{\theta })=-\dfrac{\partial {\mathbf {S}}_n (\varvec{\theta })}{\partial \varvec{\theta } }, \end{aligned}$$
    (11)

    where \({\mathbf {S}}_n (\varvec{\theta })\) is defined in (2).

  • The expected Fisher information matrix about \(\varvec{\theta _0}\) is

    $$\begin{aligned} {\mathbf {D}}_n(\varvec{\theta _0}) =E[{\mathbf {F}}_n(\varvec{\theta _0})]=E[{\mathbf {S}}_n(\varvec{\theta _0}){\mathbf {S}}_n(\varvec{\theta _0})^{\top }]=Cov({\mathbf {S}}_n(\varvec{\theta _0})). \end{aligned}$$
    (12)
  • We define the region \(\mathbf {N}_n(\varvec{\theta _0})\) for fixed \(A \ge 1,\) as

    $$\begin{aligned} \mathbf {N}_n(\varvec{\theta _0})=\{ \varvec{\theta } :\varvec{ \theta } \in \Theta ,\,\, (\varvec{\theta } - \varvec{\theta _0} )^{\top } {\mathbf {D}}_n(\varvec{\theta }_o) (\varvec{\theta } - \varvec{\theta _0} ) \le A^2 \}. \end{aligned}$$
    (13)

Elements of Observed Information Matrix

Note that the observed information matrix as defined in (11) can be written as

$$\begin{aligned} {\mathbf {F}}_n(\varvec{\theta })=\sum _{i=1}^n \mathbf {{\mathscr {F}}}_{i}(\varvec{\theta }) \end{aligned}$$

where

$$\begin{aligned} \mathbf {{\mathscr {F}}}_{i}(\varvec{\theta })= \begin{bmatrix} f_{i}^{11}(\varvec{\theta }) &{}\quad f_{i}^{12}(\varvec{\theta }) &{}\quad f_{i}^{13}(\varvec{\theta })\\ f_{i}^{21}(\varvec{\theta }) &{}\quad f_{i}^{22}(\varvec{\theta }) &{}\quad f_{i}^{23}(\varvec{\theta })\\ f_{i}^{31}(\varvec{\theta }) &{}\quad f_{i}^{32}(\varvec{\theta }) &{}\quad f_{i}^{33}(\varvec{\theta })\\ \end{bmatrix}. \end{aligned}$$
(14)

Here,

$$\begin{aligned} f_{i}^{11}(\varvec{\theta })&= \frac{z_{i}(2-\delta _{i})}{\lambda _1^2} - \frac{(1-z_{i})pt_{i}^2(1-p){\mathrm{{e}}}^{-(\lambda _{1}+\lambda _{2})t_{i}}}{(1-p+p{\mathrm{{e}}}^{-(\lambda _{1}+\lambda _{2})t_{i}})^2}, \end{aligned}$$
(15)
$$\begin{aligned} f_{i}^{22}(\varvec{\theta })&= \frac{z_{i}(\delta _{i}-1)}{\lambda _2^2} - \frac{(1-z_{i})pt_{i}^2(1-p){\mathrm{{e}}}^{-(\lambda _{1}+\lambda _{2})t_{i}}}{(1-p+p{\mathrm{{e}}}^{-(\lambda _{1}+\lambda _{2})t_{i}})^2}, \end{aligned}$$
(16)
$$\begin{aligned} f_{i}^{33}(\varvec{\theta })&= \frac{z_{i}}{p^2} + \frac{(1-z_{i})(1-{\mathrm{{e}}}^{-(\lambda _{1}+\lambda _{2})t_{i}})^2}{(1-p+p{\mathrm{{e}}}^{-(\lambda _{1}+\lambda _{2})t_{i}})^2}, \end{aligned}$$
(17)
$$\begin{aligned} f_{i}^{12}(\varvec{\theta })&= f_{i}^{12}(\varvec{\theta }) = - \frac{(1-z_{i})pt_{i}^2(1-p){\mathrm{{e}}}^{-(\lambda _{1}+\lambda _{2})t_{i}}}{(1-p+p{\mathrm{{e}}}^{-(\lambda _{1}+\lambda _{2})t_{i}})^2}, \end{aligned}$$
(18)
$$\begin{aligned} f_{i}^{13}(\mathbf {\theta })&= f_{i}^{31}(\varvec{\theta }) = f_{i}^{23}({\theta }) = f_{i}^{32}(\varvec{\theta }) = \frac{(1-z_{i})t_{i}{\mathrm{{e}}}^{-(\lambda _{1}+\lambda _{2})t_{i}}}{(1-p+p{\mathrm{{e}}}^{-(\lambda _{1}+\lambda _{2})t_{i}})^2}. \end{aligned}$$
(19)

Fisher Information Matrix

Note that we can write the Fisher information matrix as defined in (12), as

$$\begin{aligned} {\mathbf {D}}_n(\varvec{\theta }_o) =\sum _{i=1}^n \mathbf {{\mathscr {D}}}_{i}(\varvec{\theta }_o), \end{aligned}$$
(20)

where \( \mathbf {{\mathscr {D}}}_{i}(\varvec{\theta }_o) = \begin{bmatrix} d_{i}^{11}(\varvec{\theta }_o) &{} d_{i}^{12}(\varvec{\theta }_o)&{} d_{i}^{13}(\varvec{\theta }_o)\\ d_{i}^{21}(\varvec{\theta }_o) &{} d_{i}^{22}(\varvec{\theta }_o) &{} d_{i}^{23}(\varvec{\theta }_o)\\ d_{i}^{31}(\varvec{\theta }_o) &{} d_{i}^{32}(\varvec{\theta }_o) &{} d_{i}^{33}(\varvec{\theta }_o)\\ \end{bmatrix} \) and \( d_i^{kl}(\varvec{\theta }_o)=E[f_i^{kl}(\varvec{\theta _0})],\, k,l=1,2,3.\) The expressions for the elements of \(\mathbf {{\mathscr {D}}}_{i}(\varvec{\theta _0}) \) can be obtained using Lemma 6.1.

Lemma 6.1

Let R(.) be a non-negative measurable function on \(\big [0, \infty \big )\). Then, for all \(1\le i\le n,\, \lambda _{10}>0, \, \lambda _{20}>0\) and \(p_0 \in (0,1 \big ],\)

$$\begin{aligned} E\big [(1-Z_i) R(T_i)\big ]= E\big [\{1-p_0 +p_0{\mathrm{{e}}}^{-(\lambda _{10}+\lambda _{20})C_i}\}R(C_i)\big ] \end{aligned}$$

provided the expectation on the right exists.

Proof

For any \(u>0\),

$$\begin{aligned} P\{(1-Z_{i})R(T_{i})>u\}& = p_{o}P\{(1-Z_{i})R(T_{i})>u,Z_{i}=0|Y_{i}=1\}\\&+ (1-p_{o})P\{(1-Z_{i})R(T_{i})>u,Z_{i}=0|Y_{i}=0\}. \end{aligned}$$

Since \(Y_{i}\) is independent of \(C_{i},\) we have

$$\begin{aligned} P\{(1-Z_{i})R(T_{i})>u\}& = p_{o}P\{R(C_{i})>u,T_{i}^{*}>C_{i}|Y_{i}=1\}+(1-p_{o})P\{R(C_{i})>u\}\\& = \displaystyle \int _{c:R(c)>u}^{}\Big [p_{o}P\{T_{i}^{*}>c|Y_{i}=1\}+1-p_{o}\Big ]{\mathrm{d}}P\{C_{i}\le c\}\\& = \displaystyle \int _{c:R(c)>u}^{}(1-p_{o}+p_{o}{\mathrm{{e}}}^{-(\lambda _{10}+\lambda _{20})c})\,{\mathrm{d}}P\{C_{i}\le c\}. \end{aligned}$$

Hence,

$$\begin{aligned} E\{(1-Z_{i})R(T_{i})\}& = \displaystyle \int _{0}^{\infty }P\{(1-Z_{i})R(T_{i})>u\}{\mathrm{d}}u \\& = \displaystyle \int _{0}^{\infty }\displaystyle \int _{c:R(c)>u}^{}(1-p_{o}+p_{o}{\mathrm{{e}}}^{-(\lambda _{10}+\lambda _{20})c})\,{\mathrm{d}}P\{C_{i}\le c\}\,{\mathrm{d}}u. \end{aligned}$$

By Fubini’s theorem, we have

$$\begin{aligned} E\{(1-Z_{i})R(T_{i})\}& = \displaystyle \int _{0}^{\infty }\displaystyle \int _{c:R(c)>u}^{}du(1-p_{o}+p_{o}{\mathrm{{e}}}^{-(\lambda _{10}+\lambda _{20})c})\,{\mathrm{d}}P\{C_{i}\le c\}\\& = E\{(1-p_{o}+p_{o}{\mathrm{{e}}}^{-(\lambda _{10}+\lambda _{20})C_{i}})R(C_{i})\}, \end{aligned}$$

provided the expectation on the right-side exists. \(\square \)

The following Lemma 6.2 summarizes the expressions for the elements of \(\mathbf {{\mathscr {D}}}_{i}(\varvec{\theta _0}) \).

Lemma 6.2

For all \(1\le i\le n,\, \lambda _{10}>0, \, \lambda _{20}>0\) and \(p_0 \in (0,1 ],\) the following are finite:

$$\begin{aligned} d_{i}^{11}(\varvec{\theta }_o)&=p_0E\left[ \frac{1-{\mathrm{{e}}}^{-(\lambda _{10}+\lambda _{20})C_i}}{\lambda _{10}(\lambda _{10}+\lambda _{20})} - \frac{C_i^2(1-p_0){\mathrm{{e}}}^{-(\lambda _{10}+\lambda _{20})C_i}}{1-p_0+p_0{\mathrm{{e}}}^{-(\lambda _{10}+\lambda _{20})C_i}}\right], \end{aligned}$$
(21)
$$\begin{aligned} d_{i}^{22}(\varvec{\theta }_o)&=p_0E\left[ \frac{1-{\mathrm{{e}}}^{-(\lambda _{10}+\lambda _{20})C_i}}{\lambda _{20}(\lambda _{10}+\lambda _{20})} - \frac{C_i^2(1-p_0){\mathrm{{e}}}^{-(\lambda _{10}+\lambda _{20})C_i}}{1-p_0+p_0{\mathrm{{e}}}^{-(\lambda _{10}+\lambda _{20})C_i}}\right], \end{aligned}$$
(22)
$$\begin{aligned} d_{i}^{33}(\varvec{\theta }_o)&=\frac{1}{p_0}E\left[ \frac{1-{\mathrm{{e}}}^{-(\lambda _{10}+\lambda _{20})C_i}}{1-p_0+p_0{\mathrm{{e}}}^{-(\lambda _{10}+\lambda _{20})C_i}} \right], \end{aligned}$$
(23)
$$\begin{aligned} d_{i}^{12}(\varvec{\theta }_o)&=d_{i}^{21}(\varvec{\theta }_o)=-\frac{1}{\lambda _{10}+\lambda _{20}}E\left[ \frac{p_0(1-p_0)C_i^2{\mathrm{{e}}}^{-(\lambda _{10}+\lambda _{20})C_i}}{\{1-p_0+p_0{\mathrm{{e}}}^{-(\lambda _{10}+\lambda _{20})C_i}\}^2} \right], \end{aligned}$$
(24)
$$\begin{aligned} d_{i}^{13}(\varvec{\theta }_o)&= d_{i}^{31}(\varvec{\theta }_o) = d_{i}^{23}(\varvec{\theta }_o) = d_{i}^{32}(\varvec{\theta }_o) = \frac{1}{\lambda _{10}+\lambda _{20}}\nonumber \\&\quad E\Big [\frac{(\lambda _{10}+\lambda _{20})C_i{\mathrm{{e}}}^{-(\lambda _{10}+\lambda _{20})C_i}}{1-p+p{\mathrm{{e}}}^{-(\lambda _{1}+\lambda _{2})C_{i}}}\Big ]. \end{aligned}$$
(25)

Proof

The proof follows easily applying Lemma 6.1. \(\square \)

Remarks

Type-I censoring (termination of the experiment at time \(\tau >0\)) turns out to be a particular case when \(P(C_i=\tau )=1 ,\, i=1,2,\ldots ,n.\) Then, all the expectations will exist since

$$\begin{aligned} E\big [(1-Z_i) R(T_i)\big ]&= E\big [\{1-p_0 +p_0{\mathrm{{e}}}^{-(\lambda _{10}+\lambda _{20})C_i}\}R(C_i)\big ]\\& = 1-p_0 +p_0{\mathrm{{e}}}^{-(\lambda _{10}+\lambda _{20})\tau }R(\tau ) \end{aligned}$$

will always exist.

Lemma 6.3

If the censored times \(C_i\)’s for \(i=1,\ldots ,n,\) are iid random variables whose common distribution does not degenerate at origin, then

$$\begin{aligned} \liminf \limits _{n\rightarrow \infty }\,\frac{1}{n} \lambda _{\mathrm{min}}\{{\mathbf {D}}_n(\varvec{\theta _0})\} >0 \end{aligned}$$
(26)

will hold, i.e., there exists some \(\eta >0\) such that \(\lambda _{\mathrm{min}}\{{\mathbf {D}}_n(\varvec{\theta _0})\} \ge n\eta \) for n large enough.

Proof

When \(C_i\)’s are iid random variables, \(\lambda _{\mathrm{min}}\{ {\mathbf {D}}_n (\varvec{\theta _0})\} = n \lambda _{\mathrm{min}} \{ \mathbf {{\mathscr {D}}}_{1}(\varvec{\theta _0}) \}.\) Therefore, it is enough to show that \( \mathbf {{\mathscr {D}}}_{1}(\varvec{\theta _0}) \) is positive definite.

From (21), (22), (23), it is clear that \(d^{11}_1(\varvec{\theta _0})\), \(d^{22}_1(\varvec{\theta _0})\) and \(d^{33}_1(\varvec{\theta _0})\) all are 0 if and only if \(C_1=0\). This is true, since, for \(\lambda _{10}>0\), \(\lambda _{20}>0,\) \(p_0 \in (0,1),\) and \(C\ge 0\), the following functions of C defined as

$$\begin{aligned} g_1 (C)&= 1-{\mathrm{{e}}}^{-(\lambda _{10}+\lambda _{20})C}-\frac{C^2 (1-p_0){\mathrm{{e}}}^{-(\lambda _{10}+\lambda _{20})C} \lambda _{10}(\lambda _{10}+\lambda _{20})}{1-p_0+p_0{\mathrm{{e}}}^{-(\lambda _{10}+\lambda _{20})C}}, \end{aligned}$$
(27)
$$\begin{aligned} g_2 (C)&= 1-{\mathrm{{e}}}^{-(\lambda _{10}+\lambda _{20})C}-\frac{C^2 (1-p_0){\mathrm{{e}}}^{-(\lambda _{10}+\lambda _{20})C} \lambda _{20}(\lambda _{10}+\lambda _{20})}{1-p_0+p_0{\mathrm{{e}}}^{-(\lambda _{10}+\lambda _{20})C}}, \end{aligned}$$
(28)
$$\begin{aligned} g_3 (C)&= \frac{1-{\mathrm{{e}}}^{-(\lambda _{10}+\lambda _{20})C}}{1-p_0+p_0{\mathrm{{e}}}^{-(\lambda _{10}+\lambda _{20})C}}. \end{aligned}$$
(29)

are non-negative, increasing, see for example Ghitany and Maller [8]) and at \(C=0,\)

\(g_{i}(C)=0,\,i=1,2,3\). Thus, from (12), we have

$$\begin{aligned} Var(s_{11}(\varvec{\theta _0}) )= d^{11}_1(\varvec{\theta _0})>0, Var(s_{12}(\varvec{\theta _0}) )= d^{11}_2(\varvec{\theta _0})>0, Var(s_{13}(\varvec{\theta _0}) )= d^{11}_3(\varvec{\theta _0})>0, \end{aligned}$$

where \(s_{ij}(\varvec{\theta _0})'s\) are defined in (3), (4) and (5). We will prove the result by contradiction. Let us first assume that the determinant of \( \mathbf {{\mathscr {D}}}_1(\varvec{\theta _0})\) is 0. Then, \(s_{11}(\varvec{\theta _0}),\) \(s_{12}(\varvec{\theta _0})\) and \(s_{13}(\varvec{\theta _0})\) are almost surely linearly related. We can write

$$\begin{aligned} a_1 s_{11}(\varvec{\theta _0} ) + a_2 s_{12}( \varvec{\theta _0} ) + a_3 s_{13}(\varvec{\theta _0})=0 \end{aligned}$$
(30)

for some \(a_1,a_2, a_3.\) When \(C_1\ne 0\) almost surely, then \(P(Z_1=1 )>0\) and \(P(\Delta _1=1) >0.\) Now plugging in \(z_i=1\) and \(\delta _i=1\) in (30), we obtain \(t^{*}_{11}(a_1+a_2)=\frac{a_1}{\lambda _{10}}+\frac{a_3}{p_0},\) almost surely on a set, say \({\mathcal {B}}\), with positive probability. This implies that on \({\mathcal {B}}\), \(t^{*}_{11}\) is degenerate which is impossible, since on \({\mathcal {B}}\), \(t^{*}_{11}\) is exponential. Therefore, \(\mathbf {{\mathscr {D}}}_1 \) is non-singular, hence positive definite. \(\square \)

Lemma 6.4

For \(\lambda _{10}, \lambda _{20} >0\) and \(p_0 \in (0,1)\), \( | f^{rs}_i( \varvec{\theta _0} ) | <M(\varvec{\theta _0} )\) for all \(i=1, \ldots ,n\) and \(r,s=1,2,3\) where \(f^{rs}_i( \varvec{\theta } )'s\) are defined in (15), (16), (17), (18), and (19) and \(M(\varvec{\theta _0} )\) is a positive finite quantity depending on \(\varvec{\theta _0}\).

Proof

At \(\varvec{\theta }= \varvec{\theta _0}\), \(f^{rs}_i(\varvec{\theta _0})\)’s are continuous function of \(t_i\) for \(r,s=1,2,3.\) Again for all \(i=1, \ldots ,n\),

$$\begin{aligned} \lim _{t_i \rightarrow 0} f^{11}_i(\varvec{\theta _0})& = \frac{z_i(2-\delta _i)}{\lambda ^2_{1o} }; \lim _{t_i \rightarrow 0} f^{22}_i(\varvec{\theta _0} ) = \frac{z_i(\delta _i-1)}{\lambda ^2_{2o} }; \\ \lim _{t_i \rightarrow 0} f^{33}_i( \varvec{\theta _0} )& = \frac{z_i}{p_0} ; \lim _{t_i \rightarrow 0} f^{12}_i( \varvec{\theta _0} )= 0 ;\\ \lim _{t_i \rightarrow 0} f^{13}_i( \varvec{\theta _0} )& = 0 \quad \text {and}\\ \lim _{t_i \rightarrow \infty } f^{11}_i(\varvec{\theta _0} )& = \frac{z_i(2-\delta _i)}{\lambda ^2_{1o} } ; \lim _{t_i \rightarrow \infty } f^{22}_i(\varvec{\theta _0} ) = \frac{z_i(\delta _i-1)}{\lambda ^2_{2o} } ;\\ \lim _{t_i \rightarrow \infty } f^{33}_i( \varvec{\theta _0} )& = \frac{z_i}{p_0} + \frac{(1-z_i)}{(1-p_0)}; \\ \lim _{t_i \rightarrow \infty } f^{12}_i(\varvec{\theta _0})& = 0; \lim _{t_i \rightarrow \infty } f^{13}_i(\varvec{\theta _0} ) = 0. \end{aligned}$$

\(z_i\) and \(\delta _i\) being binary variables all the \(f^{rs}_i(\varvec{\theta _0})\)’s have finite limit at 0 and \(\infty \). It implies that \( |f^{rs}_i(\varvec{\theta _0} )| < M^{rs}(\varvec{\theta _0} )\) where \( M^{rs}(\varvec{\theta _0})\) is a positive quantity for \(r,s=1,2,3.\)

Let \(M(\varvec{\theta _0}) =\max _{\begin{array}{c} r,s \end{array}} M^{rs}(\varvec{\theta _0})\). Therefore, \( |f^{rs}_i(\varvec{\theta _0})| < M( \varvec{\theta _0} )\) for all \(r,s=1,2,3\) and \(i = 1, \ldots ,n.\) \(\square \)

Lemma 6.5

For any \(\varvec{\theta } \in \mathbf {N}_n(\varvec{\theta _0})\) and (26) holds, \( | w^{rs}_{ij}(\varvec{\theta }) |<M_1(\varvec{\theta }) <M_2(\varvec{\theta _0}) \) for all \(i=1, \ldots ,n\) and \(j, r,s=1,2,3,\) where, \(w_i^{rs}(\varvec{\theta })= \frac{\partial f_i^{rs}(\varvec{\theta } )}{\partial \varvec{\theta } }=( w_{i1}^{rs}( \varvec{\theta }), w_{i2}^{rs}( \varvec{\theta }), w_{i3}^{rs}( \varvec{\theta }) ).\) Here, \(M_1(\varvec{\theta })\) is a positive finite quantity depending on \(\varvec{\theta }\) and \(M_2( \varvec{\theta _0} )\) is a positive finite quantity depending on \(\varvec{\theta _0} \).

Proof

For fixed \(\varvec{\theta }\), and for all \(i=1, \ldots ,n\), \(j=1,2,3\), each of \(w^{rs}_{ij}(\varvec{\theta })\) is a continuous function of \(t_i\) with finite limit at 0 and \(\infty .\) Hence, for each \(j,r,s=1,2,3\), there exists a positive quantity \(M^{rs}_{j} (\varvec{\theta })\) such that \( | w^{rs}_{ij}(\varvec{\theta } )| \le M^{rs}_{j} (\varvec{\theta })\). For \(\lambda _{10}, \lambda _{20} >0\) and \(p_0 \in (0,1)\), each \(M^{rs}_{j} (\varvec{\theta })\) is continuous function of \(\varvec{\theta }\). Again when \(\varvec{\theta } \in \mathbf {N}_n(\varvec{\theta _0})\), using (26) we obtain

$$\begin{aligned} {| \varvec{\theta } - \varvec{\theta _0} |}^2 \le \frac{A^2}{ \lambda _{\mathrm{min}}\{{\mathbf {D}}_n(\varvec{\theta }_0)\} } < \frac{A^2}{ n \eta } \rightarrow 0 \quad \text {as} \quad n \rightarrow \infty , \end{aligned}$$

where \(\eta >0\) is a suitable constant and this implies that for each \(j,r,s=1,2,3\) there exist a quantity \(N^{rs}_{j} ( \varvec{\theta _0})\) such that \(M^{rs}_{j} (\varvec{\theta }) \le N^{rs}_{j} (\varvec{\theta _0} )\). Let \(\max _{\begin{array}{c} j,r,s \end{array}} M^{rs}_{j}( \varvec{\theta } )=M_1( \varvec{\theta } )\) and \(\max _{\begin{array}{c} j,r,s \end{array}} N^{rs}_{j}(\varvec{\theta _0} )=M_2( \varvec{\theta _0} )\). Therefore, for each \(i=1,\ldots ,n\) and for each \(j,r,s=1,2,3,\) \(| w^{rs}_{ij}( \varvec{\theta } ) |< M_1(\varvec{\theta } ) < M_2( \varvec{\theta _0})\). \(\square \)

Lemma 6.6

For \(\lambda _{10}, \lambda _{20} >0\) and \(p_0 \in (0,1),\) if the condition (26) holds, \({\mathbf {D}}_n(\varvec{\theta _0})\) is a positive definite matrix and

$$\begin{aligned} \sup \limits _{\begin{array}{c} \theta \in \mathbf {N}_n(\varvec{\theta _0}) \end{array}} \Vert {\mathbf {D}}^{-\frac{1}{2}}_n(\varvec{\theta _0}) {\mathbf {F}}_n( \varvec{\theta }) {\mathbf {D}}^{-\frac{1}{2}}_n(\varvec{\theta _0}) - \mathbf {I} \Vert \rightarrow 0 \ \text {as} \ n \rightarrow \infty \ \text { in probability, } \end{aligned}$$
(31)

will hold along with \(\lambda _{\mathrm{min}} \{{\mathbf {D}}_n(\varvec{\theta _0})\} \rightarrow \infty \) as \(n \rightarrow \infty \).

Proof

Condition (26) implies that there exist a \(\eta >0\) such that \(\lambda _{\mathrm{min}}\{{\mathbf {D}}_n(\varvec{\theta _0})\} \ge n\eta \) for n large enough. Therefore, all the eigenvalues of \({\mathbf {D}}_n(\varvec{\theta _0})\) are positive for large n which implies \({\mathbf {D}}_n(\varvec{\theta _0})\) is a positive definite matrix for large n. Now, we can define \({\mathbf {D}}^{-\frac{1}{2}}_n(\varvec{\theta _0})\), the symmetric square root of \({\mathbf {D}}^{-1}_n(\varvec{\theta _0}).\)

Now to show that (31) will hold, we decompose \({\mathbf {D}}^{-\frac{1}{2}}_n(\varvec{\theta _0}) {\mathbf {F}}_n( \varvec{\theta }) {\mathbf {D}}^{-\frac{1}{2}}_n(\varvec{\theta _0})\) as

$$\begin{aligned} {\mathbf {D}}^{-\frac{1}{2}}_n(\varvec{\theta _0}) {\mathbf {F}}_n( \varvec{\theta }) {\mathbf {D}}^{-\frac{1}{2}}_n(\varvec{\theta _0})& = \mathbf {I} + {\mathbf {D}}^{-\frac{1}{2}}_n(\varvec{\theta _0})\{{\mathbf {F}}_n( \varvec{\theta _0} )-{\mathbf {D}}_n(\varvec{\theta _0}) \} {\mathbf {D}}^{-\frac{1}{2}}_n(\varvec{\theta _0}) \\&+ {\mathbf {D}}^{-\frac{1}{2}}_n (\varvec{\theta _0})\{{\mathbf {F}}_n( \varvec{\theta } ) - {\mathbf {F}}_n( \varvec{\theta _0} ) \} {\mathbf {D}}^{-\frac{1}{2}}_n(\varvec{\theta _0}). \end{aligned}$$

Hence, it is sufficient to prove that for any unit vector \({\mathbf {u}}\) and for any \(\varvec{\theta } \in \mathbf {N}_n(\varvec{\theta _0})\) ,

$$\begin{aligned} e_n^{(1)}( \varvec{\theta _0} )& = {\mathbf {u}}^{\top }{\mathbf {D}}^{-\frac{1}{2}}_n(\varvec{\theta _0}) \{{\mathbf {F}}_n( \varvec{\theta _0} )-{\mathbf {D}}_n(\varvec{\theta _0}) \} {\mathbf {D}}^{-\frac{1}{2}}_n(\varvec{\theta _0}) {\mathbf {u}} = o_{p}(1), \end{aligned}$$
(32)
$$\begin{aligned} e_n^{(2)}( \varvec{\theta } )& = {\mathbf {u}}^{\top } {\mathbf {D}}^{-\frac{1}{2}}_n(\varvec{\theta _0}) \{{\mathbf {F}}_n( \varvec{\theta } ) - {\mathbf {F}}_n( \varvec{\theta _0} ) \} {\mathbf {D}}^{-\frac{1}{2}}_n(\varvec{\theta _0}) {\mathbf {u}} = o_{p}(1). \end{aligned}$$
(33)

To show \(e_n^{(1)}( \varvec{\theta _0}) = o_{p}(1)\), we need the results of Lemma 6.4. Now, \(f^{rs}_i(\varvec{\theta _0}) \) is a sequence of independent random variables and based on Lemma 6.4, we have \( | f^{rs}_i(\varvec{\theta _0}) | <M(\varvec{\theta _0}),\) \(\,r,s=1,2,3.\) Hence,

$$\begin{aligned} \frac{1}{n^2}\displaystyle \sum _{i=1}^{n} {\mathrm{Var}}\{ f^{rs}_i( \varvec{\theta _0}) \} \le \frac{M^2( \varvec{\theta _0})}{n} \rightarrow 0\,\,\,\,\,\text { as }\,\, n \rightarrow \infty . \end{aligned}$$

Therefore, by the weak law of large numbers, for all \(r,s=1,2,3,\)

$$\begin{aligned} \frac{1}{n}\sum _{i=1}^{n}\{f^{rs}_i(\varvec{\theta _0}) -d^{rs}_i( \varvec{\theta _0})\}= o_{p}(1). \end{aligned}$$

For any unit vector \({\mathbf {u}}\), let \({\mathbf {u}}_n = \frac{ {\mathbf {D}}^{-\frac{1}{2}}_n( \varvec{\theta _0}) {\mathbf {u}} }{\sqrt{{\mathbf {u}}^{\top }{\mathbf {D}}^{-1}_n( \varvec{\theta _0}) {\mathbf {u}} } }\). Then, \({\mathbf {u}}_n \) is also a unit vector and hence

$$\begin{aligned} | {\mathbf {u}}^{\top }_n M_n {\mathbf {u}}_n | \le \sum \limits _{\begin{array}{c} r,s=1,2,3 \end{array}} | m_{n}^{rs} | \end{aligned}$$

for any matrix \(M_n=(m_{n}^{rs}), \, r,s=1,2,3\). Thus, we have

$$\begin{aligned} | {\mathbf {u}}^{\top }_n\{ {\mathbf {F}}_n( \varvec{\theta _0})-{\mathbf {D}}_n(\varvec{\theta _0}) \} {\mathbf {u}}_n | \le \sum \limits _{\begin{array}{c} r,s=1,2,3 \end{array}} \left| \sum _{i=1}^{n}\{f^{rs}_i(\varvec{\theta _0}) -d^{rs}_i(\varvec{\theta _0} )\} \right| . \end{aligned}$$

Using the result in (26), we have for \(\eta >0\), \({\mathbf {u}}^{\top } {\mathbf {D}}^{-1}_n(\varvec{\theta _0}) {\mathbf {u}} \le \lambda _{max}\{{\mathbf {D}}^{-1}_n(\varvec{\theta _0}) \} \le \frac{1}{n\eta }.\) Then,

$$\begin{aligned} | e_n^{(1)}(\varvec{\theta _0}|& = | {\mathbf {u}}^{\top }_n({\mathbf {F}}_n( \varvec{\theta _0})-{\mathbf {D}}_n(\varvec{\theta _0}) ) {\mathbf {u}}_n| | {\mathbf {u}}^{\top } {\mathbf {D}}^{-1}_n (\varvec{\theta _0}) {\mathbf {u}}| \\& \le \frac{1}{n \eta } \sum \limits _{\begin{array}{c} r,s=1,2,3 \end{array}} \left| \sum _{i=1}^{n}(f^{rs}_i(\varvec{\theta _0}) -d^{rs}_i(\varvec{\theta _0})) \right| \\& =o_p(1). \end{aligned}$$

To show \(e_n^{(2)}(\varvec{\theta }) \overset{P}{\rightarrow } 0 \), we need the following Taylor series expansion of \(f_i^{rs}(\varvec{\theta } )\) about \(\varvec{\theta _0}. \)

$$\begin{aligned} f_i^{rs}(\varvec{\theta } ) - f_i^{rs}( \varvec{\theta _0} ) = ( \varvec{\theta } - \varvec{\theta _0} ) ^{\top } w_i^{rs}( \varvec{{\tilde{\theta }}}), \end{aligned}$$

where \(\varvec{{\tilde{\theta }}}\) is any point between \(\varvec{\theta }\) and \(\varvec{\theta _0}\), \(w_i^{rs}(\varvec{\theta })= \frac{\partial f_i^{rs}(\varvec{\theta } )}{\partial \varvec{\theta } }=( w_{i1}^{rs}( \varvec{\theta }), w_{i2}^{rs}( \varvec{\theta }), w_{i3}^{rs}( \varvec{\theta }) ),\ \,i=1,2,\ldots ,n,\,r,s=1,2,3.\) For fixed (rs),  \(w_i^{rs}(\varvec{\theta })\) is a sequence of independent random vectors. The expressions of the elements of \(w_i^{rs}(\varvec{\theta })\) are as follows.

$$\begin{aligned} w_{i1}^{11}(\varvec{\theta })& = -\frac{2}{\lambda _1^3}\Big \{z_i (2-\delta _i)-\frac{(1-z_i)p(1-p)(1-p-p{\mathrm{{e}}}^{-(\lambda _1+\lambda _2)t_i}\lambda _1^3 t_i^3 e{-(\lambda _1+\lambda _2)t_i}}{2(1-p+p{\mathrm{{e}}}^{-(\lambda _1+\lambda _2)t_i})^3}\Big \} ,\\ w_{i2}^{11}(\varvec{\theta })& = \frac{(1-z_i)p(1-p)t_i^3(1-p-p{\mathrm{{e}}}^{-(\lambda _1+\lambda _2)t_i} {\mathrm{{e}}}^{-(\lambda _1+\lambda _2)t_i}}{(1-p+p{\mathrm{{e}}}^{-(\lambda _1+\lambda _2)t_i})^3},\\ w_{i3}^{11}(\varvec{\theta })& = -\frac{(1-z_i)t_i^2{\mathrm{{e}}}^{-(\lambda _1+\lambda _2)t_i}(1-p-p{\mathrm{{e}}}^{-(\lambda _1+\lambda _2)t_i})}{(1-p+p{\mathrm{{e}}}^{-(\lambda _1+\lambda _2)t_i})^3},\\ w_{i2}^{22}(\varvec{\theta })& = -\frac{2}{\lambda _2^3}\Big \{z_i(\delta _i-1)-\frac{(1-z_i)p(1-p)(1-p-p{\mathrm{{e}}}^{-(\lambda _1+\lambda _2)t_i}\lambda _2^3 t_i^3 {\mathrm{{e}}}^{-(\lambda _1+\lambda _2)t_i}}{2(1-p+p{\mathrm{{e}}}^{-(\lambda _1+\lambda _2)t_i})^3}\Big \},\\ w_{i3}^{13}(\varvec{\theta })& = \frac{2(1-z_i)t_i{\mathrm{{e}}}^{-(\lambda _1+\lambda _2)t_i}(1-{\mathrm{{e}}}^{-(\lambda _1+\lambda _2)t_i})}{(1-p+p{\mathrm{{e}}}^{-(\lambda _1+\lambda _2)t_i})^3}, \\ w_{i3}^{33}(\varvec{\theta })& = \frac{z_i}{p^2}+\frac{(1-z_i)(1-{\mathrm{{e}}}^{-(\lambda _1+\lambda _2)t_i})^2}{(1-p+p{\mathrm{{e}}}^{-(\lambda _1+\lambda _2)t_i})^2}, \\ w_{i2}^{11}(\varvec{\theta })& = w_{i1}^{22}(\varvec{\theta }) =w_{i1}^{12}(\varvec{\theta }) =w_{i2}^{12}(\varvec{\theta }) =w_{i3}^{12}(\varvec{\theta }) =w_{i2}^{21}(\varvec{\theta }) =w_{i3}^{21}(\varvec{\theta }) =w_{i1}^{21}(\varvec{\theta }),\\ w_{i3}^{11}(\varvec{\theta })& = w_{i3}^{22}(\varvec{\theta }) =w_{i1}^{13}(\varvec{\theta }) =w_{i2}^{13}(\varvec{\theta }) =w_{i1}^{23}(\varvec{\theta }) =w_{i1}^{31}(\varvec{\theta }) =w_{i1}^{32}(\varvec{\theta }) \\& = w_{i2}^{32}(\varvec{\theta })=w_{i2}^{31}(\varvec{\theta }) =w_{i2}^{23}(\varvec{\theta }) ,\\ w_{i3}^{13}(\varvec{\theta })& = w_{i3}^{31}(\varvec{\theta }) =w_{i3}^{32}(\varvec{\theta }) =w_{i3}^{23}(\varvec{\theta }) =w_{i2}^{33}(\varvec{\theta }) =w_{i1}^{33}(\varvec{\theta }). \end{aligned}$$

For further development we need the result of Lemma 6.5. As a result of Lemma 6.5,

$$\begin{aligned} \sup \limits _{\begin{array}{c} \varvec{\theta } \in \mathbf {N}_n( \varvec{\theta _0}) \end{array}} \left\| \frac{1}{n} \displaystyle \sum _{i=1}^{n}w_i^{rs}( \varvec{{\tilde{\theta }}}) \right\| =O_p(1), \end{aligned}$$

where \(\Vert .\Vert \) denotes the norm of a vector.

Also, when \( \varvec{\theta } \in \mathbf {N}_n( \varvec{\theta _0} )\), \({| \varvec{\theta } - \varvec{\theta _0 } |}^2 \le \frac{A^2}{\lambda _{\mathrm{min}}\{{\mathbf {D}}_n(\varvec{\theta _0 })\}} \le \frac{A^2}{n\eta } \rightarrow 0\) as \(n \rightarrow \infty \).

Since \(\varvec{ {\tilde{\theta }}} \in \mathbf {N}_n( \varvec{\theta _0} )\), it follows that

$$\begin{aligned} \sup \limits _{\begin{array}{c} \varvec{\theta } \in \mathbf {N}_n( \varvec{\theta _0} ) \end{array}} \left| \frac{1}{n} ( \varvec{\theta } - \varvec{\theta _0} )^{\top } \sum _{i=1}^{n}w_i^{rs}(\varvec{ {\tilde{\theta }} }) \right|& \le \sup \limits _{\begin{array}{c} \varvec{\theta } \in \mathbf {N}_n( \varvec{\theta _0}) \end{array}} \Vert (\varvec{\theta } -\varvec{\theta _0}) \Vert \left\| \frac{1}{n} \sum _{i=1}^{n}w_i^{rs}(\varvec{{\tilde{\theta }}}) \right\| \\& \le \frac{A}{\sqrt{n\eta }} \sup \limits _{\begin{array}{c} \varvec{\theta } \in \mathbf {N}_n(\varvec{\theta _0}) \end{array}} \left\| \frac{1}{n} \sum _{i=1}^{n}w_i^{rs}(\varvec{{\tilde{\theta }}}) \right\| \\& = o_{p}(1). \end{aligned}$$

Therefore,

$$\begin{aligned} \sup \limits _{\begin{array}{c} \varvec{\theta } \in \mathbf {N}_n(\theta _0) \end{array}} | e_n^{(2)}(\varvec{\theta } ) |& = \sup \limits _{\begin{array}{c} \theta \in \mathbf {N}_n(\varvec{\theta _0} ) \end{array}} | \mathbf {u_n}^{\top } \{{\mathbf {F}}_n( \varvec{\theta } ) - {\mathbf {F}}_n( \varvec{\theta _0} ) \} \mathbf {u_n} | | {\mathbf {u}}^{\top } {\mathbf {D}}^{-1}_n {\mathbf {u}} | \\& \le \frac{1}{n \eta } \sup \limits _{\begin{array}{c} \varvec{\theta } \in \mathbf {N}_n( \varvec{\theta _0} ) \end{array}} \sum \limits _{\begin{array}{c} r,s=1,2,3 \end{array}} \left| \sum _{i=1}^{n}\{f^{rs}_i(\varvec{\theta }) -f^{rs}_i( \varvec{\theta _0} )\} \right| \\& \le \frac{1}{n \eta } \sum \limits _{\begin{array}{c} r,s=1,2,3 \end{array}} \sup \limits _{\begin{array}{c} \varvec{\theta } \in \mathbf {N}_n( \varvec{\theta _0} ) \end{array}} \left| ( \varvec{\theta } - \varvec{\theta _0} )^{\top } \sum _{i=1}^{n} w_i^{rs}(\varvec{{\tilde{\theta }}}) \right| \\& = o_{p}(1). \end{aligned}$$

This concludes the proof of Lemma (6.6). \(\square \)

Lemma 6.7

For \(\lambda _{10}, \lambda _{20} >0\) and \(p_0 \in (0,1),\) if condition (26) holds, a unique MLE of \(\varvec{\theta }_o\), say \(\widehat{\varvec{\theta _n}}\), will exist in \(\mathbf {N}_n(\varvec{\theta }_o)\) with probability 1 as \(n \rightarrow \infty \).

Proof

If the condition (26) holds, from Lemma 6.6, \({\mathbf {D}}_n(\varvec{\theta _0})\) is positive definite matrix and a symmetric square root of \({\mathbf {D}}^{-1}_n(\varvec{\theta _0})\) will always exist. Let \({\mathbf {D}}^{-\frac{1}{2}}_n(\varvec{\theta _0})\) be the symmetric square root of \({\mathbf {D}}^{-1}_n(\varvec{\theta _0})\). Further consider the Taylor-series expansion of the log-likelihood function \({\mathbf{l}}_n(\varvec{\theta })\) about the true parameter value \(\varvec{\theta _0}\) as follows:

$$\begin{aligned} {\mathbf{l}}_n(\varvec{\theta }) - {\mathbf{l}}_n(\varvec{\theta _0}) = (\varvec{\theta } - \varvec{\theta _0} )^{\top } {\mathbf {S}}_n(\varvec{\theta _0} ) - \frac{1}{2} (\varvec{\theta } - \varvec{\theta _0} )^{\top } {\mathbf {F}}_n( \varvec{{\bar{\theta }} }) (\varvec{\theta } - \varvec{\theta _0} ), \end{aligned}$$
(34)

where \(\varvec{{\bar{\theta }}}\) is any point on the line segment joining \(\varvec{\theta _0}\) and \(\varvec{\theta }\).

Let \(\varvec{\theta _n}\) be on the boundary, \(\partial \mathbf {N}_n(\varvec{\theta _0}) \) of \(\mathbf {N}_n(\varvec{\theta _0})\). We define \(v_n= \dfrac{ {\mathbf {D}}^{\frac{1}{2}}_n(\varvec{\theta _0}) (\varvec{\theta _n} - \varvec{\theta _0}) }{A}\). Then, for \(\varvec{\theta _n} \in \partial \mathbf {N}_n(\varvec{\theta _0})\), \(v_n\) is a unit vector. For any arbitrary \(\epsilon \in (0,1)\),

$$\begin{aligned} P\left( \frac{1}{2} (\varvec{\theta _n} - \varvec{\theta _0})^{\top } {\mathbf {F}}_n( \varvec{{\bar{\theta }}}) (\varvec{\theta _n} - \varvec{\theta _0} )> \frac{\epsilon A^2}{2}\right)& = P( v^{\top }_n {\mathbf {D}}^{-\frac{1}{2}}_n(\varvec{\theta _0}) {\mathbf {F}}_n( \varvec{{\bar{\theta }}}) {\mathbf {D}}^{-\frac{1}{2}}_n(\varvec{\theta _0}) v_n> \epsilon )\\&\ge {} P( \lambda _{\mathrm{min}}\{ {\mathbf {D}}^{-\frac{1}{2}}_n(\varvec{\theta _0}) {\mathbf {F}}_n( \varvec{{\bar{\theta }}}) {\mathbf {D}}^{-\frac{1}{2}}_n(\varvec{\theta _0}) \} >\epsilon ) \end{aligned}$$

Again if the condition (26) holds, from Lemma 6.6, we get the eigenvalues of \({\mathbf {D}}^{-\frac{1}{2}}_n(\varvec{\theta _0}) {\mathbf {F}}_n( \varvec{{\bar{\theta }}}) {\mathbf {D}}^{-\frac{1}{2}}_n(\varvec{\theta _0})\) converging to 1 in probability. Therefore, \(P(\frac{1}{2} ( \varvec{\theta _n} - \varvec{\theta _0} )^{\top } {\mathbf {F}}_n( \varvec{{\bar{\theta }}}) (\varvec{\theta _n} - \varvec{\theta _0} \le \frac{\epsilon A^2}{2}) = o(1)\).

As \( ( \varvec{\theta _n} - \varvec{\theta _0})^{\top } {\mathbf {S}}_n(\varvec{\theta _0})\) has mean 0, applying Chebyshev’s inequality we obtain,

$$\begin{aligned} P\left( ( \varvec{\theta _n}- \varvec{\theta _0})^{\top } {\mathbf {S}}_n(\varvec{\theta _0}) > \frac{\epsilon A}{2}\right) \le \frac{ 4( \varvec{\theta _n} - \varvec{\theta _0})^{\top } {\mathbf {D}}_n (\varvec{\theta _0})(\varvec{\theta _n} - \varvec{\theta _0)} }{\epsilon ^2 A^2} =\frac{4}{\epsilon ^2 } \end{aligned}$$

Thus,

$$\begin{aligned}&P\big ( (\varvec{\theta _n} - \varvec{\theta _0})^{\top } {\mathbf {S}}_n( \varvec{\theta _0} )>\frac{1}{2} ( \varvec{\theta _n} - \varvec{\theta _0} )^{\top } {\mathbf {F}}_n( \varvec{{\bar{\theta }} } ) ( \varvec{\theta _n} - \varvec{\theta _0} ) \big )\\&\quad \le P \left( (\varvec{\theta _n} - \varvec{\theta _0} )^{\top } {\mathbf {S}}_n( \varvec{\theta _0}) > \frac{\epsilon A}{2} \right) + P\left(\frac{1}{2} ( \varvec{\theta _n} - \varvec{\theta _0} )^{\top } {\mathbf {F}}_n( \varvec{{\bar{\theta }} } ) ( \varvec{\theta _n} - \varvec{\theta _0} ) \le \frac{\epsilon A^2}{2}\right )\\&\quad \le \frac{4}{\epsilon ^2 } + o(1). \end{aligned}$$

Therefore, for A and n large enough, \( {\mathbf{l}}_n(\varvec{\theta )} < {\mathbf{l}}_n( \varvec{\theta _0})\) for any \( \varvec{\theta } \) lying on \(\partial {\mathbf {N}_n( \varvec{\theta _0)} }\). As \({\mathbf {D}}_n(\varvec{\theta _0})\) is positive definite, by (31), \({\mathbf {F}}_n( \varvec{\theta } )\) is positive definite. Therefore, \({\mathbf{l}}_n( \varvec{\theta } )\) is a concave function in \(\mathbf {N}_n(\varvec{\theta _0} )\) which is an ellipsoid. \({\mathbf{l}}_n( \varvec{\theta } )\) does not have a maximum on the boundary of \(\mathbf {N}_n( \varvec{\theta _0} )\) and hence it has a local maximum inside \(\mathbf {N}_n( \varvec{\theta _0} )\) which is a global maximum because of its concavity. Therefore, an MLE \(\varvec{\widehat{\theta }_n}\) of \(\varvec{\theta _0} \) exists inside the ellipsoid \(\mathbf {N}_n(\varvec{\theta _0})\) with probability 1 as \(n \rightarrow \infty \) and it is unique. \(\square \)

Proof of Theorem 4.1

Based on Lemmas 6.3 and 6.7, the proof immediately follows. \(\square \)

Lemma 6.8

For \(\lambda _{10}, \lambda _{20} >0\) and \(p_0 \in (0,1),\) if the condition (26) holds, \(\lambda _{\mathrm{min}} \{{\mathbf {D}}_n\} \rightarrow \infty \) as \(n \rightarrow \infty \), and \(\varvec{\widehat{\theta }_n}\) is a consistent estimator of \(\varvec{{\theta }_o}.\)

Proof

Based on Lemma 6.7, and as the MLE \(\varvec{\widehat{\theta }_n} \in \mathbf {N}_n(\varvec{\theta _0})\), \(( \varvec{\widehat{\theta }_n} - \varvec{\theta _0} )^{\top } {\mathbf {D}}_n(\varvec{\theta }_o) \big ( \varvec{\widehat{\theta }_n} - \varvec{\theta _0} \big ) \le A^2\). Again \(( \varvec{\widehat{\theta }_n} - \varvec{\theta _0} )^{2} \lambda _{\mathrm{min}}\{ {\mathbf {D}}_n(\varvec{\theta }_o)\} \le ( \varvec{\widehat{\theta }_n} - \varvec{\theta _0})^{'} {\mathbf {D}}_n (\varvec{\theta }_o)\big ( \varvec{\widehat{\theta }_n} - \varvec{\theta _0} \big ).\) If in addition, \(\lambda _{\mathrm{min}} \{{\mathbf {D}}_n\} \rightarrow \infty \) as \(n \rightarrow \infty \), \(\varvec{\widehat{\theta }_n} \rightarrow \varvec{\theta _0}\) in probability. \(\square \)

Proof of Theorem 4.2

Based on Lemmas 6.3 and 6.8, the proof immediately follows. \(\square \)

Lemma 6.9

For \(\lambda _{10}, \lambda _{20} >0\) and \(p_0 \in (0,1)\) there exists a finite positive quantity \(G(\varvec{\theta _0})\) such that \(E{|s_i(\varvec{\theta _0})|}^4 \le G(\varvec{\theta _0}).\)

Proof of Theorem 4.2

Along the same lines as the previous two lemmas, Lemmas 6.4 and 6.5, we can show that \(s_i(\varvec{\theta _0})\) is also bounded, and hence, the moments are bounded. \(\square \)

Lemma 6.10

For \(\lambda _{10}, \lambda _{20} >0\) and \(p_0 \in (0,1),\) if the condition (26) holds, \( {\mathbf {D}}^{1/2}_n(\varvec{\theta }_o)\big (\widehat{\varvec{\theta _n}}-\varvec{\theta }_o\big ) \) is asymptotically normally distributed with mean vector \( {\varvec{0}}\) and covariance matrix \(\mathbf {I}.\)

Proof

Considering the element wise Taylor series expansion of \( {\mathbf {S}}_n(\varvec{\theta _0} )\) about \(\varvec{\widehat{\theta }_n}\), we obtain

$$\begin{aligned} {\mathbf {S}}_n( \varvec{\theta _0} ) - {\mathbf {S}}_n(\varvec{\widehat{\theta }_n}) = \mathbf {K}_n( \varvec{{\widetilde{\theta }}_n}) (\varvec{\widehat{\theta }_n} - \varvec{\theta _0}), \end{aligned}$$

where \(\varvec{{\widetilde{\theta }}_n} = ( \varvec{\theta ^{*}_n} , \varvec{\theta ^{**}_n}, \varvec{\theta ^{***}_n})\), \(\varvec{\theta ^{*}_n}\), \(\varvec{\theta ^{**}_n}\) and \(\varvec{\theta ^{***}_n}\) are three points on the line segment joining \(\varvec{\theta _0}\) and \(\varvec{\widehat{\theta }_n}\), and

$$\begin{aligned} \mathbf {K}_n(\varvec{{\widetilde{\theta }}_n}) = \begin{bmatrix} {\sum }_{i=1}^{n} f^{11}_i( \varvec{\theta ^{*}_n} ) &{}\quad {\sum }_{i=1}^{n} f^{12}_i( \varvec{\theta ^{*}_n} ) &{}\quad {\sum }_{i=1}^{n} f^{13}_i( \varvec{\theta ^{*}_n} )\\ {\sum }_{i=1}^{n} f^{21}_i( \varvec{\theta ^{**}_n} ) &{}\quad {\sum }_{i=1}^{n} f^{22}_i( \varvec{\theta ^{**}_n} ) &{}\quad {\sum }_{i=1}^{n} f^{23}_i(\varvec{\theta ^{**}_n} )\\ {\sum }_{i=1}^{n} f^{31}_i( \varvec{\theta ^{***}_n} ) &{}\quad {\sum }_{i=1}^{n} f^{32}_i( \varvec{\theta ^{***}_n} ) &{}\quad {\sum }_{i=1}^{n} f^{33}_i(\varvec{\theta ^{***}_n} )\\ \end{bmatrix}. \end{aligned}$$

As the likelihood function is maximized at MLE \(\varvec{\widehat{\theta }_n}\), \({\mathbf {S}}_n(\varvec{\widehat{\theta }}_n)={\varvec{0}},\) and we can write

$$\begin{aligned} {\mathbf {D}}^{-\frac{1}{2}}_n(\varvec{\theta }_o) {\mathbf {S}}_n(\varvec{\theta _0} ) ={\mathbf {D}}^{-\frac{1}{2}}_n(\varvec{\theta }_o) \mathbf {K}_n(\varvec{{\widetilde{\theta }}_n}) (\varvec{\widehat{\theta }_n} - \varvec{\theta _0}) = A_n(\varvec{{\widetilde{\theta }}_n}){\mathbf {D}}^{\frac{1}{2}}_n(\varvec{\theta }_o) ( \varvec{\widehat{\theta }_n} - \varvec{\theta _0} ) \end{aligned}$$

where \( A_n(\varvec{{\widetilde{\theta }}_n}) = {\mathbf {D}}^{-\frac{1}{2}}_n (\varvec{\theta }_o) \mathbf {K}_n(\varvec{{\widetilde{\theta }}_n}){\mathbf {D}}^{-\frac{1}{2}}_n(\varvec{\theta }_o)\). To prove the asymptotic normality of \( {\mathbf {D}}^{\frac{1}{2}}_n(\varvec{\theta }_o) ( \varvec{\widehat{\theta }_n} - \varvec{\theta _0} )\), it is enough to show that for any \(\varvec{\theta ^{*}_n}\), \(\varvec{\theta ^{**}_n}\) and \(\varvec{\theta ^{***}_n} \) \(\in \mathbf {N}_n(\varvec{\theta _0} )\), \(A_n(\varvec{{\widetilde{\theta }}_n}) \overset{P}{\rightarrow } \mathbf {I}\) and \({\mathbf {D}}^{-\frac{1}{2}}_n(\varvec{\theta }_o) {\mathbf {S}}_n(\varvec{\theta _0})\) converges in distribution to a standard normal random variable. Now

$$\begin{aligned} A_n(\varvec{{\widetilde{\theta }}_n})& = \mathbf {I} + {\mathbf {D}}^{-\frac{1}{2}}_n(\varvec{\theta }_o)\big \{ {\mathbf {F}}_n( \varvec{\theta _0}) - {\mathbf {D}}_n(\varvec{\theta }_o) \big \} {\mathbf {D}}^{-\frac{1}{2}}_n(\varvec{\theta }_o)\\&+ {\mathbf {D}}^{-\frac{1}{2}}_n(\varvec{\theta }_o) \big \{ \mathbf {K}_n(\varvec{{\widetilde{\theta }}_n}) - {\mathbf {F}}_n(\varvec{\theta _0} \big \} ) {\mathbf {D}}^{-\frac{1}{2}}_n(\varvec{\theta }_o). \end{aligned}$$

Applying (32), \( {\mathbf {D}}^{-\frac{1}{2}}_n(\varvec{\theta }_o)\big \{ {\mathbf {F}}_n(\varvec{\theta _0}) - {\mathbf {D}}_n(\varvec{\theta }_o)\big \} {\mathbf {D}}^{-\frac{1}{2}}_n(\varvec{\theta }_o) \) converges in probability to the zero matrix. Also for any unit vector \({\mathbf {u}}\) and \(\varvec{\theta ^{*}_n}, \varvec{\theta ^{**}_n}, \varvec{\theta ^{***}_n} \in \mathbf {N}_n(\varvec{\theta _0} )\), we can prove that

\({\mathbf {u}}^{'} {\mathbf {D}}^{-\frac{1}{2}}_n(\varvec{\theta }_o) \big \{ \mathbf {K}_n(\varvec{{\widetilde{\theta }}_n}) - {\mathbf {F}}_n(\varvec{\theta _0} \big \} ) {\mathbf {D}}^{-\frac{1}{2}}_n(\varvec{\theta }_o) {\mathbf {u}} = o_{p}(1)\). The proof follows exactly along the same lines as the proof of (33). Hence, we have \(A_n(\varvec{{\widetilde{\theta }}_n}) \overset{P}{\rightarrow } \mathbf {I}\). \({\mathbf {D}}^{-\frac{1}{2}}_n(\varvec{\theta }_o) {\mathbf {S}}_n(\varvec{\theta _0})\) can be written as

$$\begin{aligned} {\mathbf {D}}^{-\frac{1}{2}}_n(\varvec{\theta }_o) {\mathbf {S}}_n(\varvec{\theta _0})={\mathbf {D}}^{-\frac{1}{2}}_n(\varvec{\theta }_o) \sum _{i=1}^{n}( {\mathbf {s}}_{i}(\varvec{\theta _0}). \end{aligned}$$

Let \(X_{in}(\varvec{\theta _0})= {\mathbf {u}}^{\top } {\mathbf {D}}^{-\frac{1}{2}}_n(\varvec{\theta }_o){\mathbf {s}}_{i}(\varvec{\theta _0}),\,i=1,2,\ldots ,n,\) where \({\mathbf {u}}\) is a unit vector. Then, \(E(X_{in}(\varvec{\theta _0}))=0\) and \(\sum _{i=1}^{n} {\mathrm{Var}}(X_{in}(\varvec{\theta _0}) ) =1.\) To prove that \( {\mathbf {D}}^{-\frac{1}{2}}_n(\varvec{\theta }_o) {\mathbf {S}}_n(\varvec{\theta _0})\) converges to the standard normal distribution, it is enough to show that the sequence \(\{ X_{in}(\varvec{\theta _0}) \}\) follows Lindberg condition, i.e., for every \(\xi >0\),

$$\begin{aligned} \sum _{i=1}^{n} \int \limits _{\begin{array}{c} |x| \ge \xi \end{array}} x^2 {\mathrm{d}}P(X_{in}(\varvec{\theta _0}) \le x) \rightarrow 0 \quad \text { as } \quad n \rightarrow \infty \end{aligned}$$

We need Lemma 6.9 to prove the Lindberg condition. Based on Lemma 6.9,

$$\begin{aligned} E{|X_{in} (\varvec{\theta _0})|}^4& \le E{|s_{i}( \varvec{\theta _0 } )|}^4 | {\mathbf {u}}^{\top } {\mathbf {D}}^{-1}_n(\varvec{\theta }_o) {\mathbf {u}} |^2\nonumber \\& \le \frac{1}{n^2 \eta ^2} E{|s_i(\varvec{\theta _0 } )|}^4 \nonumber \\& \le \frac{G^2}{n^2 \eta ^2}. \end{aligned}$$
(35)

Let \(I({\mathscr {A}})\) denote the indicator variable on a set \({\mathscr {A}}.\) By (35), Cauchy Schwartz’s inequality and Chebyshev’s inequality, we obtain

$$\begin{aligned} \sum _{i=1}^{n} \int \limits _{\begin{array}{c} |x| \ge \xi \end{array}} x^2 {\mathrm{d}}P(X_{in}(\varvec{\theta _0}) \le x)& = \sum _{i-1}^{n} E\big \{X^2_{in}(\varvec{\theta _0}) I (|X_{in}(\varvec{\theta _0})| \ge \xi ) \big \}\\& \le \frac{G}{n\eta } \sum _{i=1}^{n} \{ P(| X_{in}(\varvec{\theta _0})|\ge \xi ) \}^{\frac{1}{2}} \\& \le \frac{G}{n\eta \xi } \displaystyle \sum _{i=1}^{n} {\{{\mathrm{Var}}(X_{in} (\varvec{\theta _0})\}}^{\frac{1}{2}} \rightarrow 0 \quad \text { as } \quad n \rightarrow \infty. \end{aligned}$$

\(\square \)

Proof of Theorem 4.3

Based on Lemmas 6.3 and 6.10, the proof immediately follows. \(\square \)

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Pal, A., Mondal, S. & Kundu, D. A Cure Rate Model for Exponentially Distributed Lifetimes with Competing Risks. J Stat Theory Pract 15, 21 (2021). https://doi.org/10.1007/s42519-020-00145-0

Download citation

Keywords

  • Cure rate model
  • Long-term survivors
  • Competing risk
  • EM algorithm
  • Asymptotic normality
  • Consistency
  • Maximum likelihood estimator

Mathematics Subject Classification

  • Primary 62F12
  • secondary 62N05