Abstract
It is often of interest in clinical trials and reliability studies to estimate the remaining lifetime of a subject or a device given that it survived up to a given period of time, that is commonly known as the so-called mean residual life function (mrlf). There have been several attempts in literature to estimate the mrlf nonparametrically ranging from empirical estimates to more sophisticated smooth estimation. Given the well known one-to-one relation between survival function and mrlf, one can plug-in any known estimates of the survival function (e.g., Kaplan–Meier estimate) into the functional form of mrlf to obtain an estimate of mrlf. In this chapter, we present a scale mixture representation of mrlf and use it to obtain a smooth estimate of the mrlf under right censoring. Asymptotic properties of the proposed estimator are also presented. Several simulation studies and a real data set are used for investigating the empirical performance of the proposed method relative to other well-known estimates of mrlf. A comparative analysis shows computational advantages of the proposed estimator in addition to somewhat superior statistical properties in terms of bias and efficiency.
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References
Abdous, B., and A. Berred. 2005. Mean residual life estimation. Journal of Statistical Planning and Inference 132: 3–19.
Agarwal, S.L., and S.L. Kalla. 1996. A generalized gamma distribution and its application in reliability. Communications in Statistics—Theory and Methods 25: 201–210.
Bhattacharjee, M.C. 1982. The class of mean residual lives and some consequences. SIAM Journal of Algebraic Discrete Methods 3: 56–65.
Chaubey, Y.P., and P.K. Sen. 1999. On smooth estimation of mean residual life. Journal of Statistical Planning and Inference 75: 223–236.
Chaubey, Y.P., and A. Sen. 2008. Smooth estimation of mean residual life under random censoring. In Ims collections—beyond parametrics in interdisciplinary research: Festschrift in honor of professor pranab k. Sen 1: 35–49.
Elandt-Johnson, R.C., and N.L. Johnson. 1980. Survival models and data analysis. New York: Wiley.
Feller. 1968. An introduction to probability theory and its applications. vol. I, 3rd ed. New York: Wiley.
Ghorai, J., A. Susarla., V, Susarla., and Van-Ryzin, J. 1982. Nonparametric Estimation of Mean Residual Life Time with Censored Data. In Nonparametric statistical inference. vol. I, Colloquia Mathematica Societatis, 32. North-Holland, Amsterdam-New York, 269-291.
Guess, F, and Proschan, F. (1988). Mean residual life theory and applications. In Handbook of statistics 7, reliability and quality control, ed. P.R. Krishnaiah., and Rao, C.R. 215–224.
Gupta, R.C., and D.M. Bradley. 2003. Representing the mean residual life in terms of the failure rate. Mathematical and Computer Modelling 37: 1271–1280.
Gupta, R.C., and S. Lvin. 2005. Monotonicity of failure rate and mean residual life function of a gamma-type model. Applied Mathematics and Computation 165: 623–633.
Hall, W.J., and J.A. Wellner. 1981. Mean residual life. In Statistics and related topics, ed. M. Csorgo, D.A. Dawson, J.N.K. Rao, and AKMdE Saleh, 169–184. Amsterdam, North-Holland.
Hille, E. (1948). Functional analysis and semigruops. American Mathematical Society vol.31.
Kalla, S.L., H.G. Al-Saqabi, and H.G. Khajah. 2001. A unified form of gamma-type distributions. Applied Mathematics and Computation 118: 175–187.
Kaplan, E.L., and P. Meier. 1958. Nonparametric estimation from incomplete observations. Journal of American Statistical Association 53: 457–481.
Kopperschmidt, K., and U. Potter. 2003. A non-parametric mean residual life estiamtor: An example from market research. Developmetns in Applied Statistics 19: 99–113.
Kulasekera, K.B. 1991. Smooth Nonparametric Estimation of Mean Residual Life. Microelectronics Reliability 31 (1): 97–108.
Lai, C.D., L. Zhang, and M. Xie. 2004. Mean residual life and other properties of weibull related bathtub shape failure rate distributions. International Journal of Reliability, Quality and Safety Engineering 11: 113–132.
Liu, S. (2007). Modeling mean residual life function using scale mixtures NC State University Dissertation. http://www.lib.ncsu.edu/resolver/1840.16/3045.
McLain, A., and S.K. Ghosh. 2011. Nonparametric estimation of the conditional mean residual life function with censored data. Lifetime Data Analysis 17: 514–532.
Morrison, D.G. 1978. On linearly increasing mean residual lifetimes. Journal of Applied Probability 15: 617–620.
Petrone, S., and P. Veronese. 2002. Non parametric mixture priors based on an exponential random scheme. Statistical Methods and Applications 11 (1): 1–20.
Ruiz, J.M., and A. Guillamon. 1996. nonparametric recursive estimator of residual life and vitality funcitons under mixing dependence condtions. Communcation in Statistics–Theory and Methods 4: 1999–2011.
Swanepoel, J.W.H., and F.C. Van Graan. 2005. A new kernel distribution function estimator based on a non-parametric transformation of the data. The Scadinavian Journal of Statistics 32: 551–562.
Tsang, A.H.C., and A.K.S. Jardine. 1993. Estimation of 2-parameter weibull distribution from incomplete data with residual lifetimes. IEEE Transactions on Reliability 42: 291–298.
Yang, G.L. 1978. Estimation of a biometric function. Annals of Statistics 6: 112–116.
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Appendices
Appendix
A: Characterization of MRLF
Lemma 8.1
Let h(t) be the hazard function of a lifetime random variable T. Then the mrlf, m(t) of T is differentiable and is obtained by solving the following differential equation:
where \(m^\prime (\cdot )\) denotes the derivative of \(m(\cdot )\).
Proof
From (8.1), it follows that \(m(t)S(t)=\int _t^\infty S(u) du\). The result follows by differentiating both sides of the previous identity with respect to (wrt) t and using the definition of \(h(t)=-S^\prime (t)/S(t)\).
It thus follows (8.7) that given h(t) we can obtain
and given m(t) we can obtain
Also under mild regularity conditions, an mrlf completely determines the distribution as can be seen by the following characterization result:
Theorem 8.3
(Hall-Wellner 1981) Let \(m:\mathbb {R}^+\rightarrow \mathbb {R}^+\) be a function that satisfies the following conditions:
-
(i)
m(t) is right continuous and \(m(0)>0\);
-
(ii)
\(e(t)\equiv m(t)+t\) is non-decreasing;
-
(iii)
if \(m(t-)=0\) for some \(t=t_0\), then \(m(t)=0\) on \([t_0,\infty )\);
-
(iv)
if \(m(t-)>0\) for all t, then \(\int _0^\infty 1/m(u)du=\infty .\)
Let \(\tau \equiv \inf \{t: m(t-)=0\}\le \infty \), and define
Then \(F(t)=1-S(t)\) is a cdf on \(\mathbb {R}^+\) with \(F(0)=0\) and \(\tau =\inf \{t: F(t)=1\}\), finite mean m(0) and mrlf m(t).
Proof
See Hall and Wellner (1981), p. xxx.
B: Chaubey-Sen’s (1999, 2008) Estimator
Given a sample of n iid positive values random variables, \(T_i{\mathop {\sim }\limits ^{iid}}F(\cdot )\) for \(i=1,\ldots ,n\), the empirical survival function \(S_n\) is defined by
Define a set of nonnegative valued Poisson weights as
where \(\lambda _n=n/T_{(n)}\) and \(T_{(n)}=\max _{1\le i\le n}T_i\). Notice that \(\lambda _n\) is chosen data-dependent, which makes the weight function stochastic. A smoothed empirical survival function can be obtained as
Plugging the smoothed empirical survival function into (8.3), the smooth estimator of the mrlf is given by
Chaubey and Sen (1999) proved that \(\tilde{m}_n(t)\) is a consistent estimator of m(t) and \(\lambda _n^{1/2}(\tilde{m}_n(t)-m(t)){\mathop {\rightarrow }\limits ^{d}}N\left( 0, {m(t)\over S(t)}\right) \) pointwise. Later Chaubey and Sen (2008) extended their methodology for the censored case and derived the asymptotic properties.
C: The Details of the Calculation of \(\hat{m}_m(t)\)
\(\hat{m}_m(t)\), the smooth estimator of m(t) for censored data (complete data are special case of censored data where weight function \(w_j={1\over n}\)), can be calculated with a closed form. The details of the calculation are given here.
where \(w_j=\hat{S}(X_j)-\hat{S}(X_j-)\) and \(\hat{S}(\cdot )\) is the KM estimate.
D: Proof of Theorem 8.2
(a) Proof of consistency
First we prove part (i) of Theorem 8.2 by showing that \(\hat{m}_m(t)\) is pointwise consistent for estimating m(t). The main tool to prove consistency is based on the following well known approximation result originally due to Feller [7], p. xxx but extended by Petrone and Veronese [22] for a wider application:
Lemma 8.2
Feller Approximation Lemma (Petrone and Veronese [22]):
Let \(g(\cdot )\) be a bounded and right continuous function on \(\mathbb {R}\) for each t. Let \(Z_k(t)\) be a sequence of random variables for each t such that \(\mu _k(t)\equiv E[Z_k(t)]\rightarrow t\) and \(\sigma ^2_k(t)\equiv Var[Z_k(t)]\rightarrow 0\) as \(k\rightarrow \infty \). Then
First, notice that by definition, \(\hat{m}_e(\cdot )\) is a right continuous function on \([0, T_{(n)}]\) and \(\hat{m}_e(t)=0\) for \(t>T_{(n)}\) and hence \(\hat{m}_e(\cdot )\) is a bounded function on \([0, \infty )\).
Let \(Z_{n}(t)\sim Ga(k_n,{t\over k_n})\) for \(t>0\), where \(Ga(k_n,{t\over k_n})\) denotes a Gamma distribution with mean \(\mu _n(t)=t\) and variance \(\sigma ^2_{n}(t)={t^2\over k_n}\) and the density function is given by
It easily follows that we can write our scale mixture estimator as
Thus, by Feller approximation result in Lemma 8.2 it follows that \(E[m(Z_n(t))]\) converges (pointwise) to m(t) if we choose the sequence \(k_n\) such that \(k_n\rightarrow \infty \) as \(n\rightarrow \infty \). Next, by the consistency of \(\hat{m}_e(t)\) and Dominated Convergence Theorem it follows that \(E[\hat{m}_e(Z_n(t))-m(Z_n(t))]\rightarrow 0\) and \(n\rightarrow \infty \). Hence it follows that \(\hat{m}_m(t)\) converges (pointwise) in probability to m(t) as \(k_n\rightarrow \infty \). This completes the proof of (i) in Theorem 8.2.
(b) Proof of Asymptotic Normality
First, notice that we can write,
where it is known from previous literature that \(\sqrt{n}(\hat{m}_e(t)-m(t))\sim GP(0,\sigma (\cdot ,\cdot ))\) and expression of the covariance function \(\sigma (\cdot ,\cdot )\) are as given in the statement of Theorem 8.2. Thus, it is sufficient to establish that first term in Eq. 8.12 converges in probability to zero as \(n\rightarrow \) by choosing a suitable growth rate of the \(k_n\) sequence as a function of n.
Using the scale mixture formulation it follows that \(\sqrt{n}(\hat{m}_m(t)-\hat{m}_e(t))=\sqrt{n}E[{\hat{m}_e(tY_n)\over Y_n}-\hat{m}_e(t)]\) where \(Y_n\sim Ga(k_n+1,1/k_n)\)
We use a first order Taylor’s expansion around t to write
where \(t_n^*{\mathop {\rightarrow }\limits ^{P}}t\) as \(n\rightarrow \infty \) because \(Y_n{\mathop {\rightarrow }\limits ^{p}} 1\). It can be seen from (8.5) that \(\hat{m}_e(t)\) is differentiable if \(t\in (X_l,X_{l+1}),l=1,\ldots ,n-1\) and \(\hat{m}_e(t)\) is not differentiable if \(t=X_l,l=1,\ldots ,n-1\). Thus, \(\hat{m}_e^\prime (t_n^*)\) exists for \(t_n^*\in [0,\infty )\backslash \{X_1,\ldots X_n\}\).
Second, we substitute the Taylor’s expansion into \(\sqrt{n}E[{\hat{m}_e(tY_n)\over Y_n}-\hat{m}_e(t)]\)
In order to complete the proof it is now sufficient to show that \(\sqrt{n}\left( {1\over Y_n}-1\right) {\mathop {\rightarrow }\limits ^{P}}0\) as \(n\rightarrow \infty \). However, as \(Y_n\sim Ga(k_n+1, 1/k_n)\), it follows that
and
Thus id we assume that \(k_n/n\rightarrow \infty \), i.e., \(k_n=cn^{1+\epsilon }\) where \(\epsilon >0\), it can be seen that \(var\left[ \sqrt{n}\left( {1\over Y_n}-1\right) \right] \rightarrow 0\) as \(n\rightarrow \infty \). Therefore, \(\sqrt{n}\left( {1\over Y_n}-1\right) {\mathop {\rightarrow }\limits ^{P}}0\) as \(n\rightarrow \infty \) and this completes the proof of Theorem 8.2.
Remark 8.1
Notice that the asymptotic properties of \(\hat{m}_m(t)\) are the same as those of the estimator of Yang [26] for complete data and the same as those of the estimator of Ghorai et al. [8] for censored data.
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Ghosh, S.K., Liu, S. (2017). Nonparametric Estimation of Mean Residual Life Function Using Scale Mixtures. In: Adhikari, A., Adhikari, M., Chaubey, Y. (eds) Mathematical and Statistical Applications in Life Sciences and Engineering. Springer, Singapore. https://doi.org/10.1007/978-981-10-5370-2_8
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