A new upside-down bathtub shaped hazard rate model for survival data analysis

doi:10.1016/j.amc.2014.04.048

Applied Mathematics and Computation

Volume 239, 15 July 2014, Pages 242-253

https://doi.org/10.1016/j.amc.2014.04.048 Get rights and content

Abstract

In medical, engineering besides demography and other applied disciplines, it is pronounced in some applications that the hazard rate of the data initially increased to a pick in the beginning age, declined abruptly till it stabilized. In statistics literature, such hazard rate is known as the upside-down bathtub shaped hazard rate and propound in the various survival studies. In this paper, we proposed a transmuted inverse Rayleigh distribution, which possesses the upside-down bathtub shape for its hazard rate. The fundamental properties such as mean, variance, mean deviation, order statistics, Renyi entropy and stress–strength reliability of the proposed model are explored here. Further, three methods of estimation namely maximum likelihood, least squares and maximum product spacings methods are used for estimating the unknown parameters of the transmuted inverse Rayleigh distribution, and compared through the simulation study. Finally, the applicability of the proposed distribution is shown for a set of real data representing the times between failures of the secondary reactor pumps.

Introduction

The survival/reliability analysis being popular among statisticians, engineers and other applied scientists, is a very important and interesting branch of statistics in which the lifetimes of the human beings as well as other man made systems/items are studied. Due to inherent randomness in nature, the failure behavior of the system may vary from system to system depending upon the nature of the system and can be regarded as a random phenomena. It seems logical to consider that the failure mechanism of the system can be explained through a probabilistic model. Keeping this in mind, the several lifetimes models have been proposed in statistics literature to modeling the survival data. The most popular lifetime model is Weibull distribution which has been widely exploited lifetime model in survival studies. But it has been realized that the applicability of Weibull distribution and some other two parameter exponential family models is restricted to non-monotone hazard rate (bathtub and upside-down bathtub (UBT)) data.

There are various real applications where the data show the UBT shapes for their hazard rates. [7] have studied the data of 3878 cases of breast carcinoma first seen in Edinburgh from 1954 to 1964 and noticed that the mortality was initially low in the first year of follow-up, reaching to a peak in the upcoming years, and then declining slowly. Here, the associated hazard rate is upside-down bathtub (UBT) hazard rate. For more applications related to UBT data, readers may be referred to [4], [6], [11].

In this article, we proposed the generalization of inverse Rayleigh distribution using the functional composition of the cumulative distribution function, called as a quadratic Rank Transmutation Map (QRTM), discussed by [13]. [2], [3] have used this idea of the QRTM to generalize the extreme values distribution and Weibull distribution respectively. The transmuted versions of the Rayleigh and Lindley distributions have also been, respectively proposed by [9], [8] using QRTM.

The organization of rest of the paper is as follows. In Section 2, the pdf and cdf of transmuted inverse Rayleigh distribution are derived using the quadratic Rank Transmutation Map. The statistical properties, mean, variance, mean deviation, quantile function, order statistics and entropy of transmuted inverse Rayleigh distribution are discussed in Section 3. In Section 4, the explicit derivation of stress–strength reliability of a system is derived where the strength of the system and stress under system is operated, follow the transmuted inverse Rayleigh distribution. In Section 5, the maximum likelihood estimators, least squares estimators and maximum product spacings estimators have been constructed for the estimation of the unknown parameters. To compare the behavior of the above said estimators, simulation study has also been conducted in Section 6. A set of real data has been considered to illustrate the applicability of the proposed distribution and its associated inferences in Section 7. Finally, the conclusions have been given in Section 8.

Section snippets

Model genesis

A quadratic Rank Transmutation Map, (QRTM), has the simple quadratic form $G_{R_{12}} (u) = u + λ u (1 - u), | λ | ⩽ 1,$ form which, we will also have the cumulative density function, (cdf), $G (x) = (1 + λ) F (x) - λ F^{2} (x),$ which on differentiation yields probability density function, (pdf), $g (x) = (1 + λ) f (x) - 2 λ f (x) F (x),$ where $(f (x), F (x))$ are the pdf and cdf of the baseline model, and $(g (x), G (x))$ respectively represent the pdf and cdf of the transmuted version of the baseline model. Here, the inverse Rayleigh distribution is considered as a baseline model. The inverse

Mean and mean deviation

The mean of TIR distribution is given by $μ = [X] = [1 + λ (1 - \sqrt{2})] \sqrt{(\frac{π}{2 σ^{2}})} .$ The second order moment is $E [X^{2}] = \frac{1 - λ}{2 σ^{2}} \int_{0}^{\infty} \frac{e^{- t}}{t} dt .$

The integral contained in the above expression does not converge and hence its explicit algebraic expression is not available. The mean of TIR distribution is plotted in Fig. 5. Clearly, the mean is increasing function in $σ$ and decreasing function in $λ$ for given values of $λ$ and $σ$ , respectively. The mean deviations about mean and median are defined by $M_{1} (X) = \int_{0}^{\infty} | x - μ | g (x) dx$ and $M_{2} (X) = \int_{0}^{\infty} | x - \tilde{μ} | g (x) dx,$

Stress–strength reliability

The stress–strength reliability is the measure of system performance under the harsher conditions and can be conceptually defined as $R = Probability that the systems strength is greater than environmental stress applied on that syatem = P [X > Y] .$ Where, X denotes the strength of the system and Y denotes the stress under which system performs. Let, the strength of the system follows TIR distribution with parameters $λ_{1}$ and $σ_{1}$ , and stress $Y \sim TIR (λ_{2}, σ_{2})$ . The stress–strength reliability can be calculated by the

Maximum likelihood estimates

Let $x_{1}, x_{2}, \dots, x_{n}$ be a (iid) random sample of size n from TIR distribution. Then, the likelihood function based on observed sample is defined as $\begin{matrix} L (\underset{\sim}{x}, σ, λ) = \frac{1}{σ^{2 n}} \prod_{i = 1}^{n} x_{i}^{- 3} e^{- \frac{1}{2 σ^{2}} \sum_{i = 1}^{n} x_{i}^{- 2}} \prod_{i = 1}^{n} [1 + λ (1 - 2 e^{- \frac{x_{i}^{- 2}}{2 σ^{2}}})] . \end{matrix}$ The log-likelihood function corresponding to (16), is given by $\log L = - 2 n \log (σ) - 3 \sum_{i = 1}^{n} \log (x_{i}) - \frac{\sum_{i = 1}^{n} x_{i}^{- 2}}{2 σ^{2}} + \sum_{i = 1}^{n} \log [1 + λ (1 - 2 e^{- \frac{x_{i}^{- 2}}{2 σ^{2}}})] .$ The maximum likelihood estimates (MLEs) $\hat{σ}$ and $\hat{λ}$ of $σ$ and $λ$ , respectively can be obtained as the simultaneous solution of the following likelihood equations $- \frac{2 n}{σ} + \frac{\sum_{i = 1}^{n} x_{i}^{- 2}}{σ^{3}} - \sum i = 1$

Comparison study

In this section, we compared the performances of the discussed estimators in therm of their mean squared errors. For this purpose, we generated ten thousandss random samples of different sizes for various choices of the parameters using the following Lemma defined as:

Lemma 1

If U be a standard uniform variate i.e. $U \sim U (0, 1)$ . Then, the random variable $x = {[- 2 σ^{2} \log \{\frac{1 + λ - \sqrt{{(1 + λ)}^{2} - 4 λ U}}{2 λ}\}]}^{- \frac{1}{2}},$ is said to be come from the transmuted inverse Rayleigh distribution with parameters $σ$ and $λ$ .

We took the following combinations of $n$

Model compatibility

In this section, we analyzed a real data set of times between failure of secondary reactor pumps taken from [12] to demonstrate the applicability of the proposed TIR distribution. The data are as follows: $\begin{matrix} 2.160, 0.150, 4.082, 0.746, 0.358, 0.199, 0.402, 0.101, \\ 0.605, 0.954, 1.359, 0.273, 0.491, 3.465, 0.070, 6.560, \\ 1.060, 0.062, 4.992, 0.614, 5.320, 0.347, 1.921 . \end{matrix}$ First, we plotted the total time on test (TTT) plot ([1]) to identify the shape of the hazard rate of the above data. Fig. 6 indicates that the data set

Conclusion

In this paper, we have introduced a generalization of the inverse Rayleigh distribution, called the transmuted inverse Rayleigh distribution, which is quite flexible to analyzing the lifetime data in engineering, medical and other areas where the data show the upside-bathtub shapes for their hazard rates. Some basic properties were derived. The plots of PDF, cdf, survival function and hazard functions were presented to show the flexibility of this distribution with wide variety of shapes. The

Acknowledgment

Authors thank to editor-in-chief Dr. Melvin Scott and two anonymous referees for their constructive comments/suggestions that greatly improved the last version of this manuscript. The corresponding author (Vikas Kumar Sharma) would like to thank University Grant Commission, New Delhi, for financial support.

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A new upside-down bathtub shaped hazard rate model for survival data analysis

Abstract

Introduction

Section snippets

Model genesis

Mean and mean deviation

Stress–strength reliability

Maximum likelihood estimates

Comparison study

Model compatibility

Conclusion

Acknowledgment

Nonlinear Anal.

How to identify a bathtub hazard rate

IEEE Trans. Reliab.

Transmuted weibull distribution: a generalization of the weibull probability distribution

Eur. J. Pure Appl. Math.

Log-logistic regression models for survival data

Appl. Stat.

Estimating parameters in continuous univariate distributions with a shifted origin

J. R. Stat. Soc. Ser. B Methodol.

Logistic regression, survival analysis, and the Kaplan–Meier curve

J. Am. Stat. Assoc.