Proportional Hazard Inverse Weibull Distribution and Associated Inference

Corresoonding Author: Suja M. Aboukhamseen Department of Statistics and Operations Research, Faculty of Science, Kuwait University, P.O. Box 5969, Safat 13060, Kuwait Email: saboukhamseen@yahoo.com Abstract: In this study, we have considered the proportional hazard version of the inverse Weibull distribution. It has been shown that the density and hazard rate functions are unimodal and the mean residual life function is bathtub shaped. For the considered model, many functions of the parameters, such as the mean, variance, coefficient of variation and the critical points of the density, hazard rate and mean residual life functions cannot be given in explicit form. As a result, the variances of the maximum likelihood estimators of such functions cannot be given in explicit form to construct the usual asymptotic confidence interval. In this study, we use the percentile bootstrap estimation method to estimate such variances to construct the asymptotic confidence intervals. The asymptotic confidence intervals are compared with the bootstrap confidence intervals. Simulation studies are carried out to examine the performance of the maximum likelihood and bootstrap estimators. An example is provided to illustrate the procedure.


Introduction
The proportional hazard model has been widely used to analyze survival data. More specifically, let Y be a random variable with survival function S 1 (y) and hazard rate function λ 1 (y). Then a random variable X with survival function S(x) = [S 1 (x)] α , α >0, has proportional hazard rate function λ(x) = αλ 1 (x). The parameter α is called a fraility parameter, see for example Marshall and Olkin (2007).
The inverse Weibull (IW) distribution with survival function given by can be used to model a variety of failure characteristics such as infant mortality, useful life and wear out periods. For the structural properties of the IW distribution, we refer the reader Khan et al. (2008a;2008b). The maximum likelihood and the least square estimations of the inverse Weibull distribution have been studied by Calabria and Pulcini (1990).
In various situations, it is more appropriate to employ the proportional hazard model than the original model. In this study, we are interested in the Proportional hazard Inverse Weibull (PHIW) distribution with survival function given by Note that when α = 1, the PHIW distribution reduces to the IW distribution. Also, when β = 1, the PHIW distribution reduces to the Generalized Inverted Exponential (GIE) distribution studied by Abouammoh and Alshingiti (2009).
It may be noted that analogues to the proportional hazard model, the proportional reversed hazard rate model has also been studied in the literature. In this case, the power of the distribution function is considered instead of the power of the survival function. For more information on the proportional reversed hazard rate model, we refer the reader to Gupta and Gupta (2007).
The main purpose of this paper is to study the bootstrap estimation of the parameters of the PHIW distribution and the associated functions including the mean, variance, coefficient of variation, the probability density, the hazard rate and mean residual life functions. The paper is organized as follows: In section 2, we consider the shape of the density, hazard rate and mean residual life functions. The moments (including the negative moments) and the associated measures are given in section 3. In section 4, we study the maximum likelihood and bootstrap estimation of functions of the parameters. Simulation studies are carried out in section 5 to investigate the performance of the estimators of various functions of the parameters. An application is provided in section 6. Finally, some conclusion and comments are given.

Probability Density Function
The Probability Density Function (PDF) of the PHIW distribution, with survival function (2), is given by: The following theorem investigates the shape of the PDF of the PHIW distribution.

Theorem 1
The

Proof.
It is clear that f(0) = f(∞) = 0. The first derivative of f(x) can be written in the form In the following, we show that w(x) changes sign from negative to positive. Since Since w(0) = −β and w(∞) = αβ + 1, it follows that w(x) changes sign from negative to positive. This implies that ϕ′(x) also changes sign from negative to positive, since −y′(x) >0. Finally, since ϕ(0) = ∞ and ϕ(∞) = 0, ϕ(x) decreases from infinity to a negative value then increases to zero, i.e., ϕ(x) changes sign from positive to negative. That is, f′ (x) changes sign from positive to negative proving the unimodality of f(x).

Hazard Rate Function
The Hazard Rate Function (HRF) of the PHIW distribution, with survival function (2) and PDF (3), is given by The following theorem investigates the shape of the HRF of the PHIW distribution.

Mean Residual Life Function
Using the substitution y = λx −β , the mean of the PHIW distribution is given by For 0 < αβ ≤ 1, using the inequality 1 − e −y ≥ y e −y , for all y ≥ 0, we have ( ) For all x > 0, the Mean Residual Life Function (MRLF) of the PHIW distribution, with survival function (2) and PDF (3), is given by For α = 1, 2, 3, . . . and β > 1, using the substitution z = λy −β , we have For the special case, α = 1, i.e., IW distribution, we have Before we determine the shape of µ(x), we need the following lemma. Lemma 1 Gupta and Akman (1995) If h(x) is upside-down bathtub shape with a unique critical point x h and f(0)µ < 1, then µ(x) is bathtub shape with a unique critical point x m . The critical point x m is the solution of the non-linear equation Moreover, x m < x h . Using the condition αβ > 1 and Lemma 1, we are now able to determine the shape of the MRLF of the PHIW distribution as follows.

Moments and Associated Measures
The kth raw moment of the PHIW distribution is given by For the special case, α = 1, i.e., IW distribution, we have: In this case, the mean, variance and coefficient of variation, respectively, are Note that if X ∼ PHIW(α, β, γ), then Z = 1/X has the exponentiated Weibull distribution (Mudholkar and Srivastava 1993) with cumulative distribution function Now the negative moments of the PHIW distribution are given by where, for any real number c: is the Pochhammer symbol, for more details see Choudhury (2005).

Maximum Likelihood Estimation
Let x 1 , x 2 , ..., x n be a random sample of size n from the PHIW distribution with PDF (3). The log-likelihood function is given by Note that (8) The solutions of the last two nonlinear equations provide the MLEs β and λ of the parameters β and λ, from which the MLE ( ), α ξ β λ = of the parameter α is obtained.
The observed information matrix is Under fairly mild regularity conditions, see Lehmann and Casella (1998) For a real-valued function of the parameters τ (θ), we have The asymptotic 100(1-α)% confidence interval for τ (θ) is given by where, z α/2 is the upper α/2 percentile of the standard normal distribution. When the function τ (θ) has no explicit expression in the parameters, as the case in this study, then has also no explicit expression and is usually estimated using simulation-based methods such as percentile bootstrap, see for example Davison and Hinkley (1997), where B random samples are selected with replacement from the given data set. For the jth bootstrap sample, the

Simulation Studies
Simulation of an observation x from the PHIW distribution is obtained by solving the equation S(x) = 1u where u is an observation from the uniform distribution over (0,1). It follows that

Analysis of the Simulation Studies
• For the mean, variance and coefficient of variation (Table 2), both confidence intervals are performing almost the same in terms of the coverage probability as well as their width • In the case of the estimation of critical points of the PDF, HRF and MRLF (Table 3), the coverage probability of both confidence intervals is the same while the bootstrap intervals are outperforming the asymptotic intervals in terms of their width Thus, overall, the simulation studies indicate that both kind of confidence intervals are performing similar in terms of the coverage probabilities as well as their width. Figure 4 shows approximately normal distribution for the sampling distribution of the MLEs of the mean, variance and coefficient of variation, indicating that the corresponding constructed asymptotic confidence intervals are appropriate. On the other hand, Figure 5 shows skewed sampling distribution of the MLEs of the critical point of the PDF, HRF and MRLF, indicating that the constructed asymptotic confidence intervals are inappropriate.

Illustrative Example
In this section, we present an illustrative example based on a real-life data set representing 98 measurements of the annual maxima of river flows (m 3 s −1 ) from 1900 to 1997 at BSWN station in Shunde area, Pearl River basin, China (Shao et al., 2004 A summary of the basic descriptive sample statistics of this data set is given in Table 4. Table 5 shows the MLEs of the parameters, the estimated log-likelihood, Kolomogrov-Smirnov and Anderson-Darling goodness-of-fit tests of the fitted PHIW model. Both tests indicate that the fitted PHIW model is suitable for the given data set. This conclusion is also supported by the diagnostic plots presented in Fig. 6. Figure 7 shows the location of the percentile bootstrap estimates of the critical points of the PDF, HRF and MRLF of the fitted PHIW model. The histograms of the percentile bootstrap estimates of the mean, variance, coefficient of variation and the critical points of the PDF, HRF and MRLF are presented in Fig.  8 which indicates that the histograms of the critical points are heavily skewed to the right.      Table 6 compares the estimates of the considered functions of the parameters, their standard error and the corresponding confidence intervals based of asymptotic and percentile bootstrap estimation procedures.

Conclusion and Comments
In this study, we have studied the proportional hazard inverse Weibull distribution and the maximum likelihood/bootstrap estimation of various functions of its parameters. It has been shown that the density and hazard rate functions of the proposed model are unimodal for all values of the parameters and its mean residual life function is bathtub shaped under certain condition on two of its parameters. The simulation studies indicate that the asymptotic and bootstrap confidence intervals of various functions of the parameters perform similarly in terms of the coverage probabilities as well as the length of the confidence intervals. We hope that our investigation will provide an alternative way to analyze data which exhibit the characteristics similar to the proposed model.