Compound distributions motivated by linear failure rate

Statistics & Operations Research Transactions SORT 40 (1) January-June 2016, 177-200 Statistics & Operations Research Transactions © Institut d’Estad´ıstica de Catalunya [email protected] ISSN: 1696-2281 eISSN: 2013-8830 www.idescat.cat/sort/ Compound distributions motivated by linear failure rate Narjes Gitifar1, Sadegh Rezaei1and Saralees Nadarajah2 Abstract Motivated by three failure data sets (lifetime of patients, failure time of hard drives and failure time of a product), we introduce three different three-parameter distributions, study basic mathematical properties, address estimation by the method of maximum likelihood and investigate finite sample performance of the estimators. We show that one of the new distributions provides a better fit to each data set than eight other distributions each having three parameters and three distributions each having two parameters. MSC: 62E15. Keywords: Linear failure rate distribution, maximum likelihood estimation, Poisson distribution. 1. Introduction Systems or components having linear failure rates are common in real life. Examples include concrete under multiaxial states of stress (Donida and Mentrasti, 1982), composite laminates with transverse shear (Reddy and Reddy, 1992) and load-sharing systems (Sutar and Naik-Nimbalkar, 2014). There are also many real data sets that exhibit approximately linear failure rates at least in the upper tails. We present three examples. The first data set, due to Dispenzieri et al. (2012), consists of the number of days from visit to clinic until death of 100 patients. The data result from a study of the relationship between serum free light chain and mortality. The 100 patients were selected randomly from a total of 7874 patients, including patients who had not died. The patients who had died were diagnosed with monoclonal gammapothy. 1Amirkabir University of Technology, Tehran, IRAN, email: srez[email protected] 2University of Manchester, Manchester M13 9PL, UK Received: July 2015 Accepted: April 2016 178 Compound distributions motivated by linear failure rate Table 1: Summary statistics of the three data sets. Statistic Data set 1 Data set 2 Data set 3 minimum 0.0054 0.0053 0.0035 first quartile 0.3368 0.3977 0.318 median 0.4774 0.7770 0.4211 third quartile 0.7412 0.9304 0.5581 maximum 0.9514 1.4040 0.6878 0.0 0.2 0.4 0.6 0.8 0 2 4 6 8 Failure time / 5000 Failure rate function Figure 1: Kaplan-Meier estimate of the failure rate function of the patient data of Dispenzieri et al. (2012). The second data set from https://www.backblaze.com/hard-drive-test-data.html is one hundred failure times in days of hard drives. The data were selected randomly from a total of 52422 hard drives, which included hard drives which had not failed. The data were collected by a large backup storage provider over two years. On each day, the SelfMonitoring, Analysis, and Reporting Technology (SMART) statistics of operational drives were recorded. When a hard drive was no longer operational, it was marked as a failure and removed. The third data set due to Hong and Meeker (2013) is one hundred failure data in weeks of a product called Product D2 that is used in offices or residences. Product D2 is “similar to a high-end copying machine connected to the Internet and installed with a Narjes Gitifar, Sadegh Rezaei and Saralees Nadarajah 179 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0 1 2 3 4 5 6 Failure time / 1000 Failure rate function Figure 2: Kaplan-Meier estimate of the failure rate function of the hard drive failure data. smart chip to record the number of pages that have been printed, as a function of time” (Hong and Meeker, 2013, page 136). The one hundred data were selected randomly from a total of 1800 observations. All three data sets are presented in the appendix. Kaplan-Meier estimates of the failure rate function (FRF) of the three data sets are shown in Figures 1, 2 and 3. We can see that the FRFs are approximately linear at least in the upper tails. The histogram of the three data sets are shown in Figures 8, 9 and 10. Some summary statistics of the three data sets are shown in Table 1. We suppose that the patient’s body or the hard drive or the product D2 is made of a number of components say Nworking independently in series. The assumption of the series structure is more reasonable than a parallel structure because it is unlikely that a patient’s body will fail if and only if all its components fail or that a hard drive will break if and only if all its components break or that a product will fail if and only if all its components fail. It is more likely that a patient’s body will fail if and only if any of its components fails or that a hard drive will break if and only if any of its components breaks or that a product will fail if and only if any of its components fails. However, in practice the components may not work independently. The distribution of the failure 180 Compound distributions motivated by linear failure rate 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0 5 10 15 Failure time / 100 Failure rate function Figure 3: Kaplan-Meier estimate of the failure rate function of the failure data of Hong and Meeker (2013). time may not have a closed form if we assume that the components are dependent, see (2) below and its discussion. We shall suppose independence for simplicity. The number Nmay vary from one patient to another or one hard drive to another or one product to another. It may depend on the type of hard drive, type of patient, type of product, weight, length, and so on. So, we may take Nas a random variable. The failure time can be written as X=min(Y1,Y2,...,YN), where Y1,Y2,...,YNdenote the failure times of the Ncomponents. Standard models for Nare the geometric, zero truncated Poisson, logarithmic, zero truncated negative binomial and zero truncated binomial distributions. For simplicity, we shall consider only the first three since each of them has one parameter. The last two distributions have two parameters each. That is, we take Nto have one of the following probability mass functions (PMFs): Pr(N=n) = (1−λ)λn−1 for 0 <λ<1 and n=1,2,...; Pr(N=n) = λn (eλ−1)n! Narjes Gitifar, Sadegh Rezaei and Saralees Nadarajah 181 for λ>0 and n=1,2,...; or Pr(N=n) = −1 ln(1−λ) λn n for 0 <λ<1 and n=1,2,.... Since the failure rate for the three data sets is approximately linear at least in the upper tail (see Figures 1, 2 and 3), we shall suppose Y1,Y2,... too follow a distribution that has a linear FRF. The distribution characterized by a linear failure rate is actually known as the linear failure rate (LFR) distribution due to Bain (1974). Its probability density function (PDF) and cumulative distribution function (CDF) are specified by fY(y;γ,β) = (β+γy)exp−βy−γ 2y2 and FY(y;γ,β) = 1−exp−βy−γ 2y2, respectively, for y>0, β≥0, γ≥0 and β+γ>0. It is easy to see that the FRF is hY(y;γ,β) = β+γy, a linear function of y. Both parameters, βand γ, are referred to as scale parameters. The distribution of X=min(Y1,Y2,...,YN)can now be derived given the assumptions that Nis either geometric, Poisson or logarithmic and Y1,Y2,... are independent LFR random variables independent of N. In the general case, the CDF and the PDF of Xcan be derived as FX(x) = Pr[min(Y1,Y2,...,YN)<x] = 1−Pr[min(Y1,Y2,...,YN)>x] =1− ∞ X n=1 Pr[min(Y1,Y2,...,Yn)>x|N=n]Pr(N=n) =1− ∞ X n=1 Pr[Y1>x,Y2>x,...,Yn>x]Pr(N=n) =1− ∞ X n=1 Prn[Y>x]Pr(N=n) = 1− ∞ X n=1 [1−FY(x)]nPr(N=n) and fX(x) = fY(x) ∞ X n=1 n[1−FY(x)]n−1Pr(N=n), 182 Compound distributions motivated by linear failure rate respectively. In the case Nis geometric, we obtain fX(x;λ,γ,β) = (1−λ)(β+γx)exp−βx−γ 2x2 h1−λexp−βx−γ 2x2i2, which we shall refer to as the linear failure rate geometric (LFRG) distribution and write X∼LFRG(λ,γ,β)for 0 <λ<1, β≥0, γ≥0 and β+γ>0. In the case Nis zero truncated Poisson, we obtain fX(x;λ,γ,β) = λ1−e−λ−1(β+γx)exp−λ−βx−γ 2x2exphλexp−βx−γ 2x2i, (1) which we shall refer to as the linear failure rate Poisson (LFRP) distribution and write X∼LFRP(λ,γ,β)for λ>0, β≥0, γ≥0 and β+γ>0. In the case Nis logarithmic, we obtain fX(x;λ,γ,β) = − λ(β+γx)exp−βx−γ 2x2 ln(1−λ)h1−λexp−βx−γ 2x2i, which we shall refer to as the linear failure rate logarithmic (LFRL) distribution and write X∼LFRL(λ,γ,β)for 0 <λ<1, β≥0, γ≥0 and β+γ>0. These distributions do not have linear failure rates. But hX(y;λ,γ,β)∼hY(y;γ,β)∼γyas y→∞. So, the assumption of linear failure rate for Y1,Y2,... guarantees that linear failure rate holds for Xtoo at least in the upper tail. The limiting cases of the LFRG, LFRP and LFRL distributions as λ↓0 is the LFR distribution. The LFRG and LFRL distributions limit to a degenerate distribution as λ↑1. If Y1,Y2,...are dependent random variables then the CDF of Xcan only be expressed as FX(x) = 1− ∞ X n=1 Pr[Y1>x,Y2>x,...,Yn>x]Pr(N=n).(2) This cannot be reduced to a closed form unless the joint dependence of (Y1,Y2,...,Yn) takes a very simple form. In the rest of this section, Section 2 and Section 3, we shall focus on the LFRP distribution. The details for the LRFG and LRFL distributions can be derived similarly. One of the most popular models for counts is the zero truncated Poisson distribution. Some of its recent applications can be found in van der Heijden et al. (2003), Elhai et al. (2008), Ginebra and Puig (2010) and Xu and Hu (2011). Narjes Gitifar, Sadegh Rezaei and Saralees Nadarajah 183 Figure 4: Probability density function of the LFRP distribution for (a) γ=0.5and β=1, (b) γ=1and β=0.5, (c) β=0.05 and λ=3, (d) γ=2and λ=1. Possible shapes of (1) are shown in Figure 4. We see that both monotonically decreasing and unimodal shapes are possible. The mode of (1) is the root of γ β+γx−β−γx=λ(β+γx)exp−βx−γ 2x2. Furthermore, fX(0) = λβ/1−e−λand fX(x)∼λγ 1−e−λ−1 xexp−λ−βx−γ 2x2 as x→∞. The lower tail of the PDF has a fixed point while its upper tail decays exponentially. The CDF and FRF of X∼LFRP(λ,γ,β)are FX(x) = 1 eλ−1neλ−exphλexp−βx−γ 2x2io and hX(x) = (β+γx)λexp−βx−γ 2x2 1−exph−λexp−βx−γ 2x2i,(3) 184 Compound distributions motivated by linear failure rate Figure 5: Failure rate function of the LFRP distribution for (a) γ=0.5and β=1, (b) γ=1and β=0.5, (c) β=0.05 and λ=3, (d) λ=3and γ=0.5. respectively, for x>0, λ>0, β≥0, γ≥0 and β+γ>0. Figure 5 shows possible shapes of (3) for different parameter values. We see that the LFRP distribution can exhibit increasing, decreasing and upside down bathtub shapes for the failure rate. The LFR distribution can exhibit only increasing or constant failure rates. Reliability and survival analysis often encounter upside down bathtub failure rates. Examples can be found in redundancy allocations in systems (Singh and Misra, 1994) and mortality modelling (Silva et al., 2010). The mode or the anti-mode of (3) is the root of γ β+γx−β−γx=−λ(β+γx)exp−βx−γ 2x2nexphλexp−βx−γ 2x2i−1o−1. Furthermore, hX(0) = λβ/1−e−λand hX(x)∼γxas x→∞. The lower tail of the FRF has a fixed point. As already noted, the upper tail of the FRF of the LFRP distribution behaves in the same manner as that of the LFR distribution. Yet the former does exhibit upside down bathtub failure rates while the latter does not. The qth quantile of X∼LFRP(λ,γ,β)say xqdefined by FX(xq) = qis xq=−β γ+sβ2 γ2−2 γlnnln[eλ−q(eλ−1)] 1 λo. Narjes Gitifar, Sadegh Rezaei and Saralees Nadarajah 185 In particular, the median of Xis Median(X) = −β γ+v u u tβ2 γ2−2 γln(lneλ−1 2(eλ−1)1 λ). Quantiles are useful for estimation and simulation. Several other distributions have been introduced in the literature by taking X= min(Y1,Y2,...,YN), where Nis a geometric, zero truncated Poisson or a logarithmic random variable: By taking Nto be a geometric random variable and Y1,Y2,... to be independent and identical Weibull random variables, Barreto-Souza et al. (2011) introduced the three-parameter Weibull geometric (WG) distribution given by the PDF f(x) = (1−λ)βγ−βxβ−1exph−(x/γ)βi n1−λexph−(x/γ)βio2 for x>0, 0 <λ<1, β>0 and γ>0; By taking Nto be a zero truncated Poisson random variable and Y1,Y2,...to be independent and identical Weibull random variables, Lu and Shi (2012) introduced the three-parameter Weibull Poisson (WP) distribution given by the PDF f(x) = λβγ−βxβ−1expn−(x/γ)β+λexph−(x/γ)βio exp(λ)−1 for x>0, λ>0, β>0 and γ>0; By taking Nto be a logarithmic random variable and Y1,Y2,... to be independent and identical Weibull random variables, Ciumara and Preda (2009) introduced the three-parameter Weibull logarithmic (WL) distribution given by the PDF f(x) = − (1−λ)βγ−βxβ−1exph−(x/γ)βi lnλn1−(1−λ)exph−(x/γ)βio for x>0, 0 <λ<1, β>0 and γ>0; By taking Nto be a geometric random variable and Y1,Y2,... to be independent and identical generalized exponential random variables, Mahmoudi and Jafari (2012) introduced the three-parameter generalized exponential geometric (GEG) distribution given by the PDF f(x) = (1−λ)βγ exp(−γx)[1−exp(−γx)]β−1 nλ[1−exp(−γx)]β−1o2 192 Compound distributions motivated by linear failure rate 1. the biases for each parameter are generally positive; 2. the biases for each parameter decrease to zero as n→∞; 3. the biases appear smallest for the parameter, λ; 4. the mean squared errors for each parameter decrease to zero as n→∞; 5. the mean squared errors appear smallest for the parameter, λ; 6. the mean squared errors appear largest for the parameter, β; 7. the biases and mean squared errors for each parameter appear reasonably small for all n≥60. We have presented results for only one choice for (λ,β,γ), namely that (λ,β,γ) = (1,1,1). But the results were similar for a wide range of other choices. In particular, the biases and mean squared errors for each parameter appeared reasonably small for all n≥60. The three real data sets in Section 4 each has a sample size greater than or equal to sixty. So, we can expect the estimates in Section 4 to be reasonable. 4. Real data applications Here, we return to the three data sets to illustrate the applicability of the LFRP distribution. The following distributions were fitted to each data: the LFR, LFRG, LFRP, LFRL, WG, WP, WL, GEG, GEP and GEL distributions. We also fitted the Weibull and gamma distributions given by the PDFs f(x) = βxβ−1 γβexp"−x γβ# and f(x) = xβ−1 γβΓ(β)exp−x γ, respectively, for x>0, α>0 and β>0. Each distribution was fitted by the method of maximum likelihood. The parameter estimates, standard errors, −lnL, AIC values and BIC values are given in Tables 2, 3 and 4. The standard errors were computed by inverted the observed information matrices. We see that the LFRP distribution yields the smallest −lnL, the smallest AIC and the smallest BIC for each data set. It provides a significantly better fit than the LFR distribution for each data set, as judged by the likelihood ratio test. The standard errors for the LFRP distribution appear reasonable, as they are smaller than the parameter estimates. Narjes Gitifar, Sadegh Rezaei and Saralees Nadarajah 193 Table 2: Parameter estimates, standard errors, log-likelihood, AIC and BIC for the twelve distributions fitted to the patient data of Dispenzieri et al. (2012). Distribution b λSE b βSE bγSE −lnLAIC BIC LFR 0.348 0.176 5.071 0.739 7.747 19.494 24.704 LFRG 0.001 0.000 0.348 0.176 5.069 0.738 7.751 21.503 29.318 LFRP 1.894 0.851 1.132 0.661 5.591 1.212 4.960 15.921 23.736 LFRL 0.001 0.000 0.342 0.173 5.063 0.734 7.750 21.500 29.315 WG 0.999 0.000 1.839 0.156 0.561 0.032 11.838 29.676 37.491 WP 2.230 0.910 1.434 0.204 0.394 0.065 8.818 23.637 31.452 WL 0.001 0.000 1.848 0.157 0.563 0.032 11.841 29.682 37.498 GEG 0.999 0.000 2.012 0.285 2.925 0.307 21.182 48.365 56.180 GEP 3.850 1.032 1.095 0.326 3.947 0.377 11.689 29.377 37.193 GEL 0.001 0.000 2.011 0.285 2.923 0.307 21.184 48.368 56.183 Weibull 1.839 0.156 0.561 0.032 11.837 27.674 32.885 Gamma 2.068 0.272 0.245 0.036 19.387 42.774 47.985 Table 3: Parameter estimates, standard errors, log-likelihood, AIC and BIC for the twelve distributions fitted to the hard drive failure data. Distribution b λSE b βSE bγSE −lnLAIC BIC LFR 0.296 0.138 2.530 0.393 42.043 88.087 93.297 LFRG 0.000 0.000 0.292 0.135 2.465 0.384 42.072 90.143 97.959 LFRP 1.753 0.691 0.776 0.375 2.841 0.572 38.849 83.698 91.514 LFRL 0.000 0.000 0.296 0.138 2.530 0.393 42.044 90.088 97.904 WG 0.999 0.000 1.751 0.152 0.774 0.046 46.993 99.986 107.801 WP 1.831 0.738 1.484 0.189 0.584 0.079 43.848 93.695 101.511 WL 0.001 0.000 1.772 0.154 0.774 0.045 46.984 99.967 107.783 GEG 0.999 0.000 1.842 0.257 2.017 0.216 55.885 117.769 125.585 GEP 3.569 1.080 1.016 0.327 2.664 0.256 47.407 100.814 108.629 GEL 0.000 0.000 1.876 0.263 2.035 0.217 55.887 117.774 125.589 Weibull 1.772 0.154 0.775 0.045 46.982 97.964 103.174 Gamma 1.902 0.249 0.369 0.055 54.304 112.608 117.818 The parameter estimates and the log-likelihood values of the LFRG and LFRL distributions are very close for all three data sets. This suggests that the likelihood surfaces for the LFRG and LFRL distributions attain their maximum points along the border corresponding to λ=0. We noted earlier LFRG and LFRL distributions reduce to the LFR distribution as λ↓0. So, the fits of LFRG and LFRL distributions do not improve on the fit of the LFR distribution for the three data sets. 194 Compound distributions motivated by linear failure rate Table 4: Parameter estimates, standard errors, log-likelihood, AIC and BIC for the twelve distributions fitted to the failure data of Hong and Meeker (2013). Distribution b λSE b βSE bγSE −lnLAIC BIC LFR 0.028 0.061 9.349 0.968 −32.148 −60.296 −55.086 LFRG 0.000 0.000 0.047 0.081 9.523 1.000 −32.069 −58.138 −50.323 LFRP 5.023 1.719 1.361 1.458 15.188 3.681 −48.555 −91.111 −83.295 LFRL 0.000 0.000 0.019 0.052 9.389 0.967 −32.133 −58.267 −50.451 WG 0.999 0.000 3.149 0.256 0.482 0.016 −44.743 −83.485 −75.670 WP 4.940 1.837 1.703 0.313 0.287 0.054 −46.938 −87.876 −80.061 WL 0.000 0.000 3.146 0.255 0.483 0.016 −44.745 −83.489 −75.674 GEG 0.003 0.000 4.552 0.476 0.753 0.133 −29.354 −52.708 −44.893 GEP 8.160 1.966 1.859 0.584 7.352 0.588 −42.532 −79.064 −71.249 GEL 2.082×10−50.000 5.546 0.918 5.304 0.442 −25.126 −44.253 −36.437 Weibull 3.146 0.255 0.483 0.016 −44.745 −85.489 −80.279 Gamma 5.371 0.735 0.081 0.012 −31.814 −59.629 −54.418 Failure time / 5000 Fitted PDFs 0.0 0.2 0.4 0.6 0.8 0.5 1.0 1.5 2.0 0.0 0.2 0.4 0.6 0.8 0.5 1.0 1.5 2.0 0.0 0.2 0.4 0.6 0.8 0.5 1.0 1.5 2.0 0.0 0.2 0.4 0.6 0.8 0.5 1.0 1.5 2.0 0.0 0.2 0.4 0.6 0.8 0.5 1.0 1.5 2.0 0.0 0.2 0.4 0.6 0.8 0.5 1.0 1.5 2.0 0.0 0.2 0.4 0.6 0.8 0.5 1.0 1.5 2.0 0.0 0.2 0.4 0.6 0.8 0.5 1.0 1.5 2.0 0.0 0.2 0.4 0.6 0.8 0.5 1.0 1.5 2.0 0.0 0.2 0.4 0.6 0.8 0.5 1.0 1.5 2.0 0.0 0.2 0.4 0.6 0.8 0.5 1.0 1.5 2.0 0.0 0.2 0.4 0.6 0.8 0.5 1.0 1.5 2.0 0.0 0.2 0.4 0.6 0.8 0.5 1.0 1.5 2.0 LFR LFRG LFRP LFRL WG WP WL GEG GEP GEL Weibull Gamma Figure 8: Density plots for the twelve distributions fitted to the patient data of Dispenzieri et al. (2012). Narjes Gitifar, Sadegh Rezaei and Saralees Nadarajah 195 Failure time / 1000 Fitted PDFs 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.5 1.0 1.5 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.5 1.0 1.5 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.5 1.0 1.5 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.5 1.0 1.5 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.5 1.0 1.5 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.5 1.0 1.5 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.5 1.0 1.5 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.5 1.0 1.5 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.5 1.0 1.5 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.5 1.0 1.5 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.5 1.0 1.5 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.5 1.0 1.5 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.5 1.0 1.5 LFR LFRG LFRP LFRL WG WP WL GEG GEP GEL Weibull Gamma Figure 9: Density plots for the twelve distributions fitted to the hard drive failure data. The density plots for the fit of the distributions for the three data sets are shown in Figures 8 to 10. The fitted PDFs of the LFRP distribution captures the observed histograms better than others. Hence, we can say that the LFRP distribution provides the best fit for at least three real data sets. The parameter estimates of the best fitting LFRP distribution for the three data sets can be interpreted as follows: •the patient’s body can be modelled as a series system having an average of b λ/h1−e−b λi=2.2 components with the 95 percent confidence interval (0.37,4.09), where the failure rate of each component is linear with an intercept of 1.132 and a slope of 5.591. That is, the failure rate of each component at time zero is 1.132 and the failure rate increases by 5.591 for every unit increase in time; •the hard drive can be modelled as a series system having an average of b λ/h1−e−b λi=2.1 components with the 95 percent confidence interval (1.26,2.98), where the failure rate of each component is linear with an intercept of 0.776 and 196 Compound distributions motivated by linear failure rate a slope of 2.841. That is, the failure rate of each component at time zero is 0.776 and the failure rate increases by 2.841 for every unit increase in time; •the product D2 can be modelled as a series system having an average of b λ/h1−e−b λi=5.1 components with the 95 percent confidence interval (−1.97,12.08), where the failure rate of each component is linear with an intercept of 1.361 and a slope of 15.188. That is, the failure rate of each component at time zero is 1.361 and the failure rate increases by 15.188 for every unit increase in time. Note that λ/1−e−λis the expected value of a zero truncated Poisson random variable. The stated confidence intervals were obtained by the delta method. Failure time / 100 Fitted PDFs 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 LFR LFRG LFRP LFRL WG WP WL GEG GEP GEL Weibull Gamma Figure 10: Density plots for the twelve distributions fitted to the failure data of Hong and Meeker (2013). 5. Conclusions We have proposed three distributions motivated by three failure data sets: the linear failure rate geometric, linear failure rate Poisson and linear failure rate logarithmic distributions. Each of these distributions has three parameters. Narjes Gitifar, Sadegh Rezaei and Saralees Nadarajah 197 We have studied mathematical properties and estimation issues for the linear failure rate Poisson distribution. We have shown in particular that its failure rate function can be decreasing, increasing and upside down bathtub shaped, more varied than the failure rate function of the linear failure rate distribution. Among the twelve distributions fitted to the three data sets, the linear failure rate Poisson distribution gave the best fit. The adequacy of fits was assessed in terms AIC values, BIC values and density plots. A future work is to estimate the parameters of the linear failure rate Poisson distribution by the method of percentiles, the method of probability weighted moments, the method of least squares, the method of weighted least squares, the method of generalized moments, and other methods. Another future work is to propose bivariate and multivariate generalizations of the linear failure rate Poisson distribution. Appendix: Three data sets The first data is 0.1102 0.2390 0.4598 0.7146 0.2608 0.0838 0.8746 0.1578 0.3358 0.0198 0.7192 0.7916 0.4486 0.4080 0.6048 0.3686 0.4686 0.5418 0.3760 0.8684 0.1572 0.4860 0.0118 0.4732 0.5450 0.8982 0.5674 0.2602 0.4330 0.3608 0.3648 0.5124 0.1360 0.7548 0.8960 0.4816 0.0818 0.3268 0.9514 0.8650 0.3372 0.5438 0.5392 0.5750 0.3672 0.6694 0.3068 0.2536 0.3756 0.3962 0.4690 0.3416 0.6430 0.9104 0.4426 0.7280 0.7370 0.7666 0.6420 0.2000 0.3588 0.6632 0.8752 0.8934 0.6526 0.1370 0.5222 0.7746 0.9230 0.6422 0.3298 0.7286 0.0054 0.3754 0.2448 0.9466 0.3256 0.3726 0.0516 0.4496 0.7850 0.8670 0.0758 0.5174 0.7742 0.5464 0.6152 0.7594 0.8310 0.4036 0.8954 0.7970 0.3638 0.0142 0.7998 0.1658 0.4572 0.7540 0.9220 0.3688 For computational stability with fitting distributions, we have divided each observation by 5000. The second data is 1.293458333 0.251375000 1.265458333 1.404000000 1.280416667 1.201500000 1.193458333 0.340333333 1.101166667 1.059250000 1.360541667 1.245125000 1.098041667 1.049875000 1.167875000 1.271500000 1.182000000 0.925916667 0.963333333 1.119666667 0.867791667 0.845375000 0.803416667 0.323500000 1.165083333 1.065958333 1.103583333 1.035583333 1.173958333 0.886916667 0.789958333 0.671791667 0.782666667 0.534125000 0.691000000 0.813750000 0.773416667 0.629291667 0.520291667 0.635000000 198 Compound distributions motivated by linear failure rate 0.695041667 0.712625000 0.428000000 0.423208333 0.615541667 0.254416667 0.160791667 0.125083333 0.416791667 0.215416667 0.214958333 0.185375000 0.228458333 0.206958333 0.228833333 0.190083333 0.205000000 0.007458333 0.192750000 0.227666667 0.155916667 0.179791667 0.018625000 0.169458333 0.066416667 0.005333333 0.115416667 0.080375000 0.495833333 0.854916667 0.498750000 0.902875000 0.967958333 0.786916667 0.920583333 0.943875000 0.807666667 0.761708333 0.733583333 1.043833333 0.893583333 0.746500000 0.736583333 0.880500000 0.889708333 0.780666667 0.668041667 0.861291667 0.711916667 0.718500000 0.863041667 0.908000000 0.833791667 0.671416667 0.826083333 0.823000000 0.784375000 0.667833333 0.669750000 0.835750000 For computational stability with fitting distributions, we have divided each observation by 1000. The third data is 0.222673061 0.257639905 0.328155859 0.515672484 0.583401130 0.642256077 0.621521735 0.587506929 0.594755485 0.316753044 0.550884304 0.312962380 0.516646945 0.546445582 0.600493703 0.297813235 0.332441913 0.333245894 0.364800151 0.429097225 0.627439232 0.313363071 0.579554283 0.391397547 0.125167305 0.541816854 0.665764686 0.398880874 0.402492151 0.423982077 0.428143776 0.341767913 0.514537781 0.686683383 0.333088363 0.249962985 0.226748439 0.286643595 0.645490088 0.584664074 0.397377064 0.609634794 0.353187577 0.536304985 0.406031202 0.586163204 0.648786836 0.516497130 0.318475607 0.494774308 0.436782434 0.245923132 0.618409876 0.255245760 0.464312202 0.454133994 0.387982016 0.218311879 0.526363495 0.418258490 0.272839591 0.151997829 0.492728139 0.290973052 0.471553883 0.363069573 0.668371780 0.501805967 0.600306622 0.477109810 0.515188714 0.283784543 0.600625759 0.299420135 0.368553098 0.653382502 0.687845701 0.379423961 0.279504337 0.407995757 0.685695223 0.259685231 0.514854899 0.501119729 0.003522425 0.672089253 0.630145059 0.310811342 0.384073475 0.388312955 0.268080935 0.437408445 0.634243302 0.239656858 0.391844012 0.347107733 0.499160234 0.325770026 0.290634387 0.371908794 For computational stability with fitting distributions, we have divided each observation by 100. 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