5.2 - Gamma Distribution

Gamma Distribution App

SMRD::smrd_app("distribution_gamma")

5.3 - Generalized Gamma Distribution

GENG Distribution App

SMRD::smrd_app("distribution_geng")

5.4 - Extendend Generalized Gamma Distribution

EGENG Distribution App

SMRD::smrd_app("distribution_egeng")

5.6 - Inverse Gaussian Distribution

IGAU Distribution App

SMRD::smrd_app("distribution_igau")

5.7 - Birnbaum-Saunders Distribution

Birnbaum-Saunders Distribution App

SMRD::smrd_app("distribution_bisa")

5.8 - Gompertz-Makeham Distribution

Gompertz-Makeham Distribution App

SMRD::smrd_app("distribution_goma")

5.10 - Distributions With a Threshold Parameter

5.10.1 - Background

• Many of the distributions introduced up to this point are defined on \(\mathcal{R}^+\)

  • Most log-location scale distributions
  • Some of the more complex distributions

• For these distributions, \(F(t)\) begins increasing at \(t=0\)

• Each of these distributions can be generalized by adding a threshold parameter,\(\gamma\ge0\)

• If \(\gamma>0\) the entire distribution is shifted by a value equal to \(\gamma\)

  • Denotes that failures cannot occur \(\forall t< \gamma\)
  • This can be useful for fitting distribution that are highly right-skewed

Figure 5.6

par(lwd = 2, family = 'serif', las = 1, 
    mgp = c(2.25,0.7, 0), tcl = -0.3, 
    font.lab = 2, font = 2, font.axis = 2, 
    las = 1, tck = 0.015)

curve(dlnorm(x, 0, 0.5), 
      col = 1, 
      n = 300, 
      xlim = c(0,6), 
      ylim = c(0,1.05), 
      xlab = 't', 
      ylab = 'f(t)')
curve(dlnorm(x-1,0,0.5), col = 2, n = 300, add = T)
curve(dlnorm(x-2,0,0.5), col = 3, n = 300, add = T)
curve(dlnorm(x-3,0,0.5), col = 4, n = 300, add = T)

text(x = seq(.797,3.797),
     y = rep(1,4), 
     labels = c(parse(text = 'gamma==0'),
                parse(text = 'gamma==1'),
                parse(text = 'gamma==2'),
                parse(text = 'gamma==3')), 
                col = 1:4, 
                font = rep(2,4))

5.11.2 Generalized Threshold-Scale cdf & pdf

Functional form of the CDF

• The form of the CDF for the generalized threshold-scale (GETS) distribution is

\[ F(t,\alpha, \sigma,\xi)=\begin{cases}\Phi\left[\log(1+\sigma z)^{1/\sigma}\right] &\mbox{for } \sigma>0, z>-1/\sigma\\\\ \Phi(z) &\mbox{for } \sigma=0,z\in(\infty,\infty)\\\\ 1-\Phi\left[\log(1+\sigma z)^{1/|\sigma|}\right] &\mbox{for } \sigma<0, z<-1/\sigma\end{cases} \]

• where

\[ \begin{aligned} \Phi &\equiv \text{ Form of generic cdf with standard parameter values}\\\\ z&=\frac{t-\alpha}{\xi}\\ \sigma &\in(-\infty,\infty)\\\\ \alpha &\in (-\infty,\infty)\\\\ \xi&>0 \end{aligned} \]

Functional form of the PDF

• The form of the PDF for the generalized threshold-scale (GETS) distribution is

\[ f(t,\alpha, \sigma,\xi)=\begin{cases}\phi\left[\log(1+\sigma z)^{1/|\sigma|}\right]\times \frac{1}{\xi(1+\sigma z)} &\mbox{for } \sigma\ne 0\\\\ \phi(z) \times \frac{1}{\xi} &\mbox{for } \sigma=0\\ \end{cases} \]

• Finally, inverting the GETS cdf gives the quantile function

\[ t_{p}=\alpha +\xi\times w(\sigma,p) \]

• where

\[ w(\sigma,p)=\begin{cases}\frac{\exp\left[\sigma\Phi^{-1}(p)\right]-1}{\sigma} &\mbox{for } \sigma>0\\\\ \Phi^{-1}(p) &\mbox{for } \sigma=0\\\\ \frac{\exp\left[|\sigma|\Phi^{-1}(1-p)\right]-1}{\sigma} &\mbox{for } \sigma<0\end{cases} \]

5.11.4 Special Cases of the GETS Distribution

• Recall from Chapter 4, the location-scale form of the normal, SEV, and LEV distributions

\[ \begin{aligned} F(t|\alpha,0,\xi)&=\Phi_{nor}\left[\frac{t-\alpha}{\xi}\right]\\\\ F(t|\alpha,0,\xi)&=\Phi_{sev}\left[\frac{t-\alpha}{\xi}\right]\\\\ F(t|\alpha,0,\xi)&=\Phi_{lev}\left[\frac{t-\alpha}{\xi}\right] \end{aligned} \]

• The threshold log-location-scale form of the NOR-GETS, SEV-GETS, and LEV-GETS distributions are

\[ \begin{aligned} F(t|\alpha,0,\xi)&=\Phi_{nor}\left[\frac{\log(t-\gamma)-\alpha}{\xi}\right]\\\\ F(t|\alpha,0,\xi)&=\Phi_{sev}\left[\frac{\log(t-\gamma)-\alpha}{\xi}\right]\\\\ F(t|\alpha,0,\xi)&=\Phi_{lev}\left[\frac{\log(t-\gamma)-\alpha}{\xi}\right] \end{aligned} \]

• where \(\gamma=\alpha-\xi/\sigma \text{ and } \mu=\log(\xi/\sigma)\)

5.11.4

• Running the code in the code chunks below will produce shiny applications that demonstrate the properties of three generalized threshold distributions
• NOR-GETS Distribution

SMRD::smrd_app("distribution_norgets")

• SEV-GETS Distribution

SMRD::smrd_app("distribution_sevgets")

• LEV-GETS Distribution

SMRD::smrd_app("distribution_levgets")

5.12 - Other Methods of Deriving Failure-Time Distributions

Three primary types

• The basic idea behind these methods is to model the physical/chemical process that cause failure

• Mixture distributions

• Power Distributions

• Physical/Chemical Degradation Distributions (Chapters 13 & 18)

Mixture Distributions

• Arise if a data set contains observations from two or more distinct processes

• Can involve mixtures can be finite or continuous

• Mixture distributions result if

  • A population of products are manufactured using different processes
  • A population of products are manufactured using different raw materials
  • A population of products operates in distinctly different environments

Finite Mixture Distributions

• Suppose we are interested in testing the fracture toughness of a new glass material to be used in future touchscreens on smartphones

  • Fracture toughness: total energy that a material can absorb before fracture

  • Greater toughness implies that if a phone is dropped the screen could flex and absorb the impact energy without cracking

  • Test: apply specified load to material samples every second until fracture occurs

  • Response of interest: number of load cycles at fracture \(N\)

  • Distribution failure cycles will be compared to legacy materials

• Further suppose that

  • 60% of the specimens were manufactured using materials from Supplier A
  • The other 40% were manufactured using materials from Supplier B
  • The Supplier B specimens contain contaminants, thus \((E[N_{A}] \ne E[N_{B}])\)

• The failure time distribution of entire sample could be expressed as shown below

  • If the failure time distributions of both samples have the same form (come from the same distribution family)
  • \(\mu_{A},\mu_{B},\;\text{and}\;\sigma\) are estimated separately

\[ F(N|\mu,\sigma)=0.4\times F(N|\mu_{A},\sigma) + 0.6\times F(N|\mu_{B},\sigma) \]

Continuous Mixture Distributions

• It’s unlikely that the level of contaminant in the specimens will only have two levels

  • We’d expect the level of contaminant to fall on a continuous spectrum
  • We’d also expect the level of contaminant across the population of specimens to follow some distribution

• This more general case results in a continuous mixture distribution, where

\[ \begin{aligned} F(N|\mu,\sigma,\mathbf{\theta})=P(N\le n)&=\int_{-\infty}^{\infty}P(N\le n|\mu=\mu_{1},\sigma)f_{\mu_{1}}(\mu_1|\mathbf{\theta})d\mu_{1}\\\\ &=\int_{-\infty}^{\infty}F_{N|\mu=\mu_{1}}(n|\mathbf{\mu},\sigma,\mathbf{\theta})f_{\mu_{1}}(\mu_1|\mathbf{\theta})d\mu_{1} \end{aligned} \]

• and

\[ f(N|\mu,\sigma,\mathbf{\theta})=\int_{-\infty}^{\infty}f_{N|\mu=\mu_{1}}(n|\mathbf{\mu},\sigma,\mathbf{\theta})f_{\mu_{1}}(\mu_1|\mathbf{\theta})d\mu_{1} \]

Power law distributions

background

• Distributions of minima and maxima of \(iid\) random variables have important applications

• Are closely related to reliability block diagrams (series/parallel components)

Power law distributions: Minimum-Type Distributions

• For components in series, the system fails when ANY component fails and the system cdf \(F_{system}(t)\) can be expressed as

\[ P(T_{system}\le t)=1-P(T_{A}>t)\cap P(T_{B}>t)\cap P(T_{C}>t)\cap P(T_{D}>t)\cap P(T_{E}>t) \]

• BUT, if we can reasonably assume independence among the components…i.e.

\[ P(T_{A}>t)\perp P(T_{B}>t)\perp P(T_{C}>t)\perp P(T_{D}>t)\perp P(T_{E}>t) \]

• Then, \(F_{system}(t)\) becomes

\[ P(T_{system}\le t)=1-P(T_{A}>t)\times P(T_{B}>t)\times P(T_{C}>t)\times P(T_{D}>t)\times P(T_{E}>t) \]

• For example: \(T_{A,B,C,D,E} \sim i.i.d EXP(\theta = 2)\)

\[ \begin{aligned} F_{system}(T)&=1-\exp\left[-0.5t\right]\exp\left[-0.5t\right]\exp\left[-0.5t\right]\exp\left[-0.5t\right]\exp\left[-0.5t\right]\\\\ &=1-\exp\left[-(0.5+0.5+0.5+0.5+0.5)t\right]\\\\ &=1-\exp\left[-2.5t\right] \end{aligned} \]

• Therefore, we see that for a system with

  • \(n = 1,2,...\) components - arranged in series
  • Where the lifetime of each component \(T_{i}\sim EXP(\theta_{i}), \;\;i= 1,2,...n\)

• The system lifetime \(T_{system}\sim EXP\left(\sum_{i=1}^n\theta_{i}\right)\)

For items in series, \(S(t_{i})\le\min S_{i}(t_{i})\)

par(family = "serif", bg = NA)
curve(pexp(x,0.5), yaxt = "n", xlab = "Time", ylab = "F(t)", lwd = 2, ylim = c(0,1))
curve(pexp(x,1.0), col = 2, lwd = 2, add = TRUE)
curve(pexp(x,1.5), col = 3, lwd = 2, add = TRUE)
curve(pexp(x,2.0), col = 4, lwd = 2, add = TRUE)
curve(pexp(x,2.5), col = 5, lwd = 2, add = TRUE)
axis(side = 2, las = 1)
box(lwd = 1.5)
legend("topleft", c(expression(theta==0.5),expression(theta==1.0),
                    expression(theta==1.5),expression(theta==2.0), 
                    expression(theta==2.5)), 
       lwd = 2, col = 1:5, bty = "n")

Power law distributions: Maximum-Type Distributions

background

• For components in parallel, the system fails when ALL of the components fail and the system cdf \(F_{system}(t)\) for components in parallel can be expressed as

\[ P(T_{system}\le t)=P(T_{B}\le t)\cap P(T_{C}\le t)\cap P(T_{D}\le t) \]

• AGAIN, if we can assume independence among the components…i.e.

\[ P(T_{B}\le t)\perp P(T_{C}\le t)\perp P(T_{D}\le t) \]

• Then, \(F_{system}(t)\) becomes

\[ \begin{aligned} P(T_{system}\le t)&=P(T_{B}\le t)\times P(T_{C}\le t)\times P(T_{D}\le t)\\\\ &=\prod_{i=1}^n P(T_{i}\le t) \end{aligned} \]

• Example: \(T_{B,C,D} \sim i.i.d EXP(\theta = 2)\)

\[ \begin{aligned} F_{system}(T)&=(1-\exp\left[-0.5t\right])(1-\exp [-0.5t])(1-\exp \left[-0.5t\right])\\\\ &=1 - 2\exp [-0.5t]+ \exp [-t] (1-\exp [-0.5t])\\\\ &=1-3\exp\left[-0.5t\right]+3\exp [-t]-\exp [-1.5t] \end{aligned} \]

• Therefore, we see that for a system with

  • \(n = 1,2,...\) components arranged in parallel
  • Where the lifetime of each component \(T_{i}\sim EXP(\theta_{i}), \;\;i= 1,2,...n\)

• The system lifetime \(T_{system}\) is not exponential

For items in parallel, \(S(t_{i})\ge\max S_{i}(t_{i})\)

par(family = "serif", las = 1)
curve(pexp(x,0.5), 
      xlab = "Time", 
      ylab = "F(t)", 
      lwd = 2, 
      ylim = c(0,1))

curve(1-2*(1-pexp(x,0.5))+(1-pexp(x,1.0)), 
      col = 2, 
      lwd = 2, 
      add = TRUE)
curve(pexp(x,0.5)*(1-2*(1-pexp(x,0.5))+(1-pexp(x,1.0))), 
      col = 3, 
      lwd = 2, 
      add = TRUE)
legend("topleft", 
       c(expression("F(t)"[1]),
         expression("F(t)"[2]),
         expression("F(t)"[3])), 
       lwd = 2, col = 1:5, bty = "n")