OVERVIEW

This Chapter Explains…

• Models for continuous failure-time processes

• Models used for the discrete data from these continuous failure-time processes

• Common censoring mechanisms that restrict our ability to observe all of the failure times that might occur in a reliability study

• Principles of likelihood and how likelihood relates to the probability of the observed data

• How likelihood ideas can be used to make inferences from reliability data

2.1.1 - Failure Time Distribution Functions

Overview of Probability Functions

• The probability_functions app from the teachingApps package includes the six probability functions used most in reliability studies and highlights:

  • Expressions for each function

  • Mathematical relationships between the functions

  • How to compute values for each function in R - for a variety of distributions

• The most current published version of the teachingApps package can be installed from CRAN by pasting the code below into the R console

install.packages('teachingApps')

• The development version of the teachingApps package (which may contain some newer features) can be installed from GitHub by pasting the code below into the R console

if(!'devtools'%in%installed.packages()[,1]) {
  
   install.packages('devtools')

  }

devtools::install_github('Auburngrads/teachingApps')

• Once the package is installed, the app can be rendered by pasting the code below into the R console

teachingApps::teachingApp('probability_functions')

Figure 2.1 Distribution Plots For \(T \sim WEIB(1.7,1)\)

• The text introduces the probability functions using the \(WEIB(1.7,1)\) distribution as an example and illustrates four functions in Figure 2.1

• The SMRD package includes the distribution plot( ) function, that can be used to reproduce this figure or similar figures for many other distributions

  • If single values are given for both parameters distribution.plot( ) returns a plot showing \(F(t), f(t), S(t)\) and \(h(t)\)

  • If multiple values are given for either parameter distribution.plot( ) returns a plot showing \(F(t), f(t), h(t)\) and a legend of the parameter values displayed

par(family = 'serif',font = 2)

library(package = SMRD)

distribution.plot('Weibull',
                  shape = c(1.7), 
                  scale = 1,
                  prob.range = c(.000001,.99))

2.2 - MODELS FOR DISCRETE DATA FROM A CONTINUOUS PROCESS

Understanding failure

• Failure implies that a product’s ability to perform one or more functions has been either…

  • Lost completely, or

  • Degraded to the point that the product cannot meet a performance specification

• When failure occurs in fielded systems, there may not be an immediate indication to the user or the maintainers

  • Some component and sub-component failures may only be discovered after a subsystem or the entire system fails

  • Component failures in complex systems can remain undiscovered until the next inspection cycle

  • Because of this, inspection intervals should be short enough to allow maintainers multiple opportunities to discover flaws and prevent serious failures

  • Inspection intervals should be long enough to not overburden maintainers with unnecessary inspections

• In real-world testing, we typically cannot observe the exact failure time for every unit in the sample

  • Testing may occur in different geographic regions

  • Sample size may be too large

  • Test items may be removed from a test to fill a “real-world” need

  • The test period may end before every item fails

  • Items may fail before the first inspection

2.2.1 - Multinomial Failure Time Model

Real world reliability data

• We can’t constantly monitor the performance of every component when a system is in the field.

• For this reason real-world failure data are often reported as the inspection time when the failures were observed

• Thus, real-world failure data are often reported as DISCRETE values corresponding to the inspection cycle

• This occurs even if the sample space for the data is continuous \(\Omega \in \mathbb{R}\)

• To analyze this real-world failure data appropriately, we partition the test into non-overlapping intervals corresponding to the time between inspections, i.e. \[ (t_0,t_1], (t_1,t_2],...,(t_{m-1},t_m],(t_m,t_{m+1}),\;\;\;\;\; t_0=0, t_{m+1}=\infty \]

• The intervals do not need to be of equal length

  • Inspections may be made more frequently at the beginning of a test to check for “infant mortality” failures or ensure that the test was set-up correctly

  • Inspections may then become less frequent once the “bad actors” have been culled from the population

Multinomial Reliability Estimation

• Estimating reliability in this scenario is similar to estmating the likelihood of a single \(n\)-sided die landing on a specific number

• However, in the context of failure data the ‘sides’ refer to the \(n\) inspection intervals

• Think about how to estimate the probability of a 6-sided die landing on a given side

  • If the die is ‘fair’ and we rolled it many times (i.e. \(n \gtrsim 10,000\) rolls) we would expect that in the long-run \[\frac{\text{# times we roll 1,2,3,4,5,6}}{\text{# of total rolls}}\rightarrow P(X=1,2,3,4,5,6) = \frac{1}{6}\]

  • However, if \(n\) is much smaller, say 100 rolls, the probability that the die lands on a given side is modeled using the multinomial distrubution with probability mass function \[ f(n,\pi_{_{1}},...,\pi_{_{m+1}}|d_{_{1}},...,d_{_{m+1}})=\frac{n!}{d_1!,...,d_{m+1}!}\pi_{_{1}}^{d_{1}},...,\pi_{_{m+1}}^{d_{m+1}} \]

• In the context of rolling a single die with \(m+1\) sides, the variables in this equation represent

  • \(n\) - number of rolls

  • \(d_1,...,d_{m+1}\) - number of times each of the faces \(1,...,m+1\) are observed

  • \(\pi_{_{1}},...,\pi_{_{m+1}}\) - the long-run probabilities of observing faces \(1,...,m+1\)

• In the context of failures occuring in \(m+1\) discrete time intervals, these variables represent

  • \(n\) - the number of systems under test

  • \(d_{1}\) - number of systems that failed in the \(1^{st}\) interval \((t_{0},t_{1}]\)

  • \(d_{2}\) - number of systems that failed in the \(2^{nd}\) interval \((t_{1},t_{2}]\)

  • \(d_{m+1}\) - number of systems still operating at the final inspection \((t_{m},\infty)\)

  • \(\pi_{_{i}},...,\pi_{_{m+1}}\) - the long-run probability that a unit will fail in interval \(i,...,m+1\)

Figure 2.5 - Relationship between \(\pi_{i}\) and \(F(t)\)

• For a fair die and a fair toss we say that \(\pi_1=\pi_2=\pi_3=\pi_4=\pi_5=\pi_6 \approx 1/6\)

  • The value \(1/6\) is the propensity of the overall EXPERIMENT, and is a function of how the die was manufactured AND how it was tossed

  • Think of it as the underlying physical dynamics that govern the likelihood of observing a given result

  • A “weighted” die would have a different propensity represented by different values of \(\pi_1\ne\pi_2\ne\pi_3\ne\pi_4\ne\pi_5\ne\pi_6\)

  • Likewise, with some practice we might be able to toss the die in such a way as to affect the propensity

• For systems under test, the propensity is a complex relationship between the system’s probability of failure and the environment in which it operates

• For the simplest case, \(\pi_{i}=P(t_{i-1}<T\le t_{i})=F(t_{i})-F(t_{i-1})\), where

  • \(\pi_i\ge 0, \forall i\)

  • \(\displaystyle \sum_{j=1}^{m+1}\pi_j=1\)

  • \(F(t)\) is the underlying failure time distribution relating the likelihood of failure at some time \(t\)

• Figure 2.5 illustrates the relationship between \(\pi_{i}\) and \(F(t)\), where the underlying failure time distribution is assumed to be \(Weibull(1.7,1)\)

par(family = 'serif', font = 2)

y <- function(t) { pweibull(t,shape = 1.7,scale = 1) }

curve(pweibull(x,shape=1.7,scale=1),lwd = 2,
      xlab = 't', ylab = 'f(t)', 
      from = 0, to = 3,
      las = 1)

segments(c(0,.5,1,1.5,2,rep(0,5)),
         c(rep(0,5),y(c(0.01,.5,1,1.5,2))),
         c(0,.5,1,1.5,2,rep(2.2,5)),
         rep(y(c(0.01,.5,1,1.5,2)),2),
         lty=rep(2,10),col=rep(1,10))

text(2.3,(y(0.5)+y(.01))/2,expression(pi[1]),cex=1.25)
text(2.3,(y(1.0)+y(0.5))/2,expression(pi[2]),cex=1.25)
text(2.3,(y(1.5)+y(1.0))/2,expression(pi[3]),cex=1.25)
text(2.3,(y(2.0)+y(1.5))/2,expression(pi[4]),cex=1.25)

mtext(side = 3,
      expression('Figure 2.5 - Graphical interpretation of the relationship between the '*pi[i]*' values and F('*t[i]*')'),
      font = 2,line = 2)

Figure 2.5 - Graphical interpretation of the relationship between the \(\pi_{i}\) values and \(F(t_{i})\)

Multinomial Survival Function

• Based on this definition of \(F(t_{i})\), the survival function evaluated at \(t_{i}\) may be expressed as

\[ S(t_i)=P(T>t_i)=1-F(t_i)=\sum_{j=i+1}^{m+1}\pi_j \]

• Where

  • \(i\) denotes the inspection of interest

  • \(j\) denotes an arbitrary inspection \(j \ne i\)

• In words, this equation states that the probability that a system will survive to inspection \(i\) is equal to the sum of the probabilities that the system will fail at inspection \(i+1\) or later

Multinomial Failure Time Model - In-Class Example

Conditional Multinomial Distribution Function

• Motivation for this example

  • Many A/C components cannot be inspected by base-level Mx due to lack of training, facilities, tools, etc.

  • Components will only be inspected every \(t\) flight-hours at the depot

  • For each component, \(\exists\) an underlying \(F(t), t\in \mathbb{R}^{+}\) describing time to failure

  • The form of \(F(t)\) is often unknown, but may sometimes be inferred from similar components/systems

• Goal of this example

  • Estimate risk of delaying replacement until next inspection (depot cycle)

  • Balance safety risks against replacement costs

• Estimating the risk of delaying replacement until the next inspection implies that

  • The system was still operating at inspection \(i-1\)

  • We want to know how likely it is that the system will fail before inspection \(i\)

  • The text defines this quantity in (2.7) as \(p_{i}\)

  • \(p_{i}\) is the conditional probability that the system will fail prior to inspection \(i\), given that it has survived to inspection \(i-1\). \[ p_i=P(t_{i-1}<T\le t_{i}|T>t_{i-1})=\frac{F(t_i)-F(t_{i-1})}{1-F(t_{i-1})}=\frac{\pi_{i}}{S(t_{i-1})} \]

2.2.2 - Multinomial CDF

Multinomial Survival Function and CDF

• From Equation 2.7 we can define the survival function in terms of \(\mathbf{p}=(p_{1},...,p_{m})\) as \[ \displaystyle S(t_{i})=\prod_{j=1}^{i}[1-p_{j}], i=1,...,m+1. \]

• It follows that the CDF is expressed as \[ \begin{aligned} F(t_{i})&=1-\prod_{j=1}^{i}[1-p_{j}], i=1,...,m+1 \\ &=1-\prod_{j=1}^{i} 1-\frac{F(t_i)-F(t_{i-1})}{1-F(t_{i-1})}\\ &=1-\prod_{j=1}^{i} \frac{1-F(t_{i-1})}{1-F(t_{i-1})}-\frac{F(t_i)-F(t_{i-1})}{1-F(t_{i-1})}\\ &=1-\prod_{j=1}^{i} \frac{1 - F(t_i)}{1-F(t_{i-1})}\\ &=1-\prod_{j=1}^{i} \frac{S(t_i)}{S(t_{i-1})}\\ &=\sum_{j=1}^{i}\pi_{j} \end{aligned} \]

• How would you state the final expression in this equation IN WORDS

2.2.2 - Multinomial CDF - Example 2.8

Computing of \(F(t_i), S(t_i), \pi_i\) and \(p_i\)

• Table 2.1 shows the resulting values for our example in which the underlying failure time distribution is \(WEIB(1.7,1)\)

• The following steps show how the values in the table were computed for the first and second inspections

• At the first inspection \(t_{1}=0.5\) \[ \begin{aligned} F(0.5)&=1-\exp\left[-\left(\frac{0.5}{1}\right)^{1.7}\right]=\underline{0.2649275}\\\\\\ S(0.5)&=1-F(0.5)=\underline{0.7350725}\\\\\\ \pi_i&=F(t_i)-F(t_{i-1})=F(0.5)-F(0)=\underline{0.2649275}\\\\\\ p_i&=\frac{\pi_i}{S(t_{i-1})}=\frac{F(0.5)-F(0)}{S(0)}=\frac{0.2649275}{1}=\underline{0.2649275} \end{aligned} \]

• At the second inspection \(t_{2}=1.0\) \[ \begin{aligned} F(1)&=1-\exp\left[-\left(\frac{1}{1}\right)^{1.7}\right]=\underline{0.6321206}\\\\\\ S(1)&=1-F(1)=\underline{0.3678794}\\\\\\ \pi_i&=F(t_i)-F(t_{i-1})=F(1)-F(0.5)=\underline{0.3671931}\\\\\\ p_i&=\frac{\pi_i}{S(t_{i-1})}=\frac{F(1)-F(0.5)}{S(0.5)}=\frac{0.3671931}{0.7350725}=\underline{0.4995331} \end{aligned} \]

2.3 - CENSORING

Types of censoring

• Observe the event plots shown in the figure below

• The event plot on the left shows the lfp1370 data set in the SMRD package

• The event plot on the right shows the turbine data set in the SMRD package

• What type of testing do you think would produce each of these event plots?

library(SMRD)

lfp1370.ld <- 
frame.to.ld(SMRD::lfp1370,
            response.column = 1, 
            censor.column = 2, 
            case.weight.column = 3,
            time.units = 'Hours')

turbine.ld <- 
frame.to.ld(SMRD::turbine,
            response.column = 1, 
            censor.column = 2,
            case.weight.column = 3,
            time.units = 'Hundreds of Hours')

par(mfrow = c(1,2),family = 'serif',font = 2)

event.plot(lfp1370.ld)
event.plot(turbine.ld) 

Event plots for the lfp1370 (left) and turbine (right) datasets

2.3.1 - Censoring Mechanisms

Types of Right Censoring

• Type I (Time) Censoring - Occurs when time considerations limit our ability to observe the exact failure times of one or more samples

  • The total available test time is predetermined before testing begins

  • Samples still operating after the predetermined end of testing are Type-I right-censored

  • In this type of test, the number of observed failures is a random quantity

  • There’s no guarantee that any failures will be observed before testing ends

• Type II (Failure) Censoring - Occurs when time considerations limit our ability to observe the exact failure times of one or more samples

  • The number of observed failures is predetermined before testing begins

  • Samples still operating after the predetermined number of failures have been observed are Type-II right-censored

  • In this type of test the total test time is a random quantity

  • There’s no guarantee a sufficient number of failures will be observed in a reasonable timeframe

• Modeling Type II censored data is simpler than for Type I censored data

• Failure censored tests (Type II) are far less common than time-censored tests

• Why would these two statements be true?

Time censored data may be further divided according to

• The number of censoring events

  • Singly censored

  • Multiply censored

• The contribution to the likelihood function \(\S\) 2.4

  • Right censored

  • Left censored

  • Interval censored

• Remember…ALL DATA IS INTERVAL DATA

2.3.2 - Important Assumptions On Censoring Mechanisms

Independent vs. Dependent Censoring

• This course considers censoring events that are independent of the time to failure

  • The time at which testing ends is driven by real-world constraints - not because we want to ensure the test ends before an item fails

  • Inspections should not be delayed so that a longer lifetime can be reported

  • Items should not be removed from a test because it appears that they are ‘about to fail’

• Competing risks considers dependent censoring events that affect time to failure

  • Dependent censoring is common in medical survivability research

  • Dependent censoring is beyond the scope of this course

2.3.2 - Important Assumptions On Censoring Mechanisms - Example

Independent vs. Dependent Censoring

• Suppose we’re investigating the time to recurrence of symptoms in cancer patients who receive a new experimental treatment

• Further suppose that the two patients described below are involved in our study

  • Patient A moves away during the study - we can’t collect any future observations

  • Patient B is killed in a car crash during the study - we can’t collect any future observations

• How should we handle the data we’ve already collected for these patients?

  • The last observation for Patient A would be independently right censored because there’s a positive probability that the cancer symptoms could return - even though we may not be able to observe when they returned

  • The data observed for Patient B is an example dependent censoring because there is zero probability that the cancer symptoms could return

2.4 - LIKELIHOOD

The Likelihood Function \(\mathscr{L}(\theta|\mathbf{y})\) And The PDF \(f(\mathbf{y}|\theta)\)

• Suppose we want to simulate 50 observations from a specified distribution to use for an example

• How would we do it?

  1. We’d first have to choose a distribution family to simulate from (log-normal, Weibull, exponential)

  2. We’d then choose a set of parameter values to indicate the specific member of our chosen distribution family

  3. Finally, we’d use a software application to generate 50 random observations using the distribution/parameter set we chose

• So, let’s simulate 50 failure observations in the interval \([50,150]\) using the following distributions

  • \(LOGNOR(\mu = 4.5, \sigma = 0.2)\)
    rlnorm(n = 50, meanlog = 4.5, sdlog = 0.2)

  • \(WEIB(\beta = 5.5, \eta = 120)\)
    rweibull(n = 50, shape = 5.5, scale = 120)

  • \(EXP(\theta = 40)\)
    rexp(n = 50, rate = 1/40)

• The parameter values in each of these distributions were chosen to generate observations between \(50\) and \(150\)

Observing data patterns

• Paste the code in the chunk below in R to bring up the shiny app

• The app shows the result of our simulation in two separate plots

teachingApps::teachingApp('data_patterns')

• The left plot

  • Shows the pdfs of each distribution along with the 50 failure times simulated from each distribution

  • When the observations are plotted this way, it’s easy to see how the distributions differ

• The right plot

  • Shows the raw observations ordered from smallest to largest

  • In this plot we can see that the observations simulated from the exponential distribution are skewed toward lower values compared to the Weibull and lognormal distributions

  • However, it’s much harder to distinguish between the observations from the Weibull and lognormal distributions.

What’s The Point?

• Specifying the distribution family and parameter values gives information about the pattern of the data that will result from the simulation

  • Sometimes, it’s easy to distinguish different patterns, just by looking at the data visually

  • More often, graphical and numerical methods are required to distinguish between the patterns of data produced by different distributions

  • Simulating more observations can sometimes make it easier to distinguish between the patterns

• But the opposite should be true as well

  • The data should also provide information about the distribution family and parameter values that were most likely to have produced them

  • Again, sometimes it’s easy to pick out what distribution family produced a data set just by looking at the pattern of the data

  • If more observations are available, it becomes easier to distinguish which distribution family produced the data

  • In most cases, numerical methods are needed to distinguish which distribution produced a data set

  • A popular numerical method of extracting information from data to identify which probability distribution is most likely to have produced it is Maximum Likelihood Estimation

2.4.1 - Likelihood-Based Statistical Methods

Background

• Suppose that we observe \(DATA = y_{1},y_{2},...,y_{n}\) from a test event

• If these observations were produced by units that can reasonably be considered as identical, then we can say that \[ y_{1},y_{2},...,y_{n} \sim F(y|\mathbf{\theta}=\theta_{1},...,\theta_{n}) \]

• However, the true form of \(F(y|\mathbf{\theta})\) and values of \(\theta_{1},...,\theta_{n}\) are unknown

• Likelihood theory states that for each distribution family, we can find \(\mathbf{\theta}\) such that the negative log of the joint probability of the data is minimized, that is \[ \boldsymbol{\hat{\theta^*}} = \underset{\boldsymbol{\theta} \in \boldsymbol \Theta}{\text{arg}\;\min} \bigg(-\log\bigg[\Pr(y_1,\ldots,y_n|\boldsymbol{\theta})\bigg]\bigg) \]

• If these observations are mutually independent, this joint probability can be stated as

\[ \begin{aligned} \boldsymbol{\hat{\theta^*}} &= \underset{\boldsymbol{\theta} \in \boldsymbol \Theta}{\text{arg}\;\min} \bigg(-\log\bigg[\prod_{i=1}^{n}\Pr(y_{i}|\boldsymbol{\theta})\bigg]\bigg)\\\\ &= \underset{\boldsymbol{\theta} \in \boldsymbol \Theta}{\text{arg}\;\min} \bigg(-\sum_{i=1}^{n}\log\bigg[\Pr(y_{i}|\boldsymbol{\theta})\bigg]\bigg)\\\\ \end{aligned} \]

• This joint probability is a loss function in \(\theta\) defined over the interval \([0,+\infty)\)

• Under what circumstance would this loss function equal \(0\)?

Maximum likelihood estimation

• The joint probability defined above \(\Pr(y_1,\ldots,y_n|\boldsymbol{\theta})\) is also known as the Likelihood Function and is denoted by \(\mathscr{L}(\boldsymbol{\theta|y_1,\ldots,y_n)}\), thus for observation \(i\) \[ \mathscr{L}(\mathbf{\theta}|y_i)=\Pr(y_i|\mathbf{\theta}) \]

• Note that

  • \(\mathscr{L}(\theta|y_i)\) is a function of the parameter vector \(\theta\) for a given observation \(y_i\)

  • \(\Pr(y_{i}|\theta)\) is a function of the observed values \(y_i\) for a given parameter vector \(\theta\)

• For a vector of observations, \(y_1,\ldots,y_n\), the total likelihood is expressed as \[ \mathscr{L}(\boldsymbol{\theta}|y_1,\ldots,y_n)=C\prod_{i=1}^{n} \mathscr{L}(\theta|y_i)=\prod_{i=1}^{n}\Pr(t_{i}|\theta) \]

• Unsurprisingly, the log of the joint probability \(\log\bigg[\Pr(y_1,\ldots,y_n|\boldsymbol{\theta})\bigg]\) is known as the log-likelihood function \(\log[\mathscr{L}(\mathbf{\theta}|y_i)]\)

  • Thus, the value of \(\boldsymbol{\theta}\) that minimizes \(-\log\bigg[\Pr(y_1,\ldots,y_n|\boldsymbol{\theta})\bigg]\), also maximizes \(\log[\mathscr{L}(\mathbf{\theta}|y_i)]\)

  • Because \(\log[x]\) is a monotone function, the value of \(\boldsymbol{\theta}\) that maximizes \(\log[\mathscr{L}(\mathbf{\theta}|y_i)]\) also maximizes \(\log[\mathscr{L}(\mathbf{\theta}|y_i)]\)

  • This value of \(\boldsymbol{\theta}\) that maximizes the total likelihood is called the maximum likelihood estimate and is denoted \(\boldsymbol{\hat{\theta}}_{_{MLE}}\)

  • The constant \(C\) in the total likelihood expression is unimportant and may be ignored in nearly all circumstances

• In maximum likelihood estimation, the objective is to maximize the total likelihood based on two inputs:

  • An assumed functional form of the joint probability

  • A set of parameter values - \(\mathbf{\theta}=\theta_{1},...,\theta_{n}\)

• The functional form of the joint probability is typically expressed in terms of the CDF and the PDF of a candidate probability distribution

• For each candidate probability distribution we can obtain a \(\boldsymbol{\hat{\theta}}_{_{MLE}}\) that gives a distribution-specific maximum likelihood estimate \(\mathscr{L}(\boldsymbol{\hat{\theta}}_{_{MLE}}|y_1,\ldots,y_n)\)

• The candidate probability distribution with the highest maximum likelihood is the best fit distribution for the data.

2.4.3 - Contributions To The Likelihood Function

• Each observation in the sample set under evaluation contributes to the total likelihood

• The degree to which an observation contritbutes to the total likelihood depends on the underlying failure distribution \(F(t)\) and whether the width of the interval in which observation occurred

• Figure 2.6 illustrates the contributions made to the total likelihood for ‘exact’ observations, along with right, left, and interval censored observations

par(family = 'serif', font = 2)

curve(dweibull(x, shape = 1.7, scale = 1),
      lwd = 2,
      las = 1, 
      xlim = c(0,2.5),
      xlab = 'Time (t)',
      ylab = 'f(t)',
      main = 'Figure 2.6 - Likelihood contributions for different kids of censoring')

polygon(x = c(seq(0,.5,.01),.5),
        y = c(dweibull(seq(0,.5,.01),shape = 1.7,scale = 1),0),
        col = 1)
polygon(x = c(1,seq(1,1.5,.01),1.5),
        y = c(0,dweibull(seq(1,1.5,.01),shape = 1.7,scale = 1),0),
        col = 1)
polygon(x = c(2,seq(2,2.4,.01),2.4),
        y = c(0,dweibull(seq(2,2.4,.01),shape = 1.7,scale = 1),0),
        col = 1)

text(x = c(.16,1.3,2.2),
     y = c(.75,.65,.15),
     labels = c('Left Censoring',
                'Interval Censoring',
                'Right Censoring'))

• Table 2.2 expresses the contribution of each observation in terms of the underlying failure distribution

• Using the expressions in Table 2.2, the total likelihood for a data set containing exact observations, and arbitrarily censored observations is expressed as \[ \begin{aligned} \mathscr{L}(\theta|t_1,\ldots,t_n) &=C\prod_{i=1}^{n} \mathscr{L}(\theta|t_i)\\ &=C\prod_{i=1}^{m+1}[F(t_{i})]^{l_{i}}[F(t_{i})-F(t_{i-1})]^{d_{i}}[1-F(t_{i})]^{r_{i}} \end{aligned} \]

• where \(n = \sum_{j=1}^{m+1}(d_{j}+r_{j}+l_{j})\)

2.4.4 - Form Of The Constant Term \(C\)

• In general, the multinomial constant term \(C\) does not effect the value of \(\boldsymbol{\hat{\theta}}_{_{MLE}}\) that maximizes the total likelihood

• For most models used in this course \(C\) may be considered a proportionality constant (i.e. \(C=1\)) \[ \mathcal{C}=\frac{n!}{d_1!...d_{m+1}!} \]

2.4.5 - Likelihood Terms For General Reliability Data

• This section presents a more general expression for the total likelihood function when

  • The intervals in which a failure occurs overlap

  • The union of the intervals do not cover the entire time line \((0,\infty)\)

• In this course, the form of \(\mathscr{L}\) given in equation 2.14 is sufficient

2.4.6 - Other Likelihood Terms

• This section presents an even more general expression for the total likelihood function when

  • Censoring occurs at random times within the intervals according to \(f_{c}(t)\)

  • The distributions represented by \(f_{c}(t)\) and \(f(t)\) are independent

• This type of censoring can occur if multiple failure modes are present in the unit under test

• Multiple failure modes with be considered in Chapter 15

• Truncated data will be discussed in Chapter 11