In Auburngrads/SMRD: Statistical Methods For Reliability Data

Liebling & Zelen - Ball Bearing Fatigue Test

• Motivation

  • This example introduce the concept of estimating failure probability from singly right censored data using the binomial distribution

  • Recall the heatexchanger data set presented in Example 1.5

  • Suppose we inspect $100$ heat exchanger tubes at Plant 1 only, and assign each tube a value $d_i$, where

$$d_i=\begin{cases}1&\mbox{if the tube has failed}\\ 0&\mbox{if the tube has not failed}\end{cases}$$

• Data set subset(SMRD::heatexchanger, plant=='Plant1')

  • The inspection data are $100$ binary observations from a Bernoulli RV

  • The Bernoulli parameter $p \in [0,1]$ represents the probability of observing a "success" in this case a success is actually a failure

  • The trials are mutually independent (i.e. The probability of "success" is the same for each observation )

• The outcome of this inspection procedue (the total number of observed "successes") is a single observation from a binomial RV with parameters $n, p$

  • Recall, the binomial distribution models the probability of observing $x$ successes in $n$ trials where the probability of success for each trial is $p$

  • The probability mass function for a $BIN(n,p)$ random variable is

$$f(x;p,n) = \left( \begin{array}{c} n \ x \end{array} \right)(p)^{x}(1 - p)^{(n-x)} \;\;\;\;\;\; \mbox{for x = 0, 1, 2,..., n}$$

• where

  • $x$ is the number of observed "successes" (failures in this case)

  • $n$ is the total sample size

  • $p$ is the true (but unknown) long-run probability that a unit will fail

• A nonparametric estimator for $F(t)$ is the binomial parameter $p$

As discussed in Chapter 2, $\pi_{i}$ represents the multinomial probability that a failure will occur in interval $i$.

$$\hat{p}{{MLE}}=\frac{x}{n}$$

Auburngrads/SMRD documentation built on Sept. 14, 2020, 2:21 a.m.