In Auburngrads/SMRD: Statistical Methods For Reliability Data

Figure 3.1

• Figure 3.1 shows censoring events occuring at the end of years 1, 2, and 3

• The nonparametric estimates for $F(t_{i})$ computed using the simple binomial method assume that the number of items at risk does not change after each inspection

• The binomial method treats each inspection as a separate analysis with singly censored data

  • Analysis 1 $\rightarrow n=100, t_{0}=0, t_{c}=Year 1 \rightarrow$ 1 failure observed

  • Analysis 2 $\rightarrow n=100, t_{0}=0, t_{c}=Year 2 \rightarrow$ 3 failures observed

  • Analysis 3 $\rightarrow n=100, t_{0}=0, t_{c}=Year 3 \rightarrow$ 5 failures observed

• The nonparametric estimate of $F(t_{i})$ based on the binomial distribution is

$$\hat{F}(t_{i})=\frac{#\text{ of failures up to time }t_i}{n}=\frac{\sum_{j=1}^{i}d_j}{n}$$ - where

+ The sample size $n=100$

+ The number of failures $d_{j}=1\;\text{(Year 1)},\;3\;\text{(Year 2)},\;5\;\text{(Year 3)}$

• Thus, for the Plant 1 data $$\begin{aligned}\hat{F}(1)=1/100\\ \hat{F}(2)=3/100\\ \hat{F}(3)=5/100\\ \end{aligned}$$

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