In an experiment, twenty test devices are stressed to investigate dielectric breakdown strength. The test was concluded at 600 hours at which time 18 units had failed. Investigate the distribution and plot the results to determine if the weibull distribution is a reasonable fit to the data. Estimate the parameters of the distribution from the graph: the shape factor \(\beta\), and the characteristics life \(\alpha\). Estimate the parameters of the distribution using maximum likelihood estimation. Compare the results

Time<-c(0.69,0.94,1.12,6.79,9.28,9.31,9.95,12.9,12.93,21.33,64.56,69.66,108.38,124.88,157.02,190.19,250.55,552.87,600,600)
Event<-c(1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0)
#length(Event)
dat<-data.frame(Time,Event)
head(dat)
library(weibulltools)
# converting to weibull data
dat2<-reliability_data(data = dat,x=Time,status = Event)

# rank regression
# first estimate the cumulative failure f(t)
dat2_cdf<-estimate_cdf(dat2, methods = "johnson")
head(dat2_cdf)
# Estimate weibull parameters
wf<-rank_regression(dat2_cdf, distribution="weibull")

# Probability plot
xwf<-plot_prob(dat2_cdf, distribution = "weibull", plot_method = "ggplot2", 
               title_main ="weibull Probability Plot", title_x = "Time hrs", title_y = "% failure rate")
plot_mod(xwf, x = wf, title_trace = "Rank Regression")

From the plot, the weibull distribution is a good fit for the model, however there is existence of one outlier at the beginning of the test when a unit failed at 0 hours time into the experiment. Overall its a good fit for the model

# maximum likelihood estimation parameters 
maxl<-ml_estimation(dat2, distribution = "weibull")
# comparing estimations
wf
## Rank Regression
## Coefficients:
##    mu  sigma  
## 4.419  1.763
maxl
## Maximum Likelihood Estimation
## Coefficients:
##    mu  sigma  
## 4.602  1.975

The maximum likelihood parameters are larger when compared to weibull estimate parameters

plot_mod(xwf, x = maxl, title_trace = "Maximum Likelihood")

Because the shape factor \(\beta\), and the caharateristics life \(\alpha\) are close for both estimates, there is no difference in the plot