Question

Twenty motors were put on test under a high-temperature setting. The lifetimes in hours of the motors under these conditions are given below. Also, the data are in the file lifetimemotor. rda at the site listed in the Preface. Suppose we assume that the lifetime of a motor under these conditions, \(X\), has a \(\Gamma(1, \theta)\) distribution. \(\begin{array}{cccccccccc}1 & 4 & 5 & 21 & 22 & 28 & 40 & 42 & 51 & 53 \\ 58 & 67 & 95 & 124 & 124 & 160 & 202 & 260 & 303 & 363\end{array}\) (a) Obtain a histogram of the data and overlay it with a density estimate, using the code hist \((x, p r=T) ;\) lines (density \((x))\) where the \(R\) vector \(x\) contains the data. Based on this plot, do you think that the \(\Gamma(1, \theta)\) model is credible? (b) Assuming a \(\Gamma(1, \theta)\) model, obtain the maximum likelihood estimate \(\widehat{\theta}\) of \(\theta\) and locate it on your histogram. Next overlay the pdf of a \(\Gamma(1, \hat{\theta})\) distribution on the histogram. Use the \(\mathrm{R}\) function dgamma \((\mathrm{x}\), shape \(=1\), scale \(=\hat{\theta}\) ) to evaluate the pdf. (c) Obtain the sample median of the data, which is an estimate of the median lifetime of a motor. What parameter is it estimating (i.e., determine the median of \(X\) )? (d) Based on the mle, what is another estimate of the median of \(X ?\)

Short answer

The graphics will show whether the Gamma distribution is a credible model for the data. Further, both the maximum likelihood estimate of \(\theta\) and the sample median provide estimates for the median lifetime of a motor. By comparing these two estimates, we can determine how well our data fits the Gamma distribution.

Step-by-step solution

Histogram and density estimation

Using a statistical software, like R, create a histogram of the data with probability density estimation. The R code for this is hist(x, pr=T); lines(density(x)), where x is the data vector. This will display a histogram with a density line on top of it.

Model credibility assessment

Compare the histogram with the density estimate and model. If the Gamma distribution with parameters \(\Gamma(1, \theta)\) matches closely with the data, then the model is considered credible.

Maximum likelihood estimation

Calculate the maximum likelihood estimate \(\widehat{\theta}\) of \(\theta\). This is given by the sample mean of the data when the shape parameter is 1 in the Gamma distribution.

Overlaying PDF

Evaluate the Probability Density Function (PDF) using the Gamma distribution, using the evaluated \(\widehat{\theta}\). Overlay this PDF on the histogram. This can be performed by the R code dgamma(x, shape=1, scale=\(\widehat{\theta}\)).

Estimate of median parameter

To estimate the median lifetime of a motor, obtain the sample median of the data. This estimates the parameter \(\theta\), the median of the Gamma distribution when shape parameter is 1.

Alternative estimate of the median

Using the maximum likelihood estimate, we can obtain another estimate for the median of \(X\). Since \(\widehat{\theta}\) is the estimate of the median, it can be compared with the sample median to check how well our data is modeled by the Gamma distribution.