7.
Product and Process Comparisons
7.3. Comparisons based on data from two processes


Comparing two exponential distributions is to compare the means or hazard rates 
The comparison of two (or more) life distributions is a common
objective when performing statistical analyses of lifetime data.
Here we look at the oneparameter exponential distribution case.
In this case, comparing two exponential distributions is equivalent to comparing their means (or the reciprocal of their means, known as their hazard rates). 

Type II Censored data  
Definition of Type II censored data  Definition: Type II censored data occur when a life test is terminated exactly when a prespecified number of failures have occurred. The remaining units have not yet failed. If n units were on test, and the prespecified number of failures is r (where r is less than or equal to n), then the test ends at t_{r} = the time of the rth failure.  
Two exponential samples oredered by time 
Suppose we have Type II censored data from two exponential
distributions with means
_{1} and
_{2}.
We have two samples from these distributions, of sizes
n_{1} on test with r_{1} failures and
n_{2} on test with r_{2} failures,
respectively. The observations are time to failure and are therefore
ordered by time.


Test of equality of _{1} and _{2} and confidence interval for _{1}/ _{2} 
Letting
Then


Numerical example 
A numerical application will illustrate the concepts outlined
above.
For this example,
H_{a}: _{1}/ _{2} 1
Then T_{1} = 4338 and T_{2} = 3836. The estimator for _{1} is 4338 / 7 = 619.71 and the estimator for _{2} is 3836 / 5 = 767.20. The ratio of the estimators = U = 619.71 / 767.20 = .808. If the means are the same, the ratio of the estimators, U, follows an F distribution with 2r_{1}, 2r_{2} degrees of freedom. The P(F < .808) = .348. The associated pvalue is 2(.348) = .696. Based on this pvalue, we find no evidence to reject the null hypothesis (that the true but unknown ratio = 1). Note that this is a twosided test, and we would reject the null hyposthesis if the pvalue is either too small (i.e., less or equal to .025) or too large (i.e., greater than or equal to .975) for a 95% significance level test. We can also put a 95% confidence interval around the ratio of the two means. Since the .025 and .975 quantiles of F_{(14,10)} are 0.3178 and 3.5504, respectively, we have
and (.228, 2.542) is a 95% confidence interval for the ratio of the unknown means. The value of 1 is within this range, which is another way of showing that we cannot reject the null hypothesis at the 95% significance level. 