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ReliaSoft's Ranking MethodWhen analyzing interval data, it is commonplace to assume that the actual failure time occurred at the midpoint of the interval. To be more conservative, you can use the starting point of the interval or you can use the end point of the interval to be most optimistic. Weibull++ allows you to employ ReliaSoft's ranking method (RRM) when analyzing interval data. Using an iterative process, this ranking method is an improvement over the standard ranking method (SRM). This article presents an example using the twoparameter Weibull distribution to illustrate how this method is employed. This method can also be easily generalized for additional models. StepbyStep Example
Table 1: Test data. As a preliminary step, we need to provide a crude estimate of the Weibull parameters for this data. To begin, we will extract the exact timestofailure (10, 40 and 50) and append them to the midpoints of the interval failures: 50 (for the interval of 20 to 80) and 47.5 (for the interval of 10 to 85). Now, the extracted list consists of the data in Table 2.
Table 2: Union of exact timestofailure with the midpoint of the interval failures. Using the traditional rank regression, we obtain the first initial estimates: Step 1 This transforms the data into the format displayed in Table 3.
Table 3: Union of exact timestofailure with the midpoint based on parameters β and η . Step 2
Table 4: Union of exact timestofailure in ascending order. Step 3 Table 5: Computation of increments for computing a revised mean order number. In general, for left censored data: The increment term for n left censored items at time = t_{0}, with a timetofailure of t_{i}, when t_{0} t_{i1} is zero. When t_{0} > t_{i1} the contribution is: Or: Where t_{i1} is the timetofailure previous to the t_{i} timetofailure and n is the number of units associated with that timetofailure (or units in the group). Or: Step 4 Table 6: Increments solved numerically. Step 5 Table 7  Mean Order Numbers (MON). Step 6 Table 8  Mean Order Numbers with ranks for a sample size of 13 units. Step 7 Table 9  New times and median ranks for regression. Step 8 Results Table 10  The parameters after the first five iterations. Using Weibull++ with rank regression on X yields: The direct MLE solution yields:


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