Template:Biasing and unbiasing of beta camsaa

Biasing and Unbiasing of Beta
Eqn. (6) returns the biased estimate of $$\beta $$. The unbiased estimate of $$\beta $$  can be calculated by using the following relationships. For time terminated data (meaning that the test ends after a specified number of failures):


 * $$\bar{\beta }=\frac{N-1}{N}\hat{\beta }$$

For failure terminated data (meaning that the test ends after a specified test time):


 * $$\bar{\beta }=\frac{N-2}{N}\hat{\beta }$$

Example 1 Two prototypes of a system were tested simultaneously with design changes incorporated during the test. Table 5.1 presents the data collected over the entire test. Find the Crow-AMSAA parameters and the intensity function using maximum likelihood estimators. Table 5.1 - Developmental test data for two identical systems Solution For the failure terminated test, using Eqn. (amsaa6):


 * $$\widehat{\beta }=\frac{22}{22\ln 620-\underset{i=1}{\overset{22}{\mathop{\sum }}}\,\ln {{T}_{i}}}$$


 * where:


 * $$\underset{i=1}{\overset{22}{\mathop \sum }}\,\ln {{T}_{i}}=105.6355$$


 * Then:


 * $$\widehat{\beta }=\frac{22}{22\ln 620-105.6355}=0.6142$$

From Eqn. (amsaa5):


 * $$\widehat{\lambda }=\frac{22}=0.4239$$

Therefore, $${{\lambda }_{i}}(T)$$  becomes:


 * $$\begin{align}

& {{\widehat{\lambda }}_{i}}(T)= & 0.4239\cdot 0.6142\cdot {{620}^{-0.3858}} \\ & = & 0.0217906\frac{\text{failures}}{\text{hr}} \end{align}$$

Figure 4fig81 shows the plot of the failure rate. If no further changes are made, the estimated MTBF is $$\tfrac{1}{0.0217906}$$  or 46 hr.