Estimation of the Lifetime of a Tip-Tilt Platform Using a Bootstrap Method

A tip-tilt platform is a movable component of the system for maintaining imaging stability in a large-aperture space telescope. There is an ongoing test to estimate the lifetime of a tip-tilt platform to be onboard a Chinese space solar telescope. This test has only a single sampling data point by nature. The prior probability distribution of the lifetime of a tip-tilt platform needed for any Bayesian lifetime estimation is also not available due to the scarcity of relevant lifetime data. A bootstrap method is a statistical method without parameterization, and it does not require any prior distribution. A bootstrap method, which can be employed with a computer, uses resampling to simulate the statistical distribution of interest. The resampling generates samples from a template sample which can be rather small (and is usually an actual sample). This effectively creates a large-size sample from a small-size sample. Generally, a bootstrap method requires the number of data points in its template sample to be no less than five. In this paper, we combine a semi-empirical sample virtual-augmentation method into a bootstrap method to estimate confidence intervals of the lifetime of the tip-tilt platform in the ongoing test. We take the lifetime of the tip-tilt platform to statistically follow a Weibull distribution,with the shape-parameter values of the Weibull distribution inferred from previous experimental data of similar products. The variance of lifetimes of similar tip-tilt platforms can be expressed by the shape-parameter values of the Weibull distribution and the recorded time of normally working of the platform in the test. Based on the expression the sample virtual-augmentation method numerically generates more data points to meet two statistical conditions, i.e., (1) their mean is equal to the recorded time of normally working of the platform, and (2) their variance is equal to the empirical variance of lifetimes of similar products. These data points constitute a template sample in our bootsrap estimation, in which 10000 samples are generated through resampling of the template sample. The mean of each generated sample is a statistical realization of lifetime. By sorting the means of the 10000 samples as μ₁≤μ₂≤…≤μ₁₀₀₀₀ we can find the confidence interval for a confidence level 1-α as (μ_k1, μ_k2), where k₁=10000×α/2 and k₂=10000×(1-α/2). We have accelerated the lifetime test by a factor of 5.5. Now that the tip-tilt platform has been working normally for more than 1.25 years (in actual time), our method yields a confidence interval at the 95% confidence level, (3.59 years, 10.18 years). If the tip-tilt platform can work normally for 2 years (in actual time), the confidence interval at the 95% level will be (5.93 years, 16.81 years).