Three-Parameter Weibull Distribution Estimation Using Least Squares Method

The three-parameter Weibull distribution is widely used in reliability analysis and lifetime data modeling due to its flexibility. The least squares method, as a classical parameter estimation technique, efficiently solves for shape, scale, and location parameters by minimizing the sum of squared residuals between observed values and model predictions.

Core Implementation Steps: Data Sorting and Cumulative Distribution: Sort failure times and compute the empirical cumulative distribution function (CDF) as the foundation for least squares fitting. In code, this typically involves using sorting algorithms and calculating median ranks using formulas like (i-0.3)/(n+0.4) for plotting positions. Linearization Transformation: Apply double logarithm transformation to the Weibull CDF to convert it into linear equation form, enabling straightforward least squares application. This requires implementing log(-log(1-F(x))) transformations where F(x) represents the cumulative probability. Initial Parameter Estimation: Obtain initial estimates through iterative adjustment of location parameters or quantile methods to accelerate convergence. Programming implementations often use percentile-based approaches or grid search for initial guess generation. Residual Optimization: Employ numerical algorithms (such as Levenberg-Marquardt) to optimize nonlinear least squares problems, balancing computational efficiency with estimation accuracy. This involves coding gradient calculations and implementing trust-region optimization techniques.

Advantages: Compared to maximum likelihood estimation, the least squares method demonstrates lower dependency on initial values and allows intuitive assessment of goodness-of-fit (e.g., R² metric). Note that special handling is required for censored data scenarios, which may involve incorporating survival analysis techniques in the implementation.