Symmetry and Skewness in Weibull Modeling: Optimal Grouping for Parameter Estimation in Fertilizer Granule Strength

This study investigates Weibull distribution modeling for data under grouped observations. Two data grouping methods (equal-width and optimal) were compared for estimating parameters of the Weibull distribution using maximum likelihood estimation (MLE) in each case. Methodologically, our contribution is twofold: First, we derive the correct Fisher information matrix for grouped data in the two-parameter Weibull and use it to compute optimal interval boundaries. Second, we derive maximum likelihood estimators for data grouped under these optimal intervals. The fit of the assumed distributions was evaluated using chi-squared goodness-of-fit tests. We also calculated Asymptotic Relative Efficiency (ARE) to compare the precision of parameter estimates across different grouping approaches. Optimal boundaries yielded systematically higher ARE than equal-width grouping in 100% of comparisons for the shape parameter c. Gains for the scale parameter b were smaller and occurred in about 62% of cases. Optimal grouping also produced generally higher chi-squared (χ2) goodness-of-fit p-values than equal-width grouping, indicating a better fit. From a symmetry standpoint, the Weibull distribution is inherently asymmetric, with the degree of asymmetry governed by the shape parameter c. We show that the choice of grouping affects the estimate of c and, thus, the inferred skewness, further explaining why optimally designed intervals yield both higher precision and a more faithful representation of failure behavior.