My questions is: Is there any particular converging pattern for Least Square Estimators and Maximum Likelihood Estimators of Weibull distribution when we increase the sample size? is there any mathematical explanation for that? To be clear, why the MLE estimator is deceasing (looks like exponential decrease) with sample size and converging to the true value (true value =0.5), and why the LSE is increasing with sample size and converging to the true value? (Apologize for a little bit tedious). I'm working on something about comparison of Least Square methods and Maximum Likelihood Estimation on Weibull Distribution. What I'm doing is that conducting a simulation study (Monte Carlo Simulation with 5000 iterations) to compare these two methods for different sample size (from 5 to 1000). The following figure indicate that for MLE by increasing sample size $N$, the estimated shape parameter is converging to the true value from a relative larger value than true value in a decreasing pattern. While for LSE, the estimated value start from a relative smaller value than the true value and then converge to the true value with a generally increasing pattern. The true values for the parameters are: shape parameter = 0.5 and scale parameter = 1.
The paper I'm working on can be found from here
Looking forward to any recommendations and comments. If my statement is not quite clear for you please let me know.