Parametric Estimation

A sample (possibly with right-censoring or truncation) is given and you must derive the MLE for one or two parameters of a named distribution.

Write the likelihood as the product of densities for fully observed values and survival functions for right-censored values. Take the log and differentiate; set the score equal to zero. For two-parameter distributions, this gives a system you may need to solve numerically — for some distributions like exponential, the MLE has a closed form. Always verify the second-order condition (or argue concavity of the log-likelihood).

L(θ) = Π [f(x_i; θ)]^{δ_i} [S(x_i; θ)]^{1 - δ_i} for right-censoring indicator δ_i.

A sample mean and variance (or sample percentiles) are given and you must fit a two-parameter distribution.

For method of moments, set sample moments equal to theoretical moments and solve for parameters. For percentile matching, set empirical percentile values equal to theoretical percentile values and solve. Method of moments is fast but less efficient than MLE; percentile matching is useful when the question gives percentiles directly.

Method of moments: m̄ = E[X; θ], s^2 = Var(X; θ); solve simultaneously for θ.

A fitted distribution is compared against observed data and you must compute a test statistic and decide whether to reject.

For chi-square, bucket the data, compute Σ (O - E)^2 / E, and compare to a chi-square critical value with the right degrees of freedom (k - 1 - number of estimated parameters). For K-S, compute the maximum vertical distance between empirical and theoretical CDFs. Anderson-Darling weights the tails more heavily. Adjust critical values when parameters were estimated from the same data.

Chi-square statistic: Σ (O - E)^2 / E with df = k - 1 - p (p = parameters estimated).