Forgetting to check regularity conditions for MLE

Maximum likelihood estimators have nice properties (consistency, asymptotic normality) only when regularity conditions hold. If the support depends on the parameter (e.g., Uniform(0, theta)), the standard MLE theory may not apply.

Confusing the score function with the likelihood

The score function is the derivative of the log-likelihood, not the likelihood itself. Setting the likelihood equal to zero (instead of the score) is a common algebraic setup error that gives nonsensical results.

Misapplying the chi-squared test

The chi-squared goodness-of-fit test requires expected cell counts of at least 5. If some cells have small expected counts, you must combine adjacent cells before computing the test statistic.

Errors in Bayesian posterior computation

The posterior is proportional to the likelihood times the prior. Candidates sometimes forget to include all observations in the likelihood or use the wrong prior family for the specified conjugate pair.

Confusing Type I and Type II errors

Type I error is rejecting a true null hypothesis (false positive). Type II error is failing to reject a false null (false negative). Power is 1 minus the Type II error rate. These definitions must be precise on the exam.

Not recognizing sufficient statistics

By the factorization theorem, a statistic T(X) is sufficient if the likelihood factors as g(T, theta) * h(X). Candidates sometimes include extra terms in T that are not needed, overcomplicating the problem.

Misinterpreting confidence intervals

A 95% confidence interval does not mean there is a 95% probability the parameter lies in the interval. It means that 95% of such intervals, constructed from repeated samples, would contain the true parameter.

Errors in simulation variance reduction

Antithetic variates require using U and 1-U as paired inputs. Control variates require knowing E[C] exactly. Misapplying these methods (e.g., using the wrong pairing) can increase variance rather than reduce it.

Using the wrong degrees of freedom

Different tests and distributions use different degrees of freedom. For a t-test with n observations and p predictors, the degrees of freedom are n-p-1, not n-1 or n-p. Getting this wrong changes critical values.

Skipping the check for stationarity in time series

Fitting an AR or MA model to non-stationary data produces unreliable parameter estimates and forecasts. Always verify stationarity (constant mean and variance) before model fitting.

Common MAS-I Mistakes