IAI CS1 common mistakes
IAI CS1 · Understand the traps and how to avoid them.
Writing the likelihood as a product of densities and then attempting to maximise it directly, without taking logs, so the derivative algebra becomes unmanageable and the score function is wrong.
Candidates translate the definition L(θ) = Π f(xi; θ) literally and start differentiating a product of n terms; the product rule explosion obscures the structure and invites sign and exponent errors.
What to do instead
Always move to the log-likelihood before differentiating: write l(θ) = Σ log f(xi; θ), simplify the sum term by term, then differentiate. Check the score function has mean zero under the model as a sanity test.
Computing a maximum likelihood estimate but never verifying it is a maximum (no second derivative check, no boundary check), and reporting a stationary point that is actually a minimum or a saddle.
The candidate solves S(θ) = 0 and treats any root as the MLE; for many CS1 models the log-likelihood is concave only on part of the parameter space, and roots outside it are easy to produce.
What to do instead
State the second derivative and its sign at the candidate, or argue concavity of l(θ) over the admissible range. Also evaluate l at the boundaries of the parameter space before declaring a maximum.
Leaving the answer to an estimation or confidence interval question as an expression in the data (e.g. x̄, Σxi²) rather than substituting the given numeric summary statistics to produce a number.
Candidates concentrate on the derivation and then run out of time or assume the algebraic form is sufficient; mark schemes award the final numeric mark separately.
What to do instead
After deriving any estimator, immediately substitute the supplied summary values (n, Σxi, Σxi²) and give the estimate to a stated number of decimal places. Treat the numeric substitution as a compulsory final step.
Constructing a confidence interval for a proportion or small-sample mean with the normal quantile when the correct pivot is a t quantile or a variance-stabilising transform (e.g. the arcsine transform for a binomial proportion).
The normal approximation is over-practised; candidates default to 1.96 without checking whether σ is known or whether the sample is small enough that the t distribution or a transform is required.
What to do instead
Before choosing a quantile, write down explicitly whether σ is known, what the sample size is, and what the underlying distribution is. For proportions near 0 or 1 or with small n, use the arcsine transform and back-transform the limits.
In hypothesis testing, computing a test statistic but then comparing it to the wrong tail or reporting a p-value for a two-sided alternative using a one-sided critical value.
Candidates compute the statistic mechanically and then pick a critical value from memory; the direction of the alternative and the doubling of the tail area are handled carelessly under time pressure.
What to do instead
Write the hypotheses and the rejection region explicitly before computing anything, and state whether the test is one- or two-sided in words. For a two-sided test, double the one-sided tail probability to get the p-value and check it against α.
Applying the delta method or deriving the asymptotic variance of a transformed estimator while omitting the squared derivative factor, or evaluating the derivative at the wrong point.
The formula Var(g(θ̂)) ≈ [g'(θ)]² Var(θ̂) is remembered loosely, and candidates substitute the sample estimate into g' after rather than before the transformation, or drop the square.
What to do instead
Write the delta method as a chain of three explicit pieces: the function g, its derivative g' evaluated at the true or estimated parameter, and the variance of the original estimator. Compute each separately before combining, and keep the square visible.
When asked for a method of moments estimator, equating only the first raw moment even when the model has two parameters, producing one equation for two unknowns.
Candidates learn the mean-matching case first and do not register that the number of moment equations must equal the number of parameters.
What to do instead
Count the parameters first, then set up that many moment equations E[X^k] = (1/n)Σxi^k for k = 1, 2, ... Solve the system simultaneously and check the estimator lies in the parameter space.
In Bayesian questions, writing the posterior as proportional to the likelihood only, dropping the prior, or failing to recognise a conjugate kernel and instead attempting an intractable normalising integral.
The proportionality constant is treated as optional decoration, and candidates do not have the conjugate families (beta-binomial, gamma-Poisson, normal-normal) at their fingertips.
What to do instead
Always write posterior ∝ prior × likelihood with both factors visible, then match the product to a known kernel and read off the parameters. Verify the result integrates to one by identifying the distribution name and its parameters.
Fitting a linear model and interpreting a coefficient or prediction interval without checking the residual assumptions, or reporting R² as evidence that the model is adequate.
Software-style output encourages candidates to read off numbers; the CS1 emphasis on model diagnostics and the distinction between confidence and prediction intervals is under-rehearsed.
What to do instead
For any regression answer, state the residual plots or tests you would inspect, distinguish the confidence interval for the mean response from the prediction interval for a new observation, and comment on leverage and influential points where relevant.
Presenting a derivation with no stated distributional assumption or support, so that indicator functions and ranges of validity (e.g. 0 < x < 1, θ > 0) are lost and the likelihood is written over the wrong region.
Candidates treat the support as implicit and focus on the density formula; this silently invalidates later integration limits, CRLB regularity conditions and sufficiency arguments.
What to do instead
At the start of every model-based question, write the density with its support and the parameter space explicitly. Carry indicator terms through the likelihood, and check regularity conditions (support not depending on θ) before invoking the Cramér-Rao lower bound.