Maximum Likelihood Estimation for Quant Interviews

Interviewers ask candidates to derive an MLE by hand because the derivation, not the formula, shows whether the reasoning behind it is understood.

Maximum likelihood estimation picks the parameter value that makes the observed data most probable under an assumed distribution. Quant interviews ask candidates to derive it by hand for a handful of standard distributions because the derivation exposes whether the candidate understands the reasoning behind the formula, rather than only the formula itself.

The likelihood function is the joint probability of the observed sample, written as a function of the unknown parameter rather than the data. Because that joint probability is a product of many terms, taking its logarithm converts the product into a sum, which is far easier to differentiate and maximize. The parameter value that maximizes the log-likelihood is identical to the one that maximizes the likelihood itself, since the logarithm is a strictly increasing function, so nothing is lost by working with the sum instead of the product.

Setting the derivative of the log-likelihood, called the score function, equal to zero and solving for the parameter is the mechanical core of every MLE derivation. For several standard distributions that equation has a closed-form solution, which is exactly what interviewers ask candidates to reproduce on a whiteboard.

For a Bernoulli or Binomial sample, the maximum likelihood estimate of the success probability is simply the sample mean, the observed proportion of successes. For a Poisson sample, the MLE of the rate parameter is also the sample mean. For an Exponential sample, the MLE of the rate parameter is one divided by the sample mean.

For a Normal sample, the MLE of the mean is the sample mean, and it is unbiased. The MLE of the variance is the average squared deviation from the sample mean, dividing by the sample size n rather than n minus 1, and that estimator is biased downward by a factor of n minus 1 divided by n. For a sample of 10 observations, that factor works out to 0.9, meaning the MLE variance estimate runs about 10 percent low on average. This is precisely why the standard unbiased sample variance formula divides by n minus 1 instead of n, and it is a favorite interview trap: a candidate who writes down the MLE variance formula without noting the bias has missed half the question.