A default model needs a probability estimate; a return model needs estimates of its mean and variance. Sample means are one starting point. This lesson compares bias, variance and mean squared error, then derives maximum-likelihood estimates for Bernoulli defaults and normal returns. It states when large-sample standard errors are justified and when boundary cases or model misspecification invalidate the usual calculation.
Learning objectives. By the end of this note you should be able to:
- define bias, variance and MSE of an estimator and compare two estimators on MSE;
- write a likelihood and log-likelihood for an i.i.d. sample and solve the score equation for Bernoulli and normal models;
- state consistency and asymptotic normality of the MLE under regularity;
- explain why divides by (biased) while remaining a standard large-sample choice.
In this lesson
- Estimators and risk
- Likelihood and the MLE
- Worked MLEs
- Consistency and asymptotic normality
- Observed information and practical SEs
- Method of moments (brief contrast)
Before you start
The full lesson, with worked examples and exercise solutions, is for members.
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