Mathematical Statistics Lecture May 2026

Lecture Notes: Foundations of Mathematical Statistics

0:15 – 0:30: The Log-Trick & Differentiation

• Why maximize log-likelihood ( \ell(\theta) ).
• Derivative for Exponential family: ( \frac\partial \ell\partial \lambda = \fracn\lambda - \sum x_i ).
• Solve: ( \hat\lambda_MLE = 1 / \barx ).

A. Properties of a Good Estimator

How do we judge if an estimator is mathematically sound?

1. Unbiasedness An estimator $\hat\theta$ is unbiased for $\theta$ if: $$E[\hat\theta] = \theta$$ The expected value of the estimator equals the true parameter. mathematical statistics lecture

• Example: The sample mean $\barX$ is an unbiased estimator for the population mean $\mu$.
• Counter-Example: The sample variance defined as $\frac1n\sum(x_i - \barx)^2$ is biased (it underestimates variance). This is why we use the Bessel correction ($n-1$ in the denominator) to make it unbiased.

2. Efficiency There are often many unbiased estimators for the same parameter. We prefer the one with the smallest variance. Why maximize log-likelihood ( \ell(\theta) )

• If $\hat\theta_1$ and $\hat\theta_2$ are both unbiased, $\hat\theta_1$ is more efficient if $Var(\hat\theta_1) < Var(\hat\theta_2)$.

3. Consistency An estimator is consistent if it converges in probability to the true parameter as the sample size $n \to \infty$. $$\hat\theta_n \xrightarrowP \theta$$ (As we get more data, the estimate gets arbitrarily close to the truth). mathematical statistics lecture

---

Part 4: The Best Free Mathematical Statistics Lectures Online

You don't need to pay for a Master’s degree to access top-tier content. The following resources represent the gold standard for mathematical statistics lectures available for free on YouTube and academic portals.