Mathematical Statistics Lecture __exclusive__ Jun 2026

Mathematical statistics is the application of probability theory

: Formal proofs for unbiasedness , consistency , and efficiency (Cramér-Rao Lower Bound). Hypothesis Testing : Defining the Null ( H0cap H sub 0 ) and Alternative ( H1cap H sub 1 ) hypotheses, Type I/II errors, and p-values.

Do you need to keep every single data point to make an inference, or can you summarize the data? Sufficient Statistics A statistic is sufficient for if the conditional distribution of the data , does not depend on . It means captures all the information the data holds about . If you can split the density like this, is sufficient. Rao-Blackwell Theorem mathematical statistics lecture

[X̄−zα/2σn,X̄+zα/2σn]open bracket cap X bar minus z sub alpha / 2 end-sub the fraction with numerator sigma and denominator the square root of n end-root end-fraction comma space cap X bar plus z sub alpha / 2 end-sub the fraction with numerator sigma and denominator the square root of n end-root end-fraction close bracket If the variance σ2sigma squared is unknown, we substitute the sample variance S2cap S squared

Take on countable values (e.g., number of heads in coin flips). Modeled using a Probability Mass Function (PMF). Sufficient Statistics A statistic is sufficient for if

Standard curricula for this subject, such as those found at MIT OpenCourseWare and the LSE , typically follow a structured progression: Mathematical Statistics (2024): Lecture 1

is the random error term, assumed to be normally distributed with a mean of zero. Ordinary Least Squares (OLS) Rao-Blackwell Theorem [X̄−zα/2σn

. Completeness is vital for establishing the uniqueness of optimal estimators. 3. Point Estimation Theory

): Failing to reject the null hypothesis when it is actually false (False Negative). Statistical Power (