A standard 50-to-90-minute mathematical statistics lecture typically follows a strict rhythm:
The difficulty lies in the abstraction. You aren't looking at spreadsheets; you are looking at functions of random variables. mathematical statistics lecture
Before we can analyze data, we must assume a mathematical structure for where that data comes from. In mathematical statistics, we assume data arises from a Random Variable $X$. The difficulty lies in the abstraction
The lecturer circles back to plain English: "So, in a bar fight, what does 'consistency' mean? It means that if you collect enough data, the chance of your estimate being wrong goes to zero." Before we can analyze data, we must assume
The core problem: We want to find a "good" statistic to estimate $\theta$. We call this statistic an Estimator, denoted $\hat\theta$.
Let $X_1, X_2, \dots, X_n$ be a random sample from a population with probability density function (pdf) $f(x; \theta)$, where $\theta$ is an unknown parameter (or vector of parameters) belonging to a parameter space $\Theta$.
The problem: You understand sufficiency. You don't understand completeness. The fix: Completeness ensures that the sufficient statistic is minimal. In lecture, think of completeness as a "uniqueness" property. If ( E[g(T)] = 0 ) for all ( \theta ), then ( g(T) = 0 ). This prevents weird, biased estimators from sneaking in.