Estimation

Aim

One can describe the aim of estimation as to identify the unknown parameter $\theta$ using estimators which enables to identify the underlying population distribution $f_X(x;\theta )$.

Types of estimation

Point estimation

Definition

The quantity $T(X_1\dots X_n)$ obtained from the observed values $X_1\dots X_n$ is called a “point” estimate of $\theta$.

Interval Estimation

Definition

A confidence interval for parameter $\theta$ with confidence coefficient $1-\alpha$ is a random interval constructed by two statistics $L(X_1\dots X_n)<U(X_1\dots X_n)$ such that:

$$P(L(\tilde{X})\le \theta \le U(\tilde{X}))\ge (1-\alpha )$$

for $\forall \theta \in \Omega$