Date: October 5, 2019

Topic: Point estimation

Recall

Explain what a statistic is?

What is the sampling distribution fo the statistic?

What properties must the random variables that are used to compute statistics have?

What do we mean when we say something is an accurate estimator?

What do we mean when we state that something is a precise estimator?

Definition

Suppose $X_1, X_2, \dots X_n$ are all independent and identical distributed random variables that all have the same cumulative probability distribution as the random variable $X$. A sample $x_1, x_2, \dots, x_n$ from the probability distribution of $X$ can be considered as single observations of the random variables $X_1, X_2, \dots X_n$

Any function $t(x_1, x_2,\dots, x_n)$ of these sample values is called a statistic and is considered to be an observation of the random variable $T(X_1,X_2,\dots,X_n)$. The probability distribution for the random variable $T$ is then known as the sampling distribution for the statistic $t$.

We use statistics to estimate the parameters of statistical distributions. In other words, we use some function of a series of independent and identically-distributed random variables to generate estimates for the parameters of the sampled distribution.

Accuracy and biased estimators

An estimator is said to be biased if the expectation of the sampling distribution is not equal to the value of the parameter. Biased estimators are inaccurate as the values we obtain will not be clustered around the true value of the parameter.

Precise estimators

An estimator is said to be precise if the amount of variation in multiple samples of the statistics is small.

These ideas of bias/precision and accuracy are discussed in the following video:

https://www.youtube.com/watch?v=qfThUCzX4g0

<aside> 📌 SUMMARY: Statistics are functions of sets of independent and identical random variables. They are often used to estimate the parameters of distribution. A statistic is said to be accurate (or unbiased) if its expectation is equal to the value of the parameter. A statistic is said to be precise if the amount of variable in multiple samples of it is small.

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