Estimation and Types of Estimation
Let us discuss Estimation and Types of Estimation.
The procedure of making judgment or decision about a population parameter is referred to as statistical estimation or simply estimation. Statistical estimation procedures provide estimates of population parameter with a desired degree of confidence. The degree of confidence can be controlled in part,
- by the size the sample (larger sample greater accuracy of the estimate) and
- by the type of the estimate made. Population parameters are estimated from sample data because it is not possible (it is impracticable) to examine the entire population in order to make such an exact determination.
The statistical estimation of the population parameter is further divided into two types, (i) Point Estimation and (ii) Interval Estimation
Point Estimation
The objective of point estimation is to obtain a single number from the sample which will represent the unknown value of the population parameter. Population parameters (population mean, variance, etc) are estimated from the corresponding sample statistics (sample mean, variance, etc).
A statistic used to estimate a parameter is called a point estimator or simply an estimator, the actual numerical value obtained by estimator is called an estimate.
A population parameter is denoted by $\theta$ which is unknown constant. The available information is in the form of a random sample $x_1,x_2,\cdots,x_n$ of size $n$ drawn from the population. We formulate a function of the sample observation $x_1,x_2,\cdots,x_n$. The estimator of $\theta$ is denoted by $\hat{\theta}$. The different random sample provides different values of the statistics $\hat{\theta}$. Thus $\hat{\theta}$ is a random variable with its own sampling probability distribution.
Interval Estimation
A point estimator (such as sample mean) calculated from the sample data provides a single number as an estimate of the population parameter, which can not be expected to be exactly equal to the population parameter because the mean of a sample taken from a population may assume different values for different samples. Therefore we estimate an interval/ range of values (set of values) within which the population parameter is expected to lie with a certain degree of confidence. This range of values used to estimate a population parameter is known as interval estimate or estimate by a confidence interval, and is defined by two numbers, between which a population parameter is expected to lie. For example, $a<\bar{x}<b$ is an interval estimate of the population mean $\mu$, indicating that the population mean is greater than $a$ but less than $b$. The purpose of an interval estimate is to provide information about how close the point estimate is to the true parameter.
Note that the information developed about the shape of a sampling distribution of the sample mean i.e. Sampling Distribution of $\bar{x}$ allows us to locate an interval that has some specified probability of containing the population mean $\mu$.
Which of the two types of estimation do you like the most, and why?
- Point estimation is nice because it provides an exact point estimate of the population value. It provides you with the single best guess of the value of the population parameter.
- Interval estimation is nice because it allows you to make statements of confidence that an interval will include the true population value.
