Unit 2: statistical estimation

Unit TWO

Statistical estimationS

2.1 Introduction
Let X₁, X₂, ……X_n be a random sample of size n from a population of N units and x₁, x₂, ……x_n be the corresponding observed values. The values of the unknown parameter(s) should be estimated from the set of the sample observations. These estimates are often single valued which are known as point estimates. Also, many times an interval is to be estimated in which the parameter is expected to lie with a certain level of confidence.
In most statistical analysis, population parameters are usually unknown and have to be estimated form a sample. As such the methods for estimating the population parameters assume an important role in statistical analysis. The random variables (such as and ) used to estimate population parameters such as  and .
2.2 Point and interval estimation
The estimate of a population parameter may be one single value or it could be a range of values. In the former case it is referred as point estimate, whereas in the latter case it is termed as interval estimate. The investigator usually makes these two types of estimates through sampling analysis. While making estimates of population parameters, the researcher can give only the best point estimate or else he shall have to speak in terms of intervals and probabilities for he can never estimate with certainty the exact values of population parameters. In short:
Point estimate: - One value (called a point) that is used to estimate a population parameter. For example, the sample mean is the point estimator of the population mean .
Interval estimate: - States the range within which a population parameter probably lies. The interval with in which a population parameter is expected to occur is called confidence interval.
2.3 Distinction between estimator and estimate

In the theory of estimation, the population with which our final interest is associated contains certain quantities whose values are unknown. For example, a manufacturing process is producing a certain machinery part. The manufacturer likes to know what percent of his product is defective. In this example, the unknown is the proportion of defectives P, which is the unknown characteristic of the population known as parameter. Since the particular parameter under question is unknown we choose a sample of suitable size and form a suitable function of the observations. This function is known as an estimator or a statistic. The unknown parameter is estimated by the estimator obtained from the sample. The numerical value of the estimator is known as an estimate. For example, the random variables (such as and ) used to estimate population parameters such as  and are conventionally called as ‘estimators’, while specific values of these (such as = 110 or =15) are referred to as ‘estimates’ of the population parameters. The estimate of a population parameter may be one single value or it could be a range of values.