Statistical Estimation

When data are collected by sampling from a population, the main important objective of statistical analysis is to draw inferences or generalizations about the population from the information embodied in the sample. The Statistical estimation, or briefly estimation, is concerned with the method by which population characteristics are estimated from the sample information. It may be point out that the true value of the parameter is an unknown constant that can be correctly as certained only by an exhaustive study of the population. However, it is ordinarily too costly or it is infeasible to enumerate complete populations to obtain the required information. In case of limited  populations, the case of complete censuses may be prohibitive and in case of unlimited populations, the costs of complete enumerations are impossible. A realistic objective may be to obtain a guess or to estimate of the unknown true value or an interval of plausible values from the sample data and also to determine the accuracy of the procedure. The Statistical estimation procedures provide us with the means of obtaining estimates of population parameters with desired degrees of precision. With respect to estimating a parameter, the following 2  types of estimates are possible:

1. Point estimates, and

2. Interval estimates

Point estimates: The  point estimate is a single number which is used as an estimate of the unknown population parameter. The procedure in point estimation is to select a random sample of n observations, x1, x2 ....x3 from a population ƒ (x, θ) and then to use some preconceived technique to arrive from these observations at a number say θ^ (read theta hat) which we accept as an estimator of θ. The estimator θ is a single point on the random variables that create the sample and hence, it too is a random variable with its own sampling distribution.

Interval estimates: A distinguished from a point estimate that provides one single value of the parameter, an interval estimate of a population parameter is a statement of two values between which it is specified by two values, i.e. the lower one &  the upper one. In more technical terms, the interval estimation refers to the estimation of a parameter by a random interval, known as  the Confidence interval, whose end points L and U with L < U, are functions of the observed random variables such that the probability that the inequality L < θ < U is satisfied in terms of pre determined number 1 - a. L and U are known as the confidence limits and are the random end points of interval estimate. Since in an interval estimate, we determine an interval of the plausible values, and hence the name interval estimation. Therefore, on the basis of sample study if we estimate the average income of the people living in a village as $ 875 it will be a point estimate. On the other side, if we say that the average income could lie between $ 800 to $ 950, it will be an interval estimate.

On comparing these two methods of estimation we find that the point estimation has an advantage as much as it gives  and exact value for the parameter under investigation. This benefit, however, is also the defect of a point estimate. On being a single point on the real number scale, a point estimate does not tell us how close the estimator is to parameter being estimated. Moreover, in the scientific investigation it is usually not necessary to know exact value of a parameter-instead it is desirable to have some degree of the confidence that the value obtained is within a certain range. The interval estimate does give such confidence and hence interval estimate should generally be employed in practice.