Estimation

When data are collected by sampling from a population, the most important objectives of statistical analysis is to draw inferences or generalizations about that population from the information embodied in the sample data. Statistical estimation, or briefly estimation, is concerned with the methods by which population characteristics are estimated form sample information. It may be pointed out that the true value of a parameter is an unknown constant that can be correctly ascertained only by an exhaustive study of the population. However, it is ordinarily too expensive or it is infeasible to enumerate complete populations to obtain the required information. In case of finite populations, the cost of complete censuses may be prohibitive and in case of infinite population, complete enumerations are impossible. A realistic objective may be to obtain a guess or 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. Statistical estimation procedures provide us with the means of obtaining estimates of population parameters with desired degrees of precision. With the respect to estimating a parameter, the following two types of estimates are possible:

1.    Point estimates, and

2.    Interval estimates.

Point estimates.

A 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,……., x_n from a population f(x, θ) and then to use some preconceived method 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 real number scale and thus the name point estimation. θ depends on the random variables that generate the sample and hence, it too is a random variable with its own sampling distribution.

Interval estimates.

As distinguished from a point estimate which provides one single value of the parameter, an interval estimate of a population parameter is a statement of two values between which it is estimated that the parameter lies. An interval estimate would always be specified by two values, i.e., the lower one and the upper one. In more technical terms, interval estimation refers to the estimation of a parameter by a random by a random interval, called 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 predetermined number, 1 – a. L and U are called the confidence limits and are the random end-points of interval estimate. Since in an interval estimate, we determine an interval of plausible values, hence the name interval estimations.

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