Statistical inference is a process of drawing an inference about the data statistically. Statistical inference, therefore, involves testing of hypothesis and estimation.
Estimation is done by the researcher in order to determine the true value of any function or population on the basis of the observations or the sample which is collectively a part of the population or the function. For the purpose of estimation, the researcher makes use of certain statistics.
In estimation, there are two terms that are used that a researcher should understand. The two terms involved in estimation are estimator and estimate. The estimator and estimate in estimation can be explained with the help of an example. Suppose x1 x2 x3 and so on are a collection of samples from a population with ‘s’ as the parameter. Now, if the T=T(x) is some statistic, then E(T(x))= s. Here the estimation of the statistic is done. So, in this case in estimation, the estimator is the statistic T, and the estimate is the parameter called ‘s.’
In estimation, after defining estimators, the next task the researchers need to complete is to understand the properties of estimators.
In estimation, the first property of an ideal estimator is that of unbiasedness.
Unbiased estimators in estimation are those types of estimators that result to zero bias for all the values of the parameter. In statistical language in estimation, unbiased estimators are those types of estimators whose mathematical expectation or mean value comes out to be the parameter of the population. If we consider the example above, then T in estimation is said to be unbiased only if its estimate is ‘s.’
In estimation, the second property of an ideal estimator is that of consistency.
The consistent estimators in estimation are those types of estimators that give consistent estimation. In other words, consistent estimators in estimation should have higher and higher degrees of concentration around the estimate as the number of random variables increases. In estimation, there is a sufficient condition of consistency which states that an estimator is said to be a consistent estimator if, during the estimation of its expected value, it gives an unbiased estimate and the variance of the estimator should be zero. In estimation, these two conditions should be fulfilled only when the number of random variables tends to infinity.
In estimation, the third property of an ideal estimator is that of efficiency.
In estimation, there may be several estimators that abide by the sufficient condition of consistency. Therefore, the property of efficiency has been introduced in the theory of estimation. In estimation, according to the condition of the property of efficiency, the consistent estimators should be normally distributed. This property is introduced in estimation because there might be a possibility that the estimators satisfying the sufficient conditions of consistency might not be efficient estimators.
In estimation, the next property of an ideal estimator is that of sufficiency.
In estimation, an estimator is said to be sufficient only if the joint conditional distribution function of the sample (or the observation where the condition is T1 T2 T3 T4 and so on) are the values under the function of the estimator when ‘T’ is given. So, this joint conditional distribution in estimation should be independent of the parameter ‘s.’
The researcher should know the term estimator. An estimator in estimation is considered to be the best estimator only if it is a minimum variance unbiased estimator (MVUE). By minimum variance in estimation, we mean that the estimator has less variability as compared to the other estimators.
