A statistical inference is basically a process that involves the inference of the data in a statistical manner. There are basically two types of statistical inferences, namely estimation and the test of the hypothesis.
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Estimation serves the purpose of determining the true value of the population that is based on the observations or the samples that are collected by sampling. To carry out estimation, the researcher needs to utilize certain statistics.
Estimation involves the use of two popular terms that a researcher should understand. The two terms that are used extensively by the researcher in estimation are the estimator and the estimate. These two terms, called the estimator and the estimate, can be explained with the help of an example. It is assumed in estimation that x1 x2 x3 (and so on) are the collection of the sample from the population having ‘s’ as their parameter. If the T=T(x) is a statistic then E(T(x))= s is the estimation. In this manner, estimation of the statistic is done. In this case of estimation, the estimator is the statistic T, and the estimate is the parameter called ‘s.’
It is important to understand the properties of estimators in estimation theory.
In estimation theory, unbiasedness is the first property that is assumed for an ideal estimator.
The Unbiasedness property of the estimators in estimation theory is basically those types of estimators that give their outcome as zero bias for all the values of the parameter. If the researcher considers the example above, then T in the theory of estimation is said to be unbiased only if its estimate is simply ‘s.’
The second property in estimation theory is that of the consistent estimators that involve the estimation that is consistent in nature. In other words, it can also be said that the consistent estimators in the theory of estimation should have a higher degree of concentration as the value of the random variable increases. In the theory of estimation, the sufficient condition of consistency explains that an estimator is supposed to be consistent only if the estimation of its expected value gives an unbiased estimate and the variance of the estimator is zero. In estimation, these two conditions are fulfilled only when the number of random variables tends to infinity.
There is another property for the ideal estimator in the theory of estimation called efficiency. According to the condition of this property in the theory of estimation, the consistent estimators should be distributed by normal distribution. This condition is introduced in the theory of estimation because there is some possibility that the estimators, which satisfy the sufficient conditions of consistency, may not be an efficient estimator.
The last property of the ideal estimator in the theory of estimation is the property of sufficiency. An estimator in the theory of estimation is said to be sufficient only if the joint conditional distribution function of the sample or the observation falls under the condition where T1 T2 T3 T4 (and so on) are the values under the function of the estimator ‘T.’ Thus, this joint conditional distribution in estimation should be independent of the parameter‘s.’
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.
