Estimation

Quantitative Results

The method of statistically drawing an inference on data is called the statistical inference.  Thus, the testing of hypothesis and the inference are the most important factors involved.  The theory of estimation is a part of statistics that extracts parameters from observations that are corrupted with noise.

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Estimation is a division of statistics and signal processing that determines the values of parameters through measured and observed empirical data.  The process of estimation is carried out in order to measure and diagnose the true value of a function or a particular set of populations.  It is done on the basis of observations on the samples, which are a combined piece of the target population or function.  Several statistics are used to perform the task of estimation.

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There are two very important terms that are used in estimation: the estimator and the estimate.  To understand the concept of the estimator and estimate in detail, we will use an example.  Let’s say that a1, a2, a3 and so on is a collection of samples from some group of a certain population with ‘x’ as its parameter.  Here, if T= T (a) is a statistic, then E (T(a)) = x.  From these equations we can realize that an estimation of the statistic has been carried out, where the statistic T is an estimator and the parameter ‘x’ is the estimator.

Estimation has many important properties for the ideal estimator.  These properties include unbiased nature, efficiency, consistency and sufficiency.

The estimators that are unbiased while performing estimation are those that have 0 bias results for the entire values of the parameter. Going by statistical language and terminology, unbiased estimators are those where the mathematical expectation or the mean proves to be the parameter of the target population.  In the above mentioned example for estimation, T is going to be the unbiased estimator only if its estimate comes out to be equal to ‘x.’

In estimation, the estimators that give consistent estimates are said to be the consistent estimators.  As the number of random variables increase, the degree of concentration should be higher and higher around the estimate in order to make the estimator of estimation the consistent estimator.  If the estimator gives an unbiased estimate and the variance of the estimator comes out to be zero, then the estimator of estimation is called the consistent estimator.  These two conditions need to be fulfilled only if the numbers of random variables reach infinity.

In estimation there are many estimators that have ample incidences of consistent estimators, and according to the property of efficiency in estimation, the consistent estimators should be normally distributed.  This kind of property was taken into account in the theory of estimation, because there were incidents of the estimators having ample consistent estimations but were not the efficient estimators.

An estimator is known as the sufficient estimator only when the joint conditional distribution function of the sample/observation has the condition of T1 T2 T3 T4 (and so on and so forth), and are the values under the given estimator function ‘T.’  Thus, the resultant joint conditional estimation has to be absolutely sovereign of the parameter ‘x.’While carrying out the task of estimation, a researcher should always know that the best estimator is the one that is the minimum variance unbiased estimator (MVUE).  This means that the estimator has the minimum variability when it is compared to other estimators.