Point estimation

In statistics, point estimation involves the use of sample data to calculate a single value (known as a point estimate since it identifies a point in some parameter space) which is to serve as a "best guess" or "best estimate" of an unknown population parameter (for example, the population mean). More formally, it is the application of a point estimator to the data to obtain a point estimate. Point estimation can be contrasted with interval estimation: such interval estimates are typically either confidence intervals, in the case of frequentist inference, or credible intervals, in the case of Bayesian inference. More generally, a point estimator can be contrasted with a set estimator. Examples are given by confidence sets or credible sets. A point estimator can also be contrasted with a distribution estimator. Examples are given by confidence distributions, randomized estimators, and Bayesian posteriors. “Bias” is defined as the difference between the expected value of the estimator and the true value of the population parameter being estimated. It can also be described that the closer the expected value of a parameter is to the measured parameter, the lesser the bias. When the estimated number and the true value is equal, the estimator is considered unbiased. This is called an unbiased estimator. The estimator will become a best unbiased estimator if it has minimum variance. However, A biased estimator with a small variance may be more useful than an unbiased estimator with a large variance. Most importantly, we prefer point estimators that has the smallest mean square errors. If we let T = h(X1,X2, . . . , Xn) be an estimator based on a random sample X1,X2, . . . , Xn, the estimator T is called an unbiased estimator for the parameter θ if E[T] = θ, irrespective of the value of θ. For example, from the same random sample we have E(x̄) = μ (mean) and E(s2) = σ2 (variance), then x̄ and s2 would be unbiased estimators for μ and σ2. The difference E[T ] − θ is called the bias of T ; if this difference is nonzero, then T is called biased.