Editing Point estimation (section)

== Properties of point estimates ==

=== Biasedness ===
“[[Bias of an estimator|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.<ref name=":0">{{Cite book|title=A Modern Introduction to Probability and Statistics|publisher=F.M. Dekking, C. Kraaikamp, H.P. Lopuhaa, L.E. Meester|year=2005|language=English}}</ref> Most importantly, we prefer point estimators that have the smallest [[Mean squared error|mean square errors.]]

If we let T = h(X<sub>1</sub>,X<sub>2</sub>, . . . , X<sub>n</sub>) be an estimator based on a random sample X<sub>1</sub>,X<sub>2</sub>, . . . , X<sub>n</sub>, the estimator T is called an unbiased estimator for the parameter θ if E[T] = θ, irrespective of the value of θ.<ref name=":0"/> For example, from the same random sample we have E(x̄) = μ (mean) and E(s<sup>2</sup>) = σ<sup>2</sup> (variance), then x̄ and s<sup>2</sup> would be unbiased estimators for μ and σ<sup>2</sup>. The difference E[T ] − θ is called the bias of T ; if this difference is nonzero, then T is called biased.

=== Consistency ===
Consistency is about whether the point estimate stays close to the value when the parameter increases its size. The larger the sample size, the more accurate the estimate is. If a point estimator is consistent, its expected value and variance should be close to the true value of the parameter. An unbiased estimator is consistent if the limit of the variance of estimator T equals zero.

=== Efficiency ===
Let ''T''<sub>1</sub> and ''T''<sub>2</sub> be two unbiased estimators for the same parameter ''θ''. The estimator ''T''<sub>2</sub> would be called ''more efficient'' than estimator ''T''<sub>1</sub> if Var(''T''<sub>2</sub>) ''<'' Var(''T''<sub>1</sub>), irrespective of the value of ''θ''.<ref name=":0"/> We can also say that the most efficient estimators are the ones with the least variability of outcomes. Therefore, if the estimator has smallest variance among sample to sample, it is both most efficient and unbiased. We extend the notion of efficiency by saying that estimator T<sub>2</sub> is more efficient than estimator T<sub>1</sub> (for the same parameter of interest), if the MSE([[Mean squared error|mean square error]]) of T<sub>2</sub> is smaller than the MSE of T<sub>1</sub>.<ref name=":0"/>

Generally, we must consider the distribution of the population when determining the efficiency of estimators. For example, in a [[normal distribution]], the mean is considered more efficient than the median, but the same does not apply in asymmetrical, or [[Skewed distribution|skewed]], distributions.

=== Sufficiency ===
In statistics, the job of a statistician is to interpret the data that they have collected and to draw statistically valid conclusion about the population under investigation. But in many cases the raw data, which are too numerous and too costly to store, are not suitable for this purpose. Therefore, the statistician would like to condense the data by computing some statistics and to base their analysis on these statistics so that there is no loss of relevant information in doing so, that is the statistician would like to choose those statistics which exhaust all information about the parameter, which is contained in the sample. We define [[sufficient statistic]]s as follows: Let X =( X<sub>1</sub>, X<sub>2</sub>, ... ,X<sub>n</sub>) be a random sample. A statistic T(X) is said to be sufficient for θ (or for the family of distribution) if the conditional distribution of X given T is free from θ.<ref name=":1">{{Cite book|title=Estimation and Inferential Statistics|publisher=Pradip Kumar Sahu, Santi Ranjan Pal, Ajit Kumar Das|year=2015|language=English}}</ref>