Point estimation

An essential task of inferential statistics is the most accurate determination of certain qualities or quantities of a population. Since the observations are assumed to be random, the determination may not be accurate, but is done as an estimate. A point estimate is in this sense a function that assigns to present observations an estimate of the size of interest. In most applications the quantity of interest is a parameter of the probability distribution of observations (for example, the average of a normal distribution ). The point estimate is different from the interval estimation substantially by the fact that its result is a single value and thus no information about the accuracy of the estimate can be derived from it. To form interval estimates point estimates are used as a basis.

A point estimate is a function of random observations, a point estimate of the calculated value of the point estimate for the present observations.

Point estimator are particularly used for the estimation of:

• Mean values
• Variances
• Standard deviations

The quality of a point estimator is assessed on the basis of various properties. The central properties in the literature are:

• Unbiasedness ( unbiasedness, genuineness ): A point estimator is unbiased if it correctly indicates the average of the actual value of the interest size. In this sense, the estimate does not have a systematic error.
• Consistency: Consistency means clear that the point estimate tends to be closer to the actual value of the interest size for a growing number of observations.
• Efficiency: efficiency is a point estimator if its spread in comparison with other point estimates is minimized. In this sense, an efficient estimator has no unnecessary scattering.

Example: There are random observations, all of which are independent and normally distributed with unknown mean and variance. The average value can be estimated by the arithmetic mean of the observations:

The expected value of this point estimate is the true parameter value:

Therefore, the estimator is unbiased.