unbiased estimate[¦ən′bī·əst ′es·tə·mət]
an estimate of a parameter of a probability distribution that is based on observed values and is free of systematic errors. More precisely, if the distribution to be estimated depends on the parameters θ1, θ2, . . . . ,θs, then the functions θi* (x1, x2, . . . , xn) of the observational results x1, x2, . . . .xn) are called unbiased estimates for the parameters θi, if for any admissible values of the parameters θ1, θ2, . . . , θs the mathematical expectation Eθi* (x1 , x2, . . . , xn) ═ θi. For example, if x1, x2, . . . , xn are the results of n independent observations of a random variable having a normal distribution
with unknowns a (mathematical expectation) and σ2 (variance), then the arithmetic mean
(1) x̄ ═ (x1 + x2 + . . . + xn)/n
is an unbiased estimate for a. The quantity
which is often used as an estimate for the variance, is not an unbiased estimate. An unbiased estimate for σ2 is
while an unbiased estimate for the standard deviation σ has the more complicated form
The estimate (1) for the mathematical expectation and the estimate (2) for the variance are unbiased estimates in the more general case of distributions that differ from a normal distribution; the estimate (3) for the standard deviation in general (for distributions other than normal) may be biased.
The use of unbiased estimates is necessary in estimating an unknown parameter by means of a large number of series of observations, each of which consists of a small number of observations. For example, let there be k sequences
xi1, xi2, . . . , xin (i ═ 1, 2, . . . , k)
with n observations in each, and let si2 be the unbiased estimate [defined as in (2)] for σ2 computed from the i th sequence of observations. Then, for large k, from the law of large numbers we have
even when n is small. Unbiased estimates play an important role in the statistical control of mass production.
REFERENCESCramer, H. Mathematicheskie melody statistiki. Moscow, 1948. (Translated from English.)
Kolmogorov, A. N. “Nesmeshchennye otsenki.” Izv. AN SSSR. Seriia matematicheskaia, 1950, no. 4.
Gnedenko, B. V., lu. K. Beliaev, and A. D. Solov’ev. Matematicheskie melody ν teorii nadezhnosti. Moscow, 1965.
IU. V. PROKHOROV