# Unbiased Estimate

## unbiased estimate

[¦ən′bī·əst ′es·tə·mət]## Unbiased Estimate

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 θ_{1}, θ_{2}, . . . . ,θ_{s}, then the functions θ_{i}^{*} (x_{1}, x_{2}, . . . , x_{n}) of the observational results x_{1}, x_{2}, . . . .x_{n}) are called unbiased estimates for the parameters θ_{i}, if for any admissible values of the parameters θ_{1}, θ_{2}, . . . , θ_{s} the mathematical expectation Eθ_{i}^{*} (x_{1} , x_{2}, . . . , x_{n}) ═ θ_{i.} For example, if x_{1}, x_{2}, . . . , x_{n} are the results of * _{n}* independent observations of a random variable having a normal distribution

with unknowns a (mathematical expectation) and σ^{2} (variance), then the arithmetic mean

(1) x̄ ═ (*x _{1} + x_{2} + . . . + x_{n})/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 *σ ^{2}* 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

x_{i1}, x_{i2}, . . . , x_{in} (*i* ═ 1, 2, . . . , *k)*

with *n* observations in each, and let s_{i}^{2} be the unbiased estimate [defined as in (2)] for σ^{2} 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.

### REFERENCES

Cramer, 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