Difference between revisions of "Inefficient statistic"
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''inefficient estimator'' | ''inefficient estimator'' | ||
| − | A [[Statistical estimator|statistical estimator]] whose variance is greater than that of an [[Efficient estimator|efficient estimator]]. In other words, for an inefficient estimator equality in the [[Rao–Cramér inequality|Rao–Cramér inequality]] is not attained for at least one value of the parameter to be estimated. A quantitative measure of inefficiency of an inefficient estimator is the number | + | A [[Statistical estimator|statistical estimator]] whose variance is greater than that of an [[Efficient estimator|efficient estimator]]. In other words, for an inefficient estimator equality in the [[Rao–Cramér inequality|Rao–Cramér inequality]] is not attained for at least one value of the parameter to be estimated. A quantitative measure of inefficiency of an inefficient estimator is the number $ e $, |
| + | the so-called efficiency, which is the ratio of the variance of an efficient estimator to that of the statistic in question. The efficiency $ e $ | ||
| + | is non-negative and does not exceed 1. The quantity $ 1/ e $ | ||
| + | indicates by how much one has to increase the number of observations in using an inefficient estimator as compared with an efficient estimator so as to achieve equivalent results in the application of the two statistics. For example, the median $ \mu _ {n} $ | ||
| + | of an empirical distribution constructed from $ n $ | ||
| + | independent normally $ N ( \theta , \sigma ^ {2} ) $- | ||
| + | distributed random variables $ X _ {1} \dots X _ {n} $ | ||
| + | is asymptotically normally distributed with parameters $ \theta $ | ||
| + | and $ \sigma ^ {2} \pi / 2n $ | ||
| + | and is an inefficient [[Order statistic|order statistic]] estimating the expectation $ \theta $. | ||
| + | In this case an efficient estimator is given by $ \overline{X}\; = ( X _ {1} + \dots + X _ {n} ) /n $, | ||
| + | which is distributed according to the normal law $ N ( \theta , \sigma ^ {2} / n ) $. | ||
| + | The efficiency $ e $ | ||
| + | of the statistic $ \mu _ {n} $ | ||
| + | is | ||
| + | |||
| + | $$ | ||
| + | e = \ | ||
| + | |||
| + | \frac{ {\mathsf D} ( \overline{X}\; ) }{ {\mathsf D} ( \mu _ {n} ) } | ||
| − | + | = | |
| + | \frac{2} \pi | ||
| + | . | ||
| + | $$ | ||
| − | Consequently, in the use of the statistic | + | Consequently, in the use of the statistic $ \mu _ {n} $ |
| + | one has to make on the average $ \pi / 2 \approx 1.57 $ | ||
| + | more observations as compared with $ \overline{X}\; $ | ||
| + | in order to obtain the same accuracy in the estimation of the unknown mathematical expectation $ \theta $. | ||
====References==== | ====References==== | ||
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> H. Cramér, "Mathematical methods of statistics" , Princeton Univ. Press (1946)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> L.N. Bol'shev, N.V. Smirnov, "Tables of mathematical statistics" , ''Libr. math. tables'' , '''46''' , Nauka (1983) (In Russian) (Processed by L.S. Bark and E.S. Kedrova)</TD></TR></table> | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> H. Cramér, "Mathematical methods of statistics" , Princeton Univ. Press (1946)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> L.N. Bol'shev, N.V. Smirnov, "Tables of mathematical statistics" , ''Libr. math. tables'' , '''46''' , Nauka (1983) (In Russian) (Processed by L.S. Bark and E.S. Kedrova)</TD></TR></table> | ||
| − | |||
| − | |||
====Comments==== | ====Comments==== | ||
Latest revision as of 22:12, 5 June 2020
inefficient estimator
A statistical estimator whose variance is greater than that of an efficient estimator. In other words, for an inefficient estimator equality in the Rao–Cramér inequality is not attained for at least one value of the parameter to be estimated. A quantitative measure of inefficiency of an inefficient estimator is the number $ e $, the so-called efficiency, which is the ratio of the variance of an efficient estimator to that of the statistic in question. The efficiency $ e $ is non-negative and does not exceed 1. The quantity $ 1/ e $ indicates by how much one has to increase the number of observations in using an inefficient estimator as compared with an efficient estimator so as to achieve equivalent results in the application of the two statistics. For example, the median $ \mu _ {n} $ of an empirical distribution constructed from $ n $ independent normally $ N ( \theta , \sigma ^ {2} ) $- distributed random variables $ X _ {1} \dots X _ {n} $ is asymptotically normally distributed with parameters $ \theta $ and $ \sigma ^ {2} \pi / 2n $ and is an inefficient order statistic estimating the expectation $ \theta $. In this case an efficient estimator is given by $ \overline{X}\; = ( X _ {1} + \dots + X _ {n} ) /n $, which is distributed according to the normal law $ N ( \theta , \sigma ^ {2} / n ) $. The efficiency $ e $ of the statistic $ \mu _ {n} $ is
$$ e = \ \frac{ {\mathsf D} ( \overline{X}\; ) }{ {\mathsf D} ( \mu _ {n} ) } = \frac{2} \pi . $$
Consequently, in the use of the statistic $ \mu _ {n} $ one has to make on the average $ \pi / 2 \approx 1.57 $ more observations as compared with $ \overline{X}\; $ in order to obtain the same accuracy in the estimation of the unknown mathematical expectation $ \theta $.
References
| [1] | H. Cramér, "Mathematical methods of statistics" , Princeton Univ. Press (1946) |
| [2] | L.N. Bol'shev, N.V. Smirnov, "Tables of mathematical statistics" , Libr. math. tables , 46 , Nauka (1983) (In Russian) (Processed by L.S. Bark and E.S. Kedrova) |
Comments
There are also cases for which the minimal attainable variance of an estimator is larger than the Cramér–Rao bound, [a1].
References
| [a1] | C.R. Rao, "Linear statistical inference and its applications" , Wiley (1965) pp. 283 |
Inefficient statistic. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Inefficient_statistic&oldid=47337