Estimated quantity

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With the estimated quantity (it is not crowded and possesses it) in the statistics, I say amount (English: Estimate) that I estimated as a parameter (a parameter, reality cannot measure it) of the probability distribution or an estimate function (not not crowded breathe: Estimator) to express it for a function of data based on measured specimen data practically. There is various kinds of estimated quantity for each parameter. These are provided according to a different standard each, and it cannot be necessarily said that which is particularly superior.

In the estimate of the parameter, there are "point estimation" to give for one numerical value and two kinds of "the interval estimation" to give the section including the parameter stochastically, but often calls point estimation quantity with estimate quantity particularly.

There are confidence interval (I display it according to the probability that the section includes a parameter in) that is usually used to quantity of interval estimation and a trust section (a parameter displays it according to the probability that the section contains) in the Bayes statistics.

In the Japanese Industrial Standards, I define it as "a statistic to use to estimate a parameter of the population." [¹].

Table of contents

Definitions about the point estimation

Parameter${\displaystyle \theta}$ Quantity of の point estimation${\displaystyle {\widehat {\theta}}}$ として,

• ${\displaystyle {\widehat {\theta }}-\theta}$  を,${\displaystyle {\widehat {\theta}}}$  I say の error.
• ${\displaystyle {\widehat {\theta}}}$  The gap from の deflection that is a parameter ${\displaystyle B({\widehat {\theta }})=\operatorname {E} ({\widehat {\theta }})-\theta}$  It is defined と.
• For all θ${\displaystyle B({\widehat {\theta}}) = 0}$  の case (in other words, for all θ ${\displaystyle \operatorname {E} ({\widehat {\theta }})=\theta}$  For の case),${\displaystyle {\widehat {\theta}}}$  I say quantity of impartiality estimate of を parameter θ.

In var(X), dispersion, E(X) of X express expectation of X as follows.

• ${\displaystyle {\widehat {\theta}}}$  The の average square error,${\displaystyle \operatorname {MSE} ({\widehat {\theta }})=\operatorname {E} [({\widehat {\theta }}-\theta) ^{2}]}$  It is defined と.

• ${\displaystyle \operatorname {MSE} ({\widehat {\theta }})=\operatorname {var} ({\widehat {\theta }})+(B({\widehat {\theta }}))^{2}}$ 

In other words, it is mean square error = dispersion + (square of the deflection).

I say an estimated function of the standard deviation of the estimated quantity of standard deviation (the square root of dispersion) of the estimated quantity of Θ that is θ with a standard error of θ.

In addition, there is the following thing in the concept about the desirable property of the estimated quantity.

最尤推定

I see the conditional probability that data are provided under the condition that assumed a parameter a certain numerical value with the function of the parameter adversely, and it is said with the likelihood function in the data. The method for the parameter becoming biggest by a likelihood function is called maximum likelihood for data. 最尤推定量 is in this way an estimated thing for a price of the parameters.

Agreement-related (Consistency)

As the number of the specimens grows big with the agreement estimate quantity, I say ような estimate quantity to converge stochastically (dispersion converges to 0) to a certain numerical value.

In other words, it is estimated quantity${\displaystyle t_{n}}$ (in n specimen size) is how small${\displaystyle \epsilon> 0}$  に vs. even if do it:

${\displaystyle \lim _{n\to \infty }{\rm {Prob}}\left \left|t_{n}-\theta \right|<\epsilon \right = 1.}$ 

In となる case, it is a parameter${\displaystyle \theta}$  I say に vs. agreement estimate quantity to do. I say "strong agreement characteristics" (strong consistency) particularly almost surely (with probability 1) when I converge.

Effective (Efficiency)

The quality of the estimated quantity is evaluated with a mean square error generally. Dispersion (= average square error) has to choose the minimus from quantity of impartiality estimate more and says this with effective estimate quantity in some cases. Dispersion of the quantity of a certain impartiality estimate may be smaller than quantity of what kind of other impartiality estimates and should not make much only of one effective or unbiased.

Strong nature (Robustness)

I say that there are few changes of the estimated quantity by the change of the model that I supposed to be the strong nature.

Other

There is a thing that the dispersion estimate quantity that during 0 and 1 must have of quantity of probability estimate in the property that estimated quantity should meet elsewhere (natural) must be non-minus number, but impartiality estimate quantity may not meet these in some cases.

Footnote

1. It is 1999, 2.42 estimate quantity ^ JIS Z 8101-1.

References

• Yasuo Nishioka "probability statistics Ohmsha that a mathematics tutorial is plain and talks" about, 2013. ISBN 9784274214073。
• Mathematical Society of Japan "mathematics dictionary" Iwanami Shoten, 2007. ISBN 9784000803090。
• It is 1999 statistics - terms and sign - Part 1 Japan Standards Association, JIS Z 8101-1: Probability and general statistics term, http://kikakurui.com/z8/Z8101-1-1999-01.html 
• Koji Fushimi 『Probability theory and statistics theory』 River appearance Publishing, 1942. ISBN 9784874720127。

Allied item

• Probability
• Statistics
• Maximum likelihood
• Deflection

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