|
|
| (One intermediate revision by one other user not shown) |
| Line 1: |
Line 1: |
| {{Refimprove|date=August 2009}}
| | Person who wrote the review is called Roberto Ledbetter and his wife does not have to like it at nearly all. In his professional life he is literally a people manager. He's always loved living in Guam and he has everything that he could use there. The widely used hobby for him also his kids is growing plants but he's been [http://Mondediplo.com/spip.php?page=recherche&recherche=ingesting ingesting] on new things lately. He's been working on [http://www.reddit.com/r/howto/search?q=website website] for some a moment now. Check it available here: http://prometeu.net<br><br> |
|
| |
|
| In [[statistics]], '''completeness''' is a property of a [[statistic]] in relation to a model for a set of observed data. In essence, it is a condition which ensures that the parameters of the probability distribution representing the model can all be estimated on the basis of the statistic: it ensures that the distributions corresponding to different values of the parameters are distinct.
| | Here is my blog post [http://prometeu.net clash of clans hack no survey no download] |
| | |
| It is closely related to the idea of [[identifiability]], but in [[statistical theory]] it is often found as a condition imposed on a [[sufficient statistic]] from which certain optimality results are derived.
| |
| | |
| ==Definition==
| |
| Consider a [[random variable]] ''X'' whose probability distribution belongs to a [[parametric family]] of probability distributions '''''P'''''<sub>''θ''</sub> parametrized by ''θ''.
| |
| | |
| Formally, a [[statistic]] ''s'' is a [[measurable function]] of ''X''; thus, a statistic ''s'' is evaluated on a random variable ''X'', taking the value ''s''(''X''), which is itself a random variable. A given realization of the random variable ''X''(''ω'') is a data-point (datum), on which the statistic ''s'' takes the value ''s''(''X''(''ω'')).
| |
| | |
| The statistic ''s'' is said to be '''complete''' for the distribution of ''X'' if for every measurable function ''g'' (which must be independent of ''θ'') the following implication holds:<ref>Young, G. A. and Smith, R. L. (2005). Essentials of Statistical Inference. (p. 94). Cambridge University Press.</ref>
| |
| :E(''g''(''s''(''X''))) = 0 for all ''θ'' implies that '''''P'''''<sub>θ</sub>(''g''(''s''(''X'')) = 0) = 1 for all ''θ''.
| |
| The statistic ''s'' is said to be '''boundedly complete''' if the implication holds for all bounded functions ''g''.
| |
| | |
| === Example 1: Bernoulli model ===
| |
| The Bernoulli model admits a complete statistic.<ref>Casella, G. and Berger, R. L. (2001). Statistical Inference. (pp. 285-286). Duxbury Press.</ref> Let ''X'' be a [[random sample]] of size ''n'' such that each ''X''<sub>i</sub> has the same [[Bernoulli distribution]] with parameter ''p''. Let ''T'' be the number of 1s observed in the sample. ''T'' is a statistic of ''X'' which has a [[binomial distribution]] with parameters (''n'',''p''). If the parameter space for ''p'' is (0,1), then ''T'' is a complete statistic. To see this, note that
| |
| | |
| :<math> \operatorname{E}(g(T)) = \sum_{t=0}^n {g(t){n \choose t}p^{t}(1-p)^{n-t}} = (1-p)^n \sum_{t=0}^n {g(t){n \choose t}\left(\frac{p}{1-p}\right)^t} .</math>
| |
| | |
| Observe also that neither ''p'' nor 1 − ''p'' can be 0. Hence <math>E(g(T)) = 0</math> if and only if:
| |
| | |
| :<math>\sum_{t=0}^n g(t){n \choose t}\left(\frac{p}{1-p}\right)^t = 0. </math>
| |
| | |
| On denoting ''p''/(1 − ''p'') by ''r'', one gets:
| |
| | |
| :<math>\sum_{t=0}^n g(t){n \choose t}r^t = 0 .</math>
| |
| | |
| First, observe that the range of ''r'' is all positive reals. Also, E(''g''(''T'')) is a [[polynomial]] in ''r'' and, therefore, can only be identical to 0 if all coefficients are 0, that is, ''g''(''t'') = 0 for all ''t''.
| |
| | |
| It is important to notice that the result that all coefficients must be 0 was obtained because of the range of ''r''. Had the parameter space been finite and with a number of elements smaller than ''n'', it might be possible to solve the linear equations in ''g''(''t'') obtained by substituting the values of ''r'' and get solutions different from 0. For example, if ''n'' = 1 and the parametric space is {0.5}, a single observation, ''T'' is not complete. Observe that, with the definition:
| |
| | |
| :<math> g(t) = 2(t-0.5), \, </math>
| |
| | |
| then, E(''g''(''T'')) = 0 although ''g''(''t'') is not 0 for ''t'' = 0 nor for ''t'' = 1.
| |
| | |
| === Example 2: Sum of normals ===
| |
| This example will show that, in a sample of size 2 from a [[normal distribution]] with known variance, the statistic X1+X2 is complete and sufficient. Suppose (''X''<sub>1</sub>, ''X''<sub>2</sub>) are [[statistical independence|independent]], identically distributed random variables, [[normal distribution|normally distributed]] with expectation ''θ'' and variance 1.
| |
| The sum
| |
| | |
| :<math>s((X_1, X_2)) = X_1 + X_2\,\!</math>
| |
| | |
| is a '''complete statistic''' for ''θ''.
| |
| | |
| To show this, it is sufficient to demonstrate that there is no non-zero function <math>g</math> such that the expectation of
| |
| | |
| :<math>g(s(X_1, X_2)) = g(X_1+X_2)\,\!</math>
| |
| | |
| remains zero regardless of the value of ''θ''.
| |
| | |
| That fact may be seen as follows. The probability distribution of ''X''<sub>1</sub> + ''X''<sub>2</sub> is normal with expectation 2''θ'' and variance 2. Its probability density function in <math>x</math> is therefore proportional to
| |
| | |
| :<math>\exp\left(-(x-2\theta)^2/4\right).</math>
| |
| | |
| The expectation of ''g'' above would therefore be a constant times
| |
| | |
| :<math>\int_{-\infty}^\infty g(x)\exp\left(-(x-2\theta)^2/4\right)\,dx.</math>
| |
| | |
| A bit of algebra reduces this to
| |
| | |
| :<math>k(\theta) \int_{-\infty}^\infty h(x)e^{x\theta}\,dx\,\!</math>
| |
| | |
| where ''k''(θ) is nowhere zero and
| |
| | |
| :<math>h(x)=g(x)e^{-x^2/4}.\,\!</math>
| |
| | |
| As a function of ''θ'' this is a two-sided [[Laplace transform]] of ''h''(''X''), and cannot be identically zero unless ''h''(''x'') is zero almost everywhere.<ref>{{cite web|last=Orloff|first=Jeremy|title=Uniqueness of Laplace Transform|url=http://web.mit.edu/jorloff/www/18.03-esg/notes/laplaceuniqueness.pdf}}</ref> The exponential is not zero, so this can only happen if ''g''(''x'') is zero almost everywhere.
| |
| | |
| ==Relation to sufficient statistics==
| |
| For some parametric families, a complete [[sufficient statistic]] does not exist. Also, a [[minimal sufficient]] statistic need not exist. (A case in which there is no minimal sufficient statistic was shown by [[R. R. Bahadur|Bahadur]] in 1957. {{Citation needed|date=September 2010}}) Under mild conditions, a minimal sufficient statistic does always exist. In particular, these conditions always hold if the random variables (associated with '''''P'''''<sub>''θ''</sub> ) are all discrete or are all continuous.{{Citation needed|date=May 2011}}
| |
| | |
| ==Importance of completeness==
| |
| The notion of completeness has many applications in statistics, particularly in the following two theorems of mathematical statistics.
| |
| | |
| ===Lehmann–Scheffé theorem===
| |
| '''Completeness''' occurs in the [[Lehmann–Scheffé theorem]],{{Citation needed|date=May 2011}}
| |
| which states that if a statistic that is unbiased, '''complete''' and [[sufficiency (statistics)|sufficient]] for some parameter θ, then it is the best mean-unbiased estimator for θ. In other words, this statistic has a smaller expected loss for any [[convex function|convex]] loss function; in many practice applications with the squared loss-function, it has a smaller mean squared error among any estimators with the same [[expected value]].
| |
| | |
| See also [[minimum-variance unbiased estimator]].
| |
| | |
| ===Basu's theorem===
| |
| '''Bounded completeness''' occurs in [[Basu's theorem]],<ref>Casella, G. and Berger, R. L. (2001). Statistical Inference. (pp. 287). Duxbury Press.</ref> which states that a statistic which is both '''boundedly complete''' and [[Sufficient statistic|sufficient]] is [[statistical independence|independent]] of any [[ancillary statistic]].
| |
| | |
| ===Bahadur's theorem===
| |
| | |
| '''Bounded completeness''' also occurs in [[Bahadur's theorem]]. If a statistic is [[sufficient]] and boundedly complete, then it is [[minimal sufficient]].
| |
| | |
| ==Notes==
| |
| {{No footnotes|date=February 2012}}
| |
| {{reflist}}
| |
| | |
| ==References==
| |
| * {{cite book|authorlink=Deb. Basu|first=D.|last=Basu|title=Statistical information and likelihood : A collection of critical essays by Dr. D. Basu|editor=J. K. Ghosh|publisher=Springer|year=1988| series=Lecture Notes in Statistics | volume=45 | isbn=0-387-96751-6|mr=953081|ref=harv}}
| |
| * {{Cite book
| |
| |last1=Bickel|first1=Peter J.|last2=Doksum|first2=Kjell A.|authorlink1=Peter J. Bickel|title=Mathematical statistics, Volume 1: Basic and selected topics|edition=Second (updated printing 2007) of the Holden-Day 1976 |mr=443141|year=2001|publisher=Pearson Prentice–Hall|isbn=0-13-850363-X|ref=harv}}
| |
| * {{cite book|first1=Lehmann|last1=E. L.|last2=Romano|first2=Joseph P.|authorlink1=Erich Leo Lehmann|title=Testing statistical hypotheses|url=http://www.springerlink.com/content/978-0-387-98864-1#section=545952&page=1|edition=Third|series=Springer Texts in Statistics|publisher=Springer|location=New York|year=2005|pages=xiv+784|isbn=0-387-98864-5|mr=2135927|ref=harv}}
| |
| * {{cite journal
| |
| |last=Lehmann |first=E.L. <!-- |authorlink=Erich Leo Lehmann -->
| |
| |last2=Scheffé |first2=H. |authorlink2=Henry Scheffé
| |
| |title=Completeness, similar regions, and unbiased estimation. I.
| |
| |journal= [[Sankhya (journal)|Sankhyā: the Indian Journal of Statistics]]
| |
| |volume=10 |issue=4 |year=1950 |pages=305–340
| |
| |mr=39201
| |
| |jstor=25048038
| |
| }}
| |
| * {{cite journal
| |
| |last=Lehmann |first=E.L. <!-- |authorlink=Erich Leo Lehmann -->
| |
| |last2=Scheffé |first2=H. |authorlink2=Henry Scheffé
| |
| |title=Completeness, similar regions, and unbiased estimation. II
| |
| |journal= Sankhyā: the Indian Journal of Statistics
| |
| |volume=15 |issue=3 |year=1955 |pages=219–236
| |
| |mr=72410
| |
| |jstor=25048243
| |
| }}
| |
| | |
| {{DEFAULTSORT:Completeness (Statistics)}}
| |
| [[Category:Statistical theory]]
| |
| [[Category:Statistical terminology]]
| |