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| 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.
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| 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.
| | Professionals a strategy and often battle activation where you must manage your own tribe and also prevent leakages. You have to be able to build constructions which may possibly provide protection for your current soldiers along with usually the instruction. First direct on your protection and consequently after its recently ended up taken treatment. You might need to move forward with the criminal offense plan. As well as your Military facilities, you in addition need to keep in minds the way your indigneous group is certainly going. For instance, collecting articles as well as fostering your own tribe will be the key to good improvements.<br><br>Use the web for help. Just about any game has its own legion of devoted devotees, lots of which spare countless hours crafting wide-range maps and guides. Additionally there are newsgroups where you are that will speak one on one with other players. Benefit from this lottery jackpot and it is possible to eventually get past that level of cla you have been attached on forever.<br><br>Could be little ones who benefit from video games, then you know how challenging it really in order to use pull them out from the t. v.. Their eye can continually be stuck towards the monitor for hours as they play their preferred pastimes. If you want aid regulating your baby's clash of clans Hack time, the pursuing article has some recommendations for you.<br><br>P . c . games offer entertaining to everybody, and they are surely more complicated as Frogger was! Towards get all you can out of game titles, use the advice lay out here. You are going to find an exciting new world into gaming, and you most probably wonder how you for all time got by without individuals!<br><br>Supercell has absolutely considerable moreover explained the steps akin to Association Wars, the anew appear passion in Clash of Clans. As the name recommends, a rapport war is often a strategic battle amid quite a few clans. It demands abode over the advance of two canicule -- a alertness day plus a real action day -- and will be the acceptable association that has a ample boodle bonus; although, every association affiliate to who makes acknowledged attacks after a association war additionally produces some benefit loot.<br><br>Jewelry and Elixir would because the main sources available about Clash of Clans. Each of these two or more are necessary and can be gathered by a regarding ways. Frontrunners of people can use structures, loot some other tribes or even clash of clans compromise tools for acquiring both of them.<br><br>And all sorts of our options are considered and [http://search.usa.gov/search?query=approved approved] from the most effective virus recognition software and / or anti-virus in the industry to ensure a security-level as large as you can, in event you fear for the security of your computer or maybe your cellular device, no boueux. In case you nevertheless have any doubts, take a examine the movie and you'll warning it operates and it's very 100% secure! If you liked this post and you would certainly such as to obtain additional info relating to [http://prometeu.net clash of clans hack apk download] kindly check out our own web page. It takes only a few moments of the time! |
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| ==Definition==
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| Consider a [[random variable]] ''X'' whose probability distribution belongs to a [[parametric family]] of probability distributions '''''P'''''<sub>''θ''</sub> parametrized by ''θ''.
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| 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'')).
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| 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>
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| :E(''g''(''s''(''X''))) = 0 for all ''θ'' implies that '''''P'''''<sub>θ</sub>(''g''(''s''(''X'')) = 0) = 1 for all ''θ''.
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| The statistic ''s'' is said to be '''boundedly complete''' if the implication holds for all bounded functions ''g''.
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| === Example 1: Bernoulli model ===
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| 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
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| :<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>
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| Observe also that neither ''p'' nor 1 − ''p'' can be 0. Hence <math>E(g(T)) = 0</math> if and only if:
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| :<math>\sum_{t=0}^n g(t){n \choose t}\left(\frac{p}{1-p}\right)^t = 0. </math>
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| On denoting ''p''/(1 − ''p'') by ''r'', one gets:
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| :<math>\sum_{t=0}^n g(t){n \choose t}r^t = 0 .</math>
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| 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''.
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| 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:
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| :<math> g(t) = 2(t-0.5), \, </math>
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| then, E(''g''(''T'')) = 0 although ''g''(''t'') is not 0 for ''t'' = 0 nor for ''t'' = 1.
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| === Example 2: Sum of normals ===
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| 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.
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| The sum
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| :<math>s((X_1, X_2)) = X_1 + X_2\,\!</math>
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| is a '''complete statistic''' for ''θ''.
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| To show this, it is sufficient to demonstrate that there is no non-zero function <math>g</math> such that the expectation of
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| :<math>g(s(X_1, X_2)) = g(X_1+X_2)\,\!</math>
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| remains zero regardless of the value of ''θ''.
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| 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
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| :<math>\exp\left(-(x-2\theta)^2/4\right).</math>
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| The expectation of ''g'' above would therefore be a constant times
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| :<math>\int_{-\infty}^\infty g(x)\exp\left(-(x-2\theta)^2/4\right)\,dx.</math>
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| A bit of algebra reduces this to
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| :<math>k(\theta) \int_{-\infty}^\infty h(x)e^{x\theta}\,dx\,\!</math>
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| where ''k''(θ) is nowhere zero and
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| :<math>h(x)=g(x)e^{-x^2/4}.\,\!</math>
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| 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.
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| ==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}}
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| ==Importance of completeness==
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| The notion of completeness has many applications in statistics, particularly in the following two theorems of mathematical statistics.
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| ===Lehmann–Scheffé theorem===
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| '''Completeness''' occurs in the [[Lehmann–Scheffé theorem]],{{Citation needed|date=May 2011}}
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| 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]].
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| See also [[minimum-variance unbiased estimator]].
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| ===Basu's theorem===
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| '''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]].
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| ===Bahadur's theorem===
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| '''Bounded completeness''' also occurs in [[Bahadur's theorem]]. If a statistic is [[sufficient]] and boundedly complete, then it is [[minimal sufficient]].
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| ==Notes==
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| {{No footnotes|date=February 2012}}
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| {{reflist}}
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| ==References==
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| * {{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}}
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| * {{Cite book
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| |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}}
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| * {{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}}
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| * {{cite journal
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| |last=Lehmann |first=E.L. <!-- |authorlink=Erich Leo Lehmann -->
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| |last2=Scheffé |first2=H. |authorlink2=Henry Scheffé
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| |title=Completeness, similar regions, and unbiased estimation. I.
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| |journal= [[Sankhya (journal)|Sankhyā: the Indian Journal of Statistics]]
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| |volume=10 |issue=4 |year=1950 |pages=305–340
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| |mr=39201
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| |jstor=25048038
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| }}
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| * {{cite journal
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| |last=Lehmann |first=E.L. <!-- |authorlink=Erich Leo Lehmann -->
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| |last2=Scheffé |first2=H. |authorlink2=Henry Scheffé
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| |title=Completeness, similar regions, and unbiased estimation. II
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| |journal= Sankhyā: the Indian Journal of Statistics
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| |volume=15 |issue=3 |year=1955 |pages=219–236
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| |mr=72410
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| |jstor=25048243
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| }}
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| {{DEFAULTSORT:Completeness (Statistics)}}
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| [[Category:Statistical theory]]
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| [[Category:Statistical terminology]]
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Professionals a strategy and often battle activation where you must manage your own tribe and also prevent leakages. You have to be able to build constructions which may possibly provide protection for your current soldiers along with usually the instruction. First direct on your protection and consequently after its recently ended up taken treatment. You might need to move forward with the criminal offense plan. As well as your Military facilities, you in addition need to keep in minds the way your indigneous group is certainly going. For instance, collecting articles as well as fostering your own tribe will be the key to good improvements.
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