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| {{essay|date=September 2009}}
| | You see books, audio programs and video's provide in depth advice and nuggets of valuable, bankable information. that when used correctly can propel you to new heights.<br><br> |
| {{More footnotes|date=November 2010}}
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| A '''well-behaved statistic''' is a term sometimes used in the theory of [[statistics]] to describe part of a procedure. This usage is broadly similar to the use of [[well-behaved]] in more general mathematics. It is essentially an assumption about the formulation of an estimation procedure (which entails the specification of an [[estimator]] or [[statistic]]) that is used to avoid giving extensive details about what conditions need to hold. In particular it means that the statistic is not an unusual one in the context being studied. Due to this, the meaning attributed to ''well-behaved statistic'' may vary from context to context.
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| The present article is mainly concerned with the context of [[data mining]] procedures applied to [[statistical inference]] and, in particular, to the group of computationally intensive procedure that have been called [[algorithmic inference]].
| | While you may not think these principles apply to residential real estate investing, once you are a closer look you may well be surprised at what you find.<br><br>The govt. will place a lien on a home the spot where the owner owes back taxes and shows no intent to pay. These administration. tax lien homes eventually may go up for auction on the highest prospective buyer. The govt. tax lien homes only truly sell for your amount that the govt. must be recoup for that taxes payable. Often there can be one bidder for the home, therefore you consider the time to do a title search upon the govt. tax lien here is where hula find out the taxes due, you are able to place a small winning tender.<br><br>What will make the difference for agents that sell homes every month and may actually never have a bad week? Using a real estate business plan (also referred to as business model). Seriously. It is not because they are so much smarter much more talented than everyone different. Of course, it doesn't hurt end up being smart and enjoy talent. The idea is, utilizing a real estate business in order to get started as a broker will be what separates you from all of the other agents. You see, most agents begin selling real estate based solely on the current talents with no real direction. That won't happen to you.<br><br>If happen to be selling your home and anyone could have appliances which have been severely outdated then truly replace them before showing the family home. You can will include a little on to the cost because buyers will be very glad to see that there are brand new appliances near the property.<br><br>You're an outside-of-the-box type of person-And last but not least, industry mentoring supplies different look at [http://Ireport.cnn.com/docs/DOC-1169363 Dean Graziosi] regarding just like you-those who aren't proud status quo. Those that like to approach a problem from substitute angle.<br><br>These departments will provide you with idea within the price of this asset. Now, you learn the cost of this estate that you want to pick up. At this point, you can make a smart and right decision. Thus, these organizations will assist you in preparing introduce business. |
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| ==Algorithmic inference==
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| {{Main|Algorithmic inference}}
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| In [[algorithmic inference]], the property of a statistic that is of most relevance is the pivoting step which allows to transference of probability-considerations from the sample distribution to the distribution of the parameters representing the population distribution in such a way that the conclusion of this [[statistical inference]] step is compatible with the sample actually observed.
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| By default, capital letters (such as ''U'', ''X'') will denote random variables and small letters (''u'', ''x'') their corresponding realizations and with gothic letters (such as <math>\mathfrak U, \mathfrak X</math>) the domain where the variable takes specifications. Facing a sample <math>\boldsymbol x=\{x_1,\ldots,x_m\}</math>, given a [[Algorithmic inference#Sampling mechanism|sampling mechanism]] <math>(g_\theta,Z)</math>, with <math>\theta</math> scalar, for the random variable ''X'', we have
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| :<math>\boldsymbol x=\{g_\theta(z_1),\ldots,g_\theta(z_m)\}.</math>
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| The sampling mechanism <math>(g_\theta,\boldsymbol z)</math>, of the statistic ''s'', as a function ? of <math>\{x_1,\ldots,x_m\}</math> with specifications in <math>\mathfrak S</math> , has an explaining function defined by the master equation: | |
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| : <math>s=\rho(x_1,\ldots,x_m)=\rho(g_\theta(z_1),\ldots,g_\theta(z_m))=h(\theta,z_1,\ldots,z_m),\qquad\qquad\qquad (1)</math>
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| for suitable seeds <math>\boldsymbol z=\{z_1,\ldots,z_m\}</math> and parameter ? | |
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| ==Well-behaved==
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| In order to derive the distribution law of the parameter ''T'', compatible with <math>\boldsymbol x</math>, the statistic must obey some technical properties. Namely, a statistic ''s'' is said to be '''well-behaved''' if it satisfies the following three statements:
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| # '''monotonicity'''. A uniformly monotone relation exists between ''s'' and ? for any fixed seed <math>\{z_1,\ldots,z_m\}</math> – so as to have a unique solution of (1);
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| # '''well-defined'''. On each observed ''s'' the statistic is well defined for every value of ?, i.e. any sample specification <math>\{x_1,\ldots,x_m\}\in\mathfrak X^m</math> such that <math>\rho(x_1,\ldots,x_m)=s</math> has a probability density different from 0 – so as to avoid considering a non-surjective mapping from <math>\mathfrak X^m</math> to <math>\mathfrak S</math>, i.e. associating via <math>s</math> to a sample <math>\{x_1,\ldots,x_m\}</math> a ? that could not generate the sample itself;
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| # '''local sufficiency'''. <math>\{\breve\theta_1,\ldots, \breve\theta_N\}</math> constitutes a true T sample for the observed ''s'', so that the same probability distribution can be attributed to each sampled value. Now, <math>\breve\theta_j= h^{-1}(s,\breve z_1^j, \ldots,\breve z_m^j)</math> is a solution of (1) with the seed <math>\{\breve z_1^j,\ldots,\breve z_m^j\}</math>. Since the seeds are equally distributed, the sole caveat comes from their independence or, conversely from their dependence on ? itself. This check can be restricted to seeds involved by ''s'', i.e. this drawback can be avoided by requiring that the distribution of <math>\{Z_1,\ldots,Z_m|S=s\}</math> is independent of ?. An easy way to check this property is by mapping seed specifications into <math>x_i</math>s specifications. The mapping of course depends on ?, but the distribution of <math>\{X_1, \ldots,X_m|S=s\}</math> will not depend on ?, if the above seed independence holds – a condition that looks like a ''local [[sufficient statistics|sufficiency]]'' of the statistic ''S''.
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| ===Example===
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| For instance, for both the [[Bernoulli distribution]] with parameter ''p'' and the [[exponential distribution]] with parameter ? the statistic <math>\sum_{i=1}^m x_i</math> is well-behaved. The satisfaction of the above three properties is straightforward when looking at both explaining functions: <math>g_p(u)=1</math> if <math>u\leq p</math>, 0 otherwise in the case of the Bernoulli random variable, and <math>g_\lambda(u)=-\log u/\lambda</math> for the Exponential random variable, giving rise to statistics
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| :<math>s_p=\sum_{i=1}^m I_{[0,p]}(u_i)</math>
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| and
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| :<math>s_\lambda=-\frac{1}{\lambda}\sum_{i=1}^m \log u_i.</math> | |
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| ''Vice versa'', in the case of ''X'' following a [[uniform distribution (continuous)|continuous uniform distribution]] on <math>[0,A]</math> the same statistics do not meet the second requirement. For instance, the observed sample <math>\{c,c/2,c/3\}</math> gives
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| <math>s'_A=11/6c</math>. But the explaining function of this ''X'' is <math>g_a(u)=u a</math>.
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| Hence a master equation <math>s_A=\sum_{i=1}^m u_i a</math> would produce with
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| a ''U'' sample <math>\{0.8, 0.8, 0.8\}</math> and a solution <math>\breve a=0.76 c</math>. This conflicts with the observed sample since the first observed value should result greater than the right extreme of the ''X'' range. The statistic <math>s_A=\max\{x_1,\ldots,x_m\}</math> is well-behaved in this case.
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| Analogously, for a random variable ''X'' following the [[Pareto distribution]] with parameters ''K'' and ''A'' (see [[Algorithmic inference#The general inversion problem solving the Fisher question|Pareto example]] for more detail of this case),
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| :<math>s_1=\sum_{i=1}^m \log x_i</math>
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| and
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| :<math>s_2=\min_{i=1,\ldots,m} \{x_i\}</math>
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| can be used as joint statistics for these parameters.
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| As a general statement that holds under weak conditions, [[sufficient statistics]] are well-behaved with respect to the related parameters. The table below gives sufficient / Well-behaved statistics for the parameters of some of the most commonly used probability distributions.
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| {{Anchor|SufficientTable}}
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| {| class="wikitable"
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| |+ Common distribution laws together with related sufficient and well-behaved statistics.
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| ! Distribution !! Definition of density function !! Sufficient/Well-behaved statistic
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| |-
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| | Uniform discrete
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| | <math>f(x;n)=1/n I_{\{1,2,\ldots,n\}}(x)</math>
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| | <math>s_n=\max_i x_i</math>
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| |-
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| | Bernoulli
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| | <math>f(x;p)=p^x (1-p)^{1-x} I_{\{0,1\}}(x)</math>
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| | <math>s_P=\sum_{i=1}^m x_i</math>
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| |-
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| | Binomial
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| | <math>f(x;n,p)=\binom{n}{x}p^x (1-p)^{n-x} I_{0,1,\ldots, n}(x)</math>
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| | <math>s_P=\sum_{i=1}^m x_i</math>
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| |-
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| | Geometric
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| | <math>f(x;p)=p(1-p)^x I_{\{0,1,\ldots\}}(x)</math>
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| | <math>s_P=\sum_{i=1}^m x_i</math>
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| |-
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| | Poisson
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| | <math>f(x;\mu)=\mathrm e^{-\mu x} \mu^x / x! I_{\{0,1,\ldots\}}(x)</math>
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| | <math>s_{M}=\sum_{i=1}^m x_i</math>
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| |-
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| | Uniform continuous
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| | <math>f(x;a,b)=1/(b-a) I_{[a,b]}(x)</math>
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| | <math>s_A=\min_i x_i; s_B=\max_i x_i</math>
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| |-
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| | Negative exponential
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| | <math>f(x;\lambda)=\lambda \mathrm e^{-\lambda x} I_{[0,\infty]}(x)</math>
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| | <math>s_{\Lambda}=\sum_{i=1}^m x_i</math>
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| |-
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| | Pareto
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| | <math>f(x;a, k)= \frac{a}{k}\left(\frac{x}{k}\right)^{-a -1} I_{[k,\infty]}(x)</math>
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| | <math>s_{A}=\sum_{i=1}^m \log x_i; s_K=\min_i x_i</math>
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| |-
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| | Gaussian
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| | <math>f(x,\mu,\sigma)= 1/(\sqrt{2 \pi}\sigma) \mathrm e^{-(x-\mu^2)/(2\sigma^2)}</math>
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| | <math>s_M=\sum_{i=1}^m x_i; s_{\Sigma}=\sqrt{\sum_{i=1}^m(x_i-\bar x)^2}</math>
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| |-
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| | Gamma
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| | <math>f(x;r,\lambda)= \lambda/\Gamma(r) (\lambda x)^{r-1} \mathrm e^{-\lambda x} I_{[0,\infty]}(x)</math>
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| | <math>s_{\Lambda}=\sum_{i=1}^m x_i; s_{K}=\prod_{i=1}^m x_i</math>
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| |}
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| == Notes ==
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| {{reflist}}
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| ==References==
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| *{{cite book
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| | author=Apolloni, B
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| | coauthors=Bassis, S., Malchiodi, D., Witold, P.
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| | title=The Puzzle of Granular Computing
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| | publisher=Springer
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| | series=Studies in Computational Intelligence
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| | location=Berlin
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| | volume=138
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| | year=2008
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| }}
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| *{{cite journal
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| | doi=10.1214/aoms/1177728604
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| | author=Bahadur, R. R.
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| | authorlink=R. R. Bahadur
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| | coauthors=Lehmann, E. L.
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| | title=Two comments on Sufficiency and Statistical Decision Functions
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| | journal=Annals of Mathematical Statistics
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| | volume=26
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| | year=1955
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| | pages=139–142
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| }}
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| {{DEFAULTSORT:Well-Behaved Statistic}}
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| <!-- Categories -->
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| [[Category:Statistical inference]]
| |
You see books, audio programs and video's provide in depth advice and nuggets of valuable, bankable information. that when used correctly can propel you to new heights.
While you may not think these principles apply to residential real estate investing, once you are a closer look you may well be surprised at what you find.
The govt. will place a lien on a home the spot where the owner owes back taxes and shows no intent to pay. These administration. tax lien homes eventually may go up for auction on the highest prospective buyer. The govt. tax lien homes only truly sell for your amount that the govt. must be recoup for that taxes payable. Often there can be one bidder for the home, therefore you consider the time to do a title search upon the govt. tax lien here is where hula find out the taxes due, you are able to place a small winning tender.
What will make the difference for agents that sell homes every month and may actually never have a bad week? Using a real estate business plan (also referred to as business model). Seriously. It is not because they are so much smarter much more talented than everyone different. Of course, it doesn't hurt end up being smart and enjoy talent. The idea is, utilizing a real estate business in order to get started as a broker will be what separates you from all of the other agents. You see, most agents begin selling real estate based solely on the current talents with no real direction. That won't happen to you.
If happen to be selling your home and anyone could have appliances which have been severely outdated then truly replace them before showing the family home. You can will include a little on to the cost because buyers will be very glad to see that there are brand new appliances near the property.
You're an outside-of-the-box type of person-And last but not least, industry mentoring supplies different look at Dean Graziosi regarding just like you-those who aren't proud status quo. Those that like to approach a problem from substitute angle.
These departments will provide you with idea within the price of this asset. Now, you learn the cost of this estate that you want to pick up. At this point, you can make a smart and right decision. Thus, these organizations will assist you in preparing introduce business.