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| {{distinguish2|[[Gauss–Markov process]]}}
| | When looking for the proper hemorrhoid treatment you will want to consider a limited important factors such as, that one you think you want, when there is a great amount of healing needed, plus how long it takes to receive results. In this particular article you will learn the answers to all of these issues, giving you the answer you have to find the proper hemorrhoid treatment.<br><br>If you have been seeing blood on the bathroom paper, you may be probably inside the late stages of this ailment plus want a extremely efficient [http://hemorrhoidtreatmentfix.com/external-hemorrhoids external hemorrhoids].<br><br>H-Miracle by Holly Hayden is not a cream or topical solution yet a step by step guide to receive rid of hemorrhoids. It offers a holistic approach to treating hemorrhoids: what to consume, what not to eat, what to do plus what to not do. It also comes with a lot of freebies like books on "How to Ease Your Allergies" and "Lessons from Miracle Doctors". A great deal of consumers like it due to the easy to follow instructions and the capability of the all-natural solutions used. Plus they additionally provide a income back guarantee just inside case the program doesn't function for you or anyone inside the family that is experiencing hemorrhoids.<br><br>A healthy digestion will equally be a key for not needing to undertake any hemorrhoid cure. Great digestion ehances usual bowel movement. Exercising daily, strolling for regarding 20 to 30 minutes a day supports the digestive program.<br><br>Well, he HAD heard of a hemorrhoids home remedy or two, he said, nevertheless couldn't really remember any details regarding them. I told him I required time to think about the upcoming step plus got out of there plus into the bright sunlight as swiftly as I could. Surgery for hemorrhoids definitely wasn't my initial choice.<br><br>Just sit for threee to five minute for a bowe movement plus don't wait till the last minute because sometimes the body tend to have a temporary or false constipation plus by the time we want to, nothing comes out (I know, I have completed this before).<br><br>Undergoing with these choices can definitely expense we expensive. And for sure not all persons could afford to pay such surgery. Then there are equally hemorrhoids treatment which could be found at house. With these hemorrhoid treatments you can make sure that we will not spend too much. In most situations, folks prefer to have natural treatment whilst the hemorrhoid continues to be on its mild stage. These all-natural treatments commonly assist you inside reducing the pain and swelling. You do not have to be concerned because we apply or employ them because they are surprisingly simple plus affordable. |
| {{Redirect|BLUE|queue management algorithm|Blue (queue management algorithm)}}
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| In [[statistics]], the '''Gauss–Markov theorem''', named after [[Carl Friedrich Gauss]] and [[Andrey Markov]], states that in a [[linear regression model]] in which the errors have expectation zero and are [[uncorrelated]] and have equal [[variance]]s, the '''best linear [[bias of an estimator|unbiased]] [[estimator]]''' ('''BLUE''') of the coefficients is given by the [[ordinary least squares]] (OLS) estimator. Here "best" means giving the lowest variance of the estimate, as compared to other unbiased, linear estimates. The errors don't need to be [[normal distribution|normal]], nor do they need to be [[independent and identically distributed]] (only [[uncorrelated]] and [[homoscedastic]]). The hypothesis that the estimator be unbiased cannot be dropped, since otherwise estimators better than OLS exist. See for examples the [[James–Stein estimator]] (which also drops linearity) or [[ridge regression]].
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| == Statement ==
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| Suppose we have in matrix notation,
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| :<math> \underline{y} = X \underline{\beta} + \underline{\varepsilon},\quad (\underline{y},\underline{\varepsilon} \in \mathbb{R}^n, \beta \in \mathbb{R}^K \text{ and } X\in\mathbb{R}^{n\times K}) </math>
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| expanding to,
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| :<math> y_i=\sum_{j=1}^{K}\beta_j X_{ij}+\varepsilon_i \quad \forall i=1,2,\ldots,n</math>
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| where <math>\beta_j</math> are non-random but '''un'''observable parameters, <math> X_{ij} </math> are non-random and observable (called the "explanatory variables"), <math>\varepsilon_i</math> are random, and so <math>y_i</math> are random. The random variables <math>\varepsilon_i</math> are called the "residuals" or "noise" (will be contrasted with "errors" later in the article; see [[errors and residuals in statistics]]). Note that to include a constant in the model above, one can choose to introduce the constant as a variable <math>\beta_{K+1}</math> with a newly introduced last column of X being unity i.e., <math>X_{i(K+1)} = 1</math> for all <math> i </math>.
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| The '''Gauss–Markov''' assumptions are
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| *<math>E(\varepsilon_i)=0, </math>
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| *<math>V(\varepsilon_i)= \sigma^2 < \infty,</math>
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| (i.e., all residuals have the same variance; that is "[[homoscedasticity]]"), and
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| *<math>{\rm cov}(\varepsilon_i,\varepsilon_j) = 0, \forall i \neq j </math>
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| for <math> i\neq j</math> that is, any the noise terms are drawn from an "uncorrelated" distribution. A '''linear estimator''' of <math> \beta_j </math> is a linear combination
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| :<math>\widehat\beta_j = c_{1j}y_1+\cdots+c_{nj}y_n</math> | |
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| in which the coefficients <math> c_{ij} </math> are not allowed to depend on the underlying coefficients <math> \beta_j </math>, since those are not observable, but are allowed to depend on the values <math> X_{ij} </math>, since these data are observable. (The dependence of the coefficients on each <math> X_{ij} </math> is typically nonlinear; the estimator is linear in each <math> y_i </math> and hence in each random <math> \varepsilon </math>, which is why this is [[linear regression|"linear" regression]].) The estimator is said to be '''unbiased''' [[if and only if]]
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| :<math>E(\widehat\beta_j)=\beta_j\,</math>
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| regardless of the values of <math> X_{ij} </math>. Now, let <math>\sum_{j=1}^K\lambda_j\beta_j</math> be some linear combination of the coefficients. Then the '''[[mean squared error]]''' of the corresponding estimation is
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| :<math>E \left(\left(\sum_{j=1}^K\lambda_j(\widehat\beta_j-\beta_j)\right)^2\right);</math>
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| i.e., it is the expectation of the square of the weighted sum (across parameters) of the differences between the estimators and the corresponding parameters to be estimated. (Since we are considering the case in which all the parameter estimates are unbiased, this mean squared error is the same as the variance of the linear combination.) The '''best linear unbiased estimator''' (BLUE) of the vector <math> \beta </math> of parameters <math> \beta_j </math> is one with the smallest mean squared error for every vector <math> \lambda </math> of linear combination parameters. This is equivalent to the condition that
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| :<math>V(\tilde\beta)- V(\widehat\beta)</math>
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| is a positive semi-definite matrix for every other linear unbiased estimator <math>\tilde\beta</math>. | |
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| The '''ordinary least squares estimator (OLS)''' is the function
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| :<math>\widehat\beta=(X'X)^{-1}X'y</math>
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| of <math> y </math> and <math> X </math> (where <math>X'</math> denotes the [[transpose]] of <math> X </math>)
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| that minimizes the '''sum of squares of [[errors and residuals in statistics|residuals]]''' (misprediction amounts):
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| :<math>\sum_{i=1}^n\left(y_i-\widehat{y}_i\right)^2=\sum_{i=1}^n\left(y_i-\sum_{j=1}^K\widehat\beta_j X_{ij}\right)^2.</math>
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| The theorem now states that the OLS estimator is a BLUE. The main idea of the proof is that the least-squares estimator is uncorrelated with every linear unbiased estimator of zero, i.e., with every linear combination <math>a_1y_1+\cdots+a_ny_n</math>
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| whose coefficients do not depend upon the unobservable <math> \beta </math> but whose expected value is always zero.
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| == Proof ==
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| Let <math> \tilde\beta = Cy </math> be another linear estimator of <math> \beta </math> and let ''C'' be given by <math> (X'X)^{-1}X' + D </math>, where ''D'' is a <math>k \times n</math> nonzero matrix. As we're restricting to ''unbiased'' estimators, minimum mean squared error implies minimum variance. The goal is therefore to show that such an estimator has a variance no smaller than that of <math> \hat\beta </math>, the OLS estimator.
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| The expectation of <math> \tilde\beta </math> is:
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| :<math>
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| \begin{align}
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| E(Cy) &= E(((X'X)^{-1}X' + D)(X\beta + \varepsilon)) \\
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| &= ((X'X)^{-1}X' + D)X\beta + ((X'X)^{-1}X' + D)\underbrace{E(\varepsilon)}_0 \\
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| &= (X'X)^{-1}X'X\beta + DX\beta \\
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| &= (I_k + DX)\beta. \\
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| \end{align}
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| </math> | |
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| Therefore, <math> \tilde\beta </math> is unbiased if and only if <math> DX = 0 </math>.
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| The variance of <math> \tilde\beta </math> is
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| :<math>
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| \begin{align}
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| V(\tilde\beta) &= V(Cy) = CV(y)C' = \sigma^2 CC' \\
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| &= \sigma^2((X'X)^{-1}X' + D)(X(X'X)^{-1} + D') \\
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| &= \sigma^2((X'X)^{-1}X'X(X'X)^{-1} + (X'X)^{-1}X'D' + DX(X'X)^{-1} + DD') \\
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| &= \sigma^2(X'X)^{-1} + \sigma^2(X'X)^{-1} (\underbrace{DX}_{0})' + \sigma^2 \underbrace{DX}_{0} (X'X)^{-1} + \sigma^2DD' \\
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| &= \underbrace{\sigma^2(X'X)^{-1}}_{V(\hat\beta)} + \sigma^2DD'.
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| \end{align}
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| </math>
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| Since ''DD''' is a positive semidefinite matrix, <math> V(\tilde\beta) </math> exceeds <math> V(\hat\beta) </math> by a positive semidefinite matrix.
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| == Generalized least squares estimator ==
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| The [[generalized least squares]] (GLS) or [[Alexander Aitken|Aitken]] estimator extends the Gauss–Markov theorem to the case where the error vector has a non-scalar covariance matrix{{spaced ndash}}the Aitken estimator is also a BLUE.<ref>A. C. Aitken, "On Least Squares and Linear Combinations of Observations", ''Proceedings of the Royal Society of Edinburgh'', 1935, vol. 55, pp. 42–48.</ref>
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| ==See also==
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| *[[Independent and identically distributed random variables]]
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| *[[Linear regression]]
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| *[[Measurement uncertainty]]
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| === Other unbiased statistics ===
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| *[[Best linear unbiased prediction]] (BLUP)
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| *[[Minimum-variance unbiased estimator]] (MVUE)
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| ==Notes==
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| <references />
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| ==References==
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| {{refbegin}}
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| * {{cite journal
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| |authorlink=R. L. Plackett |last=Plackett |first=R.L.
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| |year=1950
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| |title=Some Theorems in Least Squares
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| |journal=[[Biometrika]]
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| |volume=37 |issue=1–2 |pages=149–157
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| |doi=10.1093/biomet/37.1-2.149 |mr=36980 | jstor = 2332158
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| }}
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| {{refend}}
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| ==External links==
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| *[http://jeff560.tripod.com/g.html Earliest Known Uses of Some of the Words of Mathematics: G] (brief history and explanation of the name)
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| *[http://www.xycoon.com/ols1.htm Proof of the Gauss Markov theorem for multiple linear regression] (makes use of matrix algebra)
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| *[http://emlab.berkeley.edu/GMTheorem/index.html A Proof of the Gauss Markov theorem using geometry]
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| {{Least squares and regression analysis|state=expanded}}
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| {{DEFAULTSORT:Gauss-Markov theorem}}
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| [[Category:Statistical theorems]]
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When looking for the proper hemorrhoid treatment you will want to consider a limited important factors such as, that one you think you want, when there is a great amount of healing needed, plus how long it takes to receive results. In this particular article you will learn the answers to all of these issues, giving you the answer you have to find the proper hemorrhoid treatment.
If you have been seeing blood on the bathroom paper, you may be probably inside the late stages of this ailment plus want a extremely efficient external hemorrhoids.
H-Miracle by Holly Hayden is not a cream or topical solution yet a step by step guide to receive rid of hemorrhoids. It offers a holistic approach to treating hemorrhoids: what to consume, what not to eat, what to do plus what to not do. It also comes with a lot of freebies like books on "How to Ease Your Allergies" and "Lessons from Miracle Doctors". A great deal of consumers like it due to the easy to follow instructions and the capability of the all-natural solutions used. Plus they additionally provide a income back guarantee just inside case the program doesn't function for you or anyone inside the family that is experiencing hemorrhoids.
A healthy digestion will equally be a key for not needing to undertake any hemorrhoid cure. Great digestion ehances usual bowel movement. Exercising daily, strolling for regarding 20 to 30 minutes a day supports the digestive program.
Well, he HAD heard of a hemorrhoids home remedy or two, he said, nevertheless couldn't really remember any details regarding them. I told him I required time to think about the upcoming step plus got out of there plus into the bright sunlight as swiftly as I could. Surgery for hemorrhoids definitely wasn't my initial choice.
Just sit for threee to five minute for a bowe movement plus don't wait till the last minute because sometimes the body tend to have a temporary or false constipation plus by the time we want to, nothing comes out (I know, I have completed this before).
Undergoing with these choices can definitely expense we expensive. And for sure not all persons could afford to pay such surgery. Then there are equally hemorrhoids treatment which could be found at house. With these hemorrhoid treatments you can make sure that we will not spend too much. In most situations, folks prefer to have natural treatment whilst the hemorrhoid continues to be on its mild stage. These all-natural treatments commonly assist you inside reducing the pain and swelling. You do not have to be concerned because we apply or employ them because they are surprisingly simple plus affordable.