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96-563: The theorem was named after Carl Friedrich Gauss and Andrey Markov , although Gauss' work significantly predates Markov's. But while Gauss derived the result under the assumption of independence and normality, Markov reduced the assumptions to the form stated above. A further generalization to non-spherical errors was given by Alexander Aitken . Suppose we are given two random variable vectors, X ,  Y ∈ R k {\displaystyle X{\text{, }}Y\in \mathbb {R} ^{k}} and that we want to find

192-492: A heliometer from Fraunhofer . The scientific activity of Gauss, besides pure mathematics, can be roughly divided into three periods: astronomy was the main focus in the first two decades of the 19th century, geodesy in the third decade, and physics, mainly magnetism, in the fourth decade. Gauss made no secret of his aversion to giving academic lectures. But from the start of his academic career at Göttingen, he continuously gave lectures until 1854. He often complained about

288-423: A magnetometer in 1833 and – alongside Wilhelm Eduard Weber – the first electromagnetic telegraph in 1833. Gauss was the first to discover and study non-Euclidean geometry , coining the term as well. He further developed a fast Fourier transform some 160 years before John Tukey and James Cooley . Gauss refused to publish incomplete work and left several works to be edited posthumously . He believed that

384-406: A butcher, bricklayer, gardener, and treasurer of a death-benefit fund. Gauss characterized his father as honourable and respected, but rough and dominating at home. He was experienced in writing and calculating, whereas his second wife Dorothea, Carl Friedrich's mother, was nearly illiterate. He had one elder brother from his father's first marriage. Gauss was a child prodigy in mathematics. When

480-569: A collection of short remarks about his results from the years 1796 until 1814, shows that many ideas for his mathematical magnum opus Disquisitiones Arithmeticae (1801) date from this time. Gauss graduated as a Doctor of Philosophy in 1799, not in Göttingen, as is sometimes stated, but at the Duke of Brunswick's special request from the University of Helmstedt, the only state university of

576-491: A considerable literary estate, too. Gauss referred to mathematics as "the queen of sciences" and arithmetics as "the queen of mathematics", and supposedly once espoused a belief in the necessity of immediately understanding Euler's identity as a benchmark pursuant to becoming a first-class mathematician. On certain occasions, Gauss claimed that the ideas of another scholar had already been in his possession previously. Thus his concept of priority as "the first to discover, not

672-672: A critique of d'Alembert's work. He subsequently produced three other proofs, the last one in 1849 being generally rigorous. His attempts clarified the concept of complex numbers considerably along the way. In the preface to the Disquisitiones , Gauss dates the beginning of his work on number theory to 1795. By studying the works of previous mathematicians like Fermat, Euler, Lagrange, and Legendre, he realized that these scholars had already found much of what he had discovered by himself. The Disquisitiones Arithmeticae , written since 1798 and published in 1801, consolidated number theory as

768-431: A curious feature of his working style that he carried out calculations with a high degree of precision much more than required, and prepared tables with more decimal places than ever requested for practical purposes. Very likely, this method gave him a lot of material which he used in finding theorems in number theory. Gauss refused to publish work that he did not consider complete and above criticism. This perfectionism

864-474: A decade. Therese then took over the household and cared for Gauss for the rest of his life; after her father's death, she married actor Constantin Staufenau. Her sister Wilhelmina married the orientalist Heinrich Ewald . Gauss's mother Dorothea lived in his house from 1817 until she died in 1839. The eldest son Joseph, while still a schoolboy, helped his father as an assistant during the survey campaign in

960-483: A discipline and covered both elementary and algebraic number theory . Therein he introduces the triple bar symbol ( ≡ ) for congruence and uses it for a clean presentation of modular arithmetic . It deals with the unique factorization theorem and primitive roots modulo n . In the main sections, Gauss presents the first two proofs of the law of quadratic reciprocity and develops the theories of binary and ternary quadratic forms . The Disquisitiones include

1056-682: A habit in his later years, for example, the number of paths from his home to certain places in Göttingen, or the number of living days of persons; he congratulated Humboldt in December 1851 for having reached the same age as Isaac Newton at his death, calculated in days. Similar to his excellent knowledge of Latin he was also acquainted with modern languages. At the age of 62, he began to teach himself Russian , very likely to understand scientific writings from Russia, among them those of Lobachevsky on non-Euclidean geometry. Gauss read both classical and modern literature, and English and French works in

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1152-555: A heart attack in Göttingen; and was interred in the Albani Cemetery there. Heinrich Ewald , Gauss's son-in-law, and Wolfgang Sartorius von Waltershausen , Gauss's close friend and biographer, gave eulogies at his funeral. Gauss was a successful investor and accumulated considerable wealth with stocks and securities, finally a value of more than 150 thousand Thaler; after his death, about 18 thousand Thaler were found hidden in his rooms. The day after Gauss's death his brain

1248-469: A linear form. For example, the Cobb–Douglas function —often used in economics—is nonlinear: But it can be expressed in linear form by taking the natural logarithm of both sides: This assumption also covers specification issues: assuming that the proper functional form has been selected and there are no omitted variables . One should be aware, however, that the parameters that minimize the residuals of

1344-1425: A multiple regression model with p variables. The first derivative is d d β f = − 2 X T ( y − X β ) = − 2 [ ∑ i = 1 n ( y i − ⋯ − β p x i p ) ∑ i = 1 n x i 1 ( y i − ⋯ − β p x i p ) ⋮ ∑ i = 1 n x i p ( y i − ⋯ − β p x i p ) ] = 0 p + 1 , {\displaystyle {\begin{aligned}{\frac {d}{d{\boldsymbol {\beta }}}}f&=-2X^{\operatorname {T} }\left(\mathbf {y} -X{\boldsymbol {\beta }}\right)\\&=-2{\begin{bmatrix}\sum _{i=1}^{n}(y_{i}-\dots -\beta _{p}x_{ip})\\\sum _{i=1}^{n}x_{i1}(y_{i}-\dots -\beta _{p}x_{ip})\\\vdots \\\sum _{i=1}^{n}x_{ip}(y_{i}-\dots -\beta _{p}x_{ip})\end{bmatrix}}\\&=\mathbf {0} _{p+1},\end{aligned}}} where X T {\displaystyle X^{\operatorname {T} }}

1440-494: A parameter dependent on an independent variable does not qualify as linear, for example y = β 0 + β 1 ( x ) ⋅ x {\displaystyle y=\beta _{0}+\beta _{1}(x)\cdot x} , where β 1 ( x ) {\displaystyle \beta _{1}(x)} is a function of x {\displaystyle x} . Data transformations are often used to convert an equation into

1536-414: A positive semidefinite matrix. As it has been stated before, the condition of Var ⁡ ( β ~ ) − Var ⁡ ( β ^ ) {\displaystyle \operatorname {Var} \left({\tilde {\beta }}\right)-\operatorname {Var} \left({\widehat {\beta }}\right)} is a positive semidefinite matrix

1632-474: A private scholar. He gave the second and third complete proofs of the fundamental theorem of algebra , made contributions to number theory , and developed the theories of binary and ternary quadratic forms. Gauss was instrumental in the identification of Ceres as a dwarf planet. His work on the motion of planetoids disturbed by large planets led to the introduction of the Gaussian gravitational constant and

1728-702: A scandal in public, Eugen suddenly left Göttingen under dramatic circumstances in September 1830 and emigrated via Bremen to the United States. He wasted the little money he had taken to start, after which his father refused further financial support. The youngest son Wilhelm wanted to qualify for agricultural administration, but had difficulties getting an appropriate education, and eventually emigrated as well. Only Gauss's youngest daughter Therese accompanied him in his last years of life. Collecting numerical data on very different things, useful or useless, became

1824-415: A strong calculus as the sole tasks of astronomy. At university, he was accompanied by a staff of other lecturers in his disciplines, who completed the educational program; these included the mathematician Thibaut with his lectures, the physicist Mayer , known for his textbooks, his successor Weber since 1831, and in the observatory Harding , who took the main part of lectures in practical astronomy. When

1920-445: A university chair in Göttingen, "because he was always involved in some polemic." Gauss's life was overshadowed by severe problems in his family. When his first wife Johanna suddenly died shortly after the birth of their third child, he revealed the grief in a last letter to his dead wife in the style of an ancient threnody , the most personal surviving document of Gauss. The situation worsened when tuberculosis ultimately destroyed

2016-1290: Is H = 2 [ n ∑ i = 1 n x i 1 ⋯ ∑ i = 1 n x i p ∑ i = 1 n x i 1 ∑ i = 1 n x i 1 2 ⋯ ∑ i = 1 n x i 1 x i p ⋮ ⋮ ⋱ ⋮ ∑ i = 1 n x i p ∑ i = 1 n x i p x i 1 ⋯ ∑ i = 1 n x i p 2 ] = 2 X T X {\displaystyle {\mathcal {H}}=2{\begin{bmatrix}n&\sum _{i=1}^{n}x_{i1}&\cdots &\sum _{i=1}^{n}x_{ip}\\\sum _{i=1}^{n}x_{i1}&\sum _{i=1}^{n}x_{i1}^{2}&\cdots &\sum _{i=1}^{n}x_{i1}x_{ip}\\\vdots &\vdots &\ddots &\vdots \\\sum _{i=1}^{n}x_{ip}&\sum _{i=1}^{n}x_{ip}x_{i1}&\cdots &\sum _{i=1}^{n}x_{ip}^{2}\end{bmatrix}}=2X^{\operatorname {T} }X} Assuming

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2112-677: Is un observable, β ~ {\displaystyle {\tilde {\beta }}} is unbiased if and only if D X = 0 {\displaystyle DX=0} . Then: Since D D T {\displaystyle DD^{\operatorname {T} }} is a positive semidefinite matrix, Var ⁡ ( β ~ ) {\displaystyle \operatorname {Var} \left({\tilde {\beta }}\right)} exceeds Var ⁡ ( β ^ ) {\displaystyle \operatorname {Var} \left({\widehat {\beta }}\right)} by

2208-540: Is a K × n {\displaystyle K\times n} non-zero 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 β ^ , {\displaystyle {\widehat {\beta }},} the OLS estimator. We calculate: Therefore, since β {\displaystyle \beta }

2304-409: Is a best linear unbiased estimator (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 a 1 y 1 + ⋯ + a n y n {\displaystyle a_{1}y_{1}+\cdots +a_{n}y_{n}} whose coefficients do not depend upon

2400-450: Is a linear combination in which the coefficients c i j {\displaystyle c_{ij}} are not allowed to depend on the underlying coefficients β j {\displaystyle \beta _{j}} , since those are not observable, but are allowed to depend on the values X i j {\displaystyle X_{ij}} , since these data are observable. (The dependence of

2496-883: Is equivalent to the property that the best linear unbiased estimator of ℓ T β {\displaystyle \ell ^{\operatorname {T} }\beta } is ℓ T β ^ {\displaystyle \ell ^{\operatorname {T} }{\widehat {\beta }}} (best in the sense that it has minimum variance). To see this, let ℓ T β ~ {\displaystyle \ell ^{\operatorname {T} }{\tilde {\beta }}} another linear unbiased estimator of ℓ T β {\displaystyle \ell ^{\operatorname {T} }\beta } . Moreover, equality holds if and only if D T ℓ = 0 {\displaystyle D^{\operatorname {T} }\ell =0} . We calculate This proves that

2592-928: Is indeed a global minimum. Or, just see that for all vectors v , v T X T X v = ‖ X v ‖ 2 ≥ 0 {\displaystyle \mathbf {v} ,\mathbf {v} ^{\operatorname {T} }X^{\operatorname {T} }X\mathbf {v} =\|\mathbf {X} \mathbf {v} \|^{2}\geq 0} . So the Hessian is positive definite if full rank. Let β ~ = C y {\displaystyle {\tilde {\beta }}=Cy} be another linear estimator of β {\displaystyle \beta } with C = ( X T X ) − 1 X T + D {\displaystyle C=(X^{\operatorname {T} }X)^{-1}X^{\operatorname {T} }+D} where D {\displaystyle D}

2688-552: Is not "perfect") can be present resulting in a less efficient, but still unbiased estimate. The estimates will be less precise and highly sensitive to particular sets of data. Multicollinearity can be detected from condition number or the variance inflation factor , among other tests. Carl Friedrich Gauss This is an accepted version of this page Johann Carl Friedrich Gauss (German: Gauß [kaʁl ˈfʁiːdʁɪç ˈɡaʊs] ; Latin : Carolus Fridericus Gauss ; 30 April 1777 – 23 February 1855)

2784-412: Is not invertible and the OLS estimator cannot be computed. A violation of this assumption is perfect multicollinearity , i.e. some explanatory variables are linearly dependent. One scenario in which this will occur is called "dummy variable trap," when a base dummy variable is not omitted resulting in perfect correlation between the dummy variables and the constant term. Multicollinearity (as long as it

2880-422: Is one with the smallest mean squared error for every vector λ {\displaystyle \lambda } of linear combination parameters. This is equivalent to the condition that is a positive semi-definite matrix for every other linear unbiased estimator β ~ {\displaystyle {\widetilde {\beta }}} . The ordinary least squares estimator (OLS)

2976-470: Is the data matrix or design matrix. Geometrically, this assumption implies that x i {\displaystyle \mathbf {x} _{i}} and ε i {\displaystyle \varepsilon _{i}} are orthogonal to each other, so that their inner product (i.e., their cross moment) is zero. This assumption is violated if the explanatory variables are measured with error , or are endogenous . Endogeneity can be

Gauss–Markov theorem - Misplaced Pages Continue

3072-480: Is the data vector of regressors for the i th observation, and consequently X = [ x 1 T x 2 T ⋯ x n T ] T {\displaystyle \mathbf {X} ={\begin{bmatrix}\mathbf {x} _{1}^{\operatorname {T} }&\mathbf {x} _{2}^{\operatorname {T} }&\cdots &\mathbf {x} _{n}^{\operatorname {T} }\end{bmatrix}}^{\operatorname {T} }}

3168-728: Is the design matrix X = [ 1 x 11 ⋯ x 1 p 1 x 21 ⋯ x 2 p ⋮ 1 x n 1 ⋯ x n p ] ∈ R n × ( p + 1 ) ; n ≥ p + 1 {\displaystyle X={\begin{bmatrix}1&x_{11}&\cdots &x_{1p}\\1&x_{21}&\cdots &x_{2p}\\&&\vdots \\1&x_{n1}&\cdots &x_{np}\end{bmatrix}}\in \mathbb {R} ^{n\times (p+1)};\qquad n\geq p+1} The Hessian matrix of second derivatives

3264-498: Is the eigenvalue corresponding to k {\displaystyle \mathbf {k} } . Moreover, k T k = ∑ i = 1 p + 1 k i 2 > 0 ⟹ λ > 0 {\displaystyle \mathbf {k} ^{\operatorname {T} }\mathbf {k} =\sum _{i=1}^{p+1}k_{i}^{2}>0\implies \lambda >0} Finally, as eigenvector k {\displaystyle \mathbf {k} }

3360-529: 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 β {\displaystyle \beta } of parameters β j {\displaystyle \beta _{j}}

3456-401: Is the function of y {\displaystyle y} and X {\displaystyle X} (where X T {\displaystyle X^{\operatorname {T} }} denotes the transpose of X {\displaystyle X} ) that minimizes the sum of squares of residuals (misprediction amounts): The theorem now states that the OLS estimator

3552-592: The Celestial police . One of their aims was the discovery of further planets. They assembled data on asteroids and comets as a basis for Gauss's research on their orbits, which he later published in his astronomical magnum opus Theoria motus corporum coelestium (1809). In November 1807, Gauss followed a call to the University of Göttingen , then an institution of the newly founded Kingdom of Westphalia under Jérôme Bonaparte , as full professor and director of

3648-502: The Gauss composition law for binary quadratic forms, as well as the enumeration of the number of representations of an integer as the sum of three squares. As an almost immediate corollary of his theorem on three squares , he proves the triangular case of the Fermat polygonal number theorem for n = 3. From several analytic results on class numbers that Gauss gives without proof towards

3744-493: The astronomical observatory , and kept the chair until his death in 1855. He was soon confronted with the demand for two thousand francs from the Westphalian government as a war contribution, which he could not afford to pay. Both Olbers and Laplace wanted to help him with the payment, but Gauss refused their assistance. Finally, an anonymous person from Frankfurt , later discovered to be Prince-primate Dalberg , paid

3840-472: The method of least squares , which he had discovered before Adrien-Marie Legendre published it. Gauss was in charge of the extensive geodetic survey of the Kingdom of Hanover together with an arc measurement project from 1820 to 1844; he was one of the founders of geophysics and formulated the fundamental principles of magnetism . Fruits of his practical work were the inventions of the heliotrope in 1821,

3936-550: The popularization of scientific matters. His only attempts at popularization were his works on the date of Easter (1800/1802) and the essay Erdmagnetismus und Magnetometer of 1836. Gauss published his papers and books exclusively in Latin or German . He wrote Latin in a classical style but used some customary modifications set by contemporary mathematicians. In his inaugural lecture at Göttingen University from 1808, Gauss claimed reliable observations and results attained only by

Gauss–Markov theorem - Misplaced Pages Continue

4032-409: The "explanatory variables"), ε i {\displaystyle \varepsilon _{i}} are random, and so y i {\displaystyle y_{i}} are random. The random variables ε i {\displaystyle \varepsilon _{i}} are called the "disturbance", "noise" or simply "error" (will be contrasted with "residual" later in

4128-441: The Duke was killed in the battle of Jena in 1806. The duchy was abolished in the following year, and Gauss's financial support stopped. When Gauss was calculating asteroid orbits in the first years of the century, he established contact with the astronomical community of Bremen and Lilienthal , especially Wilhelm Olbers , Karl Ludwig Harding , and Friedrich Wilhelm Bessel , as part of the informal group of astronomers known as

4224-494: The French language. Gauss was "in front of the new development" with documented research since 1799, his wealth of new ideas, and his rigour of demonstration. Whereas previous mathematicians like Leonhard Euler let the readers take part in their reasoning for new ideas, including certain erroneous deviations from the correct path, Gauss however introduced a new style of direct and complete explanation that did not attempt to show

4320-503: The Gauss–Markov theorem to the case where the error vector has a non-scalar covariance matrix. The Aitken estimator is also a BLUE. In most treatments of OLS, the regressors (parameters of interest) in the design matrix X {\displaystyle \mathbf {X} } are assumed to be fixed in repeated samples. This assumption is considered inappropriate for a predominantly nonexperimental science like econometrics . Instead,

4416-475: The act of getting there, which grants the greatest enjoyment. When I have clarified and exhausted a subject, then I turn away from it, in order to go into darkness again. The posthumous papers, his scientific diary , and short glosses in his own textbooks show that he worked to a great extent in an empirical way. He was a lifelong busy and enthusiastic calculator, who made his calculations with extraordinary rapidity, mostly without precise controlling, but checked

4512-642: The act of learning, not possession of knowledge, provided the greatest enjoyment. Gauss confessed to disliking teaching, but some of his students became influential mathematicians, such as Richard Dedekind and Bernhard Riemann . Gauss was born on 30 April 1777 in Brunswick in the Duchy of Brunswick-Wolfenbüttel (now in the German state of Lower Saxony ). His family was of relatively low social status. His father Gebhard Dietrich Gauss (1744–1808) worked variously as

4608-593: 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 β K + 1 {\displaystyle \beta _{K+1}} with a newly introduced last column of X being unity i.e., X i ( K + 1 ) = 1 {\displaystyle X_{i(K+1)}=1} for all i {\displaystyle i} . Note that though y i , {\displaystyle y_{i},} as sample responses, are observable,

4704-456: The assumptions of the Gauss–Markov theorem are stated conditional on X {\displaystyle \mathbf {X} } . The dependent variable is assumed to be a linear function of the variables specified in the model. The specification must be linear in its parameters. This does not mean that there must be a linear relationship between the independent and dependent variables. The independent variables can take non-linear forms as long as

4800-570: The best linear estimator of Y {\displaystyle Y} given X {\displaystyle X} , using the best linear estimator Y ^ = α X + μ {\displaystyle {\hat {Y}}=\alpha X+\mu } Where the parameters α {\displaystyle \alpha } and μ {\displaystyle \mu } are both real numbers. Such an estimator Y ^ {\displaystyle {\hat {Y}}} would have

4896-419: The best linear estimator would be Y ^ = σ y ( X − μ x ) σ x + μ y {\displaystyle {\hat {Y}}=\sigma _{y}{\frac {(X-\mu _{x})}{\sigma _{x}}}+\mu _{y}} since Y ^ {\displaystyle {\hat {Y}}} has

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4992-637: The best-paid professors of the university. When Gauss was asked for help by his colleague and friend Friedrich Wilhelm Bessel in 1810, who was in trouble at Königsberg University because of his lack of an academic title, Gauss provided a doctorate honoris causa for Bessel from the Philosophy Faculty of Göttingen in March 1811. Gauss gave another recommendation for an honorary degree for Sophie Germain but only shortly before her death, so she never received it. He also gave successful support to

5088-517: The birth of Louis, who himself died a few months later. Gauss chose the first names of his children in honour of Giuseppe Piazzi , Wilhelm Olbers, and Karl Ludwig Harding, the discoverers of the first asteroids. On 4 August 1810, Gauss married Wilhelmine (Minna) Waldeck, a friend of his first wife, with whom he had three more children: Eugen (later Eugene) (1811–1896), Wilhelm (later William) (1813–1879), and Therese (1816–1864). Minna Gauss died on 12 September 1831 after being seriously ill for more than

5184-701: The burdens of teaching, feeling that it was a waste of his time. On the other hand, he occasionally described some students as talented. Most of his lectures dealt with astronomy, geodesy, and applied mathematics , and only three lectures on subjects of pure mathematics. Some of Gauss's students went on to become renowned mathematicians, physicists, and astronomers: Moritz Cantor , Dedekind , Dirksen , Encke , Gould , Heine , Klinkerfues , Kupffer , Listing , Möbius , Nicolai , Riemann , Ritter , Schering , Scherk , Schumacher , von Staudt , Stern , Ursin ; as geoscientists Sartorius von Waltershausen , and Wappäus . Gauss did not write any textbook and disliked

5280-426: The coefficients on each X i j {\displaystyle X_{ij}} is typically nonlinear; the estimator is linear in each y i {\displaystyle y_{i}} and hence in each random ε , {\displaystyle \varepsilon ,} which is why this is "linear" regression .) The estimator is said to be unbiased if and only if regardless of

5376-2682: The columns of X {\displaystyle X} are linearly independent so that X T X {\displaystyle X^{\operatorname {T} }X} is invertible, let X = [ v 1 v 2 ⋯ v p + 1 ] {\displaystyle X={\begin{bmatrix}\mathbf {v_{1}} &\mathbf {v_{2}} &\cdots &\mathbf {v} _{p+1}\end{bmatrix}}} , then k 1 v 1 + ⋯ + k p + 1 v p + 1 = 0 ⟺ k 1 = ⋯ = k p + 1 = 0 {\displaystyle k_{1}\mathbf {v_{1}} +\dots +k_{p+1}\mathbf {v} _{p+1}=\mathbf {0} \iff k_{1}=\dots =k_{p+1}=0} Now let k = ( k 1 , … , k p + 1 ) T ∈ R ( p + 1 ) × 1 {\displaystyle \mathbf {k} =(k_{1},\dots ,k_{p+1})^{T}\in \mathbb {R} ^{(p+1)\times 1}} be an eigenvector of H {\displaystyle {\mathcal {H}}} . k ≠ 0 ⟹ ( k 1 v 1 + ⋯ + k p + 1 v p + 1 ) 2 > 0 {\displaystyle \mathbf {k} \neq \mathbf {0} \implies \left(k_{1}\mathbf {v_{1}} +\dots +k_{p+1}\mathbf {v} _{p+1}\right)^{2}>0} In terms of vector multiplication, this means [ k 1 ⋯ k p + 1 ] [ v 1 ⋮ v p + 1 ] [ v 1 ⋯ v p + 1 ] [ k 1 ⋮ k p + 1 ] = k T H k = λ k T k > 0 {\displaystyle {\begin{bmatrix}k_{1}&\cdots &k_{p+1}\end{bmatrix}}{\begin{bmatrix}\mathbf {v_{1}} \\\vdots \\\mathbf {v} _{p+1}\end{bmatrix}}{\begin{bmatrix}\mathbf {v_{1}} &\cdots &\mathbf {v} _{p+1}\end{bmatrix}}{\begin{bmatrix}k_{1}\\\vdots \\k_{p+1}\end{bmatrix}}=\mathbf {k} ^{\operatorname {T} }{\mathcal {H}}\mathbf {k} =\lambda \mathbf {k} ^{\operatorname {T} }\mathbf {k} >0} where λ {\displaystyle \lambda }

5472-496: The contemporary school of Naturphilosophie . Gauss had an "aristocratic and through and through conservative nature", with little respect for people's intelligence and morals, following the motto " mundus vult decipi ". He disliked Napoleon and his system, and all kinds of violence and revolution caused horror to him. Thus he condemned the methods of the Revolutions of 1848 , though he agreed with some of their aims, such as

5568-702: The duchy. Johann Friedrich Pfaff assessed his doctoral thesis, and Gauss got the degree in absentia without further oral examination. The Duke then granted him the cost of living as a private scholar in Brunswick. Gauss subsequently refused calls from the Russian Academy of Sciences in St. Peterburg and Landshut University . Later, the Duke promised him the foundation of an observatory in Brunswick in 1804. Architect Peter Joseph Krahe made preliminary designs, but one of Napoleon's wars cancelled those plans:

5664-514: The elementary teachers noticed his intellectual abilities, they brought him to the attention of the Duke of Brunswick who sent him to the local Collegium Carolinum , which he attended from 1792 to 1795 with Eberhard August Wilhelm von Zimmermann as one of his teachers. Thereafter the Duke granted him the resources for studies of mathematics, sciences, and classical languages at the University of Göttingen until 1798. His professor in mathematics

5760-412: The end of the fifth section, it appears that Gauss already knew the class number formula in 1801. In the last section, Gauss gives proof for the constructibility of a regular heptadecagon (17-sided polygon) with straightedge and compass by reducing this geometrical problem to an algebraic one. He shows that a regular polygon is constructible if the number of its sides is either a power of 2 or

5856-426: The equality holds if and only if ℓ T β ~ = ℓ T β ^ {\displaystyle \ell ^{\operatorname {T} }{\tilde {\beta }}=\ell ^{\operatorname {T} }{\widehat {\beta }}} which gives the uniqueness of the OLS estimator as a BLUE. The generalized least squares (GLS), developed by Aitken , extends

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5952-669: The first investigations, due to mislabelling, with that of the physician Conrad Heinrich Fuchs , who died in Göttingen a few months after Gauss. A further investigation showed no remarkable anomalies in the brains of both persons. Thus, all investigations on Gauss's brain until 1998, except the first ones of Rudolf and Hermann Wagner, actually refer to the brain of Fuchs. Gauss married Johanna Osthoff on 9 October 1805 in St. Catherine's church in Brunswick. They had two sons and one daughter: Joseph (1806–1873), Wilhelmina (1808–1840), and Louis (1809–1810). Johanna died on 11 October 1809, one month after

6048-440: The first to publish" differed from that of his scientific contemporaries. In contrast to his perfectionism in presenting mathematical ideas, he was criticized for a negligent way of quoting. He justified himself with a very special view of correct quoting: if he gave references, then only in a quite complete way, with respect to the previous authors of importance, which no one should ignore; but quoting in this way needed knowledge of

6144-536: The following statements and arguments including assumptions, proofs and the others assume under the only condition of knowing X i j , {\displaystyle X_{ij},} but not y i . {\displaystyle y_{i}.} The Gauss–Markov assumptions concern the set of error random variables, ε i {\displaystyle \varepsilon _{i}} : A linear estimator of β j {\displaystyle \beta _{j}}

6240-525: The health of his second wife Minna over 13 years; both his daughters later suffered from the same disease. Gauss himself gave only slight hints of his distress: in a letter to Bessel dated December 1831 he described himself as "the victim of the worst domestic sufferings". By reason of his wife's illness, both younger sons were educated for some years in Celle , far from Göttingen. The military career of his elder son Joseph ended after more than two decades with

6336-460: The history of science and more time than he wished to spend. Soon after Gauss's death, his friend Sartorius published the first biography (1856), written in a rather enthusiastic style. Sartorius saw him as a serene and forward-striving man with childlike modesty, but also of "iron character" with an unshakeable strength of mind. Apart from his closer circle, others regarded him as reserved and unapproachable "like an Olympian sitting enthroned on

6432-467: The idea of a unified Germany. As far as the political system is concerned, he had a low estimation of the constitutional system; he criticized parliamentarians of his time for a lack of knowledge and logical errors. Some Gauss biographers have speculated on his religious beliefs. He sometimes said "God arithmetizes" and "I succeeded – not on account of my hard efforts, but by the grace of the Lord." Gauss

6528-598: The mathematician Gotthold Eisenstein in Berlin. Gauss was loyal to the House of Hanover . After King William IV died in 1837, the new Hanoverian King Ernest Augustus annulled the 1833 constitution. Seven professors, later known as the " Göttingen Seven ", protested against this, among them his friend and collaborator Wilhelm Weber and Gauss's son-in-law Heinrich Ewald. All of them were dismissed, and three of them were expelled, but Ewald and Weber could stay in Göttingen. Gauss

6624-590: The observatory was completed, Gauss took his living accommodation in the western wing of the new observatory and Harding in the eastern one. They had once been on friendly terms, but over time they became alienated, possibly – as some biographers presume – because Gauss had wished the equal-ranked Harding to be no more than his assistant or observer. Gauss used the new meridian circles nearly exclusively, and kept them away from Harding, except for some very seldom joint observations. Brendel subdivides Gauss's astronomic activity chronologically into seven periods, of which

6720-411: The original languages. His favorite English author was Walter Scott , his favorite German Jean Paul . Gauss liked singing and went to concerts. He was a busy newspaper reader; in his last years, he used to visit an academic press salon of the university every noon. Gauss did not care much for philosophy, and mocked the "splitting hairs of the so-called metaphysicians", by which he meant proponents of

6816-612: The parameters are linear. The equation y = β 0 + β 1 x 2 , {\displaystyle y=\beta _{0}+\beta _{1}x^{2},} qualifies as linear while y = β 0 + β 1 2 x {\displaystyle y=\beta _{0}+\beta _{1}^{2}x} can be transformed to be linear by replacing β 1 2 {\displaystyle \beta _{1}^{2}} by another parameter, say γ {\displaystyle \gamma } . An equation with

6912-475: The problem by accepting offers from Berlin in 1810 and 1825 to become a full member of the Prussian Academy without burdening lecturing duties, as well as from Leipzig University in 1810 and from Vienna University in 1842, perhaps because of the family's difficult situation. Gauss's salary was raised from 1000 Reichsthaler in 1810 to 2400 Reichsthaler in 1824, and in his later years he was one of

7008-425: The product of a power of 2 and any number of distinct Fermat primes . In the same section, he gives a result on the number of solutions of certain cubic polynomials with coefficients in finite fields , which amounts to counting integral points on an elliptic curve . An unfinished eighth chapter was found among left papers only after his death, consisting of work done during 1797–1799. One of Gauss's first results

7104-592: The railroad system in the US for some months. Eugen left Göttingen in September 1830 and emigrated to the United States, where he joined the army for five years. He then worked for the American Fur Company in the Midwest. Later, he moved to Missouri and became a successful businessman. Wilhelm married a niece of the astronomer Bessel ; he then moved to Missouri, started as a farmer and became wealthy in

7200-437: The rank of a poorly paid first lieutenant , although he had acquired a considerable knowledge of geodesy. He needed financial support from his father even after he was married. The second son Eugen shared a good measure of his father's talent in computation and languages, but had a vivacious and sometimes rebellious character. He wanted to study philology, whereas Gauss wanted him to become a lawyer. Having run up debts and caused

7296-435: The reader the author's train of thought. Gauss was the first to restore that rigor of demonstration which we admire in the ancients and which had been forced unduly into the background by the exclusive interest of the preceding period in new developments. But for himself, he propagated a quite different ideal, given in a letter to Farkas Bolyai as follows: It is not knowledge, but the act of learning, not possession but

7392-433: The result of simultaneity , where causality flows back and forth between both the dependent and independent variable. Instrumental variable techniques are commonly used to address this problem. The sample data matrix X {\displaystyle \mathbf {X} } must have full column rank . Otherwise X T X {\displaystyle \mathbf {X} ^{\operatorname {T} }\mathbf {X} }

7488-411: The results by masterly estimation. Nevertheless, his calculations were not always free from mistakes. He coped with the enormous workload by using skillful tools. Gauss used a lot of mathematical tables , examined their exactness, and constructed new tables on various matters for personal use. He developed new tools for effective calculation, for example the Gaussian elimination . It has been taken as

7584-586: The same mean and standard deviation as Y {\displaystyle Y} , that is, μ Y ^ = μ Y , σ Y ^ = σ Y {\displaystyle \mu _{\hat {Y}}=\mu _{Y},\sigma _{\hat {Y}}=\sigma _{Y}} . Therefore, if the vector X {\displaystyle X} has respective mean and standard deviation μ x , σ x {\displaystyle \mu _{x},\sigma _{x}} ,

7680-402: The same mean and standard deviation as Y {\displaystyle Y} . Suppose we have, in matrix notation, the linear relationship expanding to, where β j {\displaystyle \beta _{j}} are non-random but un observable parameters, X i j {\displaystyle X_{ij}} are non-random and observable (called

7776-613: The shoe business in St. Louis in later years. Eugene and William have numerous descendants in America, but the Gauss descendants left in Germany all derive from Joseph, as the daughters had no children. In the first two decades of the 19th century, Gauss was the only important mathematician in Germany, comparable to the leading French ones; his Disquisitiones Arithmeticae was the first mathematical book from Germany to be translated into

7872-567: The sum. Gauss took on the directorate of the 60-year-old observatory, founded in 1748 by Prince-elector George II and built on a converted fortification tower, with usable, but partly out-of-date instruments. The construction of a new observatory had been approved by Prince-elector George III in principle since 1802, and the Westphalian government continued the planning, but Gauss could not move to his new place of work until September 1816. He got new up-to-date instruments, including two meridian circles from Repsold and Reichenbach , and

7968-577: The summer of 1821. After a short time at university, in 1824 Joseph joined the Hanoverian army and assisted in surveying again in 1829. In the 1830s he was responsible for the enlargement of the survey network to the western parts of the kingdom. With his geodetical qualifications, he left the service and engaged in the construction of the railway network as director of the Royal Hanoverian State Railways . In 1836 he studied

8064-421: The summit of science". His close contemporaries agreed that Gauss was a man of difficult character. He often refused to accept compliments. His visitors were occasionally irritated by his grumpy behaviour, but a short time later his mood could change, and he would become a charming, open-minded host. Gauss abominated polemic natures; together with his colleague Hausmann he opposed to a call for Justus Liebig on

8160-547: The transformed equation do not necessarily minimize the residuals of the original equation. For all n {\displaystyle n} observations, the expectation—conditional on the regressors—of the error term is zero: where x i = [ x i 1 x i 2 ⋯ x i k ] T {\displaystyle \mathbf {x} _{i}={\begin{bmatrix}x_{i1}&x_{i2}&\cdots &x_{ik}\end{bmatrix}}^{\operatorname {T} }}

8256-959: The unobservable β {\displaystyle \beta } but whose expected value is always zero. Proof that the OLS indeed minimizes the sum of squares of residuals may proceed as follows with a calculation of the Hessian matrix and showing that it is positive definite. The MSE function we want to minimize is f ( β 0 , β 1 , … , β p ) = ∑ i = 1 n ( y i − β 0 − β 1 x i 1 − ⋯ − β p x i p ) 2 {\displaystyle f(\beta _{0},\beta _{1},\dots ,\beta _{p})=\sum _{i=1}^{n}(y_{i}-\beta _{0}-\beta _{1}x_{i1}-\dots -\beta _{p}x_{ip})^{2}} for

8352-399: The values of X i j {\displaystyle X_{ij}} . Now, let ∑ j = 1 K λ j β j {\textstyle \sum _{j=1}^{K}\lambda _{j}\beta _{j}} be some linear combination of the coefficients. Then the mean squared error of the corresponding estimation is in other words, it

8448-421: The years since 1820 are taken as a "period of lower astronomical activity". The new, well-equipped observatory did not work as effectively as other ones; Gauss's astronomical research had the character of a one-man enterprise without a long-time observation program, and the university established a place for an assistant only after Harding died in 1834. Nevertheless, Gauss twice refused the opportunity to solve

8544-630: Was Abraham Gotthelf Kästner , whom Gauss called "the leading mathematician among poets, and the leading poet among mathematicians" because of his epigrams . Astronomy was taught by Karl Felix Seyffer , with whom Gauss stayed in correspondence after graduation; Olbers and Gauss mocked him in their correspondence. On the other hand, he thought highly of Georg Christoph Lichtenberg , his teacher of physics, and of Christian Gottlob Heyne , whose lectures in classics Gauss attended with pleasure. Fellow students of this time were Johann Friedrich Benzenberg , Farkas Bolyai , and Heinrich Wilhelm Brandes . He

8640-585: Was a German mathematician , astronomer , geodesist , and physicist who contributed to many fields in mathematics and science. He was director of the Göttingen Observatory and professor of astronomy from 1807 until his death in 1855. He is widely considered one of the greatest mathematicians ever. While studying at the University of Göttingen , he propounded several mathematical theorems . Gauss completed his masterpieces Disquisitiones Arithmeticae and Theoria motus corporum coelestium as

8736-792: Was a member of the Lutheran church , like most of the population in northern Germany. It seems that he did not believe all dogmas or understand the Holy Bible quite literally. Sartorius mentioned Gauss's religious tolerance , and estimated his "insatiable thirst for truth" and his sense of justice as motivated by religious convictions. In his doctoral thesis from 1799, Gauss proved the fundamental theorem of algebra which states that every non-constant single-variable polynomial with complex coefficients has at least one complex root . Mathematicians including Jean le Rond d'Alembert had produced false proofs before him, and Gauss's dissertation contains

8832-459: Was arbitrary, it means all eigenvalues of H {\displaystyle {\mathcal {H}}} are positive, therefore H {\displaystyle {\mathcal {H}}} is positive definite. Thus, β = ( X T X ) − 1 X T Y {\displaystyle {\boldsymbol {\beta }}=\left(X^{\operatorname {T} }X\right)^{-1}X^{\operatorname {T} }Y}

8928-701: Was deeply affected by this quarrel but saw no possibility to help them. Gauss took part in academic administration: three times he was elected as dean of the Faculty of Philosophy. Being entrusted with the widow's pension fund of the university, he dealt with actuarial science and wrote a report on the strategy for stabilizing the benefits. He was appointed director of the Royal Academy of Sciences in Göttingen for nine years. Gauss remained mentally active into his old age, even while suffering from gout and general unhappiness. On 23 February 1855, he died of

9024-517: Was in keeping with the motto of his personal seal Pauca sed Matura ("Few, but Ripe"). Many colleagues encouraged him to publicize new ideas and sometimes rebuked him if he hesitated too long, in their opinion. Gauss defended himself, claiming that the initial discovery of ideas was easy, but preparing a presentable elaboration was a demanding matter for him, for either lack of time or "serenity of mind". Nevertheless, he published many short communications of urgent content in various journals, but left

9120-484: Was likely a self-taught student in mathematics since he independently rediscovered several theorems. He solved a geometrical problem that had occupied mathematicians since the Ancient Greeks , when he determined in 1796 which regular polygons can be constructed by compass and straightedge . This discovery ultimately led Gauss to choose mathematics instead of philology as a career. Gauss's mathematical diary,

9216-512: Was removed, preserved, and studied by Rudolf Wagner , who found its mass to be slightly above average, at 1,492 grams (3.29 lb). Wagner's son Hermann , a geographer, estimated the cerebral area to be 219,588 square millimetres (340.362 sq in) in his doctoral thesis. In 2013, a neurobiologist at the Max Planck Institute for Biophysical Chemistry in Göttingen discovered that Gauss's brain had been mixed up soon after

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