By Corneliu Constantinescu
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Extra info for C* Algebras, Volume 2: Banach Algebras and Compact Operators
71). , Toutenburg and Shalabh (2001a), Shalabh and Toutenburg (2006). 12 Orthogonal Regression Method Generally when uncertainties are involved in dependent and independent variables both, then orthogonal regression is more appropriate. The least squares principle in orthogonal regression minimizes the squared perpendicular distance between the observed data points and the line in the scatter diagram to obtain the estimates of regression coeﬃcients. This is also known as major axis regression method.
40) where z1−α/2 is the (1- α/2) percentage point of the N (0, 1) distribution. 38), then we proceed as follows. We know E RSS T −2 = σ2 and RSS ∼ χ2T −2 . σ2 Further, RSS/σ 2 and b1 are independently distributed, so the statistic t01 = = when H0 is true. 41) 18 2. 41) as RSS . 41) under the condition when σ 2 is known or unknown, respectively. For example, when σ 2 is unknown, the decision rule is to reject H0 if |t01 | > tT −2,1−α/2 where tT −2,1−α/2 is the (1 − α/2) percentage point of the t-distribution with (T − 2) degrees of freedom.
15) can also be obtained as a conditional leastsquares estimator when β is subject to the restriction U β = u for a given arbitrary u. 15) satisﬁes the equation. 81. 38 3. 15). 3, we viewed the problem of the linear model y = Xβ + e as one of ﬁtting the function Xβ to y without making any assumptions on e. Now we consider e as a random variable denoted by , make some assumptions on its distribution, and discuss the estimation of β considered as an unknown vector parameter. 16) and X is a ﬁxed or nonstochastic matrix of order T × K, with full rank K.