scholarly journals Theorem about the sufficient condition for the equality of the estimates of the OLS and Aitken of the leading coefficient of quadratic regression

2021 ◽  
pp. 138-142
Author(s):  
Marta Savkina
Keyword(s):  
Regression Model ◽  
Covariance Matrix ◽  
Regression Coefficient ◽  
Covariance Matrices ◽  
Sufficient Condition ◽  
Quadratic Regression ◽  
Quadratic Regression Model

At the paper in the case of heteroscedastic independent deviations a quadratic regression model is studied. A theorem is formulated that gives a sufficient condition on the variance of deviations for the coincidence of Aitken's estimate of the leading regression coefficient with his estimation of the OLS in the case of an odd number of observation points and a bisymmetric covariance matrix. On the basis of this theorem, in some cases examples of non-unit covariance matrices are constructed for which the indicated estimations of the leading coefficient of the quadratic regression coindcide.

scholarly journals SUFFICIENT CONDITION FOR COINCIDENCE OF THE LS AND AITKEN ESTIMATIONS OF PARAMETER OF QUADRATIC REGRESSION IN CASE HETEROSCEDASTIC DEVIATIONS

2020 ◽  
pp. 45-58
Author(s):  
Marta Savkina
Keyword(s):  
Regression Model ◽  
Covariance Matrix ◽  
Line Segment ◽  
Original System ◽  
Sufficient Condition ◽  
Quadratic Regression ◽  
Unknown Parameters ◽  
Equidistant Points ◽  
Second Degree

In the paper in case heteroscedastic independent deviations a regression model whose function has the form $f(x) = ax^2+bx+c$, where $a$, $b$ and $c$ are unknown parameters, is studied. Approximate values (observations) of functions $f(x)$ are registered at equidistant points of a line segment. The theorem which is proved at the paper gives a sufficient condition on the variance of the deviations at which the Aitken estimation of parameter $a$ coincides with its estimation of the LS in the case of odd number of observation points and bisymmetric covariance matrix. Under this condition, the Aitken and LS estimations of $b$ and $c$ will not coincide. The proof of the theorem consists of the following steps. First, the original system of polynomials is simplified: we get the system polynomials of the second degree. The variables of both systems are unknown variances of deviations, each of the solutions of the original system gives a set variances of deviations at which the estimations of Aitken and LS parameter a coincide. In the next step the solving of the original system polynomials is reduced to solving an equation with three unknowns, and all other unknowns are expressed in some way through these three. At last it is proved that there are positive unequal values of these three unknowns, which will be the solution of the obtained equation. And all other unknowns when substituting in their expression these values will be positive.

Distribution of Bark Procyanidin Contents in Pinus yunnanensis Populations and Related Species

2021 ◽  
Vol 15 (1) ◽  
pp. 105-111
Author(s):  
Xiaolong Zhang ◽  
Hong Yu ◽  
Lin Chen ◽  
Chuntang Mao ◽  
Yuanbing Wang ◽  
...  
Keyword(s):  
Environmental Factors ◽  
Regression Model ◽  
Related Species ◽  
High Variability ◽  
Quadratic Regression ◽  
Quadratic Regression Model ◽  
Pinus Yunnanensis ◽  
Northwest Yunnan ◽  
Heterogeneous Habitat ◽  
Genetic And Environmental Factors

The distribution of the bark procyanidin contents in 26 populations of Pinus yunnanensis and 8 populations of P. kesiya var. langbianensis and 5 populations of P. densata was determined. The results indicated that genetic and environmental factors commonly affected the procyanidin content in the populations of P. yunnanensis and related species. The procyanidin contents in the three related species followed the order of P. densata (54.72 mg/g) > P. kesiya (43.86 mg/g) > P. yunnanensis (37.95 mg/g). Furthermore, the procyanidin content in P. yunnanensis and related species had high variability. The procyanidin contents in the three related species showed that the procyanidin contents of P. densata distributed in northwest Yunnan and P. kesiya distributed in southern Yunnan were high, the procyanidin content of the population with introgression and hybridization or a heterogeneous habitat was also high, and that of P. yunnanensis distributed in central Yunnan was low. The quadratic regression model of procyanidin content and latitude was Y = 1329.06 -100.52* Latitude +1.95 * Latitude2.

Application of quadratic regression model for Fenton treatment of municipal landfill leachate

2012 ◽  
Vol 32 (10) ◽  
pp. 1895-1902 ◽  
Author(s):  
Mehdi Ghanbarzadeh Lak ◽  
Mohammad Reza Sabour ◽  
Allahyar Amiri ◽  
Omid Rabbani
Keyword(s):  
Regression Model ◽  
Landfill Leachate ◽  
Quadratic Regression ◽  
Quadratic Regression Model ◽  
Municipal Landfill

Performance Evaluation of African Breadfruit (Treculia Africana) Seed Dehuller

2018 ◽  
Vol 22 (4) ◽  
pp. 51-60
Author(s):  
Okey Francis Obi ◽  
Clement O. Akubuo
Keyword(s):  
Performance Evaluation ◽  
Regression Model ◽  
Recent Effort ◽  
Quadratic Regression ◽  
Linear Quadratic ◽  
Quadratic Regression Model ◽  
African Breadfruit ◽  
Treculia Africana ◽  
Non Linear

AbstractThis paper reports the effect of the parboiling time on dehulled kernel out-turns (DKO) of African breadfruit seeds, and the most recent effort to upgrade an existing dehuller and its performance. Two common and readily available varieties – Treculia var. africana and var. inverse were used in the study. The seeds were parboiled for 0 (control), 2, 5, 8, 11 and 14 minutes and then dehulled. The result revealed that the parboiling time had a significant effect on the DKO of the two varieties of the seed. The DKO increased from 0 to 5 min of the treatment, after which it decreased considerably up to 14 min of the parboiling time. The obtained data were used to develop a non-linear quadratic regression model to predict the DKO as a function of the parboiling time. The performance evaluation of the breadfruit seeds dehuller revealed that it was significantly influenced by the variety.

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