www.czasopisma.uni.lodz.pl/foe/ 6(332) 2017
[87]
Acta Universitatis Lodziensis
Folia Oeconomica
ISSN 0208-6018 e-ISSN 2353-7663
DOI: http://dx.doi.org/10.18778/0208‑6018.332.06
Grzegorz Antoni Sitek
University of Economics in Katowice, Faculty of Management, Department of Statistics, Econometrics and Mathematics, grzegors12@wp.pl
Janusz Leszek Wywiał
University of Economics in Katowice, Faculty of Management, Department of Statistics, Econometrics and Mathematics, wywial@ue.katowice.pl
On Estimation of Bi‑liner Regression Parameters
Abstract: Two non‑parallel lines will be named as bi‑lines. The relationship between the definition
of the bi‑lines function and linear regression functions of the distribution mixture is considered. The bi‑lines function parameters are estimated using the least squares method for an implicit interde‑ pendence. In general, values of parameter estimators are evaluated by means of an approximation numerical method. In a particular case, the exact expressions for the parameter estimators were de‑ rived. In this particular case, the properties of the estimators are examined in details. The bi‑lines are also used to estimate the regression functions of the distribution mixture. The accuracy of the pa‑ rameter estimation is analyzed.
Keywords: bi‑lines function, linear regression, least squares method for an implicit interdependen‑
ce, mixture of probability distribution
88 Grzegorz Antoni Sitek, Janusz Leszek Wywiał
FOE 6(332) 2017 www.czasopisma.uni.lodz.pl/foe/
1. Introduction
The classic simple regression does not represent a two‑dimensional population if it is a bimodal population. Antoniewicz (1988; 2001) proposed to approximate probability distribution of a one‑dimensional random variable by means of two points. Generalizing this result he approximates a two‑dimensional distribution by means of two lines, a technique which he called “bi‑linear regression”. In sta-tistical literature, this term is rather used to define a specific linear model, see e.g. Gabriel (1998). In this paper, we will call Antoniewicz’s model simply “bi‑lines function”, because in general it leads to a two‑dimensional data spread approxima-tion by means of two lines. In the next part of this paper, the bi‑lines are compared with regressions of two‑dimensional probability mixture distributions, or more simply regressions of mixture distributions, see e.g. Quandt (1972). The mixture of Gaussian distributions has been extensively studied e.g. by McLachlan and Peel (2000) and Lindsay and Basak (1993).
The main purpose of the paper is finding differences between the bi‑lines function and regressions of mixture distributions. Moreover, we show that the es-timators of bi‑lines function parameters are biased eses-timators of appropriate pa-rameters of regressions of mixture distribution. In a particular case, the exact ex-pressions for the estimators of the bi‑lines function parameters, as well as for their variances will be derived.
Let a bivariate distribution of (X, Y) be a mixture of two bivariate distribu-tions of (X1, Y1) and (X2, Y2). Hence:
2 a specific linear model, see e.g. Gabriel (1998). In this paper, we will call Antoniewicz’s model simply “bi-lines function”, because in general it leads to a two-dimensional data spread approximation by means of two lines. In the next part of this paper, the bi-lines are compared with regressions of two-dimensional probability mixture distributions, or more simply regressions of mixture distributions, see e.g. Quandt (1972). The mixture of Gaussian distributions has been extensively studied e.g. by McLachlan and Peel (2000) and Lindsay and Basak (1993).
The main purpose of the paper is finding differences between the bi-lines function and regressions of mixture distributions. Moreover, we show that the estimators of bi-lines function parameters are biased estimators of appropriate parameters of regressions of mixture distribution. In a particular case, the exact expressions for the estimators of the bi-lines function parameters, as well as for their variances will be derived.
Let a bivariate distribution of (X, Y) be a mixture of two bivariate distributions of (X1, Y1)
and (X2, Y2). Hence: ) , ( ) 1 ( ) , ( ) , (x y pf1 x y p f2 x y f , 0 < p < 1, (1)
where p is the mixing parameter. Let:
), ( ) 1 ( ) ( ) , ( ) 1 ( ) , ( ) , ( ) (x f x y dy p f1 x y dy p f2 x y dy ph1 x p h2 x h
(2) where: hi(x)
fi(x,y)dy, i = 1, 2, ). ( ) | ( ) ( ) | ( ) ( ) 1 ( ) ( ) , ( ) 1 ( ) , ( ) | ( 1 1 2 2 2 1 2 1 f y x w x f y x w x x h p x ph y x f p y x pf x y f (3) ) ( ) ( ) ( 1 1 x phh xx w , . ) ( ) ( ) 1 ( ) ( 2 2 x hpxh x w It is obvious that if h1(x) = h2(x) then h(x) = h1(x) = h2(x) and w1(x) = p, w2(x) = 1 – p.
Moreover, in this case:
), | ( ) 1 ( ) | ( ) | (y x pf1 y x p f2 y x f (4) ). | ( ) 1 ( ) | ( ) | (Y x pE Y1 x p E Y2 x E (5)
Particularly, if h1(x) = h2(x) and E(Yi | x) = aix + bi, i = 1, 2, then expression (5) reduces to the
following: ). )( 1 ( ) ( ) | (Y x p a1x b1 p a2x b2 E (6) (1) where p is the mixing parameter. Let:
2 a specific linear model, see e.g. Gabriel (1998). In this paper, we will call Antoniewicz’s model simply “bi-lines function”, because in general it leads to a two-dimensional data spread approximation by means of two lines. In the next part of this paper, the bi-lines are compared with regressions of two-dimensional probability mixture distributions, or more simply regressions of mixture distributions, see e.g. Quandt (1972). The mixture of Gaussian distributions has been extensively studied e.g. by McLachlan and Peel (2000) and Lindsay and Basak (1993).
The main purpose of the paper is finding differences between the bi-lines function and regressions of mixture distributions. Moreover, we show that the estimators of bi-lines function parameters are biased estimators of appropriate parameters of regressions of mixture distribution. In a particular case, the exact expressions for the estimators of the bi-lines function parameters, as well as for their variances will be derived.
Let a bivariate distribution of (X, Y) be a mixture of two bivariate distributions of (X1, Y1)
and (X2, Y2). Hence: ) , ( ) 1 ( ) , ( ) , (x y pf1 x y p f2 x y f , 0 < p < 1, (1)
where p is the mixing parameter. Let:
), ( ) 1 ( ) ( ) , ( ) 1 ( ) , ( ) , ( ) (x f x y dy p f1 x y dy p f2 x y dy ph1 x p h2 x h
(2) where: hi(x)
fi(x,y)dy, i = 1, 2, ). ( ) | ( ) ( ) | ( ) ( ) 1 ( ) ( ) , ( ) 1 ( ) , ( ) | ( 1 1 2 2 2 1 2 1 f y x w x f y x w x x h p x ph y x f p y x pf x y f (3) ) ( ) ( ) ( 1 1 x phh xx w , . ) ( ) ( ) 1 ( ) ( 2 2 x hpxh x w It is obvious that if h1(x) = h2(x) then h(x) = h1(x) = h2(x) and w1(x) = p, w2(x) = 1 – p.
Moreover, in this case:
), | ( ) 1 ( ) | ( ) | (y x pf1 y x p f2 y x f (4) ). | ( ) 1 ( ) | ( ) | (Y x pE Y1 x p E Y2 x E (5)
Particularly, if h1(x) = h2(x) and E(Yi | x) = aix + bi, i = 1, 2, then expression (5) reduces to the
following: ). )( 1 ( ) ( ) | (Y x p a1x b1 p a2x b2 E (6) 2 a specific linear model, see e.g. Gabriel (1998). In this paper, we will call Antoniewicz’s model simply “bi-lines function”, because in general it leads to a two-dimensional data spread approximation by means of two lines. In the next part of this paper, the bi-lines are compared with regressions of two-dimensional probability mixture distributions, or more simply regressions of mixture distributions, see e.g. Quandt (1972). The mixture of Gaussian distributions has been extensively studied e.g. by McLachlan and Peel (2000) and Lindsay and Basak (1993).
The main purpose of the paper is finding differences between the bi-lines function and regressions of mixture distributions. Moreover, we show that the estimators of bi-lines function parameters are biased estimators of appropriate parameters of regressions of mixture distribution. In a particular case, the exact expressions for the estimators of the bi-lines function parameters, as well as for their variances will be derived.
Let a bivariate distribution of (X, Y) be a mixture of two bivariate distributions of (X1, Y1)
and (X2, Y2). Hence: ) , ( ) 1 ( ) , ( ) , (x y pf1 x y p f2 x y f , 0 < p < 1, (1)
where p is the mixing parameter. Let:
), ( ) 1 ( ) ( ) , ( ) 1 ( ) , ( ) , ( ) (x f x y dy p f1 x y dy p f2 x y dy ph1 x p h2 x h
(2) where: hi(x)
fi(x,y)dy, i = 1, 2, ). ( ) | ( ) ( ) | ( ) ( ) 1 ( ) ( ) , ( ) 1 ( ) , ( ) | ( 1 1 2 2 2 1 2 1 f y x w x f y x w x x h p x ph y x f p y x pf x y f (3) ) ( ) ( ) ( 1 1 x phh xx w , . ) ( ) ( ) 1 ( ) ( 2 2 x hpxh x w It is obvious that if h1(x) = h2(x) then h(x) = h1(x) = h2(x) and w1(x) = p, w2(x) = 1 – p.
Moreover, in this case:
), | ( ) 1 ( ) | ( ) | (y x pf1 y x p f2 y x f (4) ). | ( ) 1 ( ) | ( ) | (Y x pE Y1 x p E Y2 x E (5)
Particularly, if h1(x) = h2(x) and E(Yi | x) = aix + bi, i = 1, 2, then expression (5) reduces to the
following: ). )( 1 ( ) ( ) | (Y x p a1x b1 p a2x b2 E (6) (2) where: 2 a specific linear model, see e.g. Gabriel (1998). In this paper, we will call Antoniewicz’s model simply “bi-lines function”, because in general it leads to a two-dimensional data spread approximation by means of two lines. In the next part of this paper, the bi-lines are compared with regressions of two-dimensional probability mixture distributions, or more simply regressions of mixture distributions, see e.g. Quandt (1972). The mixture of Gaussian distributions has been extensively studied e.g. by McLachlan and Peel (2000) and Lindsay and Basak (1993).
The main purpose of the paper is finding differences between the bi-lines function and regressions of mixture distributions. Moreover, we show that the estimators of bi-lines function parameters are biased estimators of appropriate parameters of regressions of mixture distribution. In a particular case, the exact expressions for the estimators of the bi-lines function parameters, as well as for their variances will be derived.
Let a bivariate distribution of (X, Y) be a mixture of two bivariate distributions of (X1, Y1)
and (X2, Y2). Hence: ) , ( ) 1 ( ) , ( ) , (x y pf1 x y p f2 x y f , 0 < p < 1, (1)
where p is the mixing parameter. Let:
), ( ) 1 ( ) ( ) , ( ) 1 ( ) , ( ) , ( ) (x f x y dy p f1 x y dy p f2 x y dy ph1 x p h2 x h
(2) where: hi(x)
fi(x,y)dy, i = 1, 2, ). ( ) | ( ) ( ) | ( ) ( ) 1 ( ) ( ) , ( ) 1 ( ) , ( ) | ( 1 1 2 2 2 1 2 1 f y x w x f y x w x x h p x ph y x f p y x pf x y f (3) ) ( ) ( ) ( 1 1 x phh xx w , . ) ( ) ( ) 1 ( ) ( 2 2 x hpxh x w It is obvious that if h1(x) = h2(x) then h(x) = h1(x) = h2(x) and w1(x) = p, w2(x) = 1 – p.
Moreover, in this case:
), | ( ) 1 ( ) | ( ) | (y x pf1 y x p f2 y x f (4) ). | ( ) 1 ( ) | ( ) | (Y x pE Y1 x p E Y2 x E (5)
Particularly, if h1(x) = h2(x) and E(Yi | x) = aix + bi, i = 1, 2, then expression (5) reduces to the
following: ). )( 1 ( ) ( ) | (Y x p a1x b1 p a2x b2 E (6) , i = 1, 2, 2 a specific linear model, see e.g. Gabriel (1998). In this paper, we will call Antoniewicz’s model simply “bi-lines function”, because in general it leads to a two-dimensional data spread approximation by means of two lines. In the next part of this paper, the bi-lines are compared with regressions of two-dimensional probability mixture distributions, or more simply regressions of mixture distributions, see e.g. Quandt (1972). The mixture of Gaussian distributions has been extensively studied e.g. by McLachlan and Peel (2000) and Lindsay and Basak (1993).
The main purpose of the paper is finding differences between the bi-lines function and regressions of mixture distributions. Moreover, we show that the estimators of bi-lines function parameters are biased estimators of appropriate parameters of regressions of mixture distribution. In a particular case, the exact expressions for the estimators of the bi-lines function parameters, as well as for their variances will be derived.
Let a bivariate distribution of (X, Y) be a mixture of two bivariate distributions of (X1, Y1)
and (X2, Y2). Hence: ) , ( ) 1 ( ) , ( ) , (x y pf1 x y p f2 x y f , 0 < p < 1, (1)
where p is the mixing parameter. Let:
), ( ) 1 ( ) ( ) , ( ) 1 ( ) , ( ) , ( ) (x f x y dy p f1 x y dy p f2 x y dy ph1 x p h2 x h
(2) where: hi(x)
fi(x,y)dy, i = 1, 2, ). ( ) | ( ) ( ) | ( ) ( ) 1 ( ) ( ) , ( ) 1 ( ) , ( ) | ( 1 1 2 2 2 1 2 1 f y x w x f y x w x x h p x ph y x f p y x pf x y f (3) ) ( ) ( ) ( 1 1 x phh xx w , . ) ( ) ( ) 1 ( ) ( 2 2 x hpxh x w It is obvious that if h1(x) = h2(x) then h(x) = h1(x) = h2(x) and w1(x) = p, w2(x) = 1 – p.
Moreover, in this case:
), | ( ) 1 ( ) | ( ) | (y x pf1 y x p f2 y x f (4) ). | ( ) 1 ( ) | ( ) | (Y x pE Y1 x p E Y2 x E (5)
Particularly, if h1(x) = h2(x) and E(Yi | x) = aix + bi, i = 1, 2, then expression (5) reduces to the
following: ). )( 1 ( ) ( ) | (Y x p a1x b1 p a2x b2 E (6) (3) 2 a specific linear model, see e.g. Gabriel (1998). In this paper, we will call Antoniewicz’s model simply “bi-lines function”, because in general it leads to a two-dimensional data spread approximation by means of two lines. In the next part of this paper, the bi-lines are compared with regressions of two-dimensional probability mixture distributions, or more simply regressions of mixture distributions, see e.g. Quandt (1972). The mixture of Gaussian distributions has been extensively studied e.g. by McLachlan and Peel (2000) and Lindsay and Basak (1993).
The main purpose of the paper is finding differences between the bi-lines function and regressions of mixture distributions. Moreover, we show that the estimators of bi-lines function parameters are biased estimators of appropriate parameters of regressions of mixture distribution. In a particular case, the exact expressions for the estimators of the bi-lines function parameters, as well as for their variances will be derived.
Let a bivariate distribution of (X, Y) be a mixture of two bivariate distributions of (X1, Y1)
and (X2, Y2). Hence: ) , ( ) 1 ( ) , ( ) , (x y pf1 x y p f2 x y f , 0 < p < 1, (1)
where p is the mixing parameter. Let:
), ( ) 1 ( ) ( ) , ( ) 1 ( ) , ( ) , ( ) (x f x y dy p f1 x y dy p f2 x y dy ph1 x p h2 x h
(2) where: hi(x)
fi(x,y)dy, i = 1, 2, ). ( ) | ( ) ( ) | ( ) ( ) 1 ( ) ( ) , ( ) 1 ( ) , ( ) | ( 1 1 2 2 2 1 2 1 f y x w x f y x w x x h p x ph y x f p y x pf x y f (3) ) ( ) ( ) ( 1 1 x h x ph x w , . ) ( ) ( ) 1 ( ) ( 2 2 x hpxh x w It is obvious that if h1(x) = h2(x) then h(x) = h1(x) = h2(x) and w1(x) = p, w2(x) = 1 – p.
Moreover, in this case:
), | ( ) 1 ( ) | ( ) | (y x pf1 y x p f2 y x f (4) ). | ( ) 1 ( ) | ( ) | (Y x pE Y1 x p E Y2 x E (5)
Particularly, if h1(x) = h2(x) and E(Yi | x) = aix + bi, i = 1, 2, then expression (5) reduces to the
following: ). )( 1 ( ) ( ) | (Y x p a1x b1 p a2x b2 E (6)
On Estimation of Bi‑liner Regression Parameters 89
www.czasopisma.uni.lodz.pl/foe/ FOE 6(332) 2017 It is obvious that if h1(x) = h2(x) then h(x) = h1(x) = h2(x) and w1(x) = p,
w2(x) = 1 – p. Moreover, in this case:
2 a specific linear model, see e.g. Gabriel (1998). In this paper, we will call Antoniewicz’s model simply “bi-lines function”, because in general it leads to a two-dimensional data spread approximation by means of two lines. In the next part of this paper, the bi-lines are compared with regressions of two-dimensional probability mixture distributions, or more simply regressions of mixture distributions, see e.g. Quandt (1972). The mixture of Gaussian distributions has been extensively studied e.g. by McLachlan and Peel (2000) and Lindsay and Basak (1993).
The main purpose of the paper is finding differences between the bi-lines function and regressions of mixture distributions. Moreover, we show that the estimators of bi-lines function parameters are biased estimators of appropriate parameters of regressions of mixture distribution. In a particular case, the exact expressions for the estimators of the bi-lines function parameters, as well as for their variances will be derived.
Let a bivariate distribution of (X, Y) be a mixture of two bivariate distributions of (X1, Y1)
and (X2, Y2). Hence: ) , ( ) 1 ( ) , ( ) , (x y pf1 x y p f2 x y f , 0 < p < 1, (1)
where p is the mixing parameter. Let:
), ( ) 1 ( ) ( ) , ( ) 1 ( ) , ( ) , ( ) (x f x y dy p f1 x y dy p f2 x y dy ph1 x p h2 x h
(2) where: hi(x)
fi(x,y)dy, i = 1, 2, ). ( ) | ( ) ( ) | ( ) ( ) 1 ( ) ( ) , ( ) 1 ( ) , ( ) | ( 1 1 2 2 2 1 2 1 f y x w x f y x w x x h p x ph y x f p y x pf x y f (3) ) ( ) ( ) ( 1 1 x h x ph x w , . ) ( ) ( ) 1 ( ) ( 2 2 x hpxh x w It is obvious that if h1(x) = h2(x) then h(x) = h1(x) = h2(x) and w1(x) = p, w2(x) = 1 – p.
Moreover, in this case:
), | ( ) 1 ( ) | ( ) | (y x pf1 y x p f2 y x f (4) ). | ( ) 1 ( ) | ( ) | (Y x pE Y1 x p E Y2 x E (5)
Particularly, if h1(x) = h2(x) and E(Yi | x) = aix + bi, i = 1, 2, then expression (5) reduces to the
following: ). )( 1 ( ) ( ) | (Y x p a1x b1 p a2x b2 E (6) (4) 2 a specific linear model, see e.g. Gabriel (1998). In this paper, we will call Antoniewicz’s model simply “bi-lines function”, because in general it leads to a two-dimensional data spread approximation by means of two lines. In the next part of this paper, the bi-lines are compared with regressions of two-dimensional probability mixture distributions, or more simply regressions of mixture distributions, see e.g. Quandt (1972). The mixture of Gaussian distributions has been extensively studied e.g. by McLachlan and Peel (2000) and Lindsay and Basak (1993).
The main purpose of the paper is finding differences between the bi-lines function and regressions of mixture distributions. Moreover, we show that the estimators of bi-lines function parameters are biased estimators of appropriate parameters of regressions of mixture distribution. In a particular case, the exact expressions for the estimators of the bi-lines function parameters, as well as for their variances will be derived.
Let a bivariate distribution of (X, Y) be a mixture of two bivariate distributions of (X1, Y1)
and (X2, Y2). Hence: ) , ( ) 1 ( ) , ( ) , (x y pf1 x y p f2 x y f , 0 < p < 1, (1)
where p is the mixing parameter. Let:
), ( ) 1 ( ) ( ) , ( ) 1 ( ) , ( ) , ( ) (x f x y dy p f1 x y dy p f2 x y dy ph1 x p h2 x h
(2) where: hi(x)
fi(x,y)dy, i = 1, 2, ). ( ) | ( ) ( ) | ( ) ( ) 1 ( ) ( ) , ( ) 1 ( ) , ( ) | ( 1 1 2 2 2 1 2 1 f y x w x f y x w x x h p x ph y x f p y x pf x y f (3) ) ( ) ( ) ( 1 1 x phh xx w , . ) ( ) ( ) 1 ( ) ( 2 2 x hpxh x w It is obvious that if h1(x) = h2(x) then h(x) = h1(x) = h2(x) and w1(x) = p, w2(x) = 1 – p.
Moreover, in this case:
), | ( ) 1 ( ) | ( ) | (y x pf1 y x p f2 y x f (4) ). | ( ) 1 ( ) | ( ) | (Y x pEY1 x p E Y2 x E (5)
Particularly, if h1(x) = h2(x) and E(Yi | x) = aix + bi, i = 1, 2, then expression (5) reduces to the
following: ). )( 1 ( ) ( ) | (Y x p a1x b1 p a2x b2 E (6) (5)
Particularly, if h1(x) = h2(x) and E(Yi | x) = aix + bi, i = 1, 2, then expression (5) reduces to the following:
2 a specific linear model, see e.g. Gabriel (1998). In this paper, we will call Antoniewicz’s model simply “bi-lines function”, because in general it leads to a two-dimensional data spread approximation by means of two lines. In the next part of this paper, the bi-lines are compared with regressions of two-dimensional probability mixture distributions, or more simply regressions of mixture distributions, see e.g. Quandt (1972). The mixture of Gaussian distributions has been extensively studied e.g. by McLachlan and Peel (2000) and Lindsay and Basak (1993).
The main purpose of the paper is finding differences between the bi-lines function and regressions of mixture distributions. Moreover, we show that the estimators of bi-lines function parameters are biased estimators of appropriate parameters of regressions of mixture distribution. In a particular case, the exact expressions for the estimators of the bi-lines function parameters, as well as for their variances will be derived.
Let a bivariate distribution of (X, Y) be a mixture of two bivariate distributions of (X1, Y1)
and (X2, Y2). Hence: ) , ( ) 1 ( ) , ( ) , (x y pf1 x y p f2 x y f , 0 < p < 1, (1)
where p is the mixing parameter. Let:
), ( ) 1 ( ) ( ) , ( ) 1 ( ) , ( ) , ( ) (x f x y dy p f1 x y dy p f2 x y dy ph1 x p h2 x h
(2) where: hi(x)
fi(x,y)dy, i = 1, 2, ). ( ) | ( ) ( ) | ( ) ( ) 1 ( ) ( ) , ( ) 1 ( ) , ( ) | ( 1 1 2 2 2 1 2 1 f y x w x f y x w x x h p x ph y x f p y x pf x y f (3) ) ( ) ( ) ( 1 1 x h x ph x w , . ) ( ) ( ) 1 ( ) ( 2 2 x h x h p x w It is obvious that if h1(x) = h2(x) then h(x) = h1(x) = h2(x) and w1(x) = p, w2(x) = 1 – p.
Moreover, in this case:
), | ( ) 1 ( ) | ( ) | (y x pf1 y x p f2 y x f (4) ). | ( ) 1 ( ) | ( ) | (Y x pE Y1 x p E Y2 x E (5)
Particularly, if h1(x) = h2(x) and E(Yi | x) = aix + bi, i = 1, 2, then expression (5) reduces to the
following: ). )( 1 ( ) ( ) | (Y x p a1x b1 p a2x b2 E (6) (6)
The above function we will treat as a linear regression function of the mix-ture distribution while E(Yi | x) = aix + bi, i = 1, 2, are linear regression functions of the mixture distribution.
2. The least squares method for an implicit
interdependence
Let (X, Y) be a two‑dimensional random variable. Antoniewicz (1988) proposed an original method of approximation distribution by means of two lines, neither of which is parallel to the axis of the system. Parameters a, b, c and d of the lines minimize the following function:
The above function we will treat as a linear regression function of the mixture distribution while E(Yi | x) = aix + bi, i = 1, 2, are linear regression functions of the mixture distribution.
2. The least squares method for an implicit interdependence
Let (X, Y) be a two-dimensional random variable. Antoniewicz (1988) proposed an original method of approximation distribution by means of two lines, neither of which is parallel to the axis of the system. Parameters a, b, c and d of the lines minimize the following function:
. ) , , , (a EYaX c 2YbXd 2 (7)Based on the available data, the parameters a, b, c and d will be estimated. This is equivalent to finding the straight lines that give the best fit (representation) of the points in the scatter plot of the response versus the predictor variable. We estimate the parameters using the popular least squares method, which gives the lines that minimize the sum of squares of the vertical distances from each point to the lines. The vertical distances represent the errors in the response variable. The sum of squares of these distances can then be written as follows: ( , , , ) [( )( )]. 1 2
n i i i i i d x b y c x a y d c b a S (8)The values of estimators aˆ,b,ˆc,ˆdˆ which minimize S(a, b, c, d) are derived by Antoniewicz (1988). In general, values of aˆ,b,ˆc,ˆdˆ can be obtained only numerically. This problem is considered in details by Sitek (2016).
If c = d = 0, expression (8) reduces to the following: . ] ) )( [( ) , ( 1 2
n i i i i i bx y ax y b a S (9)Values of a , that minimize S(a, b) are given by the solution (roots) of the following ^, b^ nonlinear system of two equations (10).
0 2 2 0 2 2 40 2 31 22 31 2 22 13 40 2 31 22 31 2 22 13 bm a abm bm m a am m m ab abm am m b bm m (10) where
n i v i u i uv x y m 1 , u = 0, 1, 2, …, v = 0, 1, 2, …Under the assumption that a ≠ b, after appropriate transformations we have:
(7) Based on the available data, the parameters a, b, c and d will be estimated. This is equivalent to finding the straight lines that give the best fit (representation) of the points in the scatter plot of the response versus the predictor variable. We es-timate the parameters using the popular least squares method, which gives the lines that minimize the sum of squares of the vertical distances from each point to the lines. The vertical distances represent the errors in the response variable. The sum of squares of these distances can then be written as follows:
3 The above function we will treat as a linear regression function of the mixture distribution while E(Yi | x) = aix + bi, i = 1, 2, are linear regression functions of the mixture distribution.
2. The least squares method for an implicit interdependence
Let (X, Y) be a two-dimensional random variable. Antoniewicz (1988) proposed an original method of approximation distribution by means of two lines, neither of which is parallel to the axis of the system. Parameters a, b, c and d of the lines minimize the following function:
. ) , , , (a EYaX c2YbX d 2 (7)Based on the available data, the parameters a, b, c and d will be estimated. This is equivalent to finding the straight lines that give the best fit (representation) of the points in the scatter plot of the response versus the predictor variable. We estimate the parameters using the popular least squares method, which gives the lines that minimize the sum of squares of the vertical distances from each point to the lines. The vertical distances represent the errors in the response variable. The sum of squares of these distances can then be written as follows: ( , , , ) [( )( )] . 1 2
n i i i i i d x b y c x a y d c b a S (8)The values of estimators aˆ,b,ˆc,ˆdˆ which minimize S(a, b, c, d) are derived by Antoniewicz (1988). In general, values of aˆ,b,ˆc,ˆdˆ can be obtained only numerically. This problem is considered in details by Sitek (2016).
If c = d = 0, expression (8) reduces to the following: . ] ) )( [( ) , ( 1 2
n i i i i i bx y ax y b a S (9)Values of a , that minimize S(a, b) are given by the solution (roots) of the following ^, b^ nonlinear system of two equations (10).
0 2 2 0 2 2 40 2 31 22 31 2 22 13 40 2 31 22 31 2 22 13 bm a abm bm m a am m m ab abm am m b bm m (10) where
n i v i u i uv x y m 1 , u = 0, 1, 2, …, v = 0, 1, 2, …Under the assumption that a ≠ b, after appropriate transformations we have:
90 Grzegorz Antoni Sitek, Janusz Leszek Wywiał
FOE 6(332) 2017 www.czasopisma.uni.lodz.pl/foe/
The values of estimators a,ˆb,ˆc,ˆ dˆ which minimize S(a, b, c, d) are derived by Antoniewicz (1988). In general, values of a,ˆb,ˆc,ˆdˆ can be obtained only nu-merically. This problem is considered in details by Sitek (2016).
If c = d = 0, expression (8) reduces to the following:
3 The above function we will treat as a linear regression function of the mixture distribution while E(Yi | x) = aix + bi, i = 1, 2, are linear regression functions of the mixture distribution.
2. The least squares method for an implicit interdependence
Let (X, Y) be a two-dimensional random variable. Antoniewicz (1988) proposed an original method of approximation distribution by means of two lines, neither of which is parallel to the axis of the system. Parameters a, b, c and d of the lines minimize the following function:
. ) , , , (a EYaXc2 YbXd 2 (7)Based on the available data, the parameters a, b, c and d will be estimated. This is equivalent to finding the straight lines that give the best fit (representation) of the points in the scatter plot of the response versus the predictor variable. We estimate the parameters using the popular least squares method, which gives the lines that minimize the sum of squares of the vertical distances from each point to the lines. The vertical distances represent the errors in the response variable. The sum of squares of these distances can then be written as follows: ( , , , ) [( )( )] . 1 2
n i i i i i d x b y c x a y d c b a S (8)The values of estimators aˆ,b,ˆc,ˆdˆ which minimize S(a, b, c, d) are derived by Antoniewicz (1988). In general, values of aˆ,b,ˆc,ˆdˆ can be obtained only numerically. This problem is considered in details by Sitek (2016).
If c = d = 0, expression (8) reduces to the following: . ] ) )( [( ) , ( 1 2
n i i i i i bx y ax y b a S (9)Values of a , that minimize S(a, b) are given by the solution (roots) of the following ^, b^ nonlinear system of two equations (10).
0 2 2 0 2 2 40 2 31 22 31 2 22 13 40 2 31 22 31 2 22 13 bm a abm bm m a am m m ab abm am m b bm m (10) where
n i v i u i uv x y m 1 , u = 0, 1, 2, …, v = 0, 1, 2, …Under the assumption that a ≠ b, after appropriate transformations we have:
(9)
Values of a^, b^, that minimize S(a, b) are given by the solution (roots) of the following nonlinear system of two equations (10).
3 The above function we will treat as a linear regression function of the mixture distribution while E(Yi | x) = aix + bi, i = 1, 2, are linear regression functions of the mixture distribution.
2. The least squares method for an implicit interdependence
Let (X, Y) be a two-dimensional random variable. Antoniewicz (1988) proposed an original method of approximation distribution by means of two lines, neither of which is parallel to the axis of the system. Parameters a, b, c and d of the lines minimize the following function:
. ) , , , (a EYaXc2 YbXd 2 (7)Based on the available data, the parameters a, b, c and d will be estimated. This is equivalent to finding the straight lines that give the best fit (representation) of the points in the scatter plot of the response versus the predictor variable. We estimate the parameters using the popular least squares method, which gives the lines that minimize the sum of squares of the vertical distances from each point to the lines. The vertical distances represent the errors in the response variable. The sum of squares of these distances can then be written as follows: ( , , , ) [( )( )] . 1 2
n i i i i i d x b y c x a y d c b a S (8)The values of estimators aˆ,b,ˆc,ˆdˆ which minimize S(a, b, c, d) are derived by Antoniewicz (1988). In general, values of aˆ,b,ˆc,ˆdˆ can be obtained only numerically. This problem is considered in details by Sitek (2016).
If c = d = 0, expression (8) reduces to the following: . ] ) )( [( ) , ( 1 2
n i i i i i bx y ax y b a S (9)Values of a , that minimize S(a, b) are given by the solution (roots) of the following ^, b^ nonlinear system of two equations (10).
0 2 2 0 2 2 40 2 31 22 31 2 22 13 40 2 31 22 31 2 22 13 bm a abm bm m a am m m ab abm am m b bm m (10) where
n i v i u i uv x y m 1 , u = 0, 1, 2, …, v = 0, 1, 2, …Under the assumption that a ≠ b, after appropriate transformations we have:
(10)
where
3 The above function we will treat as a linear regression function of the mixture distribution while E(Yi | x) = aix + bi, i = 1, 2, are linear regression functions of the mixture distribution.
2. The least squares method for an implicit interdependence
Let (X, Y) be a two-dimensional random variable. Antoniewicz (1988) proposed an original method of approximation distribution by means of two lines, neither of which is parallel to the axis of the system. Parameters a, b, c and d of the lines minimize the following function:
. ) , , , (a EYaXc2 YbXd 2 (7)Based on the available data, the parameters a, b, c and d will be estimated. This is equivalent to finding the straight lines that give the best fit (representation) of the points in the scatter plot of the response versus the predictor variable. We estimate the parameters using the popular least squares method, which gives the lines that minimize the sum of squares of the vertical distances from each point to the lines. The vertical distances represent the errors in the response variable. The sum of squares of these distances can then be written as follows: ( , , , ) [( )( )] . 1 2
n i i i i i d x b y c x a y d c b a S (8)The values of estimators aˆ,b,ˆc,ˆdˆ which minimize S(a, b, c, d) are derived by Antoniewicz (1988). In general, values of aˆ,b,ˆc,ˆdˆ can be obtained only numerically. This problem is considered in details by Sitek (2016).
If c = d = 0, expression (8) reduces to the following: . ] ) )( [( ) , ( 1 2
n i i i i i bx y ax y b a S (9)Values of a , that minimize S(a, b) are given by the solution (roots) of the following ^, b^ nonlinear system of two equations (10).
0 2 2 0 2 2 40 2 31 22 31 2 22 13 40 2 31 22 31 2 22 13 bm a abm bm m a am m m ab abm am m b bm m (10) where
n i v i u i uv x y m 1 , u = 0, 1, 2, …, v = 0, 1, 2, …Under the assumption that a ≠ b, after appropriate transformations we have:Under the assumption that a ≠ b, after appropriate transformations we have: ) /( 0 ) ( ) ( ) (ba m22 b2a2 m31abba m40 ba 40 31 31 22 bm m bm m a . (11) (12)
After putting the right site of equation (11) into the first equation of system (10) we obtain the following quadratic equation,
. 0 ) ( ) ( 2 22 13 31 40 13 22 31 2 31 22 40 2 m m m b m m m m m m m b (12) Next, we have:
...
( ) 4( )( 2). 22 13 31 2 31 22 40 2 40 13 22 31m m m m m m m m m m muv If Δ > 0 then there are the following two distinct roots:
, ) ( 2 ) )( ( 4 ) ( ˆ 40 22 2 31 13 31 2 22 22 40 2 31 2 40 31 22 13 40 13 31 22 1 m m m m m m mm mm m m m m m m m b (13) . ) ( 2 ) )( ( 4 ) ( ˆ 40 22 2 31 13 31 2 22 22 40 2 31 2 40 31 22 13 40 13 31 22 2 m m m m m m m m m m m m m m m m m b (14)
Substituting the results into the equation (12), we get: . ˆ ˆ ˆ 40 1 31 31 1 22 1 m b m m b m a
Using the well-known Vieta’s formulas, it can be shown that a : ˆ b1 ˆ2 ), ˆ ( / ˆ ˆ ˆ 40 1 31 2 40 1 31 31 1 22 b m bm m b m m b m ), ˆ ( ˆ ˆ 40 1 31 2 31 1 22 bm b m bm m , ˆ ˆ ˆ ˆ 40 1 2 31 2 31 1 22 bm b m bbm m . ) ˆ ˆ ( ˆ ˆ 31 2 1 40 1 2 22 bbm b b m m
Let us introduce notation:
231 22 40m m m m A A uv , BB
muv m31m22m13m40,
2 22 13 31m m m m C C uv .Hence, from Viete’s formula we have:
) /( 0 ) ( ) ( ) ( 2 2 31 40 22 b a m ab b a m b a m a b 40 31 31 22 bm m bm m a . (11) (12)
After putting the right site of equation (11) into the first equation of system (10) we obtain the following quadratic equation,
. 0 ) ( ) ( 2 22 13 31 40 13 22 31 2 31 22 40 2 m m m bm m m m m m m b (12) Next, we have:
...
( ) 4( )( 2). 22 13 31 2 31 22 40 2 40 13 22 31m m m m m m m m m m muv If Δ > 0 then there are the following two distinct roots:
, ) ( 2 ) )( ( 4 ) ( ˆ 40 22 2 31 13 31 2 22 22 40 2 31 2 40 31 22 13 40 13 31 22 1 m m m m m m m m m m m m m m m m m b (13) . ) ( 2 ) )( ( 4 ) ( ˆ 40 22 2 31 13 31 2 22 22 40 2 31 2 40 31 22 13 40 13 31 22 2 m m m m m m m m m m m m m m m m m b (14)
Substituting the results into the equation (12), we get: . ˆ ˆ ˆ 40 1 31 31 1 22 1 m b m m b m a
Using the well-known Vieta’s formulas, it can be shown that a : ˆ b1 ˆ2 ), ˆ ( / ˆ ˆ ˆ 40 1 31 2 40 1 31 31 1 22 b m bm m b m m b m ), ˆ ( ˆ ˆ1 31 2 31 1 40 22 bm b m bm m , ˆ ˆ ˆ ˆ1 31 2 31 2 1 40 22 bm b m bbm m . ) ˆ ˆ ( ˆ ˆ 31 2 1 40 1 2 22 bbm b b m m
Let us introduce notation:
231 22 40m m m m A A uv , BB
muv m31m22m13m40,
2 22 13 31m m m m C C uv .Hence, from Viete’s formula we have:
(11) After putting the right site of equation (11) into the first equation of system (10) we obtain the following quadratic equation,
4 ) /( 0 ) ( ) ( ) ( 2 2 31 40 22 b a m abb a m b a m a b 40 31 31 22 bm m bm m a . (11) (12)
After putting the right site of equation (11) into the first equation of system (10) we obtain the following quadratic equation,
. 0 ) ( ) ( 2 22 13 31 40 13 22 31 2 31 22 40 2 m m m b m m m m m m m b (12) Next, we have:
...
( ) 4( )( 2). 22 13 31 2 31 22 40 2 40 13 22 31m m m m m m m m m m muv If Δ > 0 then there are the following two distinct roots:
, ) ( 2 ) )( ( 4 ) ( ˆ 40 22 2 31 13 31 2 22 22 40 2 31 2 40 31 22 13 40 13 31 22 1 m m m m m m m m m m m m m m m m m b (13) . ) ( 2 ) )( ( 4 ) ( ˆ 40 22 2 31 13 31 2 22 22 40 2 31 2 40 31 22 13 40 13 31 22 2 m m m m m m m m m m m m m m m m m b (14)
Substituting the results into the equation (12), we get: . ˆ ˆ ˆ 40 1 31 31 1 22 1 m b m m b m a
Using the well-known Vieta’s formulas, it can be shown that a : ˆ b1 ˆ2 ), ˆ ( / ˆ ˆ ˆ 40 1 31 2 40 1 31 31 1 22 b m bm m b m m b m ), ˆ ( ˆ ˆ1 31 2 31 1 40 22 bm b m bm m , ˆ ˆ ˆ ˆ1 31 2 31 2 1 40 22 bm b m bbm m . ) ˆ ˆ ( ˆ ˆ 31 2 1 40 1 2 22 bbm b b m m
Let us introduce notation:
231 22 40m m m m A A uv , BB
muv m31m22m13m40,
2 22 13 31m m m m C C uv .Hence, from Viete’s formula we have:
(12) Next, we have: 4 ) /( 0 ) ( ) ( ) (ba m22 b2a2 m31abba m40 ba 40 31 31 22 bm m bm m a . (11) (12)
After putting the right site of equation (11) into the first equation of system (10) we obtain the following quadratic equation,
. 0 ) ( ) ( 2 22 13 31 40 13 22 31 2 31 22 40 2 m m m b m m m m m m m b (12) Next, we have: ). )( ( 4 ) ( 2 22 13 31 2 31 22 40 2 40 13 22 31m m m m m m m m m m
If Δ > 0 then there are the following two distinct roots:
, ) ( 2 ˆ 40 22 2 31 40 13 31 22 1 m m m m m m m b (13) . ) ( 2 ˆ 40 22 2 31 40 13 31 22 2 m mm mmmm b (14)
Substituting the results into the equation (12), we get: . ˆ ˆ ˆ 40 1 31 31 1 22 1 m b m m b m a
Using the well-known Vieta’s formulas, it can be shown that a : ˆ1 bˆ2 ), ˆ ( / ˆ ˆ ˆ 40 1 31 2 40 1 31 31 1 22 b m bm m b m m b m ), ˆ ( ˆ ˆ1 31 2 31 1 40 22 bm b m bm m , ˆ ˆ ˆ ˆ1 31 2 31 2 1 40 22 bm b m bbm m . ) ˆ ˆ ( ˆ ˆ2 1 40 1 2 31 22 bbm b b m m
Let us introduce notation: 2
31 22
40m m
m
A , Bm31m22m13m40, Cm31m13m222 . Hence, from Viete’s formula we have:
If Δ > 0 then there are the following two distinct roots:
4 ) /( 0 ) ( ) ( ) (ba m22 b2a2 m31abba m40 ba 40 31 31 22 bm m bm m a . (11) (12)
After putting the right site of equation (11) into the first equation of system (10) we obtain the following quadratic equation,
. 0 ) ( ) ( 2 22 13 31 40 13 22 31 2 31 22 40 2 m m m b m m m m m m m b (12) Next, we have: ). )( ( 4 ) ( 2 22 13 31 2 31 22 40 2 40 13 22 31m m m m m m m m m m
If Δ > 0 then there are the following two distinct roots:
, ) ( 2 ˆ 40 22 2 31 40 13 31 22 1 m mm mmmm b (13) . ) ( 2 ˆ 40 22 2 31 40 13 31 22 2 m mm mmmm b (14)
Substituting the results into the equation (12), we get: . ˆ ˆ ˆ 40 1 31 31 1 22 1 m b m m b m a
Using the well-known Vieta’s formulas, it can be shown that a : ˆ1 bˆ2 ), ˆ ( / ˆ ˆ ˆ 40 1 31 2 40 1 31 31 1 22 b m bm m b m m b m ), ˆ ( ˆ ˆ1 31 2 31 1 40 22 bm b m bm m , ˆ ˆ ˆ ˆ1 31 2 31 2 1 40 22 bm b m bbm m . ) ˆ ˆ ( ˆ ˆ2 1 40 1 2 31 22 bbm b b m m
Let us introduce notation: 2
31 22
40m m
m
A , Bm31m22m13m40, Cm31m13m222 . Hence, from Viete’s formula we have:
(13) 4 ) /( 0 ) ( ) ( ) (ba m22 b2a2 m31abba m40 ba 40 31 31 22 bm m bm m a . (11) (12)
After putting the right site of equation (11) into the first equation of system (10) we obtain the following quadratic equation,
. 0 ) ( ) ( 2 22 13 31 40 13 22 31 2 31 22 40 2 m m m b m m m m m m m b (12) Next, we have: ). )( ( 4 ) ( 2 22 13 31 2 31 22 40 2 40 13 22 31m m m m m m m m m m
If Δ > 0 then there are the following two distinct roots:
, ) ( 2 ˆ 40 22 2 31 40 13 31 22 1 m m m m m m m b (13) . ) ( 2 ˆ 40 22 2 31 40 13 31 22 2 m mm mmmm b (14)
Substituting the results into the equation (12), we get: . ˆ ˆ ˆ 40 1 31 31 1 22 1 m b m m b m a
Using the well-known Vieta’s formulas, it can be shown that a : ˆ1 bˆ2
), ˆ ( / ˆ ˆ ˆ 40 1 31 2 40 1 31 31 1 22 b m bm m b m m b m ), ˆ ( ˆ ˆ1 31 2 31 1 40 22 bm b m bm m , ˆ ˆ ˆ ˆ1 31 2 31 2 1 40 22 bm b m bbm m . ) ˆ ˆ ( ˆ ˆ2 1 40 1 2 31 22 bbm b b m m
Let us introduce notation: 2
31 22
40m m
m
A , Bm31m22m13m40, Cm31m13m222 . Hence, from Viete’s formula we have:
