Nonlinear regression

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Problem 15.1 from Горяинова Е.Р., Панков А.Р., Платонов Е.Н. "Прикладные методы анализа статистических данных" [ГПП12]More problems in Chapter 15 "Nonlinear Regression Models"

Using the data of Table 15.4, estimate the parameters of the Cobb-Douglas production model [6] by the method of reducing the model to a linear one:

Table 15.4 (the data are given as percentages of the base year 1960, i=11i = 11)

iiYiY_{i}KiK_{i}LiL_{i}iiYiY_{i}KiK_{i}LiL_{i}
133,1533,7779,9211100,00100,00100,00
238,2037,5984,5212109,56111,5999,54
342,5642,3587,6013120,27123,93102,81
447,3546,9691,4114131,21138,10106,44
553,4652,1895,8315141,78154,31110,91
660,3458,6497,6116153,41170,42115,65
766,3365,6496,3217167,09186,13118,66
873,0172,5096,6018183,65201,73122,55
981,2980,4599,1419199,91217,68126,62
1090,5689,67100,6620215,20235,10130,61
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(1)

Y(K,L)=AeλtKαL1−αY(K, L) = A e^{\lambda t} K^{\alpha } L^{1-\alpha };

(2)

Y(K,L)=AeλtKαLβY(K, L) = A e^{\lambda t} K^{\alpha } L^{\beta }.

Problem 15.2 from Горяинова Е.Р., Панков А.Р., Платонов Е.Н. "Прикладные методы анализа статистических данных" [ГПП12]More problems in Chapter 15 "Nonlinear Regression Models"

Using the data of Table 15.1, estimate the parameters of the model

y(x)=θ1⋅θ2x y(x) = \theta _{1} \cdot \theta _{2}^{x}

Table 15.1

kk12345678910
xkx_{k}455681087116
yky_{k}1,52,01,42,32,74,02,32,56,61,7

by the Gauss-Newton method. Compare with the results of Example 15.1.

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Problem 15.3 from Горяинова Е.Р., Панков А.Р., Платонов Е.Н. "Прикладные методы анализа статистических данных" [ГПП12]More problems in Chapter 15 "Nonlinear Regression Models"

Under the conditions of Example 15.2, find exact expressions for the components of the vector Fi(m)F_{i}^{(m)}.

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Problem 15.4 from Горяинова Е.Р., Панков А.Р., Платонов Е.Н. "Прикладные методы анализа статистических данных" [ГПП12]More problems in Chapter 15 "Nonlinear Regression Models"

The dependence of the variable yy on xx has the form of a rational function

y(x)=θ1+θ2xθ3+θ4x. y(x) = \frac{\theta _{1}+\theta _{2} x}{\theta _{3}+\theta _{4} x} .

Develop an algorithm for estimating the parameters {θ1,…,θ4}⊤\left\{ \theta_{1}, \ldots , \theta_{4}\right\}^{\top } from the observations

yk=y(xk;θ)+εk,k=1,…,n y_{k} = y\left(x_{k} ; \theta \right)+\varepsilon _{k}, \quad k = 1, \ldots , n

by the Gauss-Newton method.

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Problem 6.4 from Seber G.A., Lee A.J. "Linear Regression Analysis" [SeLe03]More problems in Chapter 6 "Straight-Line Regression"

Show how to transform the following equation into a straight line so that α\alpha and β\beta can be estimated by least squares:

y=αβαsin⁡2θ+βcos⁡2θ. y = \frac{\alpha \beta }{\alpha \sin ^{2} \theta +\beta \cos ^{2} \theta } .
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References

[ГПП12] Горяинова Е.Р., Панков А.Р., Платонов Е.Н. Прикладные методы анализа статистических данных. Издательский дом Высшей школы экономики, 2012.
[SeLe03] Seber G.A., Lee A.J. Linear Regression Analysis. 2nd ed., John Wiley & Sons, 2003.