Abstract
In this paper we study the problem of homothety’s influence on the number of optimal design support points under fixed values of a regression model’s parameters. The Cobb–Douglas two-dimensional nonlinear in parameters model used in microeconomics is considered. There exist two types of optimal designs: saturated (i.e. design with the number support points equal to the number of parameters) and excess design (i.e. design with greater number of support points). The optimal designs with the minimal number of support points are constructed explicitly. Numerical methods for constructing designs with greater number of points are used.
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Atkinson AC, Donev AN, Khuri AI, Mukerjee B, Sinha BK, Tobias RD (2007) Optimum experimental design, with SAS. Oxford University Press, Oxford
Berger M, Pansu P, Berry J-P, Saint-Raymond X (1984) Problems in geometry. Springer, New York
Chaloner K, Larntz K (1989) Optimal bayesian experimental design applied to logistic regression experiments. J Stat Plan Inference 21:191–208
de la Garza A (1954) Spacing of information in polynomial regression. Ann Math Stat 25:123–130
Dette H, Melas B (2011) A note on the de la Garza phenomenon for locally optimal designs. Ann Stat 39(2):1266–1281
Douglas PH (1976) The Cobb-Douglas production function once again: its history, its testing, and some new empirical values. J Polit Econ 84(5):903–916
Ermakov SM, Kulikov DV, Leora SN (2017) Towards the analysis of the simulated annealing method in the multiextremal case. Vestn St. Petersbg Univ 50(2):132–137
Fedorov VV (1972) Theory of optimal experiment. Academic Press, New York
Filipe J, Adams FG (2005) The estimation of the Cobb-Douglas function: a retrospective view. East Econ J 31(3):427–44
Grigoriev YuD, Denisov VI (1973) The solution of a nonlinear problem of experimental regression analysis [in Russian]. The computer applying in planning and designing, collection of proceedings, Novosibirsk electrotechnical Institute, Novosibirsk
Grigoriev YuD, Melas VB, Shpilev PV (2017) Excess of locally \(D\)-optimal designs and homothetic transformations. Vestn St. Petersb Univ 50(4):329–336
Heady EO, Dillon JL (1961) Agricultural production functions. Iowa State University Press, Ames
Khuri AI, Mukerjee B, Sinha BK, Ghosh M (2006) Design issues for generalized linear models: a review. Stat Sci 21(3):376–399
Melas VB (2006) Functional approach to optimal experimental design. Springer, New York
Pukelsheim F (2006) Optimal design of experiments. SIAM, Philadelphia
Whittle P (1973) Some general points in the theory of optimal experimental design. J R Stat Soc Ser B 35:123–130
Yang M (2010) On the de la Garza phenomenon. Ann Stat 38:2499–2524
Yang M, Stufken J (2009) Support points of locally optimal designs for nonlinear models with two parameters. Ann Stat 37:518–541
Yang M, Stufken J (2012) Identifying locally optimal designs for nonlinear models: a simple extension with profound consequences. Ann Stat 40(3):1665–1681
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This work was partially supported by Russian Foundation for Basic Research (Projects Nos. 17-01-00267-a, 17-01-00161-a)
Appendix
Appendix
Throughout this section without losing generality, we can assume that \(b_1=b_2=\gamma =1\). Also, due to the fact that D-optimal design for model (3.2) doesn’t depend on the parameter \(\theta _0\), we put \(\theta _0=1.\)
Lemma 1
Under the assumptions of this section a saturated D-optimal design for model (3.2) is concentrated at points
Proof of Lemma 1
Note that by Theorem 1 D-optimality of the design \(\xi ^*\) is equivalent to
For model (3.2) we have \(p=3\) and function \(d((x_1,x_2),\xi )\) has a form:
where \(d_{ij}\) are elements of the matrix \(M^{-1}(\xi )\). An analysis of this function shows that it has no more than 4 maxima which are concentrated at points
Indeed, for any given non-singular design \(\xi \) and fixed \(x_1=x_1^{*}\in [0,1]\) function \(d((x_1^{*},x_2),\xi )\) has two local extrema at the point \(x_2=0\) and at some point \(x_2=x_2^{*}\in (0,1]\) on the interval [0, 1]. This immediately implies that function \(d((x_1,x_2),\xi )\) has maxima at points of the form (4.2). It remains to show that the number of global maxima is less than or equal to 4. To do this, we prove that the function \(d((x_1,x_2),\xi )\) can not have two global maxima at points of the form \(\breve{D}\) when \(\beta _i>0\). We prove it by reductio ad absurdum. To be more specific let
Consider the line \(x_2=ax_1+b\) that passes through the points \(X^{*}_i.\) We have
After the appropriate replacements, we obtain
This function has no more than 2 global maxima at the points \(x_1=0\) and \(x_1=x^{*}_1\in (0,1]\) on the interval [0, 1]. We have obtained a contradiction. Thus, we have proved that the saturated D-optimal design is concentrated at points
Let us prove, by contradiction, that if \(\overline{\xi }\) is optimal, then \(\beta _1=\beta _2=0\). Consider all possible combinations of sets of parameter’s values \(\{\alpha _1,\alpha _2,\beta _1,\beta _2\}\). Let’s start with the case \(\{0<\alpha _1<1,0<\alpha _2<1,0<\beta _1<1, 0<\beta _2<1\}\). Due to the necessary optimality conditions of the design \(\xi \) the unknown variables \(\alpha _i\) and \(\beta _i\) must satisfy the system of equations
This system has a unique solution: \(\alpha _1 = \frac{1}{3\theta _1}, \alpha _2 = \frac{1}{3\theta _2}, \beta _1 = \frac{2}{3\theta _1}, \beta _2 = \frac{2}{3\theta _2}.\) A direct calculation shows that for such \(\alpha _i\) and \(\beta _i\) we have
i. e. the design \(\overline{\xi }\) is not optimal by Theorem 1. The remaining cases are also checked in a similar way. For example, for \(\{\alpha _1=1,\alpha _2=1,\beta _1=1, \beta _2=1\}\) we have
Thus, we have proved that the saturated D-optimal design is concentrated at points 4.1
Lemma 1 is proved. \(\square \)
Proof of Theorem 2
Optimality of designs (3.6)–(3.4) is verified directly by Theorem 1. For example, for the design \(\overline{\xi }\) in form (3.6) we have
On the design space \(\mathcal {X}\), this function has a local minimum point \(\left( \frac{2}{(2+e^{2})\theta _1}, \frac{2}{(2+e^{2})\theta _2}\right) \), saddle point \(\left( \frac{1}{2\theta _1}, \frac{1}{2\theta _2}\right) \) and the global minimum point \(\left( 1, 1\right) \) and takes the following values at these points, correspondingly:
The analysis of the last expression allows us to conclude that \(d\left( \left( 1, 1\right) ,\overline{\xi }\right) \) reaches maximum when \(\theta _1=\theta _2=1:\)
The function \(d((t_1,t_2),\overline{\xi })\) reaches maximum at points \(A,\ B^\prime ,\ C^\prime \):
The remaining cases are verified in a similar way.
The behavior of the function \(d((t_1,t_2),\overline{\xi })\) for \(\theta _1=\theta _2=1\), \(\mathcal {X}=[0,\frac{3}{2}]\times [0,\frac{3}{2}]\) is depicted in Fig. 4.
The only thing we need to check now is that sets \(\varLambda _3\) and \(\varLambda _4\) are defined in a proper way. Let’s start with part (a). It follows from the continuity of the function \(d((t_1,t_2),\overline{\xi })\) that these sets have a common boundary and there exist two types of the optimal designs for the parameters belonging to the boundary:
such that
It follows from the previous discussion that the function \(d(D^{\prime \prime },\xi ^{(II)}_{opt})=d((t_2,1),\xi ^{(II)}_{opt})\) has no more than one maxima on the interval (0, 1].
Note that for fixed \(t_2\) function \(d(D^{\prime \prime },\xi ^{(II)}_{opt})=d((t_2,1),\xi ^{(II)}_{opt})\) is a monotonic function by \(\theta _1\) since
Thus by the implicit function theorem the following system has a unique real solution, say, \(\theta _2=\varPsi _1( \theta _1)\)
as well as the equation \(d((1,1),\xi ^{(II)}_{opt})=3\) has a unique real solution, say, \(\theta _2=\varPsi _2( \theta _1)\). Functions \(\varPsi _1\) and \(\varPsi _2\) can be obtained by solving of corresponding equations. Now the results of part (a) immediately follows from the conditions \(\theta _2 \le \theta _1\), \(t_2\le 1\) and the fact that the boundary function is \(\varPsi (\theta _1)=\min (\theta _1,\max (\varPsi _1(\theta _1),\varPsi _2(\theta _1))).\)
Parts (b) and (c) are verified in a similar way.
Theorem 2 is proved. \(\square \)
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Grigoriev, Y.D., Melas, V.B. & Shpilev, P.V. Excess of locally D-optimal designs for Cobb–Douglas model. Stat Papers 59, 1425–1439 (2018). https://doi.org/10.1007/s00362-018-1022-0
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DOI: https://doi.org/10.1007/s00362-018-1022-0