## Bayesian D-optimal Designs for the Exponential Growth Model (1994)

Venue: | J. STATIST. PLAN. INF |

Citations: | 8 - 1 self |

### BibTeX

@ARTICLE{Mukhopadhyay94bayesiand-optimal,

author = {Saurabh Mukhopadhyay and Linda M. Haines},

title = {Bayesian D-optimal Designs for the Exponential Growth Model},

journal = {J. STATIST. PLAN. INF},

year = {1994},

volume = {44},

pages = {385--397}

}

### OpenURL

### Abstract

Bayesian optimal designs for nonlinear regression models are of some interest and importance in the statistical literature. Numerical methods for their construction are well-established, but very few analytical studies have been reported. In this paper, we consider an exponential growth model used extensively in the modelling of simple organisms, and examine the explicit form of the Bayesian D-optimal designs. In particular, we show that D ` -optimal designs for this model are balanced two--point designs for all values of the parameters. We further derive explicit expressions for Bayesian D-optimal designs which are based on exactly two points of support, and provide necessary and sufficient conditions for such designs to exist. We illustrate our results by means of two examples.

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(Show Context)
Citation Context ...standing of the nature of these designs. In particular, we consider the exponential growth model which is commonly used to describe the growth of simple organisms (see Richards, 1969; Sandland, 1983; =-=Seber and Wild, 1989-=-, pp. 327--328). While this growth curve and its attendant optimal designs are of interest in their own right, the techniques used here for design construction can be readily extended to other nonline... |

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Citation Context ...Lauter, 1974, 1976), or, as suggested by Chaloner and Larntz (1989), it can be interpreted as an approximation to the expected increase in the Shannon information provided by the experiment (see also =-=Lindley, 1956-=-; Bernardo, 1979). We use the general Equivalence Theorem, introduced by Whittle (1973) for linear models, and modified by Chaloner and Larntz (1989) to accommodate nonlinear models, to verify the Bay... |

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Citation Context ...976), or, as suggested by Chaloner and Larntz (1989), it can be interpreted as an approximation to the expected increase in the Shannon information provided by the experiment (see also Lindley, 1956; =-=Bernardo, 1979-=-). We use the general Equivalence Theorem, introduced by Whittle (1973) for linear models, and modified by Chaloner and Larntz (1989) to accommodate nonlinear models, to verify the Bayesian D--optimal... |

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Citation Context ...on, depend on the unknown parameters, `. One approach to this problem is to adopt a nominal value for ` and to determine the locally '--optimal or ' ` --optimal design for that particular value of ` (=-=Chernoff, 1953-=-). However, it is quite possible for a design which is optimal for some fixed value of ` to be completely unreasonable for other values of `. A natural approach to overcoming this difficulty is to int... |

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Citation Context ...xpressions for Bayesian D--optimal designs based on more than two support points are, as yet, available, and that in such cases recourse must be made to numerical methods for design construction (see =-=Chaloner and Larntz, 1989-=-). Finally in Subsection 4.2, we consider two specific classes of prior distribution on the parameters of the exponential growth model, viz. those of the gamma and the uniform on the scale parameter, ... |

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Citation Context ...) where w(`; x) = e \Gammafl (x\Gammaffi) . Further, we consider approximate designs, which are probability measures, ��, on the Borel field of the design space, X , as candidate optimal designs (=-=see Kiefer and Wolfowitz, 1959), and denote th-=-e set of all such designs, ��, by \Xi. Then the information matrix of ` for a design, �� 2 \Xi, is given by I(`; ��) = Z X I(`; x) ��(dx): (3) 2.2 The criterion To ensure that ` is est... |

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Citation Context ... direction of a single design point, x 2 [a; a + b], and which is given, for Bayesian D--optimality, by d(��; x) = E �� h trI(`; x)I(`; ��) \Gamma1 i \Gamma 2 (4) for some prior distributi=-=on, �� (see Silvey, 1980, pp. 20--21-=-). It then follows from the Equivalence Theorem for nonlinear models given by Chaloner and Larntz (1989), that for any fixed prior, ��, a design, �� ? �� , is Bayesian D--optimal if and on... |

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Citation Context ...s referred to as D ` --optimality. The criterion of Bayesian D--optimality can be regarded, quite simply, as reflecting the experimenter's prior belief in the parameter values with respect to design (=-=Lauter, 1974-=-, 1976), or, as suggested by Chaloner and Larntz (1989), it can be interpreted as an approximation to the expected increase in the Shannon information provided by the experiment (see also Lindley, 195... |

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Citation Context ...86) that, for any fixed `, the D ` --optimal design is based on exactly two points of support, and further that it puts equal masses on those points, i.e. it is a balanced two--point design (see also =-=Haines, 1992). Now f-=-or any balanced two--point design, ��, with support points, x 1 and x 2 such that asx 1 ! x 2sa + b, straightforward calculations show that det I(`; ��) / fl 2 w(`; x 1 ) w(`; x 2 ) (x 2 \Gamm... |

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The quantitative analysis of growth," in Plant Physiology, Volume V A: Analysis of Growth: Behavior of Plants and their
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(Show Context)
Citation Context ...we aim to provide a deeper understanding of the nature of these designs. In particular, we consider the exponential growth model which is commonly used to describe the growth of simple organisms (see =-=Richards, 1969-=-; Sandland, 1983; Seber and Wild, 1989, pp. 327--328). While this growth curve and its attendant optimal designs are of interest in their own right, the techniques used here for design construction ca... |