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# 澳洲代考|广义线性模型代考Generalized linear model代考|Optimality

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## 澳洲代考|广义线性模型代考Generalized linear model代考|Number of support points

In seeking an optimal design, an important question is how many support points are needed?
The minimum number is $p$, the number of parameters. This is intuitively obvious. At least two separate points are needed to estimate the two parameters, $\beta_{0}$ and $\beta_{1}$, of a straight line, and at least three separate points are required to estimate the three parameters, $\beta_{0}, \beta_{1}$ and $\beta_{2}$, of a quadratic. See Figure $3.2$ to recall the situation that arises when there are only two points for a quadratic.
The maximum number of support points necessary for an optimal design can be shown to be $p(p+1) / 2+1$, using Carathéodory’s Theorem. The proof is beyond the scope of this book, but is given in Rockafellar (1970, p. 155) or Silvey (1980, p. 77). Pukelsheim (1993, p. 190) showed that, if our interest is in estimating all elements of the parameter vector $\boldsymbol{\beta}$, then this upper limit can be reduced to $p(p+1) / 2$.

## 澳洲代考|广义线性模型代考Generalized linear model代考|Optimality criteria

To select a design that is “optimal,” you need to decide what criteria the design must satisfy for you to consider it optimal. There are numerous criteria in common use, and a design that is optimal under some criteria need not necessarily be optimal under others. (If you have ever looked at two people in a relationship and wondered what one of them sees in the other, then the previous sentence should not surprise you at all.)

Most criteria come under the heading of “alphabet optimality” because the criterion’s name is a letter from the alphabet (e.g., D-optimality). An extensive list of alphabetical optimality criteria are considered in Atkinson \& Donev (1992, Chapter 10). That discussion relates to designs under the general linear model, and some minor adjustments would need to be made when considering criteria applying to GLMs.

Table $3.2$ provides a list of five commonly used alphabet criteria. There are numerous other criteria that are used less frequently.
Optimality
Criterion The criterion seeks a design that minimises the
A- $\quad$ average of the variances of the parameter estimates
D- volume of a confidence hyperellipsoid for the parameters
$\quad$ Ds- $\quad$ volume of a confidence hyperellipsoid for a subset of the parameters, taking into account the presence of the remaining parameters

E- maximum variance of a linear combination of parameter estimates, $\boldsymbol{a}^{\top} \hat{\boldsymbol{\beta}}$, where $\boldsymbol{a}^{\top} \boldsymbol{a}=1$
G- $\quad$ maximum value of the standardised variance that occurs at any point in the design space
Table $3.2$ Common alphabet optimality criteria.

## 澳洲代考|广义线性模型代考GENERALIZED LINEAR MODEL代考|OPTIMALITY CRITERIA

Optimality
Criterion 该标准寻求一种设计，使
A-参数估计的方差的平均值
D- 参数的置信度超椭球体的体积
Ds-考虑到剩余参数的存在，参数子集的置信度超椭圆体的体积

E- 参数估计的线性组合的最大方差，一个⊤b^， 在哪里一个⊤一个=1
G-设计空间中任意点出现的标准化方差的最大值

## Matlab代写

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