# Show statements for error residual and Taylors Law

I have some data from a lake. "Counts" is the response variable and is a number for the organism in the lake. I also have a variable "stations" that indicate every parts of the lake (there are 45 stations).

It can be assumed that: $$Y_i=\mu_i+\epsilon_i$$ with $$E\epsilon_i=0$$, $$V \epsilon_i=\psi V(\mu_i)$$

and $$\mu_i=\beta_{stations}$$ being a station specific station mean value. It will makes to use normal distribution as response distribution.

Then I be introduced to a new law the Taylor Law: It states that there is a power law relationship between mean and variance when counting organism. And specifically the variance function is given by: $$V(\mu)=\mu^b$$ for some power $$b≥0$$. And $$\xi_i=E\epsilon_i^2$$

Then I have to show that if the Taylors law holds then $$log(\xi_i)=log(\psi)+blog(\mu_i)$$

And I have to show that if $$\epsilon_i$$ is normally distributed then $$V \epsilon_i^2=2 \xi_i^2$$.

I understand it most, intuitively. The $$\epsilon$$ is the residual error and $$\mu$$ is the mean and then there is a relationship between the mean and variance for some unknown $$b≥0$$. But I'm a bit confused what $$\xi$$ and $$\psi$$ is and and how I should gather the information to show these two statements. I hope anyone can help me?

• Is it this Taylor's law? Commented Sep 30, 2021 at 18:19

Then I have to show that if the Taylors law holds then $$log(\xi_i)=log(\psi)+blog(\mu_i)$$

You have this equation

$$V\epsilon_i = \psi V(\mu_i)$$

and since $$V(\mu_i) = \mu_i^b$$ you get

$$V\epsilon_i = \psi \mu_i^b$$

if you take the logarithm

$$\log V\epsilon_i = \log \psi + b \, \log \mu_i$$

which is the equation that follows from Taylor's law. And $$V\epsilon_i = E[\epsilon_i^2] = \xi_i$$ since $$E[\epsilon_i] = 0$$ by assumption.

But I'm a bit confused what $$\xi$$ and $$\psi$$

The variable $$\xi_i$$ is equal to $$E[\epsilon_i^2]$$. The variable $$\psi$$ is this parameter in your first equation.

And I have to show that if $$\epsilon_i$$ is normally distributed then $$V \epsilon_i^2=2 \xi_i^2$$.

All these Greek letters make it complicating. What is asked is to show that if $$X$$ is normal distributed (and with mean zero), then what is the variance of $$X$$? Hint: the distribution of $$X$$ is a gamma distribution (or can be seen as a scaled chi-squared distribution).