Alternative parameterizations of a statistical model

Question

I'm currently learning about the basics of a statistical model and reparametrization through a textbook. In one of the examples, it says that a location scale normal model is usually labelled as (mu, sigma^2). However, it says that it can also be parametrized by (mu, sigma) or (mu, sigma^-2), etc.

Wouldn't this relabeling produce a different result (i.e. different graph of the density function? This "reparametrization" is very confusing for me, if someone could help me understand it I would appreciate it. Thanks.

Image of the textbook example: http://puu.sh/CT5V6/a5c6ea9b5c.png

Answer 1

Some authors follow the convention of using location and scale parameters to specify distributions. For the normal distribution that would mean using $\mathsf{Norm}(\mu, \sigma).$ So if you use the variance $\sigma^2$ or the precision $\tau = \sigma^{-2}$ instead of $\sigma,$ then you are correct that the graphs will look somewhat different.

You are also correct that this causes confusion--especially among people who do not always check to see what parameterization is used in a particular discussion. (But now you have been warned, so you will not be one of those.)

In Bayesian statistics, it is common to use the precision, because it simplifies many useful formulas.

The exponential distribution can be parameterized either by its rate (scale) or its mean (not a true location parameter for an exponential distribution). Using $\lambda$ for rate and $\mu$ for mean, the density function of an exponential distribution would be $f(x) = \lambda e^{-\lambda x}$ or $g(x) = \frac 1 \mu e^{-x/\mu},$ for $x > 0.$

In an application of an exponential distribution one may focus on the exponential mean $\mu$ or on the rate at which the relevant Poisson events occur (where $\lambda$ is the Poisson mean and $1/\lambda$ is the exponential mean).

Authors seem to choose which parameterization to use in a particular textbook or article, depending on (a) whatever choice will be most familiar to the reader, (b) whatever choice makes explanations or formulas easier to write, (c) the whim of the moment.

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In R statistical software, the normal distribution is parameterized according to the mean and the standard deviation, and the exponential distribution (and related gamma distributions) according to the rate.

Thus, the code below produces the accompanying graph of a particular exponential density function.

curve(dexp(x, .1), 0, 50, ylab="PDF", lwd=2)
  abline(v=0, col ="green2")
  abline(h=0, col="green2")

Notice that the value at $f(0) = \lambda = 0.1$ and the mean is $\mu = 1/\lambda = 10.$

Similarly, the following plot of the density function of $\mathsf{Norm}(\mu = 100,\, \sigma = 20)$ has vertical dotted lines at the inflection points $\mu \pm \sigma$ of the density curve.

curve(dnorm(x, 100, 20), 30, 170, lwd=2, ylab="PDF")
   abline(h = 0, col="green2")
   abline(v = c(80, 120), col="red", lty="dotted")