# Can’t we all use the same Parameterizations of Distributions?

This is something that’s been bothering me for a while in statistics.

$f(x|\lambda)= \lambda e^{-\lambda x} \hspace{3mm} \text{or} \hspace{3mm} f(x|\beta)=\frac{1}{\beta} e^{-x/\beta}$

depending on who you ask. Obviously they are equivalent with $\lambda = \frac{1}{\beta}$ but the discrepancies in other distribution parameterizations have wide ranging effects in statistics.

I’ll be the first to admit, I’ve accidentally used the wrong parameterization several times for the gamma functions [rgamma(), dgamma(), etc…] in R.  They use the parameterization on the right.

$f(x|\alpha,\beta)= \frac{1}{\Gamma(\alpha) \beta^\alpha } x^{\alpha-1} e^{-x/\beta} \hspace{3mm} \text{or} \hspace{3mm} f(x|\alpha,\theta)=\frac{\theta^\alpha}{\Gamma(\alpha)} x^{\alpha -1} e^{-x \theta}$

but the left parameterization is the way that the Gamma distribution is introduced in several textbooks.

Now luckily I noticed that something was going majorly awry in my Bayesian MCMC when I used rgamma() with the intent of using the left parameterization and corrected to the right parameterization but can you imagine how many people accidentally used the wrong density functions in their analysis?

Wikipedia gives a pretty good summary of some of the reasons for the different parameterizations, I know I’ve tended to stick to the parameterization with $\theta$ but I learned the $\beta$ parameterization in the classic Casella Berger. We know that $GAMMA(p/2,2) \sim \chi_p^2$ under this parameterization, but it is a $GAMMA(p/2,1/2)$ under the alternative parameterization. Is that much of a difference? I don’t think so.

As a Bayesian who uses CRAN a lot (I’m sure many of you also use dgamma) I say we should just consolidate to the parameterization with $(\alpha, \theta)$ instead of $(\alpha,\beta)$. The one thing that’s lacking is the relationship between the Gamma and Weibull distributions.

But then again, Weibull people (Reliability and Survival Gurus) instead of using this parameterization

$f(x| k, \lambda) = \frac{k}{\lambda} \left(\frac{x}{\lambda} \right)^{k-1} e^{-(x/\lambda)^k}$

$f(x| k, \theta) = k \theta^k x^{k-1} e^{-\theta^k x^k}$

and the $(\alpha, \theta)$ parameterization of the Gamma distribution? (Also PLEASE don’t use this one: $f(x|b,k)=bkx^{k-1} \exp [-bx^k]$). It also doesn’t help things that out of the 100 or so survival models available on CRAN, they use all three of these different parameterizations when a Weibull hazard is desired.

So can we all consolidate and agree to use the same parameterizations? [i.e. let’s all use  $(\alpha, \theta)$ for the Gamma, $\lambda$ for the exponential and $(k,\theta)$ for the Weibull distributions]

Then we’ll still have a functional relationship between the Gamma, Weibull, Chi-Squared and Exponential distributions, but we won’t have to deal with those silly denominators or even God forbid, having to specify a rate parameter in CRAN instead of using the default.

Thanks.

-Andrew G Chapple

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