If $X$ and $Y$ are two NON independent random normal variables, what is the distribution of $Z = \frac{X}{Y^n}$

Question

I'm working on the BMI (body mass index) indicator ($weight/height^2$) and as now it is not optimal because the correlation between height and BMI is not minimal (let's not disscuss the criterias of optimality here, thats another debate). I'm looking into the exponent $n$ in ($weight/height^n$) for a better way to "optimize" the BMI indicator. More thoroughly, I want to know the distribution of $Z$ defined by $Z = \frac{X}{Y^n}$ where $X$ and $Y$ are two NON-independent random normal variables and $n \in \mathbb R_+$. Lets take for example $H\sim N(\mu_{H}, \sigma^2_{H})$ and $\epsilon\sim N(\mu_{\epsilon}, \sigma^2_{\epsilon})$ two independent random variables following a Normal distribution with respective mean and variance. Lets define a new random variable $W = a*H + \epsilon$, where $a \in \mathbb R$. I know that $W\sim N(a\mu_{H} + \mu_{\epsilon}, a^2\sigma^2_{H} + \sigma^2_{\epsilon})$ and for simplification lets write $W\sim N(\mu_{W}, \sigma^2_{W})$. Following the example, I want to know the distribution of $Z = \frac{W}{H^n}$. Does it follow a known distribution? I have run simulations and I know approximatly how it behaves, but I want to make the generic demonstration. Thanks in advance for the response.

Answer 1

I'm not sure we can completely dismiss criteria for optimality of BMI. BMI is an artificial index which I suppose is intended to be larger for people who are obese than for people who are scrawny.

BMI may be highly correlated with some 'better' measure of obesity, for example volume of fatty tissue as a percentage of total body volume. When you have an unquestionably better measure of obesity, you might use to 'fine tune' BMI.

One method might be to set $\log(Y_i) = a + b\log(W_i) + c\log(H_i),$ where for each subject $i$ out of a large number $n$ of randomly selected subjects $Y_i$ is better measure, $W_i$ is the weight and $H_i$ is the height. Then you should do the usual regression diagnostics, to make sure the method is valid. BMI uses $a = 0,\,b = 1$ and $c = -2.$ If your regression study were to give appreciably different values for $a, b$ and $c,$ then your new 'BMI' might be more useful.

However, on dimensionality grounds, the existing BMI might make good sense. Clearly, weight is going to increase with height in 'ideally proportioned' people, and perhaps the increase is proportional to height${}^2$---probably, not as height${}^3$ because people aren't spheres. So BMI = weight/height${}^2$ seems vaguely reasonable.

As you suggest, you might explore the relationship between $W$ and $H,$ but is should be based on data from appropriately chosen people. Then you could consider the regression model $\log W = a + b\log H$. But you really need to explore this relationship between $W$ and $H$ in real data. I agree that there is no point to an an analysis based on the assumption that $W$ and $H$ are independent. I suppose that BMI works as well as it does just because it anticipates a relationship between height and weight.

You say that you "know that $W\sim N(a\mu_{H} + \mu_{\epsilon}, a^2\sigma^2_{H} + \sigma^2_{\epsilon}).$" (Do you know the $\mu$'s and $\sigma$s?) Do you have the data that gives rise to this? If so, and if the subjects were suitable, that would be the basis for the kind of regression I mentioned in the previous paragraph.

Of course, one can find the distribution of $Z = W/H^n,$ for suitably chosen or modeled data on $W$ and $H,$ but frequently such distributions of ratios of normals have bad properties, so I'm not sure that is the best approach.