Recursions for the Moments of Some Continuous Distributions

This post follows on from my recent one, Recursions for the Moments of Some Discrete Distributions. I'm going to assume that you've read the previous post, so this one will be shorter. 

What I'll be discussing here are some useful recursion formulae for computing the moments of a number of continuous distributions that are widely used in econometrics. The coverage won't be exhaustive, by any means. I provide some motivation for looking at formulae such as these in the previous post, so I won't repeat it here. 

When we deal with the Normal distribution, below, we'll make explicit use of Stein's Lemma. Several of the other results are derived (behind the scenes) by using a very similar approach. So, let's begin by stating this Lemma.

Stein's Lemma (Stein, 1973):

"If  X ~ N[θ , σ²], and if g(.) is a differentiable function such that E|g'(X)| is finite, then 

                            E[g(X)(X - θ)] = σ² E[g'(X)]."

It's worth noting that although this lemma relates to a single Normal random variable, in the bivariate Normal case the lemma generalizes to:

"If  X and Y follow a bivariate Normal distribution, and if g(.) is a differentiable function such that E|g'(Y)| is finite, then 

                            Cov.[g(Y ), X] = Cov.(X , Y) E[g'(Y)]."

In this latter form, the lemma is useful in asset pricing models.

There are extensions of Stein's Lemma to a broader class univariate and multivariate distributions. For example, see Alghalith (undated), and Landsman et al. (2013), and the references in those papers. Generally, if a distribution belongs to an exponential family, then recursions for its moments can be obtained quite easily.

Now, let's get down to business............

Central and Non-Central Distributions

Let's imagine that you're teaching an econometrics class that features hypothesis testing. It may be an elementary introduction to the topic itself; or it may be a more detailed discussion of a particular testing problem. We're not talking here about a course on Bayesian econometrics, so in all likelihood you'll be following the "classical" Neyman-Pearson paradigm.

You set up the null and alternative hypotheses. You introduce the idea of a test statistic, and hopefully, you explain why we try to find one that's "pivotal". You talk about Type I and Type II errors; and the trade-off between the probabilities of these errors occurring. 

You might talk about the idea of assigning a significance level for the test in advance of implementing it; or you might talk about p-values. In either case, you have to emphasize to the classt that in order to apply the test itself, you have to know the sampling distribution of your test statistic for the situation where the null hypothesis is true.

Why is this?

Unbiased Estimation of a Standard Deviation

Frequently, we're interested in using sample data to obtain an unbiased estimator of a population variance. We do this by using the sample variance, with the appropriate correction for the degrees of freedom. Similarly, in the context of a linear regression model, we use the sum of the squared OLS residuals, divided by the degrees of freedom, to get an unbiased estimator of the variance of the model's error term.

But what if we want an unbiased estimator of the population standard deviation, rather than the variance?

Variance Estimators That Minimize MSE

In this post I'm going to look at alternative estimators for the variance of a population. The following discussion builds on a recent post, and once again it's really directed at students. Well, for the most part.

Actually, some of the results relating to populations that are non-Normal probably won't be familiar to a lot of readers. In fact, I can't think of a reference for where these results have been assembled in this way previously. So, I think there's some novelty here. But we'll get to that in due course.

I can just imagine you smacking your lips in anticipation!

What's the Variance of a Sample Variance?

This post is really pitched at students who are taking a course or two in introductory economic statistics. It relates to a couple of estimators of the variance of a population that we all meet in such courses - plus another one that you might not have met. In addition, I'll be emphasising the fact that some "standard" results depend crucially on certain assumptions. Not surprisingly - but  not always made clear by instructors and text books.

Sums of Random Variables

I'm currently teaching first-level course in statistical inference for  (mostly) economics students. They've taken a one-semester course in descriptive (economic) statistics, and now we're dealing with sampling distributions, estimation, hypothesis testing, and simple regression analysis.