Is there a generalized concept of noncentrality of a distribution?

Question

The theory of probability distributions forms one of the pillars of statistics, and is a foundation for statistical inference. There are more than a few probability distributions, and they are neat-O. Many of the distributions folks learn about at an elementary level (introductory statistics courses, introductory probability theory courses) are symmetrical (e.g., normal, Student's $t$, Laplace, etc.), and many are asymmetrical ($\chi^{2}$, $F$, Poisson, etc.).

Many of these distributions have been generalized to "noncentral" forms. For example, there are noncentral $t$, noncentral $\chi^{2}$, noncentral $F$, etc. These distributions share a "shifting" of the distribution of probability relative to the corresponding "central" versions of the distribution, whether or not the "central" versions are symmetric or no. [I hope it is clear what I mean by putting "shifting" in double quotes here, please ask me to clarify in comments if not.] However, there are other flavors of distributions which also entails this kind of "shifting", for example, the skew normal distribution.

What does the concept of a noncentral distribution mean in an intuitive sense? The Wikipedia entry on noncentral distributions (Access date: 4-26-2021) describes them as related to

how a test statistic is distributed when the difference tested is null, noncentral distributions describe the distribution of a test statistic when the null is false (so the alternative hypothesis is true),

but while this confirms a hunch I have had from, say reading about uniformly most power tests in Wellek's textbook on equivalence and noninferiority, I still want for understanding how to get from a central distribution to a noncentral one. Why isn't any old "shifting" flavor of a "central" distribution a noncentral probability distribution?

Bonus points for providing a general set of steps that leads one from a central distribution to a noncentral one in a formal sense. For example, there is no Wikipedia entry for a noncentral normal distribution, even though the standard normal forms the basis of the $z$ test: how would we create a noncentral normal distribution (or any other noncentral distribution)?

Answer 1

It's hard to understand how to answer this question.

For any given hypothesis and any given test statistic, the distribution under an alternative hypothesis is considered a "non-central" version of the distribution of the same statistic under the null.

In some lucky cases, the test-statistic under the alternative hypothesis has a distribution which shares a parametric family with the distribution of the test statistic under the null. The Z-test is quite contrived in that regard.

Non-central chi-square, non-central F, and non-central T are in such widespread use that they are cited in much of the literature and software, and there are a few useful analytical results. If the non-central distribution is lucky enough to be available in closed form, we usually expect that the "central" counterpart belongs in the family, just like how a t-distribution is a non-central t with non-centrality parameter set to 0.

However, beyond this lies a whole cadre of distributions that are not described in the literature. Either they're too specially tooled to be of any generalizable (or didactic) use, or they aren't even available and have to be estimated numerically, i.e. simulated. In my experience, any remotely non-routine power calculation relies on simulation to identify the distribution underlying the test statistic. To the best of my knowledge, test statistics for hypotheses about fixed or random effects in mixed models, mediators in linear models, or treatment assignment in adaptive randomized tests are highly irregular when the null is false, and extensive simulation studies are as close as we can come to getting an understanding of the operating characteristics of the test.

Answer 2

I think a simple way to think about noncentral distributions is to consider how they're built from normal distribution, e.g., non central t variable is $\frac{Z+\mu}{\sqrt{V/\nu}}$, where $Z$ is standard normal and $V\sim\chi_\nu^2$. When noncentrality parameter $\mu=0$, we have the standard normal in numerator, and the distribution becomes usual [central] Student t. Other noncentral distributions are constructed similarly. So, when your Gaussian variable has non zero mean, that's when noncentrality occurs in these distribution.

Note, that the "central" version of the Student t distribution is $\frac{Z}{\sqrt{V/\nu}}$, which came up when analyzing the properties of estimated parameters of regression. The coefficients tend to be from the normal distribution with unknown variance, hence the formulation of the Student t from the normal variable in the numerator and the square root of $\chi^2$ variable in the denominator.

Skewed variant of these distributions have no connection to normal distribution explicitly. It's a generalization in a different direction so to speak.

Naturally, the only logical noncentral extension of the standard normal variable is the Gaussian variable with nonzero mean. However, this is such a trivial case, that nobody would call this distribution "noncentral normal" variable, but you could if you wish so.

Answer 3

I agree with Aksakal and AdamO, the non-central varieties are a result of investigating the power of a test. The test itself assumes a particular null hypothesis for the purposes of argument and inference using ex-post sampling probability as evidence. Power explores the ex-ante sampling probability of the test when in fact an alternative hypothesis is true. The non-centrality parameter is related to the true alternative hypothesis. For instance, think of calculating power when testing a binomial proportion by referencing a binomial CDF. The critical value is compared to a binomial sampling distribution under the alternative, and the shape of this distribution is different from the null sampling distribution. It is not a simple shift.

For, say, a Wald test we assume the standard error is known and not a function of the null hypothesis so the non-central distribution under the alternative is just a shifted normal distribution. The reason this is a simple shift is because the nuisance parameters are not profiled and are considered known. An interesting thing to note in this simple example is that the power function is the same as the p-value function. When using more complicated tests that profile nuisance parameters before treating them as known (e.g. score, LR) the p-value function works incredibly well at approximating power, meaning we can avoid non-central distributions altogether. When calculating a p-value this profiling works to account for having estimated the nuisance parameters even though they are assumed known. Here is a paper of mine that discusses approximating a power function using a p-value function.

Johnson, G. S. (2021). Decision Making in Drug Development via Inference on Power. Researchgate.net.

Answer 4

The intuitive way to grasp noncentral distributions is through their central counterparts. There are several noncentral distributions like noncentral chi-squared, noncentral F, noncentral T, noncentral beta, noncentral negative hypergeometric, noncentral Wishart, and so on. All of them can be expressed as infinite mixtures of the corresponding central distribution. The weights of the mixture are usually Poisson probabilities (as in the first four), but could also be negative binomial weights (as in noncentral negative hypergeometric). A good starting point for you is R. Chattamvelli (1995), "A note on the noncentral beta distribution function", The American Statistician, vol 49, number 3. Hope this helps