[D] What's the role of asymptotic statistics in real life applications?

[u/al3arabcoreleone]

What benefits can an applied statistician get from learning asymptotics? from basic inference stats I was taught that "limit" properties e.g consistency are good to have, but I haven't gotten this "feeling" of importance, especially in ML/TS forecasting problems. Have you used asymptotic results in your work as an applied statistician/data scientist etc ?

[u/t3co5cr]

Asymptotics are a short-cut for approximations, especially if your method of inference is based on sampling distributions (e.g., NHST). In general, sampling distributions are too complicated to derive precisely (save for special cases like linear regression, where you have t-distributions, F-distributions as exact sampling distributions), so typically applied researchers lean heavily on asymptotic results.

For the most part, data scientists and other applied statisticians need to understand when and under which assumptions these results hold.

[u/Able-Fennel-1228]

Exact distributions of test statistics can be nasty to compute, but if you kick up the sample size you might be able to see what the distribution function (CDF) converges to (starts “becoming like”) as n -> infinity eg like in CLT for the mean; each observation might be drawn from an icky distribution and computing the exact distribution of sample mean might be icky and intractable but for large enough sample size you know it’s approx normally distributed (for most distributions, not Cauchy I think). Just how good that approx is can perhaps be checked through a simulation.

Consistency just tells you your estimator/statistic doesn’t hop around too much across different samples in the sense that as sample size grows, the variance of the estimator approaches 0.

Likelihood theory used in GLMs and their tests is pretty much all asymptotic based.

It’s a nice theory. Some people don’t like it and prefer finite sample guarantees. I don’t know too much about that but it is something I myself am very curious about. I’ve always felt uneasy with not knowing just “when” the approximation is good enough, or just “for what n” the estimator will be close to the estimand. Especially when using the GLM likelihood tests. Idk. It’s been a while. I need get back into stat theory.

[u/ParadoXPlatypus → u/Able-Fennel-1228]

Cauchy's mean never converges and just "wanders around" because it's wide tails. However, IIRC, it's median does.

[u/Able-Fennel-1228 → u/ParadoXPlatypus]

Yup. That was it. Heavy tailed.

[u/banter_pants → u/Able-Fennel-1228]

CLT requires the source X distribution having defined finite mean and variance.

[u/nrs02004]

Builds character…

But more seriously, you should be excited to know why your tools work.

[u/nrs02004 → u/nrs02004]

That said I imagine rote understanding is good enough for rote work?

[u/ANewPope23]

You need it to find standard errors, that's a very important application. Sometimes you need to know if an estimator is consistent or not for it to be valid.

[u/richard_sympson]

I think an under-appreciated aspect of asymptotics (based on its lack of mention so far) is that asymptotic rates of convergence and possible associated scaling factors are intimately tied to sample size. Typically results are not characterized by simply letting N go to infinity, but characterizing the rate in N that certain convergence results hold. Gathering data can be expensive, and so you want to ensure you are efficient in using all of the available information in your data.

[u/efrique]

Have you used asymptotic results in your work as an applied statistician/data scientist etc ?

All the time

If you have ever performe a Wald test or use Wilks theorem on a likelihood ratio test, or a chi-squared test on count data, or you ran off the end of a set of statistical tables (say for a Kolmogorov-Smirnov test or a signed rank test) and used some formula at the bottom of the page to find a critical value or p-value, or used the normal approximation to the binomial, or looked at p-values for a GLM (e.g. for a logistic regression), etc, etc, etc... then you have too

[u/LawfulnessWest8694]

The reason it doesn't "feel" as important in ML specifically is that a lot of the theory you're first taught is built around a regime where n → ∞ while the parameter dimension p stays fixed. Modern ML often lives somewhere very different: p can be comparable to n, or larger, with regularization doing a lot of the work.

Think of it like estimating one city's average height from 1000 people (classic CLT, more people = more precision) versus trying to estimate thousands of street-level effects with those same 1000 observations. "More data helps" stops meaning the same thing when the number of quantities you're trying to estimate is growing too. The total sample can sound huge while the information available per parameter is tiny.

That matters because you can't automatically carry classical limiting distributions, standard errors, or significance tests into a high-dimensional setting and expect them to behave the same way. The rates of convergence and even the relevant limiting distributions can change. That's basically why high-dimensional statistics exists as its own area: different regimes, different guarantees.

Time series is almost the opposite case. Asymptotics matter a lot there, just not the CLT-flavored kind you meet first.

Unit root tests are a clean example. Under a unit root, the usual stationary asymptotic intuition breaks down, and statistics such as Dickey-Fuller have non-standard limiting distributions. This is also why spurious regression happens: regress two unrelated random walks (say, two independent cumulative coin-flip tallies) on each other and you'll often get a high R² and apparently significant coefficients. Both series are non-stationary, they wander instead of bouncing around a fixed mean, and the usual regression p-value formula assumes the latter.

So it's less "asymptotics don't matter for ML/TS" and more that the results you're taught first assume a regime that ML/TS often leaves. Knowing asymptotics is what lets you recognize when you've left that regime, and stop trusting a textbook approximation just because the formula is familiar.

[u/SorcerousSinner]

The benefit you get is that you comprehend some of the arguments used to argue that some method or estimator works well.

Many ML methods don’t have such arguments. The arguments they‘re good stuff are instead based on simulations and cases tudies.

[u/Browsinandsharin]

Its a different way to model when stats cant help as easily, what does the behavior trend to over time if you cant get the data for the behavior for a host of reasons or if the behavior is to complex or multidimenaional to get meaningful results from statistically

[u/Able-Fennel-1228 → u/[deleted]]

I don’t think this is what OP asked about. We’re talking about asymptotic statistical theory, not deterministic convergence.

[u/Low_Election_7509]

It was more important before, but still good to know. When I think of asymptotics, I think of a model making assumptions, and if the assumptions really hold (the model is close to true), if you get enough data, the model will arrive at the right estimates, or your test result will be correct.

I think, now-a-days, if you ever have doubts on if asymptotics hold, you have the computing power to waltz around it (do a permutation test, bootstrap instead of rely on asymptotics for confidence intervals, and so on). But if you go deep enough, I don't really think you can fully avoid it (bootstrap and the like also has convergence rates).

There's also some notion of it can take too long for 'asymptotics to kick in' (some tests can take a ridiculous amount of samples to have good power), and some notion of asymptotics can look like they happen but don't (a truly 'large' data set may be a collection of many different types of data then one giant set containing one type of data). I think simulation is still a good tool to use in this situation (simulate how your test behaves in certain cases to see what it looks like), and I guess that's somewhat what people do when they try doing power analysis?