Timeline for Applying the Central Limit Theorem to a Piecewise PDF
Current License: CC BY-SA 4.0
24 events
| when toggle format | what | by | license | comment | |
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| Dec 4, 2023 at 14:02 | comment | added | whuber♦ | @Accumulation. Right. Now re-read the title of this thread! | |
| Dec 3, 2023 at 19:52 | comment | added | Acccumulation | @whuber In "piecewise linear function", the word "piecewise" modifies "linear", not "function". It's not a piecewise function that's linear, it's a function that's piecewise linear. | |
| Dec 3, 2023 at 15:43 | comment | added | whuber♦ | @Acccumulation On the contrary, when one is considering functions in a certain class, a "piecewise" version is assembled from functions of that class restricted to a partition of its domain (into at most a countable number of regions). "Piecewise linear" is a standard example. Independently of any description, a piecewise linear function has an intrinsic characterization. Moreover, in the present case there is no "piecewise" element of the definition: this function is expressed as a simple linear combination of other functions. | |
| Dec 3, 2023 at 6:41 | comment | added | Acccumulation | @whuber People see the phrase "piecewise defined function" and don't notice the word "defined", and think that being "piecewise" is a property that function can have, rather than a property that a definition can have. There is no such thing as a "piecewise function". | |
| Dec 3, 2023 at 6:39 | comment | added | Acccumulation | "I know that the central limit theorem guarantees that the sum of X will have a Normal distribution" The CLT is a statement about the asymptotic behavior in the limit as the sample size goes to infinity. Strictly speaking, it doesn't say that the distribution is normal for any finite sample size. | |
| Dec 3, 2023 at 5:05 | comment | added | Zhanxiong | @FrankHarrell See my rejoinder, in particular the lognormal example (if the link you provided is a textbook, then I think it needs to be revised or at least please call it a "naive CLT"). | |
| Dec 2, 2023 at 23:01 | vote | accept | Occhima | ||
| Dec 1, 2023 at 21:17 | history | became hot network question | |||
| Dec 1, 2023 at 19:32 | comment | added | Nicolas Bourbaki | @Zhanxiong Distributions that have thick-tails will generally be very slow to conform to the CLT. | |
| Dec 1, 2023 at 17:22 | answer | added | Zhanxiong | timeline score: 10 | |
| Dec 1, 2023 at 16:46 | comment | added | Frank Harrell | Try it on the log normal example yourself. For your particular problem the thing being estimated is bounded on [0,1] so I expect the CLT approximation to work better. But you never know if it works well enough until (1) you quantify “well” and (2) you go to the trouble of getting the right answer to compare it with. | |
| Dec 1, 2023 at 16:36 | comment | added | Zhanxiong | Give me some time, I will show you as how "excellent" the approximation will be in the answer below. | |
| Dec 1, 2023 at 16:35 | comment | added | Frank Harrell | You continue to make the same mistake about CLT unless “excellent job” means “not terrible”. And you completely missed the point about CLT. You don’t know in practice that you may need to log transform. The CLT is supposed to work for this situation without having oracle knowledge of the correct transform. And transform-then-back-transform will yield the median, not the mean. | |
| Dec 1, 2023 at 16:31 | comment | added | Zhanxiong | @FrankHarrell To clarify, we are not dealing any "practical problems" here, this problem asks for approximating $P(\sum_{i = 1}^n X_i \leq 3600)$, which I am certain that CLT can do an excellent job as this is guaranteed by probability theory. I also checked your example and I think it deals with something different (compute confidence intervals). And the inaccuracy actually stems from an incorrect application of the CI formula (for log-normal parameters, the correct procedure should be transform then back-transform). | |
| Dec 1, 2023 at 16:19 | comment | added | Frank Harrell | You’re making the common mistake of assuming that a limit theorem applies to practical problems. You’re not alone. Here’s my example: hbiostat.org/bbr/htest.html#central-limit-theorem | |
| Dec 1, 2023 at 15:31 | comment | added | Zhanxiong | @FrankHarrell If you could, I would be very interested in to see your example (maybe you can post it as an answer below). It is just very counterintuitive to me if all the conditions in CLT are satisfied (i.e., i.i.d. + finite variance) while that large $n$ did not show approximate normality. | |
| Dec 1, 2023 at 15:20 | comment | added | Frank Harrell | @Zhanxiong there is nothing at all straightforward about this application of the CLT. I have a simple example from a log-normal distribution where n=50,000 is far too small to get accurate confidence intervals with CLT. | |
| Dec 1, 2023 at 15:01 | comment | added | whuber♦ | Re the title: the CLT could care less about the characteristics of the density. After all, its original applications (in the early 18th century) were to variables that have no density at all: namely, Bernoulli variables. So, piecewise linearity is not problematic. I suspect the "piecewise" in the title refers to the notation used to express the density. | |
| Dec 1, 2023 at 14:57 | answer | added | whuber♦ | timeline score: 8 | |
| Dec 1, 2023 at 14:49 | comment | added | Zhanxiong | @FrankHarrell It's hard for me to understand why you thought "$n = 1296$ is not large at all" and this is an ill-posed question. This is just a very straightforward application of the classical CLT. | |
| Dec 1, 2023 at 14:44 | comment | added | Zhanxiong | @ThànhNguyễn $\int_{\mathbb{R}}f(x)dx = \frac{1}{3}(1 + 2) = 1$. I didn't see any problem of that. | |
| Dec 1, 2023 at 14:30 | comment | added | whuber♦ | @Frank Generally that's good advice, but in this case, where $f$ does not differ much from a linear combination of a Bernoulli$(2/3)$ variable and a uniform variable, experience tells us that the CLT gives pretty good results even for $n=5$ and by the time $n=1296$ it would be pointless to compute probabilities any other way. // Responding to the first comment, clearly this is a valid pdf: it is everywhere nonnegative and integrates to unity. | |
| Dec 1, 2023 at 13:38 | comment | added | Frank Harrell | Side note: Your instructor made a possibly ill-posed question because the CLT may not apply with that sample size, depending on the level of accuracy you need from a CLT approximation. n=1296 is not necessarily large at all. If you were in a real-world problem solving mode and not a “use the theory” mode I’d dispense with all this and just use Monte-Carlo simulation to get an answer to any desired accuracy, unless you can derive the CDF analytically. | |
| Dec 1, 2023 at 13:15 | history | asked | Occhima | CC BY-SA 4.0 |