Commonly used example data sets Is there a place where one can get standard example data for various statistics tools to try on? For example, if one is learning about ARIMA models, where would one get data that can be modelled well with an ARIMA(2,1,2)x(1,1,1)12 model? Or data for a (5,1,1)x(0,0,0) model? Or multiple linear regression etc? I'm looking for something similar to this:
https://en.wikipedia.org/wiki/Lenna
 A: There are several different places you could try, but I'll highlight two.
Help Documentation in Statistical Packages
One potentially great and easy-to-access resource is the help documentation in various vignettes in R packages.  Often times the vignettes/documentation come with built in datasets meant to facilitate learning about the procedures contained in the R package.  
For example the R package twang is for the development of boosted propensity scores.  If you install the package, open it, and then type the name of a dataset associated with the package inside the "data" function call, you'll be able to see the data:
install.packages('twang')
library(twang)
data(lalonde)
> head(lalonde)
  treat age educ black hispan married nodegree re74 re75       re78
1     1  37   11     1      0       1        1    0    0  9930.0460
2     1  22    9     0      1       0        1    0    0  3595.8940
3     1  30   12     1      0       0        0    0    0 24909.4500
4     1  27   11     1      0       0        1    0    0  7506.1460
5     1  33    8     1      0       0        1    0    0   289.7899
6     1  22    9     1      0       0        1    0    0  4056.4940

How do you get the name of the dataset (in this case, it's called lalonde)?  If you want to see all the datasets in a package, simply type:
try(data(package="packagename"))

where packagename is the name of the package in which you want to look for datasets.  So for the twang example, typing:
try(data(package = "twang") )

opens a prompt that contains the following data sets in package ‘twang’:
Data sets in package ‘twang’:

    AOD                     Subset of Alcohol and Other Drug treatment data
    egsingle                US Sustaining Effects study
    iptwExLong              Example data for iptw function (long version)
    iptwExWide              Example data for iptw function (wide version)
    lalonde                 Lalonde's National Supported Work Demonstration
                            data
    lindner                 Lindner Center data on 996 PCI patients
                            analyzed by Kereiakes et al. (2000)
    mnIptwExLong            Example data for iptw function (long version,
                            more than two treatments).
    mnIptwExWide            Example data for iptw function (wide version,
                            more than two treatments)
    raceprofiling           Traffic stop data

If you wanted to see the datasets relevant for learning the rpart package, you'd type:
try(data(package = "rpart") )

and you'd get:
Data sets in package ‘rpart’:

car.test.frame          Automobile Data from 'Consumer Reports' 1990
car90                   Automobile Data from 'Consumer Reports' 1990
cu.summary              Automobile Data from 'Consumer Reports' 1990
kyphosis                Data on Children who have had Corrective Spinal
                        Surgery
solder                  Soldering of Components on Printed-Circuit
                        Boards
stagec                  Stage C Prostate Cancer

If you then want to obtain a more detailed description of the dataset and its contents, you can simply type the name of the dataset inside the help() function call.  So typing:
help(lalonde)

essentially launches your browser and opens up a description like the one here:  https://rdrr.io/cran/cobalt/man/lalonde.html
UCLA's IDRE
Another great resource for finding datasets relevant to specific analyses is the UCLA's Institute for Digital Research & Education website (IDRE).  The site guides users through different types of analyses and contains hyperlinks directly to datasets relevant to the analyses.
For example, if you wanted to obtain a SAS dataset relevant to Poisson Regression you could click the SAS hyperlink on the IDRE link (see above) and you'll see in the tutorial, a hyperlink to a SAS datset for example 3.
A: The U.S. National Institute of Standards and Technology (NIST) has a database online at https://www.itl.nist.gov/div898/strd/nls/nls_main.shtml with both test data sets and associated non-linear regression equations for use in the design and testing of non-linear regression software. Each has example fitted parameters for verification, and there are different levels of difficulty listed. There are also two sets of initial parameter values for for each equation, a "near" and a "far", which are near and far from the reference fitted values for the purpose of testing convergence from different starting points.
The creators of this online regression testing database are aware that in testing, final fitted parameters will not be exactly equal to the listed values and they state that results to within four or five decimal places of their listed values are sufficient.
I personally found these to be of immense value in the design and testing of of my pyeq3 Python fitting library, which is the core of my zunzun.com curve and surface fitting web site. These test equations and associated data sets are what gave me the (statistical) confidence I needed to put the site on the internet in the first place.
A: A good way to access data is to actually simulate data for a number of models to learn how to identify latent structure AND then introduce pulses into the data and learn how to identify data like this. Then introduce level/step shifts and/or local time trends and then learn how to identify data like this. Then simulate data where deterministic seasonal pulses are embedded in the data and learn how to identify data like this.
Then simulate data where the model parameters change over time and learn how to identify data like this.
Then simulate data where the error variance changes over time and learn how to identify data like this. Deterministic Error variance change can be detected following TSAY http://docplayer.net/12080848-Outliers-level-shifts-and-variance-changes-in-time-series.html while linkage between the expected value and the error process is remedied by Box_Cox When (and why) should you take the log of a distribution (of numbers)?
Extend this univariate discussion to include exogenous factors that not only are important contemporaneously but with lag effects .
That is what I would do and continuously do to motivate learning by data when no theory is pre-existent and to create robotic solutions to aid analyses.
In all of the above you have the knowledge  of how the data was actually constructed and you can learn about the strengths and weaknesses of alternative identification strategies
A: Here's a collection of datasets which I occasionally use in my courses:
Epina DataLab repository
The datasets at this site are available both as text files and as IDT files (which is DataLab's internal format, most convenient to be load in Epina DataLab). The datasets are suitable for a wide range of statistiscal methods, such as regression, LDA, PCA, HCA, PLS, etc.
