Editing Statistical inference (section)

==== Model-free randomization inference ====
Model-free techniques provide a complement to model-based methods, which employ reductionist strategies of reality-simplification. The former combine, evolve, ensemble and train algorithms dynamically adapting to the contextual affinities of a process and learning the intrinsic characteristics of the observations.<ref name="Dinov Palanimalai Khare Christou 2018">
{{cite journal
|last1= Dinov | first1=Ivo
|last2= Palanimalai | first2= Selvam
|last3= Khare |first3= Ashwini
|last4= Christou |first4= Nicolas
|date= 2018
|title= Randomization-based statistical inference: A resampling and simulation infrastructure
|journal= Teaching Statistics
|volume= 40
|issue= 2
|pages= 64–73
|doi= 10.1111/test.12156
| pmid=30270947
|pmc= 6155997
}}</ref><ref name="Tang model-based Model-Free 2019">
{{cite journal
|last1= Tang | first1=Ming
|last2= Gao | first2=Chao
|last3= Goutman | first3=Stephen
|last4= Kalinin | first4=Alexandr
|last5= Mukherjee | first5=Bhramar
|last6= Guan | first6=Yuanfang
|last7= Dinov | first7=Ivo
|date= 2019
|title= Model-Based and Model-Free Techniques for Amyotrophic Lateral Sclerosis Diagnostic Prediction and Patient Clustering
|journal= Neuroinformatics
|volume= 17
| issue=3
|pages= 407–421
|doi= 10.1007/s12021-018-9406-9
| pmid=30460455 | pmc= 6527505
}}</ref>

For example, model-free simple linear regression is based either on:

* a ''random design'', where the pairs of observations <math>(X_1,Y_1), (X_2,Y_2), \cdots , (X_n,Y_n)</math> are independent and identically distributed (iid),
* or a ''deterministic design'', where the variables <math>X_1, X_2, \cdots, X_n</math> are deterministic, but the corresponding response variables <math>Y_1,Y_2, \cdots, Y_n</math> are random and independent with a common conditional distribution, i.e., <math>P\left (Y_j \leq y | X_j =x\right ) = D_x(y)</math>, which is independent of the index <math>j</math>.

In either case, the model-free randomization inference for features of the common conditional distribution <math>D_x(.)</math> relies on some regularity conditions, e.g. functional smoothness. For instance, model-free randomization inference for the population feature ''conditional mean'', <math>\mu(x)=E(Y | X = x)</math>, can be consistently estimated via local averaging or local polynomial fitting, under the assumption that <math>\mu(x)</math> is smooth. Also, relying on asymptotic normality or resampling, we can construct confidence intervals for the population feature, in this case, the ''conditional mean'', <math>\mu(x)</math>.<ref name="Politis Model-Free Inference 2019">
{{cite journal
|last1= Politis | first1=D.N.
|date= 2019
|title= Model-free inference in statistics: how and why
|journal= IMS Bulletin
|volume= 48
|url= http://bulletin.imstat.org/2015/11/model-free-inference-in-statistics-how-and-why/
}}</ref>