Add documentation and citations to README.md for methods implemented under statistical_inference/.

This includes some discussion of methods to consider implementing in future.

PiperOrigin-RevId: 823564609

Author: mjwillson

weatherbenchX/statistical_inference/README.md

@@ -42,3 +42,73 @@ with further caveats and trade-offs.
 Spatial correlation can be an issue too, although we typically avoid the need
 to account for this by treating spatial locations (on a grid, for example) as
 fixed, and averaging over them before performing any statistical inference.
+
+## Methods implemented
+
+* The standard t-test.
+* The t-test with AR(2) autocorrelation correction from [^1]
+* Both the above with delta-method confidence intervals for metrics which are
+  nonlinear functions of the mean statistics.
+* The IID bootstrap.
+* A cluster bootstrap [^3] [^4] which assumes independence only between
+  clusters.
+* The stationary bootstrap of [^5], with optimal block length selection from
+  [^6] [^7] but generalized to support non-linear functions of means of
+  multivariate statistics via a delta-method trick.
+
+## Methods to consider implementing in future
+
+* Diebold-Mariano test [^8] or another of the family of tests based on HAC
+  (Heteroskedasticity and Autocorrelation Consistent) variance estimators, as a
+  a better-studied and more standard alternative to [^1]. These methods also
+  have problems at small sample sizes and/or high degrees of autocorrelation
+  however, and there are a number of choices e.g. of kernel and window length
+  selection method with different trade-offs. [^10] offers a modern review and
+  some practical recommendations.
+* One of the second-order-correct block bootstrap CI methods of [^9] for smooth
+  functions of vector means, either the studentized or the BCa-style intervals.
+  Perhaps adapted to the circular block bootstrap to avoid the need to correct
+  for endpoint bias. Unlike a naive application of studentized (bootstrap-t) or
+  BCa intervals to the block bootstrap, these methods are more principled and
+  retain the good asymptotics of studentized and BCa intervals from the IID
+  case. It remains to be seen how effective they are at practical sample sizes
+  though despite the improved asymptotic order, and they also add some
+  complexity and further choices.
+
+[^1]: A. J. Geer, Significance of changes in medium-range forecast scores.
+  Tellus A Dyn. Meterol. Oceanogr. 68, 30229 (2016).
+
+[^2]: Efron, B. Better bootstrap confidence intervals. J.A.S.A. 82, 171-185
+  (1987)
+
+[^3]: Davison, A. C. & Hinkley, D. V. Bootstrap Methods and their Application
+  (Cambridge University Press, 1997), pp.100-101.
+
+[^4]: Sherman, M. & le Cessie, Saskia, A comparison between bootstrap methods
+  and generalized estimating equations for correlated outcomes in generalized
+  linear models, Communications in Statistics - Simulation and Computation,
+  26:3, 901-925 (1997).
+
+[^5]: Politis, D. N. & Romano, J. P. The stationary bootstrap. J.A.S.A. 89,
+  1303-1313 (1994).
+
+[^6]: Politis, D. N. & White, H. Automatic Block-Length Selection for the
+  Dependent Bootstrap, Econometric Reviews, 23:1, 53-70 (2004).
+
+[^7]: Patton, A., Politis, D. N. & White, H. Correction to "Automatic
+  Block-Length Selection for the Dependent Bootstrap" by D. Politis and
+  H. White, Econometric Reviews, 28:4, 372-375 (2009).
+
+[^8]: Diebold, F. X. & Mariano, R. S. Comparing predictive accuracy. J. Bus. Econ.
+  Stat. 20, 134–144 (2002).
+
+[^9]: Götze, F. & Künsch, H. R. Second-order correctness of the blockwise
+  bootstrap for stationary observations. Ann. Stat. 24, 1914-1933 (1996).
+
+[^10]: Lazarus, E., Lewis, D. J., Stock, J. H. & Watson, M. W. HAR inference:
+  Recommendations for practice. J. Bus. Econ. Stat. 36, 541–559 (2018).
+
+
+
+
+