ENH: stats: add array API-support

[mdhaber]

Towards gh-18867

This issue tracks progress toward the addition of array-API support to scipy.stats functions. The functions listed below look ready for conversion, and I'd be happy to review PRs for them. Priority, balancing the ease and importance of the task, is roughly in the order listed.

• moment (ENH: stats.moment: add array API support #20292)
• skew (ENH: stats.skew: add array-API support #20541 - please see this PR as an example roughly similar kurtosis through directional_stats)
• kurtosis (ENH: stats.kurtosis: add array API support #20658)
• describe (ENH: stats.describe: add array API support #20667)
• entropy (ENH: stats.entropy, special.{entr, rel_entr}: add array API support #20673)
• variation (ENH: stats.variation: add array-API support #20647)
• sem (ENH: stats.sem: add array-API support #20631)
• kstat (ENH: stats: add array-API support to kstat/kstatvar #20634)*
• kstatvar (ENH: stats: add array-API support to kstat/kstatvar #20634)*
• circmean (ENH: stats.circ___: add array-API support #20595)
• circvar (ENH: stats.circ___: add array-API support #20595)
• circstd (ENH: stats.circ___: add array-API support #20595)
• directional_stats (ENH: stats: add array API support for directional_stats #20794)
• pearsonr (ENH: stats.pearsonr: add array API support #20284)
• ttest_1samp (ENH: stats.ttest_1samp: add array-API support #20545 - please see this PR as an example for ttest_rel through normaltest)
• ttest_rel (ENH: stats: rewrite ttest_rel in terms of ttest_1samp #20883)
• ttest_ind (ENH: stats.ttest_ind: add array API support #20771)
• skewtest (ENH: stats.skewtest: add array-API support #20597)
• kurtosistest (ENH: stats.kurtosistest: add array API support #20715)
• normaltest (ENH: stats.normaltest/jarque_bera: add array-API support #20736)
• jarque_bera (ENH: stats.normaltest/jarque_bera: add array-API support #20736)
• power_divergence (ENH: stats.chisquare/power_divergence: add array API support #20753)
• chisquare (ENH: stats.chisquare/power_divergence: add array API support #20753)
• combine_pvalues (ENH: stats: add array API support to combine_pvalues #20900)

After that:

• write a function for computing weighted average. (ENH: stats._xp_mean, an array API compatible mean with weights and nan_policy #20743)
• gmean
• hmean
• pmean

After that:

• Add _SimpleNormal (ENH: stats: end-to-end array-API support for normality tests #20777)
• Add _SimpleChi2 (ENH: stats: end-to-end array-API support for NHSTs with chi-squared null distribution #20782)
• Add _SimpleBeta (ENH: stats: end-to-end array-API support for NHSTs with beta null distribution #20793)
• Add _SimpleStudentT (ENH: stats: end-to-end array-API support for NHSTs with Student's t null distribution #20884)
• Dispatch to array backend stdtrit where possible

After that:

• add N-D support to _array_api.cov; consider making it public if array API won't offer it Not really necessary. We don't need something very general, so let's not get hung up on it.
• linregress: add axis and array API support
• ks_2samp: consider natively vectorizing, then adding array API support
• mode: consider natively vectorizing (e.g. see rankdata for ideas), then adding array API support
• bartlett: consider natively vectorizing, then adding array API support (ENH: stats.bartlett: add native axis and array API support #20751)
• levene: consider natively vectorizing, then adding array API support
• anderson_ksamp: might be able to vectorize, then add array API support
• wasserstein_distance: consider natively vectorizing, then adding array API support
• energy_distance: consider natively vectorizing, then adding array API support

I'd like to implement the following using _masked_array (gh-20363):

• tmean
• tvar
• tmin
• tmax
• tstd
• tsem

These functions are held up by rankdata (possibly among other things), which is itself in need of improved array-API support. See gh-20639.

• kendalltau
• mannwhitneyu
• wilcoxon
• kruskal
• cramervonmises_2samp
• friedmanchisquare
• brunnermunzel
• ansari
• fligner
• mood

These functions need median, quantile, or similar, either directly or via iqr. See data-apis/array-api#795.

• iqr
• siegelslopes
• theilslopes
• median_test
• median_abs_deviation
• epps_singleton_2samp
• levene (optional)
• fligner (optional)
• sen_seasonal_slopes

I am not interested in supporting the following functions: bayes_mvs, mvsdist, the frequency statistics, weightedtau, somersd or other tabular methods, multiscale_graphcorr.

Many other functions are not listed here because they really need special function support to be useful.

I wrote the following functions, so I'd prefer to do the upgrades on those personally.

• monte_carlo_test (ENH: stats.monte_carlo_test: add array API support #20604)
• permutation_test
• bootstrap
• goodness_of_fit
• power
• false_discovery_control
• differential_entropy

[lucascolley]

add N-D support to _array_api.cov; consider making it public if array API won't offer it

Public in SciPy seems less than ideal (my feeling is that cov belongs more on the array library level). Upstreamed into either the standard or a new library for these type of standard-extensions seems to make more sense, unless the hypothetical SciPy version of cov would have additional functionality which would be out of scope for the array libraries.