Add quantile function for Student-T distribution

[andrjohns]

Summary

This PR adds the (vectorised) student-t quantile function with signature:

reals student_t_qf(reals p, reals nu, reals mu, reals sigma)

The current implementation directly estimates the quantile using the inverse of the incomplete beta, but implementing approximations (like those used by R) are a planned future improvement

Tests

Both prim and mix tests (using the AD framework) are added

Side Effects

An additional utility type has been added: common_container_type to detect the right container type to return for vectorised inputs.

Release notes

Added quantile function for student-t distribution

Checklist

• [x] Copyright holder: Andrew Johnson

  The copyright holder is typically you or your assignee, such as a university or company. By submitting this pull request, the copyright holder is agreeing to the license the submitted work under the following licenses: - Code: BSD 3-clause (https://opensource.org/licenses/BSD-3-Clause) - Documentation: CC-BY 4.0 (https://creativecommons.org/licenses/by/4.0/)

• [x] the basic tests are passing

  • unit tests pass (to run, use: ./runTests.py test/unit)
  • header checks pass, (make test-headers)
  • dependencies checks pass, (make test-math-dependencies)
  • docs build, (make doxygen)
  • code passes the built in C++ standards checks (make cpplint)

• [x] the code is written in idiomatic C++ and changes are documented in the doxygen

• [x] the new changes are tested

[stan-buildbot]

---

Name	Old Result	New Result	Ratio	Performance change( 1 - new / old )
arma/arma.stan	0.33	0.38	0.89	-12.78% slower
low_dim_corr_gauss/low_dim_corr_gauss.stan	0.01	0.01	1.01	0.86% faster
gp_regr/gen_gp_data.stan	0.02	0.02	1.07	6.57% faster
gp_regr/gp_regr.stan	0.09	0.09	1.07	6.17% faster
sir/sir.stan	70.97	67.96	1.04	4.24% faster
irt_2pl/irt_2pl.stan	4.2	3.95	1.06	6.1% faster
eight_schools/eight_schools.stan	0.06	0.05	1.05	4.48% faster
pkpd/sim_one_comp_mm_elim_abs.stan	0.25	0.24	1.04	4.17% faster
pkpd/one_comp_mm_elim_abs.stan	20.2	18.71	1.08	7.41% faster
garch/garch.stan	0.43	0.41	1.04	4.09% faster
low_dim_gauss_mix/low_dim_gauss_mix.stan	2.75	2.63	1.04	4.23% faster
arK/arK.stan	1.82	1.74	1.05	4.46% faster
gp_pois_regr/gp_pois_regr.stan	2.94	2.75	1.07	6.69% faster
low_dim_gauss_mix_collapse/low_dim_gauss_mix_collapse.stan	8.85	9.91	0.89	-11.94% slower
performance.compilation	177.48	177.46	1.0	0.01% faster
Mean result: 1.0273234881588633

---

Jenkins Console Log Blue Ocean Commit hash: 9d0e74af54c729b15ce95fcce935a889c4306822

---

Machine information

No LSB modules are available. Distributor ID: Ubuntu Description: Ubuntu 20.04.3 LTS Release: 20.04 Codename: focal

CPU: Architecture: x86_64 CPU op-mode(s): 32-bit, 64-bit Byte Order: Little Endian Address sizes: 46 bits physical, 48 bits virtual CPU(s): 80 On-line CPU(s) list: 0-79 Thread(s) per core: 2 Core(s) per socket: 20 Socket(s): 2 NUMA node(s): 2 Vendor ID: GenuineIntel CPU family: 6 Model: 85 Model name: Intel(R) Xeon(R) Gold 6148 CPU @ 2.40GHz Stepping: 4 CPU MHz: 2400.000 CPU max MHz: 3700.0000 CPU min MHz: 1000.0000 BogoMIPS: 4800.00 Virtualization: VT-x L1d cache: 1.3 MiB L1i cache: 1.3 MiB L2 cache: 40 MiB L3 cache: 55 MiB NUMA node0 CPU(s): 0,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78 NUMA node1 CPU(s): 1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39,41,43,45,47,49,51,53,55,57,59,61,63,65,67,69,71,73,75,77,79 Vulnerability Gather data sampling: Mitigation; Microcode Vulnerability Itlb multihit: KVM: Mitigation: VMX disabled Vulnerability L1tf: Mitigation; PTE Inversion; VMX conditional cache flushes, SMT vulnerable Vulnerability Mds: Mitigation; Clear CPU buffers; SMT vulnerable Vulnerability Meltdown: Mitigation; PTI Vulnerability Mmio stale data: Mitigation; Clear CPU buffers; SMT vulnerable Vulnerability Reg file data sampling: Not affected Vulnerability Retbleed: Mitigation; IBRS Vulnerability Spec rstack overflow: Not affected Vulnerability Spec store bypass: Mitigation; Speculative Store Bypass disabled via prctl Vulnerability Spectre v1: Mitigation; usercopy/swapgs barriers and __user pointer sanitization Vulnerability Spectre v2: Mitigation; IBRS; IBPB conditional; STIBP conditional; RSB filling; PBRSB-eIBRS Not affected; BHI Not affected Vulnerability Srbds: Not affected Vulnerability Tsx async abort: Mitigation; Clear CPU buffers; SMT vulnerable Flags: fpu vme de pse tsc msr pae mce cx8 apic sep mtrr pge mca cmov pat pse36 clflush dts acpi mmx fxsr sse sse2 ss ht tm pbe syscall nx pdpe1gb rdtscp lm constant_tsc art arch_perfmon pebs bts rep_good nopl xtopology nonstop_tsc cpuid aperfmperf pni pclmulqdq dtes64 monitor ds_cpl vmx smx est tm2 ssse3 sdbg fma cx16 xtpr pdcm pcid dca sse4_1 sse4_2 x2apic movbe popcnt tsc_deadline_timer aes xsave avx f16c rdrand lahf_lm abm 3dnowprefetch cpuid_fault epb cat_l3 cdp_l3 invpcid_single pti intel_ppin ssbd mba ibrs ibpb stibp tpr_shadow vnmi flexpriority ept vpid ept_ad fsgsbase tsc_adjust bmi1 hle avx2 smep bmi2 erms invpcid rtm cqm mpx rdt_a avx512f avx512dq rdseed adx smap clflushopt clwb intel_pt avx512cd avx512bw avx512vl xsaveopt xsavec xgetbv1 xsaves cqm_llc cqm_occup_llc cqm_mbm_total cqm_mbm_local dtherm ida arat pln pts hwp hwp_act_window hwp_epp hwp_pkg_req pku ospke md_clear flush_l1d arch_capabilities

G++: g++ (Ubuntu 9.4.0-1ubuntu1~20.04) 9.4.0 Copyright (C) 2019 Free Software Foundation, Inc. This is free software; see the source for copying conditions. There is NO warranty; not even for MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.

Clang: clang version 10.0.0-4ubuntu1 Target: x86_64-pc-linux-gnu Thread model: posix InstalledDir: /usr/bin

[spinkney]

Thanks @andrjohns for tackling this! Do you happen to know any particular ranges which are problematic for the current implementation?

[andrjohns]

Thanks @andrjohns for tackling this! Do you happen to know any particular ranges which are problematic for the current implementation?

I haven't done any proper testing at this point, it was more just getting an initial implementation in to be refined. Because it's just directly estimating, I'd say that the problematic ranges for both the values and gradients are going to mostly match those for boost's incomplete beta inverse

Definitely want to move to more stable approximations for both the gradient and value at some point though!

[andrjohns]

@spinkney on a semi-related note - after being annoyed by the need to compile c++ whenever I'm validating against Boost's Math functions, I've started exposing them in an R package: https://github.com/andrjohns/boostmath

Just an FYI in case it helps with any of your validation/testing as well!

[spinkney]

That's great!