P-Square (P²) 流式分位数估计
P-Square (P²) Streaming Quantile Estimator
开始编码你正在搭建一条流式 p95 / p99 延迟监控仪表盘。延迟事件以一长串浮点数到达 —— 每个抓取周期上百万条 —— 你需要在流尾给出分位数估计，但不能把事件全部缓存下来。排序加索引可以拿到精确值，但内存随样本数线性增长；你要的是一个固定占用的估计器。经典答案就是 Jain & Chlamtac (1985) 的 P² (P-square)：5 个标记高度、5 个标记位置，O(1) 内存、每个样本 O(1) 计算量，稳态下相对误差约 1% —— 正是所有生产级遥测栈都会做的取舍。
实现 solution(values: list[float], target_quantile: float) -> float，返回 P² 估计的分位数（例如 target_quantile = 0.95 即 95% 分位）。5 个标记追踪：
| 下标 | 追踪分位 | 角色 |
|----:|:--------------:|:-----------------|
| 0 | min | 运行最小值 |
| 1 | p / 2 | 左内部插值点 |
| 2 | p | 答案就是它 |
| 3 | (1 + p) / 2 | 右内部插值点 |
| 4 | max | 运行最大值 |
每个标记还有一个*期望*实数位置，按每样本固定增量 dn' = (0, p/2, p, (1+p)/2, 1) 增长。用前 5 个排序观测初始化标记之后，后续每个值触发三步更新：
- 定位：找到
k使得q[k] ≤ x < q[k+1]（必要时钳到极值格）。 - 计数：把该格右侧所有标记的
n[i]各加 1，并把每个n'[i]各加dn'[i]。 - 调整：对每个内部标记
i ∈ {1, 2, 3}，若|n'[i] − n[i]| ≥ 1且邻居留有空间，把标记沿±1方向滑动 1 步。用三个邻点做抛物线插值重算高度；若抛物线预测会破坏单调序q[i-1] < q[i] < q[i+1]，沿移动方向做线性插值作为兜底。把n[i]也按±1更新。
流处理结束后，返回 q[2]。
对 len(values) ≤ 5，算法无法启动（需要至少 5 个标记），所以规范要求走精确回退：排序后线性插值。具体地，给定排序后的 s[0..n-1]，分位 p 处的结果为 s[lo] + frac · (s[lo+1] − s[lo])，其中 lo = floor((n−1) · p)、frac = (n−1) · p − lo（numpy 默认的 'linear' 方法，对应 Hyndman–Fan 分类里的 type-7）。
例如，solution([1.0, 2.0, 3.0, 4.0, 5.0], 0.5) 返回 3.0（n=5 走精确路径，5 个值的中位数即排序后中间那一个）；solution([1.0, 2.0, 3.0, 4.0, 5.0], 0.95) 返回 4.8（线性插值，分数索引 4 · 0.95 = 3.8，即 4 + 0.8 · 1 = 4.8）；solution([7.5] * 200, 0.95) 精确返回 7.5，因为所有标记初始化到同一高度，抛物线/线性更新分子都是零，没有标记会移动。
实践背景
P² 是 Prometheus summary、OpenTelemetry histogram-with-quantiles、DataDog agent.histogram、以及大多数商业 APM 仪表盘背后的教科书配方。原因纯属工程：延迟仪表盘通常要在每分钟数亿条事件上算 p50、p95、p99，原始样本流既存不下，agent 内存有限，网络也带不动，更别说每个抓取周期都做一遍排序。P² 给出常数内存（每条序列每个分位 5 个浮点 + 5 个计数器）与常数级单事件计算量，稳态相对误差 ≲1%，远小于延迟本身的抖动。同样的理由让它在买方/卖方遥测里也很有用：订单簿事件循环延迟、交易所往返时间、风险引擎评估耗时。两个需要心里有数（且单元测试覆盖到）的失败模式：极小流（n ≤ 5 填不满 5 个标记，因此规范有显式的排序回退路径），以及病态单调输入（标记每一步都被推动）；标准的"抛物线 + 线性兜底"更新规则把这两类都干净处理掉了。
约束条件
- 0 < `len(values)` ≤ 10000；values 为有限浮点（无 NaN、无 inf）
- `target_quantile` 是 (0, 1) 开区间内的浮点，**两端均不取**
- Empty `values` 抛 `ValueError`
- `target_quantile` 不在 (0, 1) 抛 `ValueError`
- values 全部相等时返回该值本身
- 若 `len(values) ≤ 5`，按规范走精确路径：排序后线性插值（numpy 默认的 'linear' 插值，分数索引 `(n-1) * p`）
- 若 `len(values) > 5`，运行完整的 P² 算法；输出为流处理结束时的标记 `q[2]`
- P² 是**近似**算法；比较容差 `rel_tol = 1e-2`、`abs_tol = 1e-2`
样例
Case 1 · statement-example: n=5 exact median
输入: [[1,2,3,4,5],0.5]
期望: 3
n=5 时 P^2 的 5 个标记还无法启动，按规范回退到精确分位数（先排序再线性插值）。中位数即排序后第 3 个值 3.0。
Case 2 · statement-example: n=5 q=0.95 fast path
输入: [[1,2,3,4,5],0.95]
期望: 4.8
n=5 走精确路径。线性插值的分数索引 = 4 * 0.95 = 3.8，结果 = sorted[3] + 0.8*(sorted[4]-sorted[3]) = 4 + 0.8*1 = 4.8。
Case 3 · statement-example: all-equal stream returns the constant
输入: [[7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5],0.95]
期望: 7.5
所有样本相同时，5 个标记的高度全部初始化为同一个值，后续 P^2 更新里 q[i] 的差全部为零，标记不动，最终估计就是该常数 7.5。
Case 4 · typical: gauss(0,1) n=1000 median
输入: 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期望: -0.022110311307862562
标准正态 n=1000 的真实中位数接近 0；P^2 流式估计在 1% 容差内匹配排序+线性插值的精确值。
Case 5 · boundary: sorted-ascending 1..1000 q=0.95
输入: [[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496,497,498,499,500,501,502,503,504,505,506,507,508,509,510,511,512,513,514,515,516,517,518,519,520,521,522,523,524,525,526,527,528,529,530,531,532,533,534,535,536,537,538,539,540,541,542,543,544,545,546,547,548,549,550,551,552,553,554,555,556,557,558,559,560,561,562,563,564,565,566,567,568,569,570,571,572,573,574,575,576,577,578,579,580,581,582,583,584,585,586,587,588,589,590,591,592,593,594,595,596,597,598,599,600,601,602,603,604,605,606,607,608,609,610,611,612,613,614,615,616,617,618,619,620,621,622,623,624,625,626,627,628,629,630,631,632,633,634,635,636,637,638,639,640,641,642,643,644,645,646,647,648,649,650,651,652,653,654,655,656,657,658,659,660,661,662,663,664,665,666,667,668,669,670,671,672,673,674,675,676,677,678,679,680,681,682,683,684,685,686,687,688,689,690,691,692,693,694,695,696,697,698,699,700,701,702,703,704,705,706,707,708,709,710,711,712,713,714,715,716,717,718,719,720,721,722,723,724,725,726,727,728,729,730,731,732,733,734,735,736,737,738,739,740,741,742,743,744,745,746,747,748,749,750,751,752,753,754,755,756,757,758,759,760,761,762,763,764,765,766,767,768,769,770,771,772,773,774,775,776,777,778,779,780,781,782,783,784,785,786,787,788,789,790,791,792,793,794,795,796,797,798,799,800,801,802,803,804,805,806,807,808,809,810,811,812,813,814,815,816,817,818,819,820,821,822,823,824,825,826,827,828,829,830,831,832,833,834,835,836,837,838,839,840,841,842,843,844,845,846,847,848,849,850,851,852,853,854,855,856,857,858,859,860,861,862,863,864,865,866,867,868,869,870,871,872,873,874,875,876,877,878,879,880,881,882,883,884,885,886,887,888,889,890,891,892,893,894,895,896,897,898,899,900,901,902,903,904,905,906,907,908,909,910,911,912,913,914,915,916,917,918,919,920,921,922,923,924,925,926,927,928,929,930,931,932,933,934,935,936,937,938,939,940,941,942,943,944,945,946,947,948,949,950,951,952,953,954,955,956,957,958,959,960,961,962,963,964,965,966,967,968,969,970,971,972,973,974,975,976,977,978,979,980,981,982,983,984,985,986,987,988,989,990,991,992,993,994,995,996,997,998,999,1000],0.95]
期望: 950
升序输入会让 P^2 的 95% 标记不停被推到右侧，最终落在接近 950 处，与精确值 950.05 相差不到 1%。
Case 6 · adversarial: 999 ones + 1 large outlier q=0.99
输入: [[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1000000],0.99]
期望: 1
999 个 1.0 加 1 个巨大值 1e6。99% 分位的精确值为 1.0（异常值落在 99% 分位之上），P^2 估计同样收敛到 1.0，体现出尾部估计对单点极值的鲁棒性。
最近提交
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编码区
实现 solution(...)。本地运行当前支持 Python 可见样例；服务端提交会运行可见样例和隐藏测试。
默认展示公开样例。点击「运行样例」后会在这里显示实际输出；点击「提交评测」会进入隐藏测试。
Case 1 · statement-example: n=5 exact median
输入: [[1,2,3,4,5],0.5]
期望: 3
n=5 时 P^2 的 5 个标记还无法启动，按规范回退到精确分位数（先排序再线性插值）。中位数即排序后第 3 个值 3.0。
Case 2 · statement-example: n=5 q=0.95 fast path
输入: [[1,2,3,4,5],0.95]
期望: 4.8
n=5 走精确路径。线性插值的分数索引 = 4 * 0.95 = 3.8，结果 = sorted[3] + 0.8*(sorted[4]-sorted[3]) = 4 + 0.8*1 = 4.8。
Case 3 · statement-example: all-equal stream returns the constant
输入: [[7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5,7.5],0.95]
期望: 7.5
所有样本相同时，5 个标记的高度全部初始化为同一个值，后续 P^2 更新里 q[i] 的差全部为零，标记不动，最终估计就是该常数 7.5。
Case 4 · typical: gauss(0,1) n=1000 median
输入: 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期望: -0.022110311307862562
标准正态 n=1000 的真实中位数接近 0；P^2 流式估计在 1% 容差内匹配排序+线性插值的精确值。
Case 5 · boundary: sorted-ascending 1..1000 q=0.95
输入: [[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496,497,498,499,500,501,502,503,504,505,506,507,508,509,510,511,512,513,514,515,516,517,518,519,520,521,522,523,524,525,526,527,528,529,530,531,532,533,534,535,536,537,538,539,540,541,542,543,544,545,546,547,548,549,550,551,552,553,554,555,556,557,558,559,560,561,562,563,564,565,566,567,568,569,570,571,572,573,574,575,576,577,578,579,580,581,582,583,584,585,586,587,588,589,590,591,592,593,594,595,596,597,598,599,600,601,602,603,604,605,606,607,608,609,610,611,612,613,614,615,616,617,618,619,620,621,622,623,624,625,626,627,628,629,630,631,632,633,634,635,636,637,638,639,640,641,642,643,644,645,646,647,648,649,650,651,652,653,654,655,656,657,658,659,660,661,662,663,664,665,666,667,668,669,670,671,672,673,674,675,676,677,678,679,680,681,682,683,684,685,686,687,688,689,690,691,692,693,694,695,696,697,698,699,700,701,702,703,704,705,706,707,708,709,710,711,712,713,714,715,716,717,718,719,720,721,722,723,724,725,726,727,728,729,730,731,732,733,734,735,736,737,738,739,740,741,742,743,744,745,746,747,748,749,750,751,752,753,754,755,756,757,758,759,760,761,762,763,764,765,766,767,768,769,770,771,772,773,774,775,776,777,778,779,780,781,782,783,784,785,786,787,788,789,790,791,792,793,794,795,796,797,798,799,800,801,802,803,804,805,806,807,808,809,810,811,812,813,814,815,816,817,818,819,820,821,822,823,824,825,826,827,828,829,830,831,832,833,834,835,836,837,838,839,840,841,842,843,844,845,846,847,848,849,850,851,852,853,854,855,856,857,858,859,860,861,862,863,864,865,866,867,868,869,870,871,872,873,874,875,876,877,878,879,880,881,882,883,884,885,886,887,888,889,890,891,892,893,894,895,896,897,898,899,900,901,902,903,904,905,906,907,908,909,910,911,912,913,914,915,916,917,918,919,920,921,922,923,924,925,926,927,928,929,930,931,932,933,934,935,936,937,938,939,940,941,942,943,944,945,946,947,948,949,950,951,952,953,954,955,956,957,958,959,960,961,962,963,964,965,966,967,968,969,970,971,972,973,974,975,976,977,978,979,980,981,982,983,984,985,986,987,988,989,990,991,992,993,994,995,996,997,998,999,1000],0.95]
期望: 950
升序输入会让 P^2 的 95% 标记不停被推到右侧，最终落在接近 950 处，与精确值 950.05 相差不到 1%。
Case 6 · adversarial: 999 ones + 1 large outlier q=0.99
输入: [[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1000000],0.99]
期望: 1
999 个 1.0 加 1 个巨大值 1e6。99% 分位的精确值为 1.0（异常值落在 99% 分位之上），P^2 估计同样收敛到 1.0，体现出尾部估计对单点极值的鲁棒性。