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VaR 预测回溯检验：Pinball（分位）损失评分

Pinball (Quantile) Loss for VaR Forecast Backtesting

题目类型

算法 · Array 算法 · Numerical Methods 量化 · Backtesting 量化 · Risk 适配身份 · Risk Quant

请实现 solution(realized_pnls: list[float], var_forecasts: list[float], alpha: float) -> float。风险研究团队的 VaR 仪表盘把这套例程当作严格 proper 评分规则的横向竞速指标：在 VaR 回溯检验诸指标中，pinball 比基于计数的 Kupiec POF、基于分区的 Basel 红绿灯更适合做模型横评，因为 pinball 是分位预测的严格 proper 评分规则——pinball 最低的模型即最准确的分位预报者。给定长度同为 T 的两条日序列——日实现 PnL（有符号；负为亏损）与日 VaR 预测（在置信水平 alpha 下的正亏损数值）——以及置信水平 alpha，请计算每日平均 pinball 损失：

tau       = 1 - alpha                         # 尾部概率（如 0.01）
q[t]      = -var_forecasts[t]                  # 有符号 PnL 的 tau 分位
residual  = realized_pnls[t] - q[t]
loss[t]   = tau * residual                   if residual >= 0
loss[t]   = (1 - tau) * (-residual)          if residual < 0
PinballLoss = mean(loss[t])                   # 即 sum(loss[t]) / T

等价的紧凑写法：loss[t] = max(tau * residual, -(1 - tau) * residual)——非对称 V 形（"pinball" 之名取自一长一短两臂的 V，短臂为保守便宜侧、长臂为尾部错过昂贵侧）。算法是单趟 O(T) 扫描，维护一个累加和并在末尾做一次除法。

例

solution([50000.0, -120000.0, 30000.0, -180000.0, -60000.0], [100000.0, 100000.0, 100000.0, 100000.0, 100000.0], 0.99) 返回 20440.0。5 日窗口，alpha=0.99（tau=0.01）下恒为 10万 VaR；有符号 PnL 分位 q=-100000。五日残差为 [150000, -20000, 130000, -80000, 40000]。残差 >= 0 的日子是过报（实现优于尾部），按轻权重 tau = 0.01 计：0.01 * [150000, 130000, 40000] = [1500, 1300, 400]。残差 < 0 的日子是尾部错过，按重权重 1 - tau = alpha = 0.99 计：0.99 * [20000, 80000] = [19800, 79200]。总和 1500 + 19800 + 1300 + 79200 + 400 = 102200；除以 5 日得平均 102200 / 5 = 20440.0。两个尾部错过日主导评分——正是下尾 pinball 的设计意图。

五个陷阱

第一个陷阱是 tau = 1 - alpha，而不是 tau = alpha。alpha = 0.99 的 VaR 是有符号 PnL 分布的 1% 尾部——即 tau = 0.01 分位，不是 tau = 0.99 分位。设 tau = alpha 会让 V 形两条斜率对调：过报本应乘 tau 却乘 alpha；尾部错过本应乘 (1 - tau) = alpha 却乘 tau。alpha = 0.99 下数值偏差可达 alpha / (1 - alpha) ≈ 99 倍。

第二个陷阱是有符号分位 q = -var_forecasts[t]。var_forecast 是正的亏损量级（99% 1 日 VaR 的 25 万美元写作 250000.0，不是 -250000.0）；对应的有符号 PnL 的 tau 分位是其取负。直接把 var_forecasts[t] 当分位会把有符号 PnL 与一个本应为负的正数比较，每一天的残差方向都搞错。

第三个陷阱是不对称惩罚方向。对下尾 tau 分位，标准 pinball 在 residual >= 0 一侧用 tau（过报，轻），在 residual < 0 一侧用 (1 - tau)（尾部错过，重）。residual >= 0 表示实现 PnL 优于尾部预测，即银行过报了尾部；residual < 0 表示尾部被错过，即实现劣于预测。alpha = 0.99（tau = 0.01）下，两个权重为 0.01（轻）与 0.99（重）。其经济含义是：希望 VaR 适度保守（轻微过报便宜），同时严罚错过尾部（欠报付重）——这正是 pinball 作为 tau 分位严格 proper 评分规则的所在。

第四个陷阱是残差符号约定：residual = realized_pnls[t] - q[t]，不是 q[t] - realized_pnls[t]。颠倒减法会让 V 形上下翻转——本应陡的那一侧变缓、反之亦然。这与 tau-vs-alpha 互换有时会偶然抵消通过单测，因此本题的对抗用例分别独立钉死每一陷阱。

第五个陷阱是取均不取和。输出是每日平均 sum(loss) / T，不是总和。这是严格 proper 评分规则文献中的标准约定：均值形式才能让 pinball 在不同窗口长度下直接可比（同一模型 T = 250 与 T = 1000 应给出可比分数，不应相差 4 倍）。

前哨与边界

T == 0（空窗）：返回 0.0（空均值约定；两数组保证等长，所以空 realized_pnls 对应空 var_forecasts）。除零回避之外无其他特例。

残差恰为 0（realized_pnls[t] == -var_forecasts[t]）：V 形两侧同得 loss[t] = 0。>= 0 与 < 0 的分支无差。

整窗均过报（每日 residual >= 0）：答案化为 tau * mean(residual) = (1 - alpha) * mean(residual)。

整窗均尾部错过（每日 residual < 0）：答案化为 (1 - tau) * mean(|residual|) = alpha * mean(|residual|)。

alpha = 0.5：tau = 0.5，V 对称，loss[t] = 0.5 * |residual|——pinball 化为以 q = -var_forecasts[t] 为中心的半-平均绝对误差。此为中位数评分极端。

alpha 接近 1.0（如 0.999）：过报惩罚趋近 0（tau ≈ 0），尾部错过惩罚趋近 |residual|（alpha ≈ 1）——非对称在高置信水平下最为极端。

长窗中的单日重大事件：均值会把代价稀释到 T 日，所以同样的 -100 万对 10 万 VaR 的破口在 T = 1 下得 891000，在 T = 1500 下只得 594.0。这种稀释正是该指标取均（而非取和）的目的。

实践背景

Pinball 损失是分位预测的严格 proper 评分规则经典代表（Gneiting 与 Raftery 2007，《Strictly Proper Scoring Rules, Prediction, and Estimation》）；在 VaR 回溯检验诸指标中是 VaR 研究仪表盘上首选的横向竞速指标，因为它直接根据被预测的真实分位评分，而非粗粒度的计数或分区摘要。Basel 红绿灯把整年的破口数概括为绿/黄/红三档分类；Kupiec POF 检验边际破口率是否符合名义；Christoffersen 系列检验独立性与联合覆盖率；Lopez-I 量级损失是带平方超额的破口加权计数；pinball 则是其严格 proper 评分规则的同胞——其期望恰在真实条件分位处取得最小，因此是优化 VaR 模型的"正确损失"。本题是题库中严格 proper 评分规则这一支：与 Basel 红绿灯（分类、仅计数）、破口聚簇计数（描述性）、Kupiec POF（似然比、边际率）、Christoffersen 独立性／联合覆盖率似然比、以及 Lopez-I 量级损失（破口计数加平方超额）都不同——每一个都是 VaR 仪表盘上的不同部件。

约束条件

0 <= len(realized_pnls) == len(var_forecasts) <= 1500
对所有 t，|realized_pnls[t]| <= 1e6
对所有 t，0.0 <= var_forecasts[t] <= 1e6
0.5 <= alpha < 1.0
输出：float —— 非负的每日平均 pinball 损失；rel_tol=1e-9, abs_tol=1e-9

样例

Case 1 · statement-example: 5-day window with alpha=0.99 and mixed residuals

输入: [[50000,-120000,30000,-180000,-60000],[100000,100000,100000,100000,100000],0.99]

期望: 20440

5 日窗口，alpha=0.99（tau=0.01）下恒为 10万 VaR。q=-100000，残差=[150000,-20000,130000,-80000,40000]。残差>=0 的三天（过报，轻惩）loss=tau*residual=0.01*[150000,130000,40000]=[1500,1300,400]；残差<0 的两天（尾部错过，重惩）loss=alpha*|residual|=0.99*[20000,80000]=[19800,79200]。总和 102200，平均 20440.0。

Case 2 · visible: empty backtest window

输入: [[],[],0.99]

期望: 0

T=0：空平均约定返回 0.0。无任何日，无损失。

Case 3 · visible: residual exactly zero — pinball loss is zero

输入: [[-100],[100],0.99]

期望: 0

单一日 realized_pnl=-100、var=100，q=-100。残差=0，V 形两分支都为 0。T=1 平均 0.0。陷阱演示：反向残差者本题仍得 0，但其他用例可区分。

Case 4 · visible: single tail miss at alpha=0.99 — heavy penalty alpha

输入: [[-200],[100],0.99]

期望: 99

T=1，q=-100，残差=-100（尾部被错过）。下尾标准 pinball：loss=(1-tau)*|residual|=alpha*100=0.99*100=99.0。T=1 平均 99.0。交换 tau 与 alpha 者得 1.0——相差 99 倍。

Case 5 · visible: single over-forecast at alpha=0.95 — light penalty tau

输入: [[10],[50],0.95]

期望: 3.0000000000000027

T=1，q=-50，残差=10-(-50)=60>0（实现优于尾部预测——过度预报）。loss=tau*residual=(1-alpha)*60=0.05*60=3.0。过报惩罚轻——保守成本低；同量级的尾部错过将付 0.95*60=57.0（重）。

Case 6 · visible: alpha=0.5 reduces to half mean-absolute-error

输入: [[-1,1,-2,2],[0,0,0,0],0.5]

期望: 0.75

alpha=0.5 使 tau=0.5，V 形对称：每日 loss=0.5*|residual|。v=0 则 q=0，残差=[-1,1,-2,2]，loss=[0.5,0.5,1.0,1.0]，平均 0.75。这是中位数评分函数（MAE 的一半）。

Case 7 · visible: signed-quantile pitfall — q is the NEGATION of VaR

输入: [[10],[100],0.99]

期望: 1.100000000000001

T=1，alpha=0.99，v=100。有符号 PnL 分位为 q=-100（不是 q=+100）。残差=10-(-100)=110>0（过报），loss=tau*110=0.01*110=1.1。以 q=+100 者得残差=-90、loss=alpha*90=89.1——完全错误。

Case 8 · visible: equal-magnitude residuals — tau-vs-alpha swap is invisible here

输入: [[100,-300],[100,100],0.99]

期望: 100

两日，alpha=0.99（tau=0.01）。q=-100。第0日残差=200，loss=tau*200=0.01*200=2；第1日残差=-200，loss=alpha*200=0.99*200=198。平均 100.0。等量级时 tau<->alpha 互换不可见；下题以不等量级揭示之。

Case 9 · visible: asymmetric residual magnitudes — distinguishes correct alpha-vs-tau weighting

输入: [[10,-200,30],[100,100,100],0.99]

期望: 33.8

三日，alpha=0.99。q=-100。残差=[110,-100,130]，loss=tau*[110,-,130]+alpha*[100]=[1.1,99,1.3]，平均 33.8。交换 tau 与 alpha 者得 80.0——可区分。

Case 10 · visible: average-not-sum — output is mean per day, not the total

输入: [[10,10,10,10],[0,0,0,0],0.95]

期望: 0.5000000000000004

T=4，alpha=0.95。q=0，残差=10（过报）。每日 loss=tau*10=(1-alpha)*10=0.05*10=0.5；四日平均 0.5。输出总和者得 2.0——偏差 4 倍。取均使该指标在不同窗口长度下可比。

Case 11 · visible: residual sign — realized minus quantile, not the other way

输入: [[200],[50],0.99]

期望: 2.500000000000002

T=1，alpha=0.99，v=50。q=-50，残差=realized-q=200-(-50)=250>=0（过报），loss=tau*250=0.01*250=2.5。反向残差（q-realized）者得 -250、loss=alpha*250=247.5——偏差 99 倍。

Case 12 · visible: all-tail-miss — collapses to alpha*mean(|residual|)

输入: [[-200,-300,-150],[100,100,100],0.99]

期望: 115.5

三日均为尾部错过。q=-100，残差=[-100,-200,-50]，loss=alpha*[100,200,50]=[99,198,49.5]，平均 346.5/3=115.5。这与 alpha*mean(|residual|) 一致。

Case 13 · visible: all-over-forecast — collapses to tau*mean(residual)

输入: [[0,50,100],[50,50,50],0.95]

期望: 5.000000000000004

T=3，alpha=0.95（tau=0.05）。q=-50，残差=[50,100,150]均为正。loss=tau*[50,100,150]=[2.5,5,7.5]，平均 5.0。与 tau*mean(residual)=0.05*100 一致。

Case 14 · visible: alpha very close to 1.0 — over-forecast cost approaches zero

输入: [[10],[0],0.999]

期望: 0.010000000000000009

alpha=0.999（tau=0.001），v=0 则 q=0，残差=10（过报）。loss=tau*residual=0.001*10=0.01。alpha 趋近 1 时过报惩罚趋近 0，尾部错过惩罚趋近 |residual|，高置信下保守几乎零成本。

Case 15 · visible: alpha=0.5 with non-zero VaR — symmetric V at q=-VaR

输入: [[0,0],[10,10],0.5]

期望: 5

alpha=0.5，q=-10，残差=[10,10]均为正。对称 V：loss=0.5*|residual|=5，平均 5.0。即使在 alpha=0.5 时仍需 q=-VaR；对称 V 下残差反向同值是巧合，其他用例可区分。

Case 16 · visible: single catastrophic day diluted across T=1500

输入: 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期望: 594

T=1500。第0日 realized=-1e6、v=1e5、q=-1e5，残差=-9e5（尾部错过），loss=alpha*9e5=0.99*9e5=891000。余下各日 residual=0 损失为 0。平均 891000/1500=594.0。陈列取均使同一重大事件在不同窗长下可比。

Case 17 · visible: large window with alternating over-forecast and exact match

输入: 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期望: 1.0000000000000002

T=1500。q=-100。偶数日残差=200（过报），loss=tau*200=0.01*200=2；奇数日残差=0，loss=0。平均 2/2=1.0。

Case 18 · visible: zero-residual constant window — total loss is exactly zero

输入: [[-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50,-50],[50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50,50],0.99]

期望: 0

T=1500，每日 realized=-50、v=50，q=-50，残差=0、loss=0。平均 0.0。验证长窗零损失下不产生漂移。

Case 19 · visible: 4-day typical case at alpha=0.95 with mixed residuals

输入: [[200,-150,50,-300],[100,100,100,100],0.95]

期望: 65

T=4，alpha=0.95（tau=0.05）。q=-100，残差=[300,-50,150,-200]。loss=[15,47.5,7.5,190]，平均 260/4=65.0。

Case 20 · visible: 2-day window at alpha=0.975

输入: [[100,-200],[100,100],0.975]

期望: 51.25

T=2，alpha=0.975（tau=0.025）。q=-100。第0日残差=200（过报），loss=tau*200=5；第1日残差=-100（尾部错过），loss=alpha*100=97.5。平均 51.25。

Case 21 · visible: alpha=0.9 with paired residuals

输入: [[20,-40],[10,10],0.9]

期望: 15

T=2，alpha=0.9（tau=0.1）。q=-10。第0日残差=30（过报），loss=tau*30=3；第1日残差=-30（尾部错过），loss=alpha*30=27。平均 15.0。

Case 22 · visible: cents-level magnitudes preserve scale invariance

输入: [[0.01,-0.02,0.03],[0,0,0],0.99]

期望: 0.006733333333333334

T=3，alpha=0.99，v=0 则 q=0。残差=[0.01,-0.02,0.03]，loss=[0.0001,0.0198,0.0003]，平均 ≈ 0.006733。损失与量级线性同增。

Case 23 · visible: all-zero series at alpha=0.99

输入: [[0,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0],0.99]

期望: 0

T=10，所有 PnL 与预报为 0。q=0，残差=0，loss=0。平均 0.0。平凡但可捔平公式中意外常项。

Case 24 · visible: T=4 chain of tail misses with alpha=0.99

输入: [[-50,-100,-150,-200],[10,10,10,10],0.99]

期望: 113.85

T=4，alpha=0.99，q=-10。残差=[-40,-90,-140,-190]均为负（全为尾部错过）。loss=alpha*[40,90,140,190]=[39.6,89.1,138.6,188.1]，平均 455.4/4=113.85。

Case 25 · visible: stratified T=10 — half quiet, half tail-miss

输入: [[-100,-100,-100,-100,-100,-200,-200,-200,-200,-200],[100,100,100,100,100,100,100,100,100,100],0.99]

期望: 49.5

T=10，alpha=0.99，q=-100。前 5 日残差=0 损失 0；后 5 日残差=-100（尾部错过）损失 99。平均 5*99/10=49.5。尾部错过占比通过取均被稀释。

Case 26 · visible: single $1M over-forecast at alpha=0.99

输入: [[1000000],[0],0.99]

期望: 10000.00000000001

T=1，v=0 则 q=0，残差=1e6（过报）。loss=tau*residual=0.01*1e6=10000。T=1 平均 10000.0（浮点意义下）。高 alpha 下 tau 小，过报量级即使到上限惩罚仍较轻。

Case 27 · visible: single -$1M tail miss against $100k VaR at alpha=0.99

输入: [[-1000000],[100000],0.99]

期望: 891000

T=1，q=-1e5，残差=-9e5（尾部错过）。loss=alpha*9e5=0.99*9e5=891000。T=1 平均 891000.0。尾部错过权重 alpha 体现 99% VaR 失败的高代价。

例

五个陷阱

前哨与边界

实践背景

约束条件

样例

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