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1733Why Guardrail Metrics Matter Even When the Primary Metric WinsWhy is a primary-metric win not enough to ship an experiment if latency, complaints, or cancellation rates deteriorate?统计中等essay未尝试面试订阅1735Why Ratio Metrics Need More Care Than CountsWhy do ratio metrics such as revenue per active user often require more design care than simple counts?统计中等essay未尝试面试订阅1756Backing Out Omitted Covariance From a Slope DropA desk regresses slippage Y on inventory pressure X. Without an urgency control, the OLS slope on X is 0.90. After adding a perfect measure of urgency U, the slope falls to 0.60. Suppose the structural model is Y = beta X + 0.5 U + noise and Var(X)=1. What is Cov(X,U)?统计简单derivation未尝试免费1757Recovering Signal-to-Noise From Two Attenuated SlopesA latent factor X* is measured twice: X1 = X* + e1 and X2 = X* + e2, where e1 and e2 are independent classical errors with the same variance and are independent of X*. Regressing Y on X1 alone gives slope 0.80. Regressing Y on the average (X1+X2)/2 gives slope 1.00. Under the same true structural slope beta, what is Var(X*)/Var(e)?统计简单derivation未尝试免费1759Averaging Two Noisy MeasurementsAgain the structural model is Y=2X+u with E[u\mid X]=0, but now you observe two noisy proxies: W 1=X+\eta 1, \qquad W 2=X+\eta 2, where \eta 1,\eta 2 are independent of each other and of X,u. Suppose Var (X)=4 and each noise term has variance 1. If you regress Y on the average proxy W=(W 1+W 2)/2, what is the probability limit of the slope?统计中等derivation未尝试面试订阅1760Wald Ratio With an Explicit Sign ConventionAn exchange latency shock Z in 0,1 moves the fraction of aggressive orders from 0.30 when Z=0 to 0.18 when Z=1, and average slippage from 4.2 bps when Z=0 to 5.4 bps when Z=1. Using the consistent orientation Delta Y / Delta X = (E[Y|Z=1]-E[Y|Z=0]) / (E[X|Z=1]-E[X|Z=0]), what Wald IV estimate do you get for slippage per one-unit increase in aggressive-order fraction?统计中等derivation未尝试面试订阅1761Direct and Total Effect After a Routing Channel SplitA routing signal X increases child-order fragmentation M by 2 units on average. The outcome obeys Y = 1.3 X + 0.4 M + noise, with X otherwise exogenous. What coefficient on X would you expect in a regression that controls for M, and what total effect of a one-unit increase in X would appear in a regression of Y on X alone?统计简单derivation未尝试免费1766What Remains After Controlling the Hedge ChannelA signal X changes the hedge ratio H immediately, and both X and H affect desk PnL Y. The direct effect of X on Y is +1.2 bps, while the channel through H contributes another +0.8 bps. If H is measured perfectly and you regress Y on X and H, what effect does the coefficient on X identify, and what number should it equal?统计简单derivation未尝试免费1767Weak-IV Risk From the First Stage AloneA proposed instrument shifts treatment by 0.02 with standard error 0.015 in the first stage. Even if the exclusion story sounds plausible, what first-stage F-statistic do you get, and what is the main identification concern?统计中等derivation未尝试面试订阅1776Lasso Threshold Calibration 1A standardized lasso fit has score vector (4.1, 2.3, 1.7). What is the smallest lambda that makes every coefficient exactly zero?统计中等derivation未尝试免费1778Lasso Threshold Calibration 3In an orthonormal lasso update, a coordinate has score z = 2.6 and penalty lambda = 1.1. What coefficient results after soft-thresholding?统计中等数值题未尝试免费1781One-SE Lambda Choice 1A cross-validation table for a regularized alpha model reports (lambda, mean error, standard error) = [(0.01, 0.42, 0.02), (0.1, 0.41, 0.015), (1.0, 0.423, 0.01)]. Using the one-standard-error rule, which lambda should you choose?统计简单数值题未尝试免费1790Ridge Effective Degrees of Freedom 5A standardized ridge model has singular-value squares d j 2 = [12.25, 4, 0.25] and penalty lambda = 0.25. What is the effective degrees of freedom tr(S lambda) = sum d j 2/(d j 2+lambda)?统计困难derivation未尝试面试订阅1791Scaling Before Lasso on Mixed UnitsA signal library mixes raw prices, basis-point spreads, and z-scored microstructure features. Why should the team standardize features before running Lasso?统计简单derivation未尝试免费1796Signal Stationarity Classification 1A candidate signal is defined by X t = ε t + 0.3 ε t-1 . Is it weakly stationary?统计简单essay未尝试免费1797Signal Stationarity Classification 2A candidate signal is defined by X t = 0.2 t + ε t. Is it weakly stationary?统计中等derivation未尝试免费1798Signal Stationarity Classification 3A candidate signal is defined by X t = ε t + s t where s t is a fixed deterministic day-of-week pattern. Is it weakly stationary?统计简单derivation未尝试免费1799Signal Stationarity Classification 4A candidate signal is defined by X t = A cos(ω t) + B sin(ω t), where E[A]=E[B]=0, Var(A)=Var(B), Cov(A,B)=0. Is it weakly stationary?统计中等derivation未尝试免费1801Measurement Noise Effect 1A latent stationary signal Y t has gamma(0) = 5 and gamma(1) = 2. You observe X t = Y t + eta t, where eta t is iid noise with variance 1.5 independent of Y t. What are gamma X(0) and gamma X(1)?统计简单essay未尝试免费1802Measurement Noise Effect 2A latent stationary signal Y t has gamma(0) = 8 and gamma(1) = 3. You observe X t = Y t + eta t, where eta t is iid noise with variance 2 independent of Y t. What are gamma X(0) and gamma X(1)?统计中等derivation未尝试免费