题目1719 · 统计
0.049 vs 0.051 Decision Cliff
Two backtests differ only slightly: one reports p = 0.049 and the other p = 0.051. Why is it bad practice to call one ‘real’ and the other ‘not real’ purely because one is below 0.05?
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中文题目题目3221 · 统计
A Confidence Interval Cannot Price a One-Off Launch Decision
A PM sees a frequentist 95% confidence interval for next-month strategy edge of [-0.1, 0.4] and asks, "So what is the probability the true edge is positive for this launch decision?" Why can't the interval answer that question by itself, and what Bayesian quantity would answer it
打开 →题目1768 · 统计
A Lagged Variable Is Not Automatically a Valid Instrument
Someone proposes using yesterday's order-flow imbalance as an instrument for today's imbalance in a return-impact regression. Why is this not automatically a valid instrument in financial data?
打开 →题目1653 · 统计
A Smoothed Bernoulli Estimator vs the Sample Proportion
Let $X\sim \mathrm{Binomial}(10,p)$ and consider the estimator $$\delta = \frac{X+1}{12}$$ for $p$. At the parameter value $p=0.2$, compute the bias, variance, and MSE of $\delta$, and compare its MSE with the usual sample proportion $\hat p = X/10$.
打开 →题目1762 · 统计
A Tiny First Stage Is a Weak-Instrument Warning
Two candidate rollouts have the same reduced-form impact on PnL: $$E[Y\mid Z=1]-E[Y\mid Z=0]=0.02.$$ For rollout A, the first stage is $0.20$; for rollout B, the first stage is $0.01$. Which rollout creates the weaker IV design, and why?
打开 →题目3162 · 统计
Allocate Capital Only Above a 60% Posterior
A latent profitable regime has prior probability $\frac{2}{5}$. Independent signals arrive sequentially; each `H` doubles the odds of the regime and each `T` halves them. After the signal string `HHT`, should you act if the required posterior threshold is $\frac{3}{5}$? Also repo
打开 →题目1661 · 统计
Anchor-Based Shrinkage Crossover Interval
Suppose Xbar ~ N(theta, 0.16). A desk uses delta = 0.6Xbar + 0.8. For what values of theta does delta have lower MSE than Xbar?
打开 →题目1697 · 统计
Any False Sector Winner After Within-Sector Mining
A desk has 12 sectors, each containing 5 genuinely null variants. In each sector it keeps only the smallest p-value, and it flags the sector if that winning p-value is below 1%. Assuming independence, what is the probability at least one sector is falsely flagged?
打开 →题目1689 · 统计
Approximate Fair-Coin Test with a Normal Approximation
A coin is flipped 20 times and lands heads 14 times. Use the normal approximation to test fairness at the 5% two-sided level.
打开 →题目1836 · 统计
AR(1) Forecast Error Variance 1
For the AR(1) model X_t = phi X_(t-1) + e_t with phi = 0.6 and Var(e_t) = 1, what is the h = 3 step forecast error variance?
打开 →题目1838 · 统计
AR(1) Forecast Error Variance 3
For the AR(1) model X_t = phi X_(t-1) + e_t with phi = 0.5 and Var(e_t) = 2.25, what is the h = 4 step forecast error variance?
打开 →题目1821 · 统计
AR(1) Multi-Step Forecast 1
A signal follows X_t = 0 + 0.6 X_(t-1) + e_t with Var(e_t) = 2 and current value X_t = 10. What is the h = 3 step forecast E[X_(t+3) | X_t]?
打开 →题目1822 · 统计
AR(1) Multi-Step Forecast 2
A signal follows X_t = 4 + 0.7 X_(t-1) + e_t with Var(e_t) = 1.5 and current value X_t = 8. What is the h = 2 step forecast E[X_(t+2) | X_t]?
打开 →题目1823 · 统计
AR(1) Multi-Step Forecast 3
A signal follows X_t = -1 + 0.8 X_(t-1) + e_t with Var(e_t) = 1 and current value X_t = 3. What is the h = 4 step forecast E[X_(t+4) | X_t]?
打开 →题目6026 · 统计
ARCH(1) as the Beta-Zero Special Case
A GARCH(1,1) reduces to ARCH(1) when $\beta=0$: $h_t=\omega+\alpha r_{t-1}^2$. With $\omega=0.7$ and $\alpha=0.3$, compute the unconditional variance $\bar h$ as a decimal.
打开 →题目1841 · 统计
ARMA Identification or Simplification 1
You observe the diagnostic statement: ACF tails off geometrically, PACF cuts after lag 1. What is the correct modeling conclusion?
打开 →题目1842 · 统计
ARMA Identification or Simplification 2
You observe the diagnostic statement: ACF cuts after lag 1, PACF tails off. What is the correct modeling conclusion?
打开 →题目1843 · 统计
ARMA Identification or Simplification 3
You observe the diagnostic statement: Both ACF and PACF tail off. What is the correct modeling conclusion?
打开 →题目1844 · 统计
ARMA Identification or Simplification 4
You observe the diagnostic statement: AIC prefers ARMA(2,1) but BIC prefers ARMA(1,1). What is the correct modeling conclusion?
打开 →题目1845 · 统计
ARMA Identification or Simplification 5
You observe the diagnostic statement: (1-0.5L) X_t = (1-0.5L) e_t. What is the correct modeling conclusion?
打开 →题目1883 · 统计
Average Gross Leverage at Launch
A live manager panel shows 27 low-leverage funds and 18 high-leverage funds. Survival rates for those groups were 90% and 60%, respectively. Suppose low-leverage funds average 1.2x gross leverage and high-leverage funds average 2.4x gross leverage. What was the average gross lev
打开 →题目1753 · 统计
Average Leverage and Average Residual-Maker Diagonal
A regression uses n = 25 observations and three estimated parameters including the intercept. What are (i) the average leverage and (ii) the average diagonal entry of the residual-maker matrix I - H?
打开 →题目1759 · 统计
Averaging Two Noisy Measurements
Again 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 $\operatorname{Var}(X)=4$ and each noise term has variance 1. If yo
打开 →题目1756 · 统计
Backing Out Omitted Covariance From a Slope Drop
A 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)?
打开 →题目1711 · 统计
Backtest Slide Misreads p = 0.03
A PM deck says: ‘The event-study p-value is 0.03, so there is a 97% probability the signal is real.’ What is the statistical mistake?
打开 →题目1658 · 统计
Best Blend of Two Correlated Unbiased Signals
Two unbiased estimators of the same parameter have variances 9 and 4, and their correlation is 0.5. For T(a) = aT1 + (1-a)T2, what value of a minimizes variance, and what is the resulting minimum variance?
打开 →题目1718 · 统计
Best-of-50 Reporting Without Adjustment
A researcher tests 50 candidate features and only reports the one with the smallest p-value, which happens to be 0.01. Why is it misleading to present 0.01 as if it came from a single pre-specified test?
打开 →题目1895 · 统计
Better Dataset for Failure-Rate Estimation
A vendor offers two hedge-fund datasets. Dataset A contains only funds that are currently reporting, but it includes long backfilled histories for those funds. Dataset B stores monthly reporting snapshots and preserves closed funds in the historical archive. Which dataset is be
打开 →题目1700 · 统计
BH q Range That Keeps Exactly Three Families
Five family-level winners have ordered p-values 0.004, 0.011, 0.018, 0.031, and 0.070. For what range of BH target levels q would Benjamini-Hochberg keep exactly the first three discoveries?
打开 →题目1659 · 统计
Bias Allowed for a Lower-Variance Regularized Estimate
An unbiased estimator U has variance 0.06. A regularized estimator R has variance 0.03 and constant bias b. What is the largest absolute bias |b| for which R still has smaller MSE than U?
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