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1923Required Observation Count 3A research lead uses the approximation n >= K * sigma 2 / delta 2 with K = 9, signal standard deviation 30 bp, and target detectable mean shift 4 bp. What is the minimum integer sample size?统计中等数值题未尝试免费1926Minimum Detectable Edge 1Using the same approximation n >= K * sigma 2 / delta 2, suppose K = 7.84, sigma = 25 bp, and you can collect only n = 400 observations. What is the smallest detectable mean edge delta, in basis points?统计简单essay未尝试免费1928Minimum Detectable Edge 3Using the same approximation n >= K * sigma 2 / delta 2, suppose K = 9, sigma = 30 bp, and you can collect only n = 900 observations. What is the smallest detectable mean edge delta, in basis points?统计中等数值题未尝试免费1931Effective Sample Under Serial Dependence 1A return series has 240 observations and lag-1 autocorrelation 1/5. Using the heuristic n eff = n * (1-rho)/(1+rho), what is the effective sample size?统计简单essay未尝试免费1936Paired Design Observation Ratio 1Two strategies are tested on the same names and dates, giving correlation 1/5 between their noise terms. Relative to an unpaired comparison with the same marginal variance, by what factor does the paired design reduce required sample size?统计简单数值题未尝试免费1941Power Score Design Choice 1Compare two research designs using the score delta * sqrt(n) / sigma. Design A has delta=4 bp, n=225, sigma=2 bp. Design B has delta=3 bp, n=400, sigma=1 bp. Which design has the larger score?统计简单数值题未尝试免费3066Signal Extraction from One Noisy PrintA latent scalar state has prior x\sim N(10,4). You observe y=13 through y=x+\varepsilon with \varepsilon\sim N(0,5). Compute the Kalman gain, posterior mean, and posterior variance.统计中等derivation未尝试面试订阅3068Latent Fair-Value UpdateA latent scalar state has prior x\sim N(-1,16). You observe y=3 through y=x+\varepsilon with \varepsilon\sim N(0,9). Compute the Kalman gain, posterior mean, and posterior variance.统计中等derivation未尝试面试订阅3071Local-Level Forecast Then UpdateSuppose x t=x t-1 +w t with w t\sim N(0,2), and y t=x t+v t with v t\sim N(0,3). At time t-1 the filtered state is N(7,4). You observe y t=9. Compute the predicted mean/variance and the updated mean/variance at time t.统计中等derivation未尝试面试订阅3072Random-Walk Value Filter StepSuppose x t=x t-1 +w t with w t\sim N(0,1), and y t=x t+v t with v t\sim N(0,4). At time t-1 the filtered state is N(-2,5). You observe y t=0. Compute the predicted mean/variance and the updated mean/variance at time t.统计中等derivation未尝试面试订阅3076Fusing Two Noisy Dealer QuotesA latent scalar state has prior N(0,9). Two conditionally independent sensors observe y 1=2 with noise variance 4 and y 2=-1 with noise variance 5. Compute the posterior mean and posterior variance after both observations.统计中等derivation未尝试面试订阅3077Two-Sensor Latent Level EstimateA latent scalar state has prior N(5,16). Two conditionally independent sensors observe y 1=9 with noise variance 9 and y 2=3 with noise variance 4. Compute the posterior mean and posterior variance after both observations.统计中等derivation未尝试面试订阅3078Dual Feed State CombinationA latent scalar state has prior N(-2,25). Two conditionally independent sensors observe y 1=-1 with noise variance 1 and y 2=2 with noise variance 4. Compute the posterior mean and posterior variance after both observations.统计中等derivation未尝试面试订阅3081Two Missing Days Before a Print ArrivesA local-level model satisfies x t=x t-1 +w t with w t\sim N(0,1), and observations have noise variance 2. After the last filtered state N(3,4), there are 2 consecutive missing observations. Then you observe a new value y=6. Compute the variance just before the new observation and the updated mean/variance after processing it.统计中等derivation未尝试面试订阅3082One Missing Observation Then UpdateA local-level model satisfies x t=x t-1 +w t with w t\sim N(0,3), and observations have noise variance 5. After the last filtered state N(-1,9), there are 1 consecutive missing observations. Then you observe a new value y=2. Compute the variance just before the new observation and the updated mean/variance after processing it.统计中等derivation未尝试面试订阅3086Steady-State Gain with Q=1, R=2Consider the scalar local-level model in steady state: x t=x t-1 +w t, w t\sim N(0,1), and y t=x t+v t, v t\sim N(0,2). Compute the steady-state posterior variance C and the steady-state Kalman gain K.统计困难derivation未尝试面试订阅3091Long-Run Variance of a Quiet GARCH ProcessFor a GARCH(1,1) model h t=\omega+ r t-1 2+ h t-1 with \omega= 1 10 , = 1 5 , and = 3 5 , assume + <1. Compute the unconditional variance E[h t].统计中等derivation未尝试面试订阅3093Steady Variance from Daily GARCH ParametersFor a GARCH(1,1) model h t=\omega+ r t-1 2+ h t-1 with \omega=1, = 1 10 , and = 4 5 , assume + <1. Compute the unconditional variance E[h t].统计中等derivation未尝试面试订阅3096Tomorrow Variance After a Large ShockIn a GARCH(1,1) model with \omega= 1 10 , = 1 5 , and = 7 10 , suppose the current squared return is r t 2=4 and the current conditional variance is h t=2. Compute h t+1 .统计简单derivation未尝试面试订阅3099Volatility Update from a Moderate ReturnIn a GARCH(1,1) model with \omega=1, = 3 20 , and = 3 5 , suppose the current squared return is r t 2=4 and the current conditional variance is h t=5. Compute h t+1 .统计简单derivation未尝试面试订阅