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1627Sparse Symmetric Shock Model From Variance and Fourth MomentA symmetric shock variable takes values -a, 0, and a with probabilities p, 1-2p, and p. If the sample second and fourth raw moments are m 2 and m 4, solve for a and p by method of moments.统计中等derivation未尝试免费1631Recovering Latent Regime Size from Second and Fourth MomentsA stylized one-period microstructure model writes the observed shock as Y = S a + \varepsilon, where S takes values +1 and -1 with equal probability, a>0 is an unknown regime magnitude, and \varepsilon \sim N(0, 2) is independent noise. From data, the empirical second moment is m 2 = 5 and the empirical fourth moment is m 4 = 43. Use the method of moments to estimate a and 2.统计困难derivation未尝试面试订阅1653A Smoothed Bernoulli Estimator vs the Sample ProportionLet X\sim Binomial (10,p) and consider the estimator = X+1 12 for p. At the parameter value p=0.2, compute the bias, variance, and MSE of , and compare its MSE with the usual sample proportion p = X/10.统计中等derivation未尝试面试订阅1777Lasso Threshold Calibration 2A standardized lasso fit has absolute score magnitudes (3.8, 2.5, 0.9). What is the smallest lambda that zeroes the weakest feature while leaving the other two still active?统计简单essay未尝试免费1787Ridge Effective Degrees of Freedom 2A standardized ridge model has singular-value squares d j 2 = [16, 4] and penalty lambda = 4. What is the effective degrees of freedom tr(S lambda) = sum d j 2/(d j 2+lambda)?统计中等derivation未尝试免费1794Duplicate Feature Under Pure LassoIf two predictors are exactly identical and the model uses pure Lasso, what modeling pathology should you expect?统计中等essay未尝试免费1800Signal Stationarity Classification 5A candidate signal is defined by X t = t ε t. Is it weakly stationary?统计困难derivation未尝试面试订阅1808Two-Point Sample Mean Variance 3A weakly stationary process has gamma(0) = 5 and gamma(1) = -1. What is Var((X 1 + X 2)/2)?统计中等数值题未尝试免费1814Validity or Stationarity Check 4Consider the proposal An AR(1) with phi = -0.6. Is it valid from a stationarity / autocorrelation perspective?统计困难derivation未尝试面试订阅1845ARMA Identification or Simplification 5You observe the diagnostic statement: (1-0.5L) X t = (1-0.5L) e t. What is the correct modeling conclusion?统计困难essay未尝试面试订阅2081Worst Shortfall of a Simple Hedge 11A non-traded payoff pays 4, 1, and 6 in the up, middle, and down states of a trinomial stock (120, 100, 80). A desk hedges it with cash -8 and Delta = 0.1 shares of stock. What is the worst-case shortfall of that hedge across the three states?数理金融简单数值题未尝试免费2082Worst Shortfall of a Simple Hedge 12A non-traded payoff pays 3, 5, and 1 in the up, middle, and down states of a trinomial stock (120, 100, 80). A desk hedges it with cash 7 and Delta = -0.05 shares of stock. What is the worst-case shortfall of that hedge across the three states?数理金融简单数值题未尝试免费2396Variance of an Equal-Weight Correlated EnsembleFive base models each have prediction variance 4, and every pair of model predictions has correlation 0.25. If you average the five predictions equally, what is the ensemble variance?机器学习简单derivation未尝试免费2397Sample Size Crossover Between Two Model FamiliesModel A has excess test MSE 0.04 + 18/n, while model B has excess test MSE 0.16 + 4/n, where n is sample size. At what sample size do they tie?机器学习简单derivation未尝试免费2398Bias Budget Implied by a Variance ReductionA regularization change reduces a model's variance term from 0.30 to 0.11 while leaving irreducible noise unchanged. How much extra bias squared could you add before the total MSE stops improving?机器学习中等derivation未尝试免费2399Optimal Weight on a Noisy Unbiased ModelModel A is unbiased with variance 9. Model B has variance 1.44 and fixed bias 0.6. If you blend them as P w = wA + (1-w)B and treat their errors as independent, what weight w minimizes MSE?机器学习困难derivation未尝试面试订阅2400How Many Independent Fits to Hit a Variance TargetEach independently trained model has variance 2.4 and negligible bias. How many equally weighted independent fits must you average to bring the variance term below 0.3?机器学习中等derivation未尝试免费2401Total Error After the Dataset QuadruplesA model currently has bias 2 = 0.09, variance = 0.24, and irreducible noise = 0.50. If quadrupling the dataset quarters the variance term while leaving the other two terms unchanged, what is the new test MSE?机器学习简单derivation未尝试免费2402Second Crossover With a Lower-Bias Flexible ModelA flexible model has excess error 0.02 + 24/n, while a simpler model has excess error 0.14 + 6/n. At what sample size do they tie?机器学习中等derivation未尝试面试订阅2403Variance of a Correlated Five-Model CommitteeFive models each have variance 1.6 and pairwise correlation 0.4. What is the variance of their equal-weight average?机器学习中等derivation未尝试免费