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1646MLE with Right-Censored Exponential DataA venue studies the time to the next spread-widening event. Eight observation windows are each followed for up to 5 seconds. In total, 5 windows contain an event before 5 seconds and 3 windows are right-censored at 5 seconds. The total observed exposure time across all 8 windows is 40 seconds. Model event times as i.i.d. Exp ( ) and find the MLE of .统计中等derivation未尝试面试订阅1647MLEs in a Lognormal Model from Log-SummariesSuppose positive holding-period multipliers are modeled as lognormal: if X\sim Lognormal ( , 2) then \log X\sim N( , 2). For a sample of size 12, you are given \overline \log X = 0.3, \qquad \sum i=1 12 (\log X i-0.3) 2 = 10.8. Find the MLEs of and 2, and then give the fitted median of X.统计中等derivation未尝试面试订阅1648Gaussian No-Intercept Regression as an MLE ProblemSuppose observations satisfy Y i = X i + \varepsilon i, \qquad \varepsilon i\stackrel iid \sim N(0, 2), with no intercept and known Gaussian errors. You are told that X iY i = 48, \qquad X i 2 = 16. Find the MLE of .统计中等derivation未尝试面试订阅1649MLE of a Gamma Scale with Known ShapeSuppose trade-duration observations are modeled as Gamma with known shape k=3 and unknown scale . Under this parameterization, E[X]=k .If the sample mean is 12, find the MLE of .统计简单derivation未尝试免费1650MLE with Known Normal Variance and InvarianceSuppose X 1,\dots,X 25 \sim N( ,4) i.i.d., and the sample mean is X=1.2. Find the MLE of , and then use invariance to estimate e .统计简单derivation未尝试免费1651Bias Budget for a Faster ProxyA slow benchmark estimator U is unbiased with variance 0.64. A faster proxy P has variance 0.25 but constant bias b. What is the largest absolute bias |b| for which P still has smaller MSE than U?统计中等derivation未尝试面试订阅1652Optimal Shrink Toward a Desk AnchorA desk observes X ~ N(theta, 9) and reports delta c = cX + (1-c)4. At the specific parameter value theta = 5, what choice of c minimizes MSE, and what is the minimum MSE?统计中等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1654When an Adjusted Signal Beats the Raw EstimateSuppose X ~ N(theta, 1). A desk uses the adjusted estimator delta = 0.75X + 0.5 instead of the raw signal X. For what values of theta does delta have smaller MSE than X?统计中等derivation未尝试面试订阅1655Stale Low-Noise Estimate vs Fresh Noisy EstimateA stale estimator S for today's parameter has variance 0.04 but incurs bias Delta because the regime has moved. A fresh estimator F is unbiased for today's parameter but has variance 0.25. What is the largest |Delta| for which S still has lower MSE than F?统计中等derivation未尝试面试订阅1656Sample Size Needed for a 40% MSE CutThe sample mean of n i.i.d. observations with variance sigma 2 has MSE sigma 2/n. A desk currently uses n = 10 observations. What total sample size is needed so the MSE is at most 60% of the current MSE?统计简单derivation未尝试免费1657Optimal Haircut on a Multiplicative Vol SignalAn estimator A is unbiased for theta and satisfies Var(A) = 0.3 theta 2. A risk team reports delta c = cA instead. Find the value of c that minimizes MSE, and give the minimum MSE as a multiple of theta 2.统计困难derivation未尝试面试订阅1658Best Blend of Two Correlated Unbiased SignalsTwo 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?统计简单derivation未尝试免费1659Bias Allowed for a Lower-Variance Regularized EstimateAn 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?统计中等derivation未尝试面试订阅1660Norm Implied by a 25% Positive-Part James-Stein ShrinkIn dimension p = 4 with unit noise variance, the positive-part James-Stein shrinkage factor is 0.75 for an observed vector x. What value of ||x|| 2 produced that factor?统计中等derivation未尝试面试订阅1661Anchor-Based Shrinkage Crossover IntervalSuppose 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?统计中等derivation未尝试面试订阅1662When Smaller Variance Beats a Vanishing BiasEstimator A is asymptotically unbiased with asymptotic MSE 1/n. Estimator B has asymptotic variance 0.7/n and asymptotic bias 1/n. From what smallest integer n onward does B have lower asymptotic MSE than A?统计简单derivation未尝试免费1663Sample Size for a Stable Variance EstimatorUnder a normal model, the unbiased sample variance satisfies Var(S 2) = 2 sigma 4 / (n-1). What is the smallest n for which the standard deviation of S 2 is at most 0.5 sigma 2?统计简单derivation未尝试免费1664Weight on the Noisier Signal to Hit a Variance TargetTwo independent unbiased estimators T1 and T2 have variances 1 and 4. In the blend W = wT2 + (1-w)T1, what is the smallest positive w that makes Var(W) exactly 0.9?统计简单derivation未尝试免费1665Interval Where a Fixed Benchmark Beats a Noisy Unbiased SignalA noisy unbiased signal X satisfies X ~ N(theta, 0.25). A fallback benchmark always reports 1.2. For what values of theta does the fixed benchmark have lower MSE than X?统计简单derivation未尝试免费