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1634Inferring Cross-Day Heterogeneity from Paired Signal OutcomesSuppose each trading day has an unobserved hit probability P \sim Beta ( , ). Conditional on P, two independent intraday signals H 1 and H 2 are Bernoulli(P). From data, you estimate E[H 1] = 0.60, \qquad P(H 1=1, H 2=1) = 0.42. Use the method of moments to estimate and .统计中等derivation未尝试面试订阅1635Estimating a Three-Point Shock Model from Even MomentsA stylized inventory-shock model assumes the one-step PnL jump X takes values -a, 0, and +a with probabilities p/2, 1-p, and p/2, respectively, where a>0 is unknown. From data, the empirical second moment is 2 and the empirical fourth moment is 10. Use the method of moments to estimate p and a.统计简单derivation未尝试免费1636MLE of a Bernoulli Signal Hit RateA binary trading signal was profitable on 44 of the last 80 trading days. Model each day as an independent Bernoulli(p) outcome. Find the maximum likelihood estimator of p, and then estimate the probability that the next 3 days are all profitable under the fitted model.统计简单derivation未尝试免费1637MLE of a Poisson Order-Arrival IntensityDuring a 40-minute observation window, a venue records 120 child-order arrivals. Model the arrivals as a homogeneous Poisson process with intensity arrivals per minute. Find the MLE of , and estimate the probability of seeing zero arrivals in the next minute under the fitted model.统计简单derivation未尝试免费1638MLE of an Exponential Waiting-Time ModelTen independent waiting times between mid-price changes sum to 25 seconds. Model each waiting time as Exp ( ). Find the MLE of , and under the fitted model compute the median waiting time.统计简单derivation未尝试免费1639Joint MLEs in a Normal ModelSuppose X 1,\dots,X 9 are modeled as i.i.d. N( , 2). From the sample you know that X = 5, \qquad \sum i=1 9 (X i- X) 2 = 18. Find the MLEs of and 2.统计简单derivation未尝试面试订阅1640MLE of a Uniform Upper BoundFive i.i.d. observations are modeled as Uniform (0, ). The sample maximum is 7.4. Find the MLE of , and then estimate the median of the fitted distribution.统计简单derivation未尝试面试订阅1641MLE of a Geometric Success ProbabilityA strategy is repeatedly tried until the first profitable fill. Let X be the number of attempts until the first success, with support 1,2,\ldots, and model X\sim Geometric (p). If the sample mean from many independent episodes is 4, find the MLE of p. Under the fitted model, what is the probability that at least 4 attempts are needed?统计中等derivation未尝试面试订阅1642MLE in a Three-State Multinomial ModelA market regime model has three states: calm, trending, and dislocated, with probabilities (p 1,p 2,p 3). Over 100 days, the observed counts are 20 calm days, 30 trending days, and 50 dislocated days. Find the MLE of (p 1,p 2,p 3).统计简单derivation未尝试免费1643MLE of a Pareto Tail IndexSuppose large execution slippage magnitudes are modeled as Pareto with known scale x m=1 and unknown tail index , so the density is f(x)= x - -1 , \qquad x\ge 1. If n=8 observations satisfy \sum i=1 8 \log X i = 12, find the MLE of . Then estimate P(X>10) under the fitted model.统计中等derivation未尝试面试订阅1644MLE of a Weibull Scale with Known ShapeSuppose execution delays are modeled as Weibull with known shape k=2 and unknown scale , with density f(x)= 2x 2 e -(x/ ) 2 , \qquad x>0. If n=10 observations satisfy \sum i=1 10 X i 2 = 90, find the MLE of .统计中等derivation未尝试面试订阅1645MLE of a Laplace Location ParameterSuppose short-horizon pricing errors are modeled as i.i.d. Laplace( ,b) with known scale b=2, so the density is proportional to e -|x- |/2 . The observed sample is -1,\;0,\;2,\;2,\;3,\;5,\;7. Find the MLE of .统计中等derivation未尝试面试订阅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