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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1666Exact Bootstrap Variance of a Two-Point MeanA sample is a,b . In the nonparametric bootstrap, a resample of size 2 is drawn with replacement. Derive the variance of the bootstrap sample mean.统计简单derivation未尝试免费1667Bootstrap Bias of the Sample Maximum on a Tiny SampleA sample is 1,4 . Under the nonparametric bootstrap with resample size 2, compute the bootstrap expectation of the sample maximum and the resulting bootstrap bias estimate for the original maximum statistic.统计简单derivation未尝试免费1668Bootstrap Variance of a Bernoulli Mean From the Plug-In LawA sample of size n has empirical success rate p hat. Under the nonparametric bootstrap, what is the variance of the resampled sample mean conditional on the observed data?统计简单derivation未尝试免费1669When the Bootstrap Median Is Forced to Equal One of Two ValuesA sample is 0,0,10 . Under the nonparametric bootstrap with resample size 3, what values can the bootstrap median take, and what condition determines which one occurs?统计中等derivation未尝试面试订阅1670m-out-of-n Bootstrap Variance Scaling for a MeanIf a statistic is the sample mean and you use an m-out-of-n nonparametric bootstrap instead of an n-out-of-n bootstrap, how does the conditional variance of the resampled mean scale with m?统计简单derivation未尝试免费1671Why the Naive Bootstrap Misses a BoundaryAn estimator is constrained to be nonnegative and lands exactly at 0 on the observed sample. Why can the naive nonparametric bootstrap badly misrepresent uncertainty near that boundary?统计简单derivation未尝试免费1672Pairs Bootstrap Versus Residual Bootstrap in Heteroskedastic DataWhy can a residual bootstrap be invalid for regression under heteroskedasticity while a pairs bootstrap remains defensible?统计中等derivation未尝试面试订阅