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1628Estimating a Zero-Inflated Order-Arrival ModelPer-second order arrivals are modeled as follows: with probability the market is inactive and the observed count is exactly 0; with probability 1- , the count is Poisson ( ). From data, the empirical zero frequency is 0.70 and the empirical mean count is 0.60. Use the method of moments to estimate ( , ).统计困难derivation未尝试面试订阅1629MoM for a Random Amplitude Bernoulli CountLet X=AZ where Z is Bernoulli with success probability p and the success amplitude A is a positive constant. If the sample mean is m 1 and the sample second raw moment is m 2, solve for A and p.统计中等derivation未尝试免费1630Shifted Exponential Calibration from Raw MomentsA toy latency model assumes X = c + Y, \qquad Y \sim Exp ( ), with unknown deterministic floor c>0 and unknown rate . From historical data, the empirical mean of X is 8 and the empirical second raw moment is 73. Use the method of moments to estimate c and .统计简单derivation未尝试免费1633Two-Rate Latency Mixture With Known Mixing WeightA latency variable is a 50-50 mixture of two exponential laws with rates \lambda 1 and \lambda 2. The first two raw moments are m 1 and m 2. Write the two equations that method of moments imposes on (\lambda 1,\lambda 2).统计中等derivation未尝试面试订阅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1677Why Studentization Can Improve Interval CalibrationWhy can studentizing a bootstrap statistic improve interval accuracy in skewed or scale-varying problems?统计困难essay未尝试面试订阅1958Verify the Designed Minimizer in a Quadratic-Reciprocal Objective 13Let r>0 and a>0. Show that x=r uniquely minimizes H(x)=a x 2 + 2 a r(1+r) 2 /(1+x) over x>-1.数学中等derivation未尝试免费1964Negative Exponential Tilt 19Minimize J(x)=16 e x + 4 e -2x . What x minimizes J?数学困难derivation未尝试免费1966Balanced Softplus Tradeoff 21The linear reward and saturation penalty balance exactly at a central point. The desk maximizes K(x) = 2 x - 4 ln(1+e x). What x is optimal?数学简单数值题未尝试免费1967Positive Softplus Tilt 22The linear reward is strong relative to the saturation penalty, so the optimum should be positive. The desk maximizes K(x) = 3 x - 4 ln(1+e x). What x is optimal?数学简单derivation未尝试免费1970Maximum Value at the Interior Utility-Impact Optimum 25For F(x)=4 ln(1+x)-x 2 on x>-1, what x maximizes F and what is the maximum value?数学困难derivation未尝试免费1979Minimum Objective Value Under a Total-Size Constraint 9For positive a,b,c, what is the minimum value of a x 2 + b y 2 + c z 2 subject to x+y+z=N?数学中等derivation未尝试免费1987Derive the Alpha-Maximizing Point Under a Quadratic Risk Budget 17Derive the maximizer of mu 1 x + mu 2 y subject to a x 2 + b y 2 = R 2 for positive a,b.数学中等derivation未尝试免费2026Why Dispersion Raises a Convex Penalty 6Why does randomizing Q around a fixed mean raise E[Q/(1-Q)] relative to plugging in the mean directly?数学简单derivation未尝试免费2031Backing Out the Stress State From a Reciprocal Plug-In Score 11A funding-buffer score uses phi(L)=1/(1+L). Suppose leverage L equals 0 with probability 1/2 and H with probability 1/2. If phi(E[L]) = 1/3, what is H and what is E[phi(L)]?数学简单derivation未尝试免费2034Conditional Jensen Lower Bound 14If phi is convex, what inequality holds between E[phi(X)|F] and phi(E[X|F]) almost surely?数学困难derivation未尝试面试订阅2038Universal Lower Bound for a Convex Stress Multiplier 18A convex stress multiplier is phi(x)=e x. If a signal X has mean 0.2, what lower bound does Jensen's inequality give for E[e X]?数学中等derivation未尝试免费2042Jensen Upper Bound for an Expected Log Score 22If X > -1 almost surely and E[X]=0.2, what upper bound does Jensen give for E[ln(1+X)]?数学简单derivation未尝试免费