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中文题目A 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 $(p_1,p_2,p_3)$.
打开 →Suppose trade-duration observations are modeled as Gamma with known shape $k=3$ and unknown scale $\theta$. Under this parameterization, $$E[X]=k\theta.$$If the sample mean is $12$, find the MLE of $\theta$.
打开 →A 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 $X\sim \mathrm{Geometric}(p)$. If the sample mean from many independent episodes is $4$, find the MLE of $p$. Under the
打开 →Suppose short-horizon pricing errors are modeled as i.i.d. Laplace$(\mu,b)$ with known scale $b=2$, so the density is proportional to $e^{-|x-\mu|/2}$. The observed sample is $$-1,\;0,\;2,\;2,\;3,\;5,\;7.$$ Find the MLE of $\mu$.
打开 →Suppose large execution slippage magnitudes are modeled as Pareto with known scale $x_m=1$ and unknown tail index $\alpha$, so the density is $$f(x)=\alpha x^{-\alpha-1}, \qquad x\ge 1.$$ If $n=8$ observations satisfy $$\sum_{i=1}^8 \log X_i = 12,$$ find the MLE of $\alpha$. Then
打开 →During a 40-minute observation window, a venue records 120 child-order arrivals. Model the arrivals as a homogeneous Poisson process with intensity $\lambda$ arrivals per minute. Find the MLE of $\lambda$, and estimate the probability of seeing zero arrivals in the next minute u
打开 →Five i.i.d. observations are modeled as $\mathrm{Uniform}(0,\theta)$. The sample maximum is $7.4$. Find the MLE of $\theta$, and then estimate the median of the fitted distribution.
打开 →Suppose execution delays are modeled as Weibull with known shape $k=2$ and unknown scale $\lambda$, with density $$f(x)=\frac{2x}{\lambda^2}e^{-(x/\lambda)^2}, \qquad x>0.$$ If $n=10$ observations satisfy $$\sum_{i=1}^{10} X_i^2 = 90,$$ find the MLE of $\lambda$.
打开 →Ten independent waiting times between mid-price changes sum to 25 seconds. Model each waiting time as $\mathrm{Exp}(\lambda)$. Find the MLE of $\lambda$, and under the fitted model compute the median waiting time.
打开 →Suppose $X_1,\dots,X_{25}\sim N(\mu,4)$ i.i.d., and the sample mean is $\bar X=1.2$. Find the MLE of $\mu$, and then use invariance to estimate $e^{\mu}$.
打开 →Suppose positive holding-period multipliers are modeled as lognormal: if $X\sim \mathrm{Lognormal}(\mu,\sigma^2)$ then $\log X\sim N(\mu,\sigma^2)$. 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
打开 →A 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.
打开 →A 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
打开 →Suppose observations satisfy $$Y_i = \beta X_i + \varepsilon_i, \qquad \varepsilon_i\stackrel{iid}{\sim}N(0,\sigma^2),$$ with no intercept and known Gaussian errors. You are told that $$\sum X_iY_i = 48, \qquad \sum X_i^2 = 16.$$ Find the MLE of $\beta$.
打开 →Suppose $X_1,\dots,X_9$ are modeled as i.i.d. $N(\mu,\sigma^2)$. From the sample you know that $$\bar X = 5, \qquad \sum_{i=1}^9 (X_i-\bar X)^2 = 18.$$ Find the MLEs of $\mu$ and $\sigma^2$.
打开 →A desk observes only 4 new defaults for a rare event. Using a strong historical Beta prior, the Bayesian posterior mean default rate is much lower than the sample proportion, while the frequentist MLE equals the sample proportion exactly. Explain why these two answers can legitim
打开 →For an intercept-only logistic model with n_1 positives and n_0 negatives, what fitted probability p_hat maximizes the log-likelihood?
打开 →A notebook computes PCA on the full feature matrix and then feeds the resulting components into every cross-validation fold. Why is that not a harmless speed optimization?
打开 →Why is the dummy-variable trap more than just a harmless coding oversight?
打开 →Why can a rare-event payoff have an unstable Monte Carlo estimate even when most simulated paths look harmless?
打开 →Why should changing the tradable universe be counted as another research branch rather than as harmless context?
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