题目1811 · 统计
Validity or Stationarity Check 1
Consider the proposal rho(h)=0.8^|h|. Is it valid from a stationarity / autocorrelation perspective?
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中文题目题目1812 · 统计
Validity or Stationarity Check 2
Consider the proposal rho(1)=1.2. Is it valid from a stationarity / autocorrelation perspective?
打开 →题目1813 · 统计
Validity or Stationarity Check 3
Consider the proposal An AR(1) with phi = 1.03. Is it valid from a stationarity / autocorrelation perspective?
打开 →题目1814 · 统计
Validity or Stationarity Check 4
Consider the proposal An AR(1) with phi = -0.6. Is it valid from a stationarity / autocorrelation perspective?
打开 →题目1815 · 统计
Validity or Stationarity Check 5
Consider the proposal An MA(1) claiming rho(1)=0.7. Is it valid from a stationarity / autocorrelation perspective?
打开 →题目6024 · 统计
Persistence and the Covariance-Stationarity Verdict
A GARCH(1,1) has $\alpha=0.20$, $\beta=0.75$. Compute the persistence $\alpha+\beta$ and state whether the process is covariance-stationary (i.e. has a finite, time-invariant unconditional variance). Give the persistence as a decimal.
打开 →题目1796 · 统计
Signal Stationarity Classification 1
A candidate signal is defined by X_t = ε_t + 0.3 ε_{t-1}. Is it weakly stationary?
打开 →题目1797 · 统计
Signal Stationarity Classification 2
A candidate signal is defined by X_t = 0.2 t + ε_t. Is it weakly stationary?
打开 →题目1798 · 统计
Signal Stationarity Classification 3
A candidate signal is defined by X_t = ε_t + s_t where s_t is a fixed deterministic day-of-week pattern. Is it weakly stationary?
打开 →题目1799 · 统计
Signal Stationarity Classification 4
A candidate signal is defined by X_t = A cos(ω t) + B sin(ω t), where E[A]=E[B]=0, Var(A)=Var(B), Cov(A,B)=0. Is it weakly stationary?
打开 →题目1800 · 统计
Signal Stationarity Classification 5
A candidate signal is defined by X_t = t ε_t. Is it weakly stationary?
打开 →题目4976 · 随机过程
Infer Missing Birth-Death Rate From Stationarity 11
A three-state birth-death CTMC has stationary distribution (0.5, 0.3, 0.2). The rates 0->1 and 1->2 are 0.6 and 0.4, and the rate 1->0 is 1. What rate 2->1 is needed for stationarity?
打开 →题目2905 · 概率
Ehrenfest Urn Stationarity
In the Ehrenfest urn model with $N$ balls, state $i$ means exactly $i$ balls are red. Each step, choose one ball uniformly at random and flip its color. Find the stationary distribution of the chain on $\{0,1,\dots,N\}$.
打开 →题目4979 · 随机过程
Infer Missing Birth-Death Rate From Stationarity 14
A three-state birth-death CTMC has stationary distribution (0.2, 0.5, 0.3). The rates 0->1 and 1->2 are 1.5 and 0.9, and the rate 1->0 is 0.6. What rate 2->1 is implied?
打开 →题目2902 · 概率
Laziness Does Not Change Stationarity
Suppose $\pi$ is stationary for a Markov chain with transition matrix $P$. Fix $\theta\in(0,1)$ and define a lazy version \[ P'=\theta I+(1-\theta)P. \] Show that $\pi$ is also stationary for $P'$.
打开 →题目4324 · 机器学习
Small Data With Local Stationarity
You have limited labeled data, and the target depends on local translation-equivariant patterns in a 2D signal map. Which architecture family usually brings the strongest built-in inductive bias?
打开 →题目2893 · 概率
Uniform Stationarity for a Doubly Stochastic Chain
Suppose a finite Markov chain has transition matrix $P$ whose rows and columns both sum to $1$. Show that the uniform distribution is stationary.
打开 →题目1816 · 统计
Why Differencing Helps a Trend but Not White Noise
A signal looks like deterministic trend plus short-memory noise. Why can first differencing help stationarity, while differencing plain white noise usually just injects a negative lag-1 correlation?
打开 →题目1818 · 统计
Why Ergodicity Matters
Why is ergodicity stronger than stationarity, and why do practitioners care about it when they average one long signal history?
打开 →题目2676 · 机器学习
A Tiny Tail Probability Can Dominate Average PnL
A strategy makes +0.04% on 98% of days and loses -2.5% on the remaining 2% of days. What is the unconditional average daily return?
打开 →题目1836 · 统计
AR(1) Forecast Error Variance 1
For the AR(1) model X_t = phi X_(t-1) + e_t with phi = 0.6 and Var(e_t) = 1, what is the h = 3 step forecast error variance?
打开 →题目1838 · 统计
AR(1) Forecast Error Variance 3
For the AR(1) model X_t = phi X_(t-1) + e_t with phi = 0.5 and Var(e_t) = 2.25, what is the h = 4 step forecast error variance?
打开 →题目1821 · 统计
AR(1) Multi-Step Forecast 1
A signal follows X_t = 0 + 0.6 X_(t-1) + e_t with Var(e_t) = 2 and current value X_t = 10. What is the h = 3 step forecast E[X_(t+3) | X_t]?
打开 →题目1842 · 统计
ARMA Identification or Simplification 2
You observe the diagnostic statement: ACF cuts after lag 1, PACF tails off. What is the correct modeling conclusion?
打开 →题目1844 · 统计
ARMA Identification or Simplification 4
You observe the diagnostic statement: AIC prefers ARMA(2,1) but BIC prefers ARMA(1,1). What is the correct modeling conclusion?
打开 →题目2672 · 机器学习
Autocorrelation-Corrected Sample Size
A monthly feature is observed for 60 months and behaves roughly like an AR(1) series with lag-1 autocorrelation $\rho=0.6$. Using the heuristic $n_\text{eff}\approx n(1-\rho)/(1+\rho)$, what is the effective sample size?
打开 →题目2673 · 机器学习
Average Edge Across Opposing Regimes
A signal earns +6 bps on 70% of days in calm markets and -10 bps on 30% of days in stressed markets. What is its unconditional average daily edge in bps?
打开 →题目2675 · 机器学习
Break-Even Hit Rate After Trading Costs
A directional model earns +1 unit on a correct trade and -1 unit on an incorrect trade before costs. Each round trip also pays a cost of 0.08 units regardless of outcome. What hit rate $p$ makes expected net PnL zero?
打开 →题目3111 · 统计
Does This GARCH Have a Finite Long-Run Variance?
For a GARCH(1,1) model with $\omega=\frac{1}{5}$, $\alpha=\frac{1}{4}$, and $\beta=\frac{3}{4}$, decide whether the model has a finite unconditional variance. If it does, compute it.
打开 →题目1856 · 统计
Horizon Risk Budget Ratio 1
A stationary mean-reverting spread obeys X_(t+1) = 1/2 X_t + epsilon_(t+1), where Var(epsilon_(t+1)) = 4. Starting from the current level, what fraction of the same-horizon random-walk forecast-error variance does the 4-step mean-reverting forecast-error variance represent?
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