题目2489 · 机器学习
A Constant Shift in One Feature 19
If a feature x is replaced by x+k in a regression that already includes an intercept, what happens to the slope on x and the intercept?
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中文题目题目2481 · 机器学习
Adding an Orthogonal Zero-Signal Feature 11
If a new centered feature z is orthogonal to both the existing centered design and the response y, what coefficient does OLS assign to z?
打开 →题目1843 · 统计
ARMA Identification or Simplification 3
You observe the diagnostic statement: Both ACF and PACF tail off. What is the correct modeling conclusion?
打开 →题目1753 · 统计
Average Leverage and Average Residual-Maker Diagonal
A regression uses n = 25 observations and three estimated parameters including the intercept. What are (i) the average leverage and (ii) the average diagonal entry of the residual-maker matrix I - H?
打开 →题目2494 · 机器学习
Centered Simple Regression Through the Origin 24
After centering x and y in simple regression with an intercept, what optimization problem remains for the slope?
打开 →题目2504 · 机器学习
Compute a Ridge Coefficient Numerically 9
In an orthogonal coordinate with d = 9, z = 18, and lambda = 3, what is the ridge coefficient?
打开 →题目2516 · 机器学习
Coordinate-Descent Update for a Positive Orthogonal Lasso Coordinate 21
In an orthogonal coordinate with d = 5, z = 11, and lambda = 3, what coefficient does one exact lasso coordinate-descent update return?
打开 →题目2514 · 机器学习
Equivalent L2 Radius in One Dimension 19
In one dimension, if the ridge solution equals beta_hat_lambda, what radius t makes the constrained problem min RSS(beta) subject to |beta| <= t share the same optimizer?
打开 →题目1748 · 统计
Fitted Variance From R-Squared and Response Variance
A regression with intercept has response variance 9 and R^2 = 4/9. What is the variance of the fitted values?
打开 →题目1648 · 统计
Gaussian No-Intercept Regression as an MLE Problem
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 →题目2476 · 机器学习
Infer the Slope From Covariance and Variance 6
In a simple regression with an intercept, Cov(x,y)=12 and Var(x)=16. What is the OLS slope beta_hat?
打开 →题目1749 · 统计
Intercept From Center of the Cloud
In a simple OLS regression with intercept, x_bar = 3, y_bar = 5, and the slope estimate is -0.4. What is the intercept?
打开 →题目2486 · 机器学习
Intercept From Means 17
In a simple regression with intercept, xbar = 3, ybar = 11, and beta_hat = 2. What is alpha_hat?
打开 →题目2472 · 机器学习
Intercept From Sample Means and Slope 2
Derive the OLS intercept in simple regression with an intercept once beta_hat is known.
打开 →题目2540 · 机器学习
Intercept Shift for a Deployment Prior Change
A logistic model was trained under class prior 0.5 and has intercept -0.4. At deployment the base rate falls to 0.2 while feature likelihood ratios are assumed unchanged. What adjusted intercept should be used?
打开 →题目2498 · 机器学习
Lasso Zero Threshold in an Orthogonal Coordinate 3
In an orthogonal one-feature problem with x^T x = d and score z = x^T y, for what lambdas does the lasso coefficient become exactly zero?
打开 →题目2276 · 数理金融
LSM Regression Interpretation 1
An LSM continuation regression is fit linearly from two in-the-money paths: (S,C)=(70,24) and (90,10). What continuation estimate does that line give at S = 80?
打开 →题目2277 · 数理金融
LSM Regression Interpretation 2
Two in-the-money paths used in an LSM fit are (S,C)=(65,22) and (95,13). If the desk uses a linear continuation fit, what slope is implied?
打开 →题目2278 · 数理金融
LSM Regression Interpretation 3
A linear LSM continuation fit passes through (S,C)=(85,16) and (100,7). What intercept a in C(S)=a+bS is implied?
打开 →题目2279 · 数理金融
LSM Regression Interpretation 4
An LSM continuation fit is C(S) = a + 0.2S and passes through (S,C)=(72,20). At what spot S would the continuation estimate equal 15.6?
打开 →题目2280 · 数理金融
LSM Regression Interpretation 5
A linear LSM continuation fit passes through (75,18) and gives continuation 14 at S = 90. If another in-the-money path is at S = 105, what continuation value must it have to stay on the same fitted line?
打开 →题目1833 · 统计
MA(1) Invertibility Check 3
An MA(1) execution-noise model uses theta = -0.7. Is the model invertible?
打开 →题目2525 · 机器学习
One Newton Step for an Intercept-Only Logistic Model
An intercept-only logistic model is fit to 7 positives and 3 negatives. Starting from b_0 = 0, what is one Newton step b_1 for minimizing the negative log-likelihood?
打开 →题目2480 · 机器学习
Orthogonal Features Give Coordinatewise Coefficients 9
Suppose two features x1 and x2 are centered and orthogonal. Derive the OLS coefficients in terms of x1^T y, x2^T y, ||x1||^2, and ||x2||^2.
打开 →题目2496 · 机器学习
Orthogonal-Design Ridge Coefficient 1
In a one-feature orthogonal design with x^T x = d and x^T y = s, derive the ridge coefficient as a function of lambda.
打开 →题目2487 · 机器学习
Prediction Invariance Under Equivalent Parameterizations 16
Why can two different coefficient vectors produce exactly the same OLS predictions when the design is rank-deficient?
打开 →题目2493 · 机器学习
Projection Error Is Orthogonal to the Fitted Subspace 23
Why is y - X beta_hat orthogonal to every fitted vector Xv?
打开 →题目2482 · 机器学习
Projection Interpretation of the Fitted Vector 12
In one sentence, what geometric object is X beta_hat in OLS?
打开 →题目2478 · 机器学习
Residual Orthogonality to Features 8
Why must the OLS residual vector be orthogonal to every column of the design matrix at the optimum?
打开 →题目1743 · 统计
Residual Sum of Squares From TSS and R-Squared
An OLS fit with intercept has total sum of squares TSS = 250 and R^2 = 0.84. What is the residual sum of squares?
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