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Computable exponential bounds for screened estimation and simulation
Probability · Submitted 1 Dec 2006
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim The main results are several nonasymptotic, explicit exponential bounds for the screened estimates.
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The abstract, sentence by sentence
Suppose the expectation $E(F(X))$ is to be estimated by the empirical averages of the values of $F$ on independent and identically distributed samples $\{X_i\}$. A sampling rule called the "screened" estimator is introduced, and its performance is studied. When the mean $E(U(X))$ of a different function $U$ is known, the estimates are "screened," in that we only consider those which correspond to times when the empirical average of the $\{U(X_i)\}$ is sufficiently close to its known mean. As long as $U$ dominates $F$ appropriately, the screened estimates admit exponential error bounds, even when $F(X)$ is heavy-tailed. The main results are several nonasymptotic, explicit exponential bounds for the screened estimates. A geometric interpretation, in the spirit of Sanov's theorem, is given for the fact that the screened estimates always admit exponential error bounds, even if the standard estimates do not. And when they do, the screened estimates' error probability has a significantly better exponent. This implies that screening can be interpreted as a variance reduction technique. Our main mathematical tools come from large deviations techniques. The results are illustrated by a detailed simulation example.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Suppose the expectation $E(F(X))$ is to be estimated by the empirical averages of the values of $F$ on independent and identically distributed samples $\{X_i\}$. | background 1.00 | 0.10 | 0.02 | asserted 0.62 |
| s1 | A sampling rule called the "screened" estimator is introduced, and its performance is studied. | contribution 0.99 | 0.89 | 0.05 | asserted 0.98 |
| s2 | When the mean $E(U(X))$ of a different function $U$ is known, the estimates are "screened," in that we only consider those which correspond to times when the empirical average of the $\{U(X_i)\}$ is sufficiently close to its known mean. | method 0.98 | 0.52 | 0.07 | asserted 0.99 |
| s3 | As long as $U$ dominates $F$ appropriately, the screened estimates admit exponential error bounds, even when $F(X)$ is heavy-tailed. | result 1.00 | 0.85 | 0.40 | asserted 0.99 |
| s4 | The main results are several nonasymptotic, explicit exponential bounds for the screened estimates. | result 0.79 | 0.94 | 0.70 | asserted 1.00 |
| s5 | A geometric interpretation, in the spirit of Sanov's theorem, is given for the fact that the screened estimates always admit exponential error bounds, even if the standard estimates do not. | interpretation 0.98 | 0.91 | 0.15 | asserted 1.00 |
| s6 | And when they do, the screened estimates' error probability has a significantly better exponent. | result 0.95 | 0.83 | 0.16 | asserted 1.00 |
| s7 | This implies that screening can be interpreted as a variance reduction technique. | interpretation 1.00 | 0.80 | 0.18 | asserted 0.72 |
| s8 | Our main mathematical tools come from large deviations techniques. | method 1.00 | 0.70 | 0.03 | asserted 1.00 |
| s9 | The results are illustrated by a detailed simulation example. | method 0.58 | 0.90 | 0.03 | asserted 1.00 |
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Original abstract
Suppose the expectation $E(F(X))$ is to be estimated by the empirical averages of the values of $F$ on independent and identically distributed samples $\{X_i\}$. A sampling rule called the "screened" estimator is introduced, and its performance is studied. When the mean $E(U(X))$ of a different function $U$ is known, the estimates are "screened," in that we only consider those which correspond to times when the empirical average of the $\{U(X_i)\}$ is sufficiently close to its known mean. As long as $U$ dominates $F$ appropriately, the screened estimates admit exponential error bounds, even when $F(X)$ is heavy-tailed. The main results are several nonasymptotic, explicit exponential bounds for the screened estimates. A geometric interpretation, in the spirit of Sanov's theorem, is given for the fact that the screened estimates always admit exponential error bounds, even if the standard estimates do not. And when they do, the screened estimates' error probability has a significantly better exponent. This implies that screening can be interpreted as a variance reduction technique. Our main mathematical tools come from large deviations techniques. The results are illustrated by a detailed simulation example.
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