{
 "sig-39": {
  "source": "Tibshirani, Don't Try for the Triple 20, Significance 8(1) (2011), author-hosted copy https://www.stat.cmu.edu/~ryantibs/papers/darts-signif.pdf",
  "where": "p. 48",
  "kept": "research/aim-where-your-misses-pay/sources/tibshirani2011-significance-excerpts.txt",
  "excerpt": "have a way of estimating a player’s s. This would allow us to produce a customised heatmap that instructs the player where to aim to optimise his or her average score. To estimate the inaccuracy parameter s, suppose that we had the player throw 100 darts aimed at the centre of the board (so that µ = 0). If we measured the positions of these throws, then we could use standard sta- tistical methods to find s. This would be quite easy to compute given the simple normal model for dart throws: it is the distance from the target within which 39% of his shots land."
 },
 "tpt-curl": {
  "source": "Tibshirani, Price and Taylor, A Statistician Plays Darts, JRSS A 174(1) (2011), author-hosted copy https://www.stat.cmu.edu/~ryantibs/papers/darts.pdf",
  "where": "Figure 5 caption",
  "kept": "research/aim-where-your-misses-pay/sources/tpt2011-paper-excerpts.txt",
  "excerpt": "Figure 5: Path of the optimal location µ∗ parametrized by σ. Starting at σ = 0, the optimal µ is in the center of the T20, and moves slightly up and to the left. Then it jumps to the T19 at σ = 16.4. From here it curls into the center of the board, stopping a bit lower than and the left of the bullseye at σ = 100."
 },
 "tpt-authors": {
  "source": "Tibshirani, Price and Taylor, A Statistician Plays Darts, JRSS A 174(1) (2011), author-hosted copy https://www.stat.cmu.edu/~ryantibs/papers/darts.pdf",
  "where": "sec. 2.2",
  "kept": "research/aim-where-your-misses-pay/sources/tpt2011-paper-excerpts.txt",
  "excerpt": "Author 1 and author 2 each threw 100 darts at the bullseye and recorded their scores, from which we estimate their standard deviations to be σ1 = 64.6 and σ2 = 26.9, respectively. Thus Figure 2 shows their personalized heatmaps. To maximize his expected score, author 1 should be aiming at the S8, close to the bullseye. Meanwhile, author 2 (who is a fairly skilled darts player) should be aiming at the T19."
 },
 "sig-25": {
  "source": "Tibshirani, Don't Try for the Triple 20, Significance 8(1) (2011), author-hosted copy https://www.stat.cmu.edu/~ryantibs/papers/darts-signif.pdf",
  "where": "p. 47",
  "kept": "research/aim-where-your-misses-pay/sources/tibshirani2011-significance-excerpts.txt",
  "excerpt": "Meanwhile, a player with decent accuracy, s = 25mm, should aim at the triple 19. This earns him or her an average of 15.8 points per throw, and aiming at the triple 20 gives only 14.9, nearly a full point lower."
 },
 "sig-60": {
  "source": "Tibshirani, Don't Try for the Triple 20, Significance 8(1) (2011), author-hosted copy https://www.stat.cmu.edu/~ryantibs/papers/darts-signif.pdf",
  "where": "p. 47",
  "kept": "research/aim-where-your-misses-pay/sources/tibshirani2011-significance-excerpts.txt",
  "excerpt": "For a beginner, s = 60mm, aiming somewhat lower than and to the left of the board’s centre gives the highest ex- pected score, 12.4 points per throw. On average, aiming at the triple 20 would cost this player more than two points each throw, yielding only 10.2 points."
 },
 "sig-5": {
  "source": "Tibshirani, Don't Try for the Triple 20, Significance 8(1) (2011), author-hosted copy https://www.stat.cmu.edu/~ryantibs/papers/darts-signif.pdf",
  "where": "p. 47",
  "kept": "research/aim-where-your-misses-pay/sources/tibshirani2011-significance-excerpts.txt",
  "excerpt": "For a player with near-perfect accuracy, s = 5mm, the best place to aim is indeed the triple 20; by aiming there the player can achieve an average of 42.9 points per throw."
 },
 "tpt-mm": {
  "source": "Tibshirani, Price and Taylor, A Statistician Plays Darts, JRSS A 174(1) (2011), author-hosted copy https://www.stat.cmu.edu/~ryantibs/papers/darts.pdf",
  "where": "sec. 2.1",
  "kept": "research/aim-where-your-misses-pay/sources/tpt2011-paper-excerpts.txt",
  "excerpt": "where F and F −1 denote the Fourier transform and inverse Fourier transform, respectively. Thus we can make two 2-dimensional arrays of the Gaussian density and the score function evaluated, say, on a millimeter scale across the dartboard, and rapidly compute their convolution using two FFTs (fast Fourier transform) and one inverse FFT."
 },
 "tpt-jump": {
  "source": "Tibshirani, Price and Taylor, A Statistician Plays Darts, JRSS A 174(1) (2011), author-hosted copy https://www.stat.cmu.edu/~ryantibs/papers/darts.pdf",
  "where": "sec. 4.2",
  "kept": "research/aim-where-your-misses-pay/sources/tpt2011-paper-excerpts.txt",
  "excerpt": "Shown in Figure 5 is the optimal aiming location µ∗ (σ) = argmaxµ Eµ,σ2 [s(Z)] as σ varies from 0 to 100, calculated at increments of 0.1. The path appears to be continuous except for a single jump at σ = 16.4."
 },
 "tpt-notgame": {
  "source": "Tibshirani, Price and Taylor, A Statistician Plays Darts, JRSS A 174(1) (2011), author-hosted copy https://www.stat.cmu.edu/~ryantibs/papers/darts.pdf",
  "where": "sec. 1",
  "kept": "research/aim-where-your-misses-pay/sources/tpt2011-paper-excerpts.txt",
  "excerpt": "It is important to note that we are not proposing an optimal strategy for a specific darts game."
 },
 "sig-501": {
  "source": "Tibshirani, Don't Try for the Triple 20, Significance 8(1) (2011), author-hosted copy https://www.stat.cmu.edu/~ryantibs/papers/darts-signif.pdf",
  "where": "p. 46",
  "kept": "research/aim-where-your-misses-pay/sources/tibshirani2011-significance-excerpts.txt",
  "excerpt": "Common games will often have a more complicated objective in mind (in “501”, for example, players count down from 501 points to zero and are penalised for going negative), but still involve this principle of high score."
 },
 "sig-dull": {
  "source": "Tibshirani, Don't Try for the Triple 20, Significance 8(1) (2011), author-hosted copy https://www.stat.cmu.edu/~ryantibs/papers/darts-signif.pdf",
  "where": "p. 48",
  "kept": "research/aim-where-your-misses-pay/sources/tibshirani2011-significance-excerpts.txt",
  "excerpt": "While this would certainly give an accurate estimate, no player would want to get out a ruler and measure the distances between 100 of his throws and the centre of the dartboard. This is a pretty dull task!"
 },
 "tpt-horiz": {
  "source": "Tibshirani, Price and Taylor, A Statistician Plays Darts, JRSS A 174(1) (2011), author-hosted copy https://www.stat.cmu.edu/~ryantibs/papers/darts.pdf",
  "where": "sec. 3",
  "kept": "research/aim-where-your-misses-pay/sources/tpt2011-paper-excerpts.txt",
  "excerpt": "For example, it is common for most players to have a smaller variance in the horizontal direction than in the vertical one, since the throwing motion is up-and-down with no (intentional) lateral component."
 },
 "tpt-tilt": {
  "source": "Tibshirani, Price and Taylor, A Statistician Plays Darts, JRSS A 174(1) (2011), author-hosted copy https://www.stat.cmu.edu/~ryantibs/papers/darts.pdf",
  "where": "sec. 3",
  "kept": "research/aim-where-your-misses-pay/sources/tpt2011-paper-excerpts.txt",
  "excerpt": "Also, a right-handed player may possess a different “tilt” to his or her error distribution (defined by the sign of the correlation) than a left-handed player."
 },
 "tpt-aniso": {
  "source": "Tibshirani, Price and Taylor, A Statistician Plays Darts, JRSS A 174(1) (2011), author-hosted copy https://www.stat.cmu.edu/~ryantibs/papers/darts.pdf",
  "where": "sec. 3.2",
  "kept": "research/aim-where-your-misses-pay/sources/tpt2011-paper-excerpts.txt",
  "excerpt": "The flexibility in this new model leads to some interesting results. For example, consider the case of author 2: from the scores of his 100 throws aimed at the bullseye, recall that we estimate his marginal standard deviation to be σ = 26.9 according to the simple Gaussian model. The corresponding heatmap instructs him to aim at the T19. However, under the more general Gaussian model, we estimate his x and y standard deviations to be σx = 17.9 and σy = 39.1, and the new heatmap tells him to aim slightly above the T20."
 },
 "tpt-above": {
  "source": "Tibshirani, Price and Taylor, A Statistician Plays Darts, JRSS A 174(1) (2011), author-hosted copy https://www.stat.cmu.edu/~ryantibs/papers/darts.pdf",
  "where": "sec. 3.2",
  "kept": "research/aim-where-your-misses-pay/sources/tpt2011-paper-excerpts.txt",
  "excerpt": "The flexibility in this new model leads to some interesting results. For example, consider the case of author 2: from the scores of his 100 throws aimed at the bullseye, recall that we estimate his marginal standard deviation to be σ = 26.9 according to the simple Gaussian model. The corresponding heatmap instructs him to aim at the T19. However, under the more general Gaussian model, we estimate his x and y standard deviations to be σx = 17.9 and σy = 39.1, and the new heatmap tells him to aim slightly above the T20."
 },
 "hw-pros": {
  "source": "Haugh and Wang, Play Like the Pros?, arXiv:2011.11031v2 https://arxiv.org/pdf/2011.11031",
  "where": "sec. 2",
  "kept": "research/aim-where-your-misses-pay/sources/haugh-wang-2022-arxiv-excerpt.txt",
  "excerpt": "A simple model such as (1) will not be sufficiently rich for modeling the skills of professional darts players, because these players tend to focus on (and practice throwing at) specific parts of the darts board, e.g. T20, T19, DB, etc. This means that their skill levels, as determined by Σ, are likely to depend on a, which is indeed what we observe in the data-set that we use in Section 6."
 },
 "tpt-skew": {
  "source": "Tibshirani, Price and Taylor, A Statistician Plays Darts, JRSS A 174(1) (2011), author-hosted copy https://www.stat.cmu.edu/~ryantibs/papers/darts.pdf",
  "where": "sec. 4.1",
  "kept": "research/aim-where-your-misses-pay/sources/tpt2011-paper-excerpts.txt",
  "excerpt": "As expected, the skewness parameter α is negative in both cases, meaning that there is a downwards vertical skew. However, the size of the skew is not large enough to produce heatmaps that differ significantly from Figure 4, and hence we omit them here."
 }
}