Notas De Finanzas

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Chapter 1. Bootstrap Method
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1.1

Introduction
The Practice of Statistics

Statistics is the science of learning from experience, especially experience that arrives a little bit at a time. Most people are not natural-born statisticians. Left to our own devices we are not very good at picking out patterns from a sea of noisy data. To put it another way, we all are too good at pickingout non-existent patterns that happen to suit our purposes? Statistical theory attacks the problem from both ends. It provides optimal methods for finding a real signal in a noisy background, and also provides strict checks against the overinterpretation of random patterns. Statistical theory attempts to answer three basic questions: 1. Data Collection: How should I collect my data? 2. Summary: Howshould I analyze and summarize the data that I’ve collected? 3. Statistical Inference: How accurate are my data summaries? The bootstrap is a recently developed technique for making certain kinds of statistical inferences. It is only recently developed because it requires modern computer power to simplify the often intricate calculations of traditional statistical theory.

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MotivatedExample

We now illustrate the just mentioned three basic statistical concepts using a front-page news from the New York Times of January 27, 1987. A study was done to see if small aspirin doses would prevent heart attacks in healthy middle-aged men. The data for the aspirin study were collected in a particularly efficient way: by a controlled, randomized, doubleblind study. One half of the subjectsreceived aspirin and the other half received a control substance, or placebo, with no active ingredients. The subjects were randomly assigned to the aspirin or placebo groups. Both the subjects and the supervising physicians were blind to the assignments, with the statisticians keeping a secret code of who received which substance. Scientists, like everyone else, want the subject they are workingon to succeed. The elaborate precautions of a controlled, randomized, blinded experiment guard against seeing benefits that don’t exist, while maximizing the chance of detecting a genuine positive effect.

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The summary statistics in the study are very simple: heart attacks (fatal plus non-fatal) aspirin group: placebo group: 104 189 subjects 11,037 11,034

What strikes the eye here is thelower rate of heart attacks in the aspirin group. The ratio of the two rates is ˆ 104/11037 = 0.55. θ= 189/11034 It suggests that the aspirin-takers only have 55% as many as heart attacks as placebo-takers. ˆ Of course we are not interested in θ. What we would like to know is θ, the true ratio, that is the ratio we would see if we could treat all subjects, and not just a sample of them. ˆ The toughquestion is how do we know that θ might not come out much less favorably if the experiment were run again? This is where statistical inference comes in. Statistical theory allows us to make the following inference: the true value of θ lies in the interval 0.43 < θ < 0.70 with 95% confidence. Note that ˆ ˆ ˆ θ = θ + (θ − θ) = 0.55 + [θ − θ(ω0 )], ˆ ˆ where θ and θ(ω0 ) (= 0.55) are two numbers. Instatistics, we use θ − θ(ω) to describe ˆ ˆ θ − θ(ω0 ). Since ω cannot be observed exactly, we instead study the fluctuation of θ − θ(ω) ˆ among all ω. If, for most ω, θ − θ(ω) is around zero, we can conclude statistically that θ is ˆ ˆ close to 0.55 (= θ(ω0 ). (Recall the definition of consistency.) If P (ω : |θ − θ(ω)| < 0.1) = 0.95, we claim that with 95% confidence that θ − 0.55 is no more than0.1. In the aspirin study, it also track strokes. The results are presented as the following: strokes aspirin group: placebo group: For strokes, the ratio of the two rates is ˆ 119/11037 = 1.21. θ= 98/11034 It now looks like taking aspirin is actually harmful. However, the interval for the true stroke ratio θ turns out to be 0.93 < θ < 1.59 with 95% confidence. This includes the neutral value θ =...