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Here is the definition of bootstrap t-method in book Statistics and Data Analysis for Financial Engineering with R examples page 144. But I cannot understand:

  1. Here $\bar{Y}$ is really the population mean (resampling), but random variable itself is not normal. How can equation (6.8) be a t-distribution approximately (I remember the condition of Gaussian is necessary for t-distribution)?

  2. Suppose equation (6.8) is t-distribution, then why do we use $\alpha/2$ quantiles of resamples but not use the $\alpha/2$ t-value?

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Confidence Interval for a General Parameter:

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Basic Bootstrap Interval:

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Regarding question 1: In general (without additional assumptions), this statistic does not follow a t-distribution because, as you said, neither is the mean normally distributed at arbitrary sample sizes nor does the variance follow a chi-squared distribution when normality of the sample values may not be assumed (well, put aside some asymptotic considerations that do not matter for the answer of your question). The statistic is called t mainly because it is computed using a very similar formula but it does not follow a t-distribution and the text does not imply that it should. However, if normality holds, this distribution will closely resemble a t-distribution on the long run.

Regarding your question 2: Bootstrapping essentially means resampling from your sample to get an estimate of the true unknown sampling distribution of your t-statistic. Here, you estimate the sampling distribution of your t-statistic. It will be an approximate t-distribution under conditions under which the "classic" t-test would be valid and under any other conditions, it will resemble whatever the sampling distribution is. For instance, if your variable follows a skewed distribution, the bootstrap samples of t will also be skewed. Then, the t distribution is no longer your best approximate for the sampling distribution but rather your estimated distribution is. Hence, you should compute the quantiles on this distribution instead.

Does this help?

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  • $\begingroup$ Many thanks. So except for the form similar with t-distribution, it is nothing related to t-distribution. Pls see my update, do you know 1. How could we obtain the approximation 6.12 from 6.11, is it using bias of resampling to approximate the bias of sampling? And why is bootstrap t-method more precise than Basic bootstrap (introduced at the end)? $\endgroup$ – user6703592 Jul 30 at 14:42
  • $\begingroup$ No, there's no explicit bias correction in this method which is why it works not so well for very skewed parameters (such as correlations). You simply assume that the variability across your bootstrap samples approximates the across-sample variation of the quantities - that is, you go from 6.11 to 6.12. This has also been proven somewhere but I have no comprehensive reference for you, sorry. Maybe this helps to disentangle all the crazy bootstrap variants: users.stat.umn.edu/~helwig/notes/bootci-Notes.pdf $\endgroup$ – StoryTeller0815 Jul 30 at 19:09
  • $\begingroup$ Regarding your second question, I don't know but I strongly recommend you to leave your original reference and do some research. There are many freely available texts, e.g.: m.tau.ac.il/~saharon/Boot/10.1.1.133.8405.pdf or jstor.org/stable/2345699?seq=1#page_scan_tab_contents or the bible of the bootstrap: pdfs.semanticscholar.org/9169/… (this is not freely available) $\endgroup$ – StoryTeller0815 Jul 30 at 19:14

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