Applied Data-Centric Social Sciences by Aki-Hiro Sato

Author:Aki-Hiro Sato
Language: eng
Format: epub
Publisher: Springer Japan, Tokyo

(3.168)

where .

we call this the Bootstrap sample, resampled from . Then we can approximate the sampling distribution of by the Bootstrap distribution of

(3.169)

i.e., the distribution of is induced by the random mechanism Eq. (3.168) with held fixed at its observed value. The simplest way to set probabilities to resample sequences is to put the probability at each point of .

Let be the data matrix and represent the -th data set, i.e., . In our model, we conduct the Bootstrap sequences by the following procedure.

As we express in Fig. 3.4, we select data sets at random with replacement from the data sets and call the obtained data sets . We introduce its -th data set so that where is selected with the probability . We also select data sets from and call it . Its -th content will be where is selected with the probability . We connect the obtained data sets as .

Then, we calculate the disturbance terms , and which we get by substituting data sets , and for Eqs. (3.114), (3.161) and (3.162). Using , and , we can calculate and by Eqs. (3.119) and (3.163). At the end, we get the Bootstrap variable by substituting and for Eq. (3.164). Repeating this procedure, we can estimate the sampling distribution of by means of the Bootstrap distribution.

Fig. 3.4Procedure to construct Bootstrap sequences , and from

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