Estimating parameters for multiple variables

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See also: The Bootstrap, Analyzing and using data introduction, The parametric Bootstrap, The non-parametric Bootstrap, VoseNBoot,

We are sometimes interested in estimating parameters that describe relationships between variables, for example: regression parameters and rank correlation coefficients. The Bootstrap can provide uncertainty about these estimates in an intuitive way, by Bootstrapping the paired data values.

From the perspective of ModelRisk code, the key to Bootstrapping pairs (or triplets, etc.) of data is to recognise which elements are random and which are not. Bootstrapping attempts to estimate the distributions of the random components and then resample from them.

Bootstrapping pairs (triplets, etc) of a data set is very easy to do. The method we employ has two steps:

  1. A VoseStepUniform(1,n) distribution is created to provide us with an index that will select a random pair, where we have n pairs of observations;

  2. A lookup logic then uses this index to select a value from the data set. We tend to use the INDEX or OFFSET functions for this purpose, though the VLOOKUP and HLOOKUP functions will also work

Example 1: Estimate of regression parameters

We show how the both non-parametric and parametric versions of the Bootstrap can be used for linear regression, and how the non-parametric version relaxes the assumption of Normally distributed residuals.

Example 2: Difference between two population means

We offer two models to analyse the difference between two means. The first model considers random samples for two populations and analyses what we can infer about the difference between their means. The second model considers before and after effects of some experiment, to analyse the mean effect.

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