Converting A Single Distribution To A Location-scale Family
The following shows how to implement a location-scale family in a statistical package or programming environment where only functions for the "standard" version of a distribution are available. It is designed for R but should generalize to any language and library.
The example here is of the Student's t-distribution, which is normally provided in R only in its standard form, with a single degrees of freedom parameter df. The versions below with _ls appended show how to generalize this to encompass an arbitrary location parameter mu and scale parameter sigma.
| Probability density function (PDF): | dt_ls(x, df, mu, sigma) = |
1/sigma * dt((x - mu)/sigma, df) |
| Cumulative distribution function (CDF): | pt_ls(x, df, mu, sigma) = |
pt((x - mu)/sigma, df) |
| Quantile function (inverse CDF): | qt_ls(prob, df, mu, sigma) = |
qt(prob, df)*sigma + mu |
| Generate a random variate: | rt_ls(df, mu, sigma) = |
rt(df)*sigma + mu |
Note that the generalized functions do not have standard deviation sigma since the standard t distribution does not have standard deviation 1.
Read more about this topic: Location-scale Family
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