Quantile-parameterized distribution: Difference between revisions

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</math>

and the functions <math>g_i(y)</math> are continuously differentiable and linearly independent basis functions. Here, essentially, <math>L_0</math> and <math>L_1</math> are the lower and upper bounds (if they exist) of a random variable with quantile function <math>F^{-1}(y)</math>. These distributions are called quantile-parameterized because for a given set of quantile pairs <math>\{(x_i, y_i) \mid i=1,\ldots,n\}</math>, where <math>x_i=F^{-1}(y_i)</math>, and a set of <math>n</math> basis functions <math>g_i(y)</math>, the coefficients <math>a_i</math> can be determined by solving a set of linear equations.<ref name="KeelinPowley" /> If one desires to use more quantile pairs than basis functions, then the coefficients <math>a_i</math> can be chosen to minimize the sum of squared errors between the stated quantiles <math>x_i</math> and <math>F^{-1}(y_i)</math>. Keelin and Powley<ref name="KeelinPowley" /> illustrate this concept for a specific choice of basis functions that is a generalization of quantile function of the [[normal distribution]], <math>x=\mu+\sigma \varphiPhi^{-1} (y)</math>, for which the mean <math>\mu</math> and standard deviation <math>\sigma</math> are linear functions of cumulative probability <math>y</math>:

: <math>\mu(y)=a_1+a_4 y</math>