Securing Density Estimates via Smooth Moment-Based Empirical Distribution Function Approximants

Abstract

This paper proposes an adaptive density estimation procedure that hinges on securing moment-based approximants of certain splines passing through particular points that are obtained from an appropriately adjusted and truncated empirical distribution function. More specifically, a four-parameter beta density estimate is initially fitted to the data in order to determine the endpoints of the distribution which are combined to the data points. Interpolants of the continuity-corrected empirical distribution function evaluated at these points are then approximated by smooth functions involving polynomials. As a matter of course, the density estimates are obtained by differentiation. Any quantile of the corresponding distribution can thereby be directly evaluated from the associated distribution functions. The Cramér-von Mises goodness-of-fit statistic is utilized as a measure of accuracy. Three illustrative examples are presented.

Appendix:

The Fritsch-Carlson Monotonic Cubic Interpolation

Given a set of data points $\left( x_{0}, f_{0}\right) ,\left( x_{1}, f_{1}\right) , \ldots ,\left( x_{p}, f_{p}\right)$ where $x_0< \cdots < x_p$ and $f_0< \cdots < f_p$, this approach relies on Hermite interpolation with the requirement that the derivatives at the knots be $s_1 \tau _1,\, s_2 \tau _2, \ldots , s_p \tau _p$, where

$$\begin{aligned} s_i = \left\{ \begin{array}{ll} \frac{f_2-f_1}{f_1-f_0}, &{} i=1,\\ \frac{1}{2} \left( \frac{f_{i}-f_{i-1}}{f_{i-1}-f_{i-2}} + \frac{f_{i+1}-f_{i}}{f_{i}-f_{i-1}} \right) , &{} i=2,\ldots ,p-1, \\ \frac{f_p-f_{p-1}}{f_{p-1}-f_{p-2}}, &{} i=p, \end{array} \right. \nonumber \\ \end{aligned}$$

and

$$\begin{aligned} \tau _i = \left\{ \begin{array}{ll} \min \Big ( \frac{3\,\delta _1}{\sqrt{s_1^2 + s_2^2}}, \, 1 \Big ), &{} i=1,\\ \min \Big ( \frac{3\,\delta _{i}}{\sqrt{s_{i}^2 + s_{i+1}^2}}, \, \frac{3\,\delta _{i-1}}{\sqrt{s_{i-1}^2 + s_{i}^2}}, \, 1 \Big ), &{} i=2,\ldots ,p-1, \\ \min \Big ( \frac{3\,\delta _{p-1}}{\sqrt{s_{p-1}^2 + s_p^2}}, \, 1 \Big )&{} i=p, \\ \end{array} \right. \nonumber \\ \end{aligned}$$

with

$$\begin{aligned} \delta _i = \frac{f_{i+1} - f_{i}}{f_{i} - f_{i-1}}, \quad i = 1,\ldots ,p-1, \end{aligned}$$

the Hermite interpolation function on each interval $\left[ x_{i}, x_{i+1}\right] , \ i=0,1, \ldots ,p-1,$ being

$$\begin{aligned} H(x)=&\left( 1+2 \frac{x-x_{i}}{x_{i+1}-x_{i}}\right) \left( \frac{x-x_{i+1}}{x_{i}-x_{i+1}}\right) ^{2} f_{i}+\left( 1+2 \frac{x-x_{i+1}}{x_{i}-x_{i+1}}\right) \left( \frac{x-x_{i}}{x_{i+1}-x_{i}}\right) ^{2} f_{i+1} \\&+\left( x-x_{i}\right) \left( \frac{x-x_{i+1}}{x_{i}-x_{i+1}}\right) ^{2} f_{i}^{\prime }+\left( x-x_{i+1}\right) \left( \frac{x-x_{i}}{x_{i+1}-x_{i}}\right) ^{2} f_{i+1}^{\prime } \,. \end{aligned}$$

If one wishes readily to obtain some d.f. values or to determine certain quantiles on the basis of $\{x_1,\ldots ,x_n\},$ a sample of n ordered observations, a Fritsch-Carlson third degree spline ought to provide reasonably accurate values throughout the support of the distribution once it is applied to the points

$$\begin{aligned} \big \{(\ell ,0), (x_1, {\tfrac{1}{n}}-{\tfrac{1}{2n}}),\ldots , (x_n, {\tfrac{n}{n}}-{\tfrac{1}{2n}}) , (u,1)\big \}, \end{aligned}$$

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where $\ell$ and u denote the estimated endpoints.