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.
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The financial support of the Natural Sciences and Engineering Research Council of Canada is gratefully acknowledged. We also wish to thank the reviewers for their valuable comments and suggestions.
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Appendix:
Appendix:
1.1 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
and
with
the Hermite interpolation function on each interval \(\left[ x_{i}, x_{i+1}\right] , \ i=0,1, \ldots ,p-1,\) being
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
where \(\ell\) and u denote the estimated endpoints.
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Provost, S.B., Yang, Z. & Ahmed, S.E. Securing Density Estimates via Smooth Moment-Based Empirical Distribution Function Approximants. J Indian Soc Probab Stat 23, 1–18 (2022). https://doi.org/10.1007/s41096-022-00119-4
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DOI: https://doi.org/10.1007/s41096-022-00119-4