Closed-Form Expressions for the Quantile Function of the Chi Square Distribution Using the Hybrid of Quantile Mechanics and Spline Interpolation
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Chi square distribution is a continuous probability
distribution primarily used in hypothesis testing,
contingency analysis, and construction of
confidence limits in inferential statistics but not
necessarily in the modeling of real-life phenomena.
The closed-form expression for the quantile
function (QF) of Chi square is not available because
the cumulative distribution function cannot be
transformed to yield the QF and consequently
places limitations on the use of the QF. Researchers
have over the years proposed approximations that
improve over time in terms of speed, computational
efficiency, and precision, and so on. However, most
of the available closed-form expressions (quantile approximation) fail at the extreme tails of the
distribution. This paper used the Quantile
mechanics approach to obtain second-order
nonlinear ordinary differential equations whose
solutions using the power series method yielded
initial approximates in form of series for different
values of the degrees of freedom. The initial
approximate varies with the exact (R software)
values which serve as the reference and the error
between them was minimized by the natural cubic
spline interpolation. The final approximates are
correct up to an average of 8 decimal places, have
small error, and is closer to the exact when
compared with some other results from other
researchers. The upper tail of the distribution was
considered and excellent results were obtained
which is a major improvement over the existing
results in the literature. The approach used in this
work is a hybrid of series expansions and numerical
algorithms. Computer codes can be written for the
application.
Keywords
QA Mathematics