The Asymptotic Inversion of Certain Cumulative Distribution Functions (original) (raw)

An application of Lévy's inversion formula for higher order asymptotic expansion

Journal of Statistical Planning and Inference, 2002

The asymptotic expansions for the distribution of statistics are, in general, given by applying LÃ evy's inversion formula to the characteristic function. This paper shows an inversion formula for higher order asymptotic expansion of the distribution of a scalar valued function which contains dependent statistics. The usage of the formula is illustrated by derivation of third order asymptotic expansion of the distribution of Hotelling's T 2 -statistic under the elliptical distribution as an example.

APPROXIMATION TO THE CUMULATIVE NORMAL DISTRIBUTION USING HYPERBOLIC TANGENT BASED FUNCTIONS

This paper presents a method for approximation of the standard normal distribution by using hyperbolic tangent based functions. The presented approximate formula for the cumulative distribution depends on one numerical coefficient only, and its accuracy is admissible. Furthermore, in some particular cases, closed forms of inverse formulas are derived. Numerical results of the present method are compared with those of an existing method.

Asymptotic inversion of incomplete gamma functions

Mathematics of Computation, 1992

The normalized incomplete gamma functions P(a, x) and Q(a, x) are inverted for large values of the parameter a. That is, x-solutions of the equations P(a,x)=p, Q(a,x) = q, p e [0, 1], q = 1-p, are considered, especially for large values of a. The approximations are obtained by using uniform asymptotic expansions of the incomplete gamma functions in which an error function is the dominant term. The inversion problem is started by inverting this error function term. Numerical results indicate that for obtaining an accuracy of four correct digits, the method can already be used for a = 2, although a is a large parameter. It is indicated that the method can be applied to other cumulative distribution functions.

Series representations and approximation of some quantile functions appearing in finance

2012

It has long been agreed by academics that the inversion method is the method of choice for generating random variates, given the availability of a cheap but accurate approximation of the quantile function. However for several probability distributions arising in practice a satisfactory method of approximating these functions is not available. The main focus of this thesis will be to develop Taylor and asymptotic series representations for quantile functions of the following probability distributions; Variance Gamma, Generalized Inverse Gaussian, Hyperbolic, -Stable and Snedecor’s F distributions. As a secondary matter we briefly investigate the problem of approximating the entire quantile function. Indeed with the availability of these new analytic expressions a whole host of possibilities become available. We outline several algorithms and in particular provide a C++ implementation for the variance gamma case. To our knowledge this is the fastest available algorithm of its sort.

On the Inverted Gamma Distribution

2016

If a random variable follows a particular distribution then the distribution of the reciprocal of that random variable is called inverted distribution. In this paper we studied some issues related with inverted gamma distribution which is the reciprocal of the gamma distribution. We provide forms for the characteristic function, rth raw moment, skewness, kurtosis, Shannon entropy, relative entropy and Renyi entropy function. This paper deals also with the determination of R = P[Y < X] when X and Y are two independent inverted gamma distributions (IGD) with different scale parameters and different shape parameters. Different methods to estimate inverted gamma distribution parameters are studied, Maximum Likelihood estimator, Moments estimator, Percentile estimator, least square estimator and weighted least square estimator. An empirical study is conducted to compare among these methods.

Approximate Function Inversion by Series Expansions

ForsChem Research Reports, 2020

This short report introduces a series expansion method for obtaining approximate analytical expressions for inverse functions. The method is tested using a simple, but known function, and illustrated with the inversion of two specific cumulative probability functions, corresponding to the type III standard Epanechnikov, and the type II standard integrated-normal distributions. This series expansion method is particularly useful when the original function considered cannot be easily inverted into a known analytical expression.

On a New Class of Univariate Continuous Distributions that are Closed Under Inversion

2013

Inverted probability distributions find applications in various real – life situations including econometrics, survey sampling, biological sciences and life – testing. Closure under inversion implies that the reciprocal of a continuous random variable X has the same probability function as the original random variable, allowing for a possible change in parameter values. To date, only a very few probability distributions have been found to possess the closure property. In this paper, an attempt has been made to generate a class of distributions that are closed under inversion, and to develop some statistical properties of this class of distributions.

Quantile Mechanics 3: Series Representations of some Distributions appearing in Finance

2012

It has long been agreed by academics that the inversion method is the method of choice for generating random variates, given the availability of the quantile function. However for several probability distributions arising in practice a satisfactory method of approximating these functions is not available. The main focus of this paper will be to develop Taylor and asymptotic series expansions for the quantile functions belonging to the following probability distributions; Variance Gamma, Generalized Inverse Gaussian, Hyperbolic and α-Stable. As a secondary matter, based on these analytic expressions we briefly investigate the problem of approximating the quantile function.

Approximations for Standard Normal Distribution Function and Its Invertible

arXiv (Cornell University), 2022

In this paper, we introduce a new approximation of the cumulative distribution function of the standard normal distribution based on Tocher's approximation. Also, we assess the quality of the new approximation using two criteria namely the maximum absolute error and the mean absolute error. The approximation is expressed in closed form and it produces a maximum absolute error of 4.43 × 10 −10 while the mean absolute error is 9.62 × 10 −11. In addition, we propose an approximation of the inverse cumulative function of the standard normal distribution based on Polya approximation and compare the accuracy of our findings with some of the existing approximations. The results show that our approximations surpass other the existing ones based on the aforementioned accuracy measures.

ghyp: A package on generalized hyperbolic distributions

In this document the generalized hyperbolic (GH) distribution is explained to the extent which is required to understand the internals of the R package ghyp. Essentially, the density and moment generating functions of the GH distribution and its special cases are provided together with some important properties of the GH family. In addition the algorithm underlying the fitting procedure for the multivariate GH distribution is explained. Then we shed light on how to switch between different parametrizations of the GH distribution. In the appendix we describe the generalized inverse Gaussian distribution and give some useful facts regarding the modified Bessel function of the third kind. Finally, we write on the design on the package ghyp and give some code chunks which indicate how the software can be used.