Quantile Mechanics 3: Series Representations of some Distributions appearing in Finance (original) (raw)

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.

Functions appearing in Finance

2016

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.

Quantile Function Expansion Using Regularly Varying Functions

Methodology and Computing in Applied Probability

We present a simple result that allows us to evaluate the asymptotic order of the remainder of a partial asymptotic expansion of the quantile function h(u) as u → 0 + or 1 − . This is focussed on important univariate distributions when h(·) has no simple closed form, with a view to assessing asymptotic rate of decay to zero of tail dependence in the context of bivariate copulas. The Introduction motivates the study in terms of the standard Normal. The Normal, Skew-Normal and Gamma are used as initial examples. Finally, we discuss approximation to the lower quantile of the Variance-Gamma and Skew-Slash distributions.

Quantile Approximation of the Chi–square Distribution using the Quantile Mechanics

2017

In the field of probability and statistics, the quantile function and the quantile density function which is the derivative of the quantile function are one of the important ways of characterizing probability distributions and as well, can serve as a viable alternative to the probability mass function or probability density function. The quantile function (QF) and the cumulative distribution function (CDF) of the chi-square distribution do not have closed form representations except at degrees of freedom equals to two and as such researchers devise some methods for their approximations. One of the available methods is the quantile mechanics approach. The paper is focused on using the quantile mechanics approach to obtain the quantile density function and their corresponding quartiles or percentage points. The outcome of the method is second order nonlinear ordinary differential equation (ODE) which was solved using the traditional power series method. The quantile density function w...

On generating T-X family of distributions using quantile functions

Journal of Statistical Distributions and Applications, 2014

The cumulative distribution function (CDF) of the T-X family is given by R{W(F(x))}, where R is the CDF of a random variable T, F is the CDF of X and W is an increasing function defined on [0, 1] having the support of T as its range. This family provides a new method of generating univariate distributions. Different choices of the R, F and W functions naturally lead to different families of distributions. This paper proposes the use of quantile functions to define the W function. Some general properties of this T-X system of distributions are studied. It is shown that several existing methods of generating univariate continuous distributions can be derived using this T-X system. Three new distributions of the T-X family are derived, namely, the normal-Weibull based on the quantile of Cauchy distribution, normal-Weibull based on the quantile of logistic distribution, and Weibull-uniform based on the quantile of log-logistic distribution. Two real data sets are applied to illustrate the flexibility of the distributions.

Recovery of quantile and quantile density function using the frequency moments

Statistics & Probability Letters, 2018

The problem of recovering quantiles and quantile density functions of a positive random variable via the values of frequency moments is studied. The uniform upper bounds of the proposed approximations are derived. Several simple examples and corresponding plots illustrate the behavior of the recovered approximations. Some applications of the constructions are discussed as well. Namely, using the empirical counterparts of the constructions yield the estimates of the quantiles, and the quantile density functions. By means of simulations, the average errors in terms of L 2-norm are evaluated to justify the consistency of the estimate of the quantile density function. As an application of the constructions, the question of estimating the so-called expected shortfall measure in risk models is also studied.

Asymptotic Expansions of Generalized Quantiles and Expectiles for Extreme Risks

Probability in the Engineering and Informational Sciences, 2015

Generalized quantiles of a random variable were defined as the minimizers of a general asymmetric loss function, which include quantiles, expectiles and M-quantiles as their special cases. Expectiles have been suggested as potentially better alternatives to both Value-at-Risk and expected shortfall risk measures. In this paper, we first establish the first-order expansions of generalized quantiles for extreme risks as the confidence level α↑ 1, and then investigate the first-order and/or second-order expansions of expectiles of an extreme risk as α↑ 1 according to the underlying distribution belonging to the max-domain of attraction of the Fréchet, Weibull, and Gumbel distributions, respectively. Examples are also presented to examine whether and how much the first-order expansions have been improved by the second-order expansions.

A new reduced quantile function for generating families of distributions

2024

In this paper, a variant of the T-X(Y) generator was developed by suppressing the scale parameter of the classical Lomax distribution in the quantile function. Uniquely, the reduction of the number of parameters essentially accounts for the parsimony of the attendant model. The study considered the Exponential distribution as the transformer and consequently obtained the New Reduced Quantile Exponential-G (NRQE-G) family. The Type-II Gumbel distribution was deployed as the baseline to obtain a special sub-model known as the New Reduced Quantile Exponential Type-II Gumbel (NRQE-T2G) model. Some functional properties of the distribution namely, moment and its related measures such as the mean, variance, second, third, and fourth moments were obtained. The Mode, skewness, Kurtosis, index of dispersion, coeffi cient of variation, order statistics, survival, hazard, and quantile function were also derived. The maximum likelihood estimation method was used to estimate its parameters. The model's credibility, applicability, and fl exibility were proven using two real-life datasets.

The Asymptotic Inversion of Certain Cumulative Distribution Functions

Mathematics in Industry, 2010

The inversion of cumulative distribution functions is an important topic in statistics, probability theory and econometrics, in particular for computing percentage points of the distribution functions. The numerical inversion of these distributions needs accurate starting values, and for the standard distributions powerful asymptotic formulas can be used to obtain these values.. It is explained how a uniform asymptotic expansions of a standard form representing several well-known distribution functions can be used for the asymptotic inversion of these functions. As an example we consider the inversion of the hyperbolic cumulative distribution function.