The Uniform Asymptotic Expansion of a Class of Integrals Related to Cumulative Distribution Functions (original) (raw)
Asymptotic expansions of functions of statistics
Applications of Mathematics
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Approximation of CDF of Non-Central Chi-Square Distribution by Mean-Value Theorems for Integrals
Mathematics, 2021
The cumulative distribution function of the non-central chi-square distribution χn′2(λ) of n degrees of freedom possesses an integral representation. Here we rewrite this integral in terms of a lower incomplete gamma function applying two of the second mean-value theorems for definite integrals, which are of Bonnet type and Okamura’s variant of the du Bois–Reymond theorem. Related results are exposed concerning the small argument cases in cumulative distribution function (CDF) and their asymptotic behavior near the origin.
The Annals of Statistics, 1997
Although the cumulative distribution function may be differentiated to obtain the corresponding density function, whether or not a similar relationship exists between their asymptotic expansions remains a question. We provide a rigorous argument to prove that Lugannani and Rice's asymptotic expansion for the cumulative distribution function of the mean of a sample of i.i.d. observations may be differentiated to obtain Daniels's asymptotic expansion for the corresponding density function. We then apply this result to study the relationship between the truncated versions of the two series, which establishes the derivative of a truncated Lugannani and Rice series as an alternative asymptotic approximation for the density function. This alternative approximation in general does not need to be renormalized. Numerical examples demonstrating its accuracy are included.
Integral distribution-free statistics of -type and their asymptotic comparison
Computational Statistics & Data Analysis, 2009
Generalizing the Cramér-von Mises and the Kolmogorov-Smirnov test, different integral statistics based on L p -norms are compared with respect to local approximate Bahadur efficiency. Simulation results corroborate the theoretical findings. Several examples illustrate that goodness-of-fit testing based on L p -norms should receive more attention. It is shown that, given a distribution function F 0 and a specific alternative, one can draw the plot of efficiency as a function of p and determine the value of p giving the maximum efficiency.
Some asymptotic expansions and distribution approximations outside a CLT context 1
Some asymptotic expansions non necessarily related to the central limit theorem are discussed. After observing that the smoothing inequality of Esseen implies the proximity, in the Kolmogorov distance sense, of the distributions of the random variables of two random sequences satisfying a sort of general asymptotic relation, two instances of this observation are presented. A first example, partially motivated by the the statistical theory of high precision measurements, is given by a uniform asymptotic approximation to (g(X + µn))n∈I N, where g is some smooth function, X is a random variable having a moment and a bounded density and (µ n) n∈I N is a sequence going to infinity; the multivariate case as well as the proofs and a complete set of references will be published elsewhere. We next present a second class of examples given by a randomization of the interesting parameter in some classical asymptotic formulas, namely, a generic Laplace's type integral, by the sequence (µ n X) n∈I N , X being a Gamma distributed random variable. Finally , a simulation study of this last example is presented in order to stress the quality of asymptotic approximations proposed. 1. Asymptotics for random variables The setup for the generic question studied in this paper is given by [A], the following set of conditions. A1 There is a real parameter sequence (µ n) n∈I N such that µ +∞ := lim n→+∞ µ n = +∞. A2 We are given three sequences of random variables, depending on the parameter sequence 1 , (Xn)n∈I N, (Yn = 0)n∈I N and (Zn)n∈I N, and verifying: Xn = Yn + Zn or Xn = Yn 1 + Zn Yn and lim n−→+∞ Zn Yn = 0. (1) 1 This work was partially supported by Financiamento Base 2008 ISFL-1-297 from FCT/MCTES/PT.
On New Formulas for the Cumulative Distribution Function of the Noncentral Chi-Square Distribution
Mediterranean Journal of Mathematics, 2017
The main aim of this article is to derive three new formulas for the cumulative distribution function of the noncentral chi-square distribution. The main advantage of such formulas is that they are given in terms of modified Bessel functions, leaky aquifer function and generalized incomplete gamma function which have a wide range of applications. In addition, the computational efficiency of the newly derived formulas versus already known formulas is established.
An approximation to the F distribution using the chi-square distribution
Computational Statistics & Data Analysis, 2002
For the cumulative distribution function (c.d.f.) of the F distribution, F(x; k; n), with associated degrees of freedom, k and n, a shrinking factor approximation (SFA), G( kx; k), is proposed for large n and any ÿxed k, where G(x; k) is the chi-square c.d.f. with degrees of freedom, k, and = (kx; n) is the shrinking factor. Numerical analysis indicates that for n=k ¿ 3, approximation accuracy of the SFA is to the fourth decimal place for most small values of k. This is a substantial improvement on the accuracy that is achievable using the normal, ordinary chi-square, and Sche à e-Tukey approximations. In addition, it is shown that the theoretical approximation error of the SFA, |F(x; k; n) − G( kx; k)|, is O(1=n 2 ) uniformly over x.