UDC Anderson-Darling and New Weighted Cramér-von Mises Statistics. G. V. Martynov

Transcription

1 UDC Anderson-Darling and New Weighted Cramér-von Mises Statistics G. V. Martynov Institute for Information Transmission Problems of the RAS, Bolshoy Karetny per., 19, build.1, Moscow, 12751, Russia Abstract. Anderson-Darling statistic modifies the classical empirical process in the interval [, 1] by multiplying it by a weighting function ψ(t = (t(1 t 1/2. The weighting function redistributes the test sensitivity to deviations of the alternative distribution function from the hypothetical one between different subsets of [,1]. In practice, the tests can be of interest with other weighting functions. There a new formulas was proposed for eigenfunctions of the Anderson-Darling statistics. Also, it was analyzed a statistic inverse to the Anderson-Darling statistic with the weighting function ψ(t = (t(1 t 1/2. The theory is based on the use of various special functions book [1]. In practice, could be useful the Cramér-von-Mises tests with the weighting functions, belonging to family ψ(t = t α (1 t β, α > 1, β > 1. The paper contains a table of distribution for statistics with different values of the degrees α > 1 and β > 1. The table was calculated without using the statistical simulation. Keywords: Cramér-von-Mises test, Anderson-Darling statistic, goodness-of-fit test, weighting function, eigenvalues, eigenfunctions, statistical tables. 1. Introduction: Weighted Cramér-von Mises test One-dimensional weighted Cramér-von Mises statistic is ω 2 n = n ψ 2 (t(f n (t t 2 dt, where F n (t is the empirical distribution function based on the observations X 1, X 2,..., X n of the uniformly distributed on [, 1] random variable, and ψ(t is a weighting function. The statistic (1 designed to test the hypothesis H : F (t = t, against the alternative H 1 : F (t t, t [, 1], where F (x is continuous distribution function. If the condition ψ 2 (t t (1 tdt < is fulfilled then the statistic ω 2 n converges in probability to ω 2 = ξ 2 (tdt,

2 where ξ(t, t [, 1], is the Gaussian process with zero mean and the covariance function K ψ (t, τ = ψ(tψ(τ(min(t, τ tτ. The Gaussian process ξ(t can be developed in the Karhunen-Loève series x k ϕ k (t ξ(t =, λk i=1 where x k, k = 1, 2,..., are the independent standard normal random variables, and λ k and ϕ k (t, i = 1, 2,..., are the eigenvalues and eigenfunctions of the linear operator with the kernel K(t, τ, i.e. solutions of the Fredholm integral equation ϕ(t = λ ψ(tψ(τ(min(t, τ tτϕ(τdτ. Under the contigual alternatives H 1 : F n (t = t + δ(t/ n, n = 1, 2,..., the distribution of ω 2 is the distribution of noncentral quadratic form Q = (x k + δ k 2, where δ k = λ k i=1 δ(tϕ k (tdt. By twice differentiation (2 respect to t, we obtain differential equation h (t + λψ 2 (th(t = with the conditions h( = h(1 =. Here, h(t = ϕ(t/ψ(t. Deheuvels and Martynov in article [3] described for ψ(t = t β the follows result. Let {B(t : t 1} be the Brownian bridge. Then, for each β = 1 2ν 1 > 1, the Karhunen-Loeve expansions of {ξ(t = t β B(t : < t 1} is given by ( t β x k e kb (t t 1 2ν 1 2 J ν z ν,k t 1 2ν B(t =, e B,k (t =, < t 1. λkb νjν 1 (z ν,k k=1 Here, {ω k : k 1} are i.i.d. N(, 1 random variables, and, for k = 1, 2,..., the eigenvalues are λ k = (z ν,k /2ν 2, z ν,k, k = 1, 2,..., are zeros of the Bessel functions J ν (z. It is considered also the Cramér-von Mises statistic of the form ω 2 n(a, b = n t 2a (1 t 2b (F n (t t 2 dt

3 with the weighting function ψ(t = t a (1 t b, a > 1, known case is the Anderson-Darling statistic b > 1. The well A 2 n = ωn(.5, 2.5 = n (F n (t t 2 dt t(1 t with a =.5 and b =.5. For the statistic A 2 n equation (3 is transformed to t(1 th (t + λh(t =, h( = h(1 =. Anderson and Darling in article [2] found that their statistic has λ k = k(k+ 1 and h k (t = t(1 tp k (2t 1, k = 1, 2,..., where P k(t, k = 1, 2,..., are the Legendre polynomials. The information related to the subject of this work can also be found in book [, 5, 5, 6]. 2. New formulas for eigenfunctions for the the Anderson-Darling statistic At first, we will propose a direct method for deriving expressions for considering eigenfunctions. We will find the possible solutions of the differential equation ( in the form h k (t = t(1 t(1+a 1, k t+a 2, k t a k 1, k t k 1 t(1 tv k (t, k = 1, 2,... Here, h 1 (t and h 2 (t are understood to be h 1 (t = t(1 t and h 2 (t = t(1 t(1 + a 1, 2 t. This solution satisfy to the conditions h( = h(1 =. Theorem 1 The solutions of the equation ( can be represented for each λ = λ k = k(k + 1, k = 1, 2,..., as or V k (t = 1 + β 1,k t + β 1,k β 2,k t 2 + β 1,k β 2,k β 3,k t β 1,k...β k 1,k t k 1 V k (t = ((((β k 1, k t + 1β k 2, k t β 3, k t + 1β 2, k t + 1β 1, k t + 1, where β s, k = 1 λ k /λ s = 1 k(k+1 s(s+1, s = 1, 2,..., k 1. The following theorem solution can be represents another solution with using of the hypergeometric functions. Theorem 2 The solutions of the equation ( can be represented for each λ = λ k = k(k + 1, k = 1, 2,..., as h k (t = t 2F 1 ( k, k + 1; 2; t, k = 1, 2,...

4 This result can be derived from the fact that the equation ( is particular case of the equation for hypergeometric function 2 F 1. Theorem 3 The following identity is valid: (1 tv k (t 2 F 1 ( k, k + 1; 2; t. Theorem The normalized eigenfunctions of the covariance operator corresponding to the Anderson-Darling statistic can be written as: ϕ k (t = 2 k(k + 1(2k + 1 t(1 tv k (t = 2 t k(k + 1(2k + 1 (1 t 2 F 1 ( k, k + 1; 2; t = 2r + 1 = 2 t(1 tp k(k + 1 k (2t 1, k = 1, 2,..., t [, 1]. 3. Statistic with ψ(t = t(1 t Here, we will consider the statistic Cramér-von Mises with ψ(t = t(1 t. It is inverse for Anderson-Darling statistic The equation (3 have the form h (t + λt(1 th(t = with the conditions h( = and h(1 =. Its solution is h(t = C 1 1F 1 ( 1 a 16, 1 2, +C 2 1F 1 ( 3 a 16, 3 2, a( 1 + 2x 2 a( 1 + 2x 2 exp( ax(1 x/2 ( 1 + 2x exp( ax(1 x/2. Here, 1 F 1 (a; b; z is the Kummer confluent hypergeometric function. With applying the conditions h( = h(1 =, the following equations can be derived h( = C 1 1F 1 ( 1 a 16, 1 2, h(1 = C 1 1F 1 ( 1 a 16, 1 2, a a C 2 1F 1 ( 3 a 16, 3 2, + C 2 1F 1 ( 3 a 16, 3 2, a =, a =.

5 Hence, the equation for eigenvalues is the determinant of the previous equation ( 1 a 1F 1 16, 1 ( a 3 a 2, 1F 1 16, 3 a 2, =. This equation can be resolved by numerical methods. First zeros of the left multiplier are: First zeros of the right multiplier are: Another weighted statistics The equation (3 with ψ(t = (t(1 t 3/2 has the solution with Heun triconfluent function. The same equation with ψ 2 (t = 1 cos(πt has the solution with Mathieu function. 5. Table of the distribution for statistics with ψ(t = (t(1 t α In the Table 1 we present the quantiles of the ω 2 (α, β distribution with α = β. More detailed tables can be found in article [7]. Quantiles of ω 2 (α, α/s Table 1 p/α S

6 Acknowledgments The work was carried out at IITP RAS and supported by Russian Science Foundation (grant RSF No References 1. Abramowitz M., Stegun I. A. Handbook of Mathematical Function. National Bureau of Standards. Applied Nathematics Series Vol Anderson T. W., Darling D. A. Asymptotic theory of certain Goodness of Fit criteria based on stochastic processes // The Annals of Mathematical Statistics Vol. 23, no. 2. P Deheuvels P., Martynov G. Karhunen-Loève expansions for weighted Wiener processes and Brownian bridges via Bessel functions // Progress in Probability. Birkhäuser, Basel/Switzerland, 23. Vol. 55. P Durbin J., Knott M. Components of the Cramér-von-Mises statistics. I. // J. R. Statist. Soc., Ser. B, Vol. 3. P Martynov G. V. The Omega Square Tests. M.: Nauka, (in Russian. 6. Martynov G. V. Statistical tests based on empirical processes and related questions // J. Soviet. Math Vol. 61. P (Original Russian Text published in Progress in Science and Technology, Series on Probability Theory, Mathematical Statistics, Theoretical Cybernetics (Itogi Nauki i Tekhniki, VINITI Acad. of Sc. USSR, Moscow, 199. Vol. 3. P Martynov G. V. Anderson Darling statistic and its inverse // Journal of Communications Technology and Electronics Vol. 61, no. 6. P ( Original Russian Text published in Informatsionnye Protsessy Vol. 15, no. 3. P