Entropy based test of goodness of fit is proposed for inverse-gaussian distribution

V. Mergel

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Mathematical Communications 4(1999), 191-195 191Test of goodness of fit for the inverse-gaussiandistributionVictor Mergel∗Abstract. Entropy based test of goodness of fit is proposed forinverse-gaussian distribution.Key words: entropy, statistical estimate, inverse-gaussian distri-bution, test of goodness of fitAMS subject classifications: 60G07, 60G17Received January 18, 1999 Accepted May 6, 19991. IntroductionDenote by R1 a real line with the distance ρ(x, y) = |x− y| between x ∈ R1 andy ∈ R1. Then let v(y, r) = {x ∈ R1 : ρ(x, y) < r}.Consider now the random variable ξ with unknown density function f(x), x ∈R1. Denote by supp(f) = {x ∈ R1 : f(x) > 0}. The problem is to estimate theentropyH = −∫R1f(x) ln f(x)dx <∞ (1)based on the independent identically distributed sample X1, . . . , XN , N ≥ 2, ofrandom variable ξ.For a fixed observation Xi, i ∈ {1, ..., N}, and fixed k ∈ {1, ..., N−1}, we definerandom variables ρi,k as follows:ρi,1 := min{ρ (Xi, Xj) , j ∈ {1, ..., N}\ {i}} = ρ (Xi, Xj1),ρi,2 := min{ρ (Xi, Xj) , j ∈ {1, ..., N}\ {i, j1}} = ρ (Xi, Xj2),...ρi,k := min{ρ (Xi, Xj) , j ∈ {1, ..., N}\ {i, j1, . . . , jk−1}} = ρ (Xi, Xjk),...ρi,N−1 := max{ρ (Xi, Xj) , j ∈ {1, ..., N} \ {i}} = ρ(Xi, XjN−1),where Xjk−1 isa vector such that with probability 1:ρ(Xi, Xjk−2)< ρ(Xi, Xjk−1)< ρ (Xi, Xjk)∗Department of Mathematics, Taras Shevchenko University of Kyiv, Vladimirskaya 64, 252601Kyiv, Ukraine, e-mail: VIK@g-auto.pul.kiev.ua192 V. MergelConsequently with probability 1, we haveρ (Xi, Xj1) < ρ (Xi, Xj2) < . . . < ρ(Xi, XjN−1).For a fixed k ∈ {1, . . . , N − 1}, we letρ¯k = {N∏i=1ρi,k}1/N (2)denote the geometric mean of random variables ρ1,k, . . . , ρN,k. We define the sta-tistical estimate of (1)Hk,N = ln ρ¯k + ln2γ(N − 1),where k ∈ {1, 2, . . . , N −1} and γ=exp{− ∫∞0e−t ln tdt} is an Euler constant. Thefollowing Theorems are proved in [5]Theorem 1. Let k ∈ {1, . . . , N −1} be fixed and suppose that there exists an ε > 0such that ∫R1| ln f(x)|1+εf(x)dx <∞. (3)ThenlimN→∞EHk,N = H,where entropy H is defined in (2).Theorem 2. Let k, ε > 0 as in Theorem 1 and∫R1| ln f(x)|2+εf(x)dx <∞. (4)Then the entropy estimate Hk,N given in (2), is a consistent estimate of H, asN →∞.Note that the estimate H1,N was considered by Kozachenko and Leonenko (see[9]), but our conditions are weaker than those proposed in [9] even in the case k = 1.2. Maximum entropy principle for the inverse - gaussian dis-tributionWe present first the following:Theorem 3. Let K be a class of probability density functions f(x), x ∈ R1 withsupp{f} = (0,∞) which satisfy conditions (3) and (4). Note that the densityf∗ (x) =√λ√2pix3exp{− λ2m2(x−m)2x}I(0,∞) (x) (5)of an inverse-gaussian distribution IG(m,λ) belongs to the class K. Then amongall densities f from class K such that∞∫0xf(x)dx = m (6)Test of goodness of fit for the inverse-gaussian distribution 193∞∫0logxf(x)dx = 2λme(λm− 1λ ) logm+ 2eλm√λm√2pi(Kδ(1λ))′δ∣∣∣∣∣δ=− 12(7)∞∫0f(x)xdx =1m+1λ(8)the entropy is maximized by the density of the inverse-gaussian distribution (Kδ (λ)is a modified Bessel function).Proof. To prove this theorem we use the Maximum-Entropy Principle (see [8]).The Lagrangian in this case will be:L = −∞∫0f(x) log f(x)dx − α(∞∫0f(x)dx− 1)− µ1(∞∫0xf(x)dx−m)−µ2(∞∫0logxf(x)dx−(2 λme(λm− 1λ ) logm + 2eλm√λm√2pi(Kδ(1λ))′δ∣∣∣δ=− 12))−µ3(∞∫0f(x)xdx− ( 1m+ 1λ)).Then∂L∂f= −1− logf (x)− α− µ1x− µ2 logx− µ3x= 0orf (x) = A exp{−µ1x− µ2 logx− µ3x}.To determie A, µ1, µ2, µ3 we use (6) , (7) , (8) and∞∫0f(x)dx = 1. Som = A∞∫0exp{−µ1x− (µ2 − 1) logx− µ3x},2λme(λm− 1λ ) logm + 2eλm√λm√2pi(Kδ(1λ))′δ∣∣∣∣∣δ=− 12= A∞∫0logx exp{−µ1x− µ2 logx− µ3x},1m+1λ= A∞∫0exp{−µ1x− (µ2 + 1) logx− µ3x}and1 = A∞∫0exp{−µ1x− µ2 logx− µ3x}.194 V. MergelFrom here it is easy to get that: µ1 = λ2m2 , µ2 =32, µ3 = λ2 , A =√λ√2piexp{λm}andf∗ (x) =√λ√2piexp{λm}exp{−32 logx− λ2m2x− λ2x} I(0,∞) (x)=√λ√2pix3exp{− λ2m2(x−m)2x}I(0,∞) (x)that isH (f) ≤ H (f∗) = −∞∫0f(x) log f(x)dx= −∞∫0√λ√2pix3exp{− λ2m2(x−m)2x}(12log λ2pix3− λ2m2(x−m)2x)dx= −∞∫012f∗ (x) log λ2pidx+32∞∫0f∗ (x) logxdx+ λ2m2∞∫0f∗ (x)xdx− λm∞∫0f∗ (x)dx+ λ2∞∫0f∗(x)x dx =12 log2piλ +12+3 λm e( λm− 1λ ) logm+ 3eλm√λm√2pi(Kδ(1λ))′δ∣∣∣δ=− 1223. Tests of goodness of fit for the inverse-gaussian distribu-tionWe propose the entropy based test of goodness of fit for the so-called inverse-gaussian distribution (see, for example [10]). This distribution is important forboth turbulence theory and finance (see, for example [1] and [2]).The main idea for the construction of tests of goodness of fit is based on themaximum entropy principle (see [8]). Consider a class of densities satisfying certainrestrictions. Find a consistent estimator of entropy for the members of the class.Next, using the so-called maximun entropy principle (see, for example [8] or formore recent developments [4]),determine a member of the class maximizing entropyand find its parametric consistent estimator. Finally, take a function of the aboveestimators as a test statistics of goodness fit for the member maximizing the entropy.Tests of goodness of fit based on the sample entropy was proposed by Vasicek(see [11]) for one-dimensional normal distribution and by Dudewicz and Van derMeuler (see [3]) for uniform distribution, among the others. The essential differencebetween the tests proposed by these authors and ours lies in the choice of entropyestimator. In our work we have broaded this list of entropy-based tests, proposedby Goria, Leonenko and Mergel (see [5]).Let X1, X2, . . .XN , N ≥ 2, be an independent sample from a member of K .Let X¯N , S2N denote the sample mean and variance, respectively, then∧mN= X¯N∧1λN=S2N∧m3NTest of goodness of fit for the inverse-gaussian distribution 195are consistent estimators of parameters m, λ, respectively.Under the null hypothesis H0 : X1, X2, . . .XN are a sample from the inverse-gaussian distribution IG(m,λ), we have that for each fixed k ∈ {1, . . . , N − 1}ζN = log(√2pie SN√∧m3N)+ 3∧m2N log∧mNS2Nexp{ ∧m2NS2N− S2N∧m3N}+ 3√∧mNSN√2piexp{ ∧m2NS2N} (Kδ(S2N∧m3N))′δ∣∣∣∣∣δ=− 12−Hk,N → 0in probability N → ∞. While under the alternative H1 : X1, X2, . . .XN are asample from any other distribution from class K with finite second order moment,we obtain that ζN → const 6= 0 as N → ∞ in probability for every fixed k ∈{1, 2, . . . , N − 1}. This means that this test is consistent for such alternatives.References[1] D.E. Barndorff-Nielsen, Hyperbolic distributions and distributions on hy-perbolas, Scan. J. Statist. 5(1978), 151–157.[2] D.E. Barndorff-Nielsen, Models for non-Gaussian variation: with appli-cations to turbulence, Proc. R. Soc. London A368, 501–520.[3] E. I. Dudewicz, E. C.Van derMeuler, Entropy based test of uniformity,JASA 76(1981), 974–987.[4] M.N.Frontini, A.Tagliani, Entropy-Convergence in Stieltjes and ham-burger moment problem, Appl. Math. Comput. 88(1997), 39–51.[5] M.N.Goria, N. N. Leonenko, V.V.Mergel, Statistical Estimators of En-tropy of Random Vector and its Applications in Testing of Statistical Hypothe-ses, Bernoulli, submitted.[6] D.V. Iliesku, V.G.Voda, On the inverse-gaussian distribution, Bull. Math.de la Soc. Sci. Math. de la R. S. de Roumanic 25(73)(1981), 1–15.[7] A.M.Kagan, Yu. V. Linnik, C. B. Rao, Characterization Problems inMathematical Statistics, John Wiley, New York, 1973.[8] J.N.Kapur, Maximum Entropy Models in Science and Engineering, JohnWiley, New York, 1989.[9] L. F.Kosachenko, N.N. Leonenko, On statistical estimation of entropy ofrandom vector, Prob. Inf. Trans. 23(1987), 9–16 (in Russian).[10] V.Seshardi, The Inverse-Gaussian Distribution, Clarendon Press, Oxford,1993.[11] O.Vasicek, A test of normality based on sample entropy, J. Roy. Stat.Soc.,Ser. B 38(1976), 254–256.

Test of goodness of fit for the inverse-gaussian distribution

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