In the present paper we prove a general theorem which gives the rates of

convergence in distribution of asymptotically normal statistics based on samples of random size. The proof of the theorem uses the rates of convergences

in distribution for the random size and for the statistics based on samples of

nonrandom size

Bening, V. E.

Galieva, N. K.

Korolev, R. A.

EKE Repository of Publications

On rate of convergence in distribution ofasymptotically normal statistics based onsamples of random sizeV. E. Beninga, N. K. Galievab, R. A. KorolevcaLomonosov Moscow State UniversityFaculty of Computational Mathematics and CyberneticsInstitute of Informatics Problems of the Russian Academy of Sciences (IPI RAN)bening@yandex.rubKazakhstan Branch of Lomonosov Moscow State UniversitycPeoples’ Friendship University of RussiaDedicated to Mátyás Arató on his eightieth birthdayAbstractIn the present paper we prove a general theorem which gives the rates ofconvergence in distribution of asymptotically normal statistics based on sam-ples of random size. The proof of the theorem uses the rates of convergencesin distribution for the random size and for the statistics based on samples ofnonrandom size.Keywords: sample of random size; asymptotically normal statistic; transfertheorem; rate of convergence; mixture of distributions; Laplace distribution;Student’s distribution1. IntroductionAsymptotic properties of distributions of sums of random number of random vari-ables are subject of many papers (see e.g. Gnedenko&Fahim, 1969; Gnedenko,1989; Kruglov&Korolev, 1990; Gnedenko&Korolev, 1996; Bening&Korolev, 2002;vonChossy&Rappl, 1983). This kind of sums are widely used in insurance, eco-nomics, biology, etc. (see Gnedenko, 1989; Gnedenko, 1998; Bening&Korolev,2002). However, in mathematical statistics and its applications, there are commonAnnales Mathematicae et Informaticae39 (2012) pp. 17–28Proceedings of the Conference on Stochastic Models and their ApplicationsFaculty of Informatics, University of Debrecen, Debrecen, Hungary, August 22–24, 201117statistics that are not sums of observations. Examples are the rank statistics, U-statistics, linear combinations of order statistics, etc. In this case the statistics areoften situations when the sample size is not predetermined and can be regardedas random. For example, in reliability testing the number of failed devices at aparticular time is a random variable.Generally, in most cases related to the analysis and processing of experimentaldata, we can assume that the number of random factors, influencing the observedvalues, is itself random and varies from observation to observation. Therefore,instead of different variants of the central limit theorem, proving the normality ofthe limiting distribution of classical statistics, in such situations we should rely ontheir analogues for samples of random size. This makes it natural to study theasymptotic behavior of distributions of statistics of general form, based on samplesof random size. For example, Gnedenko (1989) examines the asymptotic propertiesof the distributions of sample quantiles constructed from samples of random size.In this paper we estimate the rate of convergence of distribution functions ofasymptotically normal statistics based on samples of random size. The estima-tions depend on the rates of convergences of distributions of the random size ofsample and the statistic based on sample of nonrandom size. Such statements areusually called transfer theorems. In the present paper we prove transfer theoremsconcerning estimates of convergence rate.In this paper we use the following notation and symbols: R as real numbers,N as positive integers, Φ(x), ϕ(x) as standard normal distribution function anddensity.In Section 2 we give a sketch of the proof of a general transfer theorem, Sections3, 4 and 5 contain the main theorems, their proofs and examples.Consider random variables N1, N2, . . . and X1, X2, . . . defined on a commonmeasurable space (Ω,A,P). The random variables X1, X2, . . . Xn denote observa-tions, n is a nonrandom size of sample, the random variable Nn denotes a randomsize of sample and depends on a natural parameter n ∈ N. Suppose that therandom variables Nn take on positive integers for any n ≥ 1, that is Nn ∈ N,and do not depend on X1, X2, . . .. Suppose that X1, X2, . . . are independent andidentically distributed observations having a distribution function F (x).Let Tn = Tn(X1, . . . , Xn) be some statistic, that is a real measurable functionon observations X1, . . . , Xn. The statistic Tn is called asymptotically normal withparameters (µ, 1/σ2), µ ∈ R, σ > 0, ifP(σ√n(Tn − µ) < x) −→ Φ(x), n→∞, x ∈ R, (1.1)where Φ(x) is the standard normal distribution function.The asymptotically normal statistics are abundant. Recall some examples ofasymptotically normal statistics: the sample mean (assuming nonzero variances),the maximum likelihood estimators (under weak regularity conditions), the centralorder statistics and many others.For any n ≥ 1 define the random variable TNn byTNn(ω) ≡ TNn(ω)(X1(ω), . . . , XNn(ω)(ω)), ω ∈ Ω. (1.2)18 V. E. Bening, N. K. Galieva, R. A. KorolevTherefore, TNn is a statistic constructed from the statistic Tn and from the sampleof random size Nn.In Gnedenko&Fahim (1969) and Gnedenko (1989), the first and second transfertheorems are proved for the case of sums of independent random variables andsample quantiles.Theorem 1.1 (Gnedenko, 1989). Let X1, X2, . . . be independent and identicallydistributed random variables and Nn ∈ N denotes a sequence of random variableswhich are independent of X1, X2, . . .. If there exist real numbers bn > 0, an ∈ Rsuch that1. P( 1bnn∑i=1(Xi − an) < x)−→ Ψ(x), n→∞and2. P(Nnn< x)−→ H(x), H(0+) = 0, n→∞,where Ψ(x) and H(x) are distribution functions, then, as n→∞,P( 1bnNn∑i=1(Xi − an) < x)−→ G(x), n→∞,where the distribution function G(x) is defined by its characteristic functiong(t) =∞∫0(ψ(t))z dH(z)and ψ(t) is the characteristic function of Ψ(x).The proof of the theorem can be read in Gnedenko (1998).Theorem 1.2 (Gnedenko, 1989). Let X1, X2, . . . be independent and identicallydistributed random variables and Nn ∈ N is a sequence of random variables whichare independent of X1, X2, . . ., and let Xγ:n be the sample quantile of order γ ∈(0, 1) constructed from sample X1, . . . , Xn. If there exist real numbers bn > 0,an ∈ R such that1. P( 1bn(Xγ:n − an) < x)−→ Φ(x), n→∞and2. P(Nnn< x)−→ H(x), H(0+) = 0, n→∞,On rate of convergence in distribution of asymptotically normal statistics . . . 19where H(x) is a distribution function, then, as n→∞,P( 1bn(Xγ:Nn − an) < x)−→ G(x), n→∞where the distribution function G(x) is a mixture of normal distribution with themixing distribution HG(x) =∞∫0Φ(x√y) dH(y).In Bening&Korolev (2005), the following general transfer theorem is proved forasymptotically normal statistics (1.1).Theorem 1.3. Let {dn} be an increasing and unbounded sequence of positive in-tegers. Suppose that Nn →∞ in probability as n→∞. Let Tn(X1, . . . , Xn) be anasymptotically normal statistics, that isP(σ√n(Tn − µ) < x) −→ Φ(x), n→∞.Then a necessary and sufficient condition for a distribution function G(x) to satisfyP(σ√dn(TNn − µ) < x) −→ G(x), n→∞,is that there exists a distribution function H(x) with H(0+) = 0 satisfyingP(Nn < dnx) −→ H(x), n→∞, x > 0,and G(x) has a formG(x) =∞∫0Φ(x√y) dH(y), x ∈ R,that is the distribution G(x) is a mixture of the normal law with the mixing distri-bution H.Now, we give a brief sketch of proof of Theorem 1.3 to make references later.2. Sketch of proof of Theorem 1.3The proof of Theorem 1.3 is closely related to the proof of Theorems 6.6.1 and6.7.3 for random sums in Kruglov&Korolev (1990).By the formula of total probability, we haveP(σ√dn(TNn − µ) < x)−G(x)20 V. E. Bening, N. K. Galieva, R. A. Korolev=∞∑k=1P(Nn = k)P(σ√k(Tk − µ) <√k/dnx)−G(x)=∞∑k=1P(Nn = k)(Φ(√k/dnx)−G(x))+∞∑k=1P(Nn = k)(P(σ√k(Tk − µ) <√k/dnx)− Φ(√k/dnx))≡ J1n + J2n. (2.1)From definition of G(x) the expression for J1n can be written in the formJ1n =∞∫0Φ(x√y) dP(Nn < dny)−∞∫0Φ(x√y) dH(y)=∞∫0Φ(x√y) d(P(Nn < dny)−H(y)).Using the formula of integration by parts for Lebesgue integral (see e.g. Theorem2.6.11 in Shiryaev, 1995) yieldsJ1n = −∞∫0(P(Nn < dny)−H(y))dΦ(x√y). (2.2)By the condition of the present theorem,P(Nn < dny)−H(y) −→ 0, n→∞for any fixed y ∈ R, therefore, by the dominated convergence theorem (see e.g.Theorem 2.6.3 in Shiryaev, 1995), we haveJ1n −→ 0, n→∞.Consider J2n. For simplicity, instead of the condition for the statistic Tn to beasymptotically normal (see (1.1)), we suggest a stronger condition which describesthe rate of convergence of distributions of Tn to the normal law. Suppose that thefollowing condition is satisfied.Condition 1. There exist real numbers α > 0 and C1 > 0 such thatsupx∣∣∣P(σ√n(Tn − µ) < x)− Φ(x)∣∣∣ ≤ C1nα, n ∈ N.From the condition we obtain estimates for J2n. We have|J2n| =∣∣∣ ∞∑k=1P(Nn = k)(P(σ√k(Tk − µ) <√k/dnx)− Φ(√k/dnx))∣∣∣On rate of convergence in distribution of asymptotically normal statistics . . . 21≤ C1∞∑k=1P(Nn = k)1kα= C1E(Nn)−α =C1dαnE(Nn/dn)−α. (2.3)Since, by the condition of theorem, the random variables Nn/dn have a weak limit,then the expectation E(Nn/dn)−α is typically bounded. Because dn → ∞, fromthe last inequality it follows thatJ2n −→ 0, n→∞.3. The main resultsSuppose that the limiting behavior of distribution functions of the normalized ran-dom size is described by the following condition.Condition 2. There exist real numbers β > 0, C2 > 0 and a distribution H(x)with H(0+) = 0 such thatsupx≥0∣∣∣P(Nnn< x)−H(x)∣∣∣ ≤ C2nβ, n ∈ N.Theorem 3.1. If for the statistic Tn(X1, . . . , Xn) condition 1 is satisfied, for therandom sample size Nn condition 2 is satisfied, then the following inequality holdssupx∣∣∣P(σ√n(TNn − µ) < x)−G(x)∣∣∣ ≤ C1EN−αn + C22nβ ,where the distribution G(x) has the formG(x) =∞∫0Φ(x√y) dH(y), x ∈ R.Corollary 3.2. The statement of the theorem remains valid if the normal law isreplaced by any limiting distribution.Corollary 3.3. If the moments E(Nn/n)−α are bounded uniformly in n, that isE(Nnn)−α≤ C3, C3 > 0, n ∈ N,then the right side of the inequality in the statement of the theorem has the formC1C3nα+C22nβ= O(n−min(α,β)).22 V. E. Bening, N. K. Galieva, R. A. KorolevCorollary 3.4. By Hölder’s inequality for 0 < α ≤ 1, the following estimate holdsEN−αn ≤(E1Nn)α,which is useful from practical viewpoint. In this case, the right side of the inequalityhas the formC1(E1Nn)α+C22nβ.Corollary 3.5. Note that, condition 2 means that the random variables Nn/nconverge weakly to V which has the distribution H(x). From the definition of weakconvergence with function x−α, x ≥ 1, for Nn ≥ n, n ∈ N, it follows thatE(Nnn)−α−→ E 1V α, n→∞,that is the moments E(Nn/n)−α are bounded in n and, therefore, the estimate fromCorollary 3.3 holds.The case Nn ≥ n appears when the random variable Nn takes on valuesn, 2n, . . . , kn with equal probabilities 1/k for any fixed k ∈ N. In this case, therandom variables Nn/n do not depend on n and, therefore, converge weakly to Vwhich takes values 1, 2, . . . , k with equal probability 1/k.Corollary 3.6. From the proof of the theorem it follows that skipping of conditions1 and 2 yields the following statementsupx∣∣∣P(σ√n(TNn − µ) < x)−G(x)∣∣∣≤∞∑k=1P(Nn = k)supx∣∣∣P(σ√k(Tk − µ) < x)− Φ(x)∣∣∣+12supx≥0∣∣∣P(Nnn< x)−H(x)∣∣∣.Following the proof of Theorem 3.1 (see Section 2 and 4), we can formulatemore general result.Theorem 3.7. Let a random element Xn in some measurable space and randomvariable Nn be defined on a common measurable space and independent for anyn ∈ N. Suppose that a real-valued statistic Tn = Tn(Xn) and the random variableNn satisfy the following conditions.1. There exist real numbers α > 0, σ > 0, µ ∈ R, C1 > 0 and a sequence0 < dn ↑ +∞, n→∞, such thatsupx∣∣∣P(σ√dn(Tn − µ) < x)− Φ(x)∣∣∣ ≤ C1nα, n ∈ N.On rate of convergence in distribution of asymptotically normal statistics . . . 232. There exist a number C2 > 0, a sequence 0 < δn ↓ 0, n → ∞ and adistribution function H(x) with H(0+) = 0 such thatsupx≥0∣∣∣P(Nndn< x)−H(x)∣∣∣ ≤ C2δn, n ∈ N.Then the following inequality holdssupx∣∣∣P(σ√dn(TNn − µ) < x)−G(x)∣∣∣ ≤ C1EN−αn + C22 δn,where the distribution function G(x) has the formG(x) =∞∫0Φ(x√y) dH(y), x ∈ R.4. Proof of Theorem 3.1Suppose x ≥ 0. Using formulas (2.1)–(2.3) with dn = n yieldssupx≥0∣∣∣P(σ√n(TNn − µ) < x)−G(x)∣∣∣ ≤ I1n + I2n, (4.1)whereI1n = supx≥0∞∫0∣∣P(Nn < ny)−H(y)∣∣ dΦ(x√y), (4.2)I2n =∞∑k=1P(Nn = k) supx≥0∣∣∣P(σ√k(Tk − µ) <√k/nx)− Φ(√k/nx)∣∣∣. (4.3)To estimate the variable I1n we use equality (4.2) and condition 2,I1n ≤ C2nβsupx≥0∞∫0dΦ(x√y) =C22nβ. (4.4)The series in I2n (see (4.3)) is estimated by using condition 1.I2n ≤ C1∞∑k=11kαP(Nn = k) = C1EN−αn . (4.5)Note that the estimate (4.5) is valid for x < 0. For I1n and negative x, we have(see (2.1) and (2.2))I1n = supx<0∣∣∣ ∞∫0(P(Nn < ny)−H(y))dΦ(x√y)∣∣∣24 V. E. Bening, N. K. Galieva, R. A. Korolev= supx<0∣∣∣ ∞∫0(P(Nn < ny)−H(y))dΦ(|x|√y)∣∣∣≤ supx≥0∞∫0∣∣P(Nn < ny)−H(y)∣∣dΦ(x√y),and we can use (4.4) again. The statement of the theorem follows from (4.1), (4.4)and (4.5). The theorem is proved.5. ExamplesWe consider two examples of use of Theorem 3.1 when the limiting distributionfunction G(x) is known.5.1. Student’s distributionBening&Korolev (2005) shows that if the random sample size Nn has the negativebinomial distribution with parameters p = 1/n and r > 0, that is (in particular,for r = 1, it is the geometric distribution)P(Nn = k) =(k + r − 2) · · · r(k − 1)!1nr(1− 1n)k−1, k ∈ N,then, for an asymptotically normal statistic Tn the following limiting relationshipholds (see Corollary 2.1 in Bening&Korolev, 2005)P(σ√n(TNn − µ) < x) −→ G2r(x√r), n→∞, (5.1)where G2r(x) is Student’s distribution with parameter γ = 2r, having densitypγ(x) =Γ((γ + 1)/2)√piγΓ(γ/2)(1 +x2γ)−(γ+1)/2, x ∈ R,where Γ(·) is the gamma function, and γ > 0 is a shape parameter (if the parameterγ is a positive integer, then it is called the number of degrees of freedom). In oursituation the parameter may be arbitrary small, and we have typical heavy-taileddistribution. If γ = 2, that is r = 1, then the distribution function G2(x) can befound explicitlyG2(x) =12(1 +x√2 + x2), x ∈ R.For r = 1/2, we obtain the Cauchy distribution.Bening et al. (2004) gives an estimate of rate of convergence for random samplesize, for 0 < r < 1,supx≥0∣∣∣P( NnENn< x)−Hr(x)∣∣∣ ≤ Crnr/(r+1), Cr > 0, n ∈ N, (5.2)On rate of convergence in distribution of asymptotically normal statistics . . . 25whereHr(x) =rrΓ(r)x∫0e−ryyr−1 dy, x ≥ 0,for r = 1, the right side of the inequality can be replaced by 1/(n− 1). So, Hr(x)is a distribution with parameter r ∈ (0, 1], andENn = r(n− 1) + 1. (5.3)From(1 + x)γ =∞∑k=0γ(γ − 1) · · · (γ − k + 1)k!xk, |x| < 1, γ ∈ R,we haveEN−1n =1(n− 1)(1− r)( 1nr−1− 1)= O(n−r), 0 < r < 1, n ∈ N. (5.4)If the Berry-Esseen estimate is valid for the rate of convergence of distribution ofTn, that issupx∣∣∣P(σ√n(Tn − µ) < x)− Φ(x)∣∣∣ = O( 1√n), n ∈ N, (5.5)then from Theorem 3.1 with α = 1/2, β = r/(r+ 1), from relations (5.1)–(5.4) andCorollary 3.4, we have the following estimatesupx∣∣∣P(σ√n(TNn − µ) < x)−G2r(x√r)∣∣∣= O( 1nr/2)+O( 1nr/(r+1))= O( 1nr/2), r ∈ (0, 1), n ∈ N. (5.6)5.2. Laplace distributionConsider Laplace distribution with distribution function Λγ(x) and densityλγ(x) =1γ√2exp{−√2|x|γ}, γ > 0, x ∈ R.Bening&Korolev (2008) gives a sequence of random variables Nn(m) which dependson the parameter m ∈ N. Let Y1, Y2, . . . be independent and identically distributedrandom variables with some continuous distribution function. Let m be a positiveinteger andN(m) = min{i ≥ 1 : max1≤j≤mYj < maxm+1≤k≤m+iYk}.It is well-known that such random variables have the discrete Pareto distributionP(N(m) ≥ k) = mm+ k − 1 , k ≥ 1. (5.7)26 V. E. Bening, N. K. Galieva, R. A. KorolevNow, let N (1)(m), N (2)(m), . . . be independent random variables with the samedistribution (5.7). Define the random variableNn(m) = max1≤j≤nN (j)(m),then Bening&Korolev (2008) shows thatlimn→∞P(Nn(m)n< x)= e−m/x, x > 0, (5.8)and, for an asymptotically normal statistic Tn, the following relationship holdsP(σ√n(TNn(m) − µ) < x) −→ Λ1/m(x), n→∞,where Λ1/m(x) is the Laplace distribution function with parameter γ = 1/m.Lyamin (2010) gives the estimate for the rate of convergence for (5.8),supx≥0∣∣∣P(Nn(m)n< x)− e−m/x∣∣∣ ≤ Cmn, Cm > 0, n ∈ N. (5.9)If the Berry-Esseen estimate is valid for the rate of convergence of distribution forthe statistic (see (5.5)), then from Corollary 3.4 for α = 1/2, β = 1 and frominequality (5.9), we havesupx∣∣∣P(σ√n(TNn(m)−µ) < x)−Λ1/m(x)∣∣∣ = O((EN−1n (m))1/2)+O(n−1). (5.10)Consider the variable EN−1n (m). From definition of Nn(m) and inequality (5.7),we haveP(Nn(m) = k) =( km+ k)n−( k − 1m+ k − 1)n= mnk∫k−1xn−1(m+ x)n+1dx,therefore,EN−1n (m) =∞∑k=11kP(Nn(m) = k) = mn∞∑k=11kk∫k−1xn−1(m+ x)n+1dx≤ mn∞∑k=1k∫k−1xn−2(m+ x)n+1dxmn∞∫0xn−2(m+ x)n+1dx.To calculate the last integral we use the following formula (see formula 856.12 inDwight, 1961)∞∫0xm−1(a+ bx)m+ndx =Γ(m)Γ(n)anbmΓ(m+ n)a, b,m, n > 0.On rate of convergence in distribution of asymptotically normal statistics . . . 27We haveEN−1n (m) ≤ mnΓ(n− 1)Γ(2)m2Γ(n+ 1)=1m(n− 1) = O(n−1).Now, by this formula and (5.10), we obtainsupx∣∣∣P(σ√n(TNn(m) − µ) < x)− Λ1/m(x)∣∣∣ = O( 1√n).References[1] Bening,V. E.&KorolevV.Y. (2002). Generalized Poisson Models and Their Applica-tions in Insurance and Finance, VSP Press, Netherlands.[2] Bening,V. E.&Korolev,V.Y. (2005). On an application of the Student distributionin the theory of probability and mathematical statistics. Theory Probab. Appl., 49(3),377–391.[3] Bening,V. E.&Korolev,V.Y. (2008). Some statistical problems related to theLaplace distribution. Informatics and its Applications, IPI RAN, 2(2), 19–34 [inRussian].[4] Bening,V. E.,Korolev,V.Y.&UDa (2004). Estimates of rates of convergences of dis-tributions of some statistics to the Student distribution. J. People’s FriendshipUniver. Russia, 1(12), 59–74.[5] VonChossy,R.&Rappl,G. (1983). Some approximation methods for the distributionof random sums. Insurance: Mathematics and Economics, 2, 251–270.[6] Dwight,H.B. (1961). Tables of Integrals and Other Mathematical Data, 4th edn.Macmillan, USA.[7] Gnedenko, B.V. (1989). On estimation of the unknown parameters of distributionsfrom a random number of independent observations. Proceedings of Tbilisi Math.Inst., AN GSSR, 92, 146–150 [in Russian].[8] Gnedenko, B.V. (1998). Theory of Probability, 6th edn. CRC Press, USA.[9] Gnedenko, B.V.&Fahim,G. (1969). On a transfer theorem. Soviet Math. Dokl., 10,769–772.[10] Gnedenko, B.V.&KorolevV.Y. (1996). Random Summation. Limit Theorems andApplications, CRC Press, USA.[11] Kruglov,V.M.&KorolevV.Y. (1990). Limit Theorems for Random Sums, MoscowUniversity Press, Moscow [in Russian].[12] Lyamin,O.O. (2011). On the rate of convergence of the distributions of certain statis-tics to the Laplace and Student distributions. Moscow University ComputationalMathematics and Cybernetics, 35(1), 37–46.[13] Shiryaev,A.N. (1995). Probability, 2nd edn. Springer, USA.28 V. E. Bening, N. K. Galieva, R. A. Korolev

On rate of convergence in distribution of asymptotically normal statistics based on samples of random size

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