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1

On the Bahadur Representation of Quantiles for a Sample from ρ*-Mixing Structure Population

Dudek Dagmara, Kuczmaszewska Anna

Advances in Science and Technology. Research Journal

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2022

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Vol. 16, no 3

316--330

EN

In this paper, we establish the strong consistency and the Bahadur representation of sample quantiles for ρ*-mixing random variables. Additionally, the asymptotic normality and the Berry-Esseen bound of sample quantiles for ρ*-mixing random variables are presented. Additionally, we provide the rate of convergence of sample quantiles to population counterparts. Moreover, numerical simulation is presented to ilustrate and verify obtained results.

2

On the number of reflexive and shared nearest neighbor pairs in one-dimensional uniform data

Bahadir S., Ceyhan E.

Probability and Mathematical Statistics

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2018

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Vol. 38, Fasc. 1

123--137

EN

For a random sample of points in R, we consider the number, of pairs whose members are nearest neighbors (NNs) to each other and the, number of pairs sharing a common NN. The pairs of the first type are called, reflexive NNs, whereas the pairs of the latter type are called shared NNs. In, this article, we consider the case where the random sample of size n is from, the uniform distribution on an interval. We denote the number of reflexive NN pairs and the number of shared NN pairs in the sample by Rn and Qn, respectively. We derive the exact forms of the expected value and the variance for both Rn and Qn, and derive a recurrence relation for Rn which may also be used to compute the exact probability mass function (pmf) of Rn. Our approach is a novel method for finding the pmf of Rn and agrees with the results in the literature. We also present SLLN and CLT results for both Rn and Qn as n goes to infinity.

3

M-estimation of the mixed-type generalized linear model

Dong Y., Song L., Wang M., Amin M.

Probability and Mathematical Statistics

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2018

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Vol. 38, Fasc. 1

209--223

EN

To investigate the features of the individual from the mixed-type model, a novel model, named the mixed-type generalized linear model, is proposed firstly in this work, which is verified to be realistic and useful. We consider the robustness of M-estimation to estimate the unknown parameters of the mixed-type generalized linear model. By applying the law of large numbers and the central limit theorem, the consistency and asymptotic normality of the M-estimation for the mixed-type generalized linear model are proved with regularity assumptions. At last, in order to evaluate the finite sample performance of the estimator for the new model, several applied instances are presented, which show the good performance of the estimator.

4

Asymptotic behaviour of linear rank statistics for the two-sample problem

Inglot T.

Probability and Mathematical Statistics

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2012

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Vol. 32, Fasc. 1

93--116

EN

Applying the strong approximation technique we present a unified approach to asymptotic results for multivariate linear rank statistics for the two-sample problem. We reprove asymptotic normality of these statistics under the null hypothesis and under local alternatives convergent at a moderate rate to the null hypothesis. We also provide a moderate deviation theorem for these statistics under the null hypothesis. Proofs are short and use natural argumentation.

5

An improved two-stage estimator in a partly linear model.

Qian W., Mammitzsch V.

Badania Operacyjne i Decyzje

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2000

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Nr 3-4

91-116

EN

Consider the partly linear model y/sub i/=x/sub i//sup T/ beta +g(t/sub i/)+e/sub i/, 1[left angle bracket]or=i[left angle bracket]or=n Chai et al. (1995) suggested a two-stage estimator of beta . In this paper, an improved two-stage estimator beta /sub n/ of beta and an estimator g(.) of the unknown function g(.) are established. The strong consistency and consistency rate of beta /sub n/ to beta is given and the asymptotic normality of beta /sub n/ is studied. We also obtain the optimal uniform convergence rate of g/sub n/ to g under rather weak assumptions.

PL

Rozpatrzono model częściowo liniowy y1=xTi B + g(ti) + ei, 1<=I<=n, zamieszczony w pracy Chai G.X. i współautorów (1995), w którym sugerowano zastosowanie dwustopniowego estymatora parametru B. Zaproponowano poprawiony dwustopniowy estymator Bn parametru B, a także estymator gn(.) nieznanej funkcji g(.). Wykazano silną zgodność, a także stopień zbieżności estymatora Bn z B oraz asymptotyczną normalność estymatorów Bn. Uzyskano ponadto optymalny stopień jednostajnej zbieżności gn do g przy bardzo słabych założeniach.