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Artykuły w czasopismach na temat „AR(1) model”

Autor: Grafiati

Data publikacji: 4 czerwca 2021

Data aktualizacji: 25 stycznia 2023

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1

Chan, K. S., Joseph D. Petruccelli, H. Tong, and Samuel W. Woolford. "A multiple-threshold AR(1) model." Journal of Applied Probability 22, no. 2 (June 1985): 267–79. http://dx.doi.org/10.2307/3213771.

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We consider the model Zt = φ (0, k)+ φ(1, k)Zt–1 + at (k) whenever rk−1<Zt−1≦rk, 1≦k≦l, with r0 = –∞ and rl =∞. Here {φ (i, k); i = 0, 1; 1≦k≦l} is a sequence of real constants, not necessarily equal, and, for 1≦k≦l, {at(k), t≧1} is a sequence of i.i.d. random variables with mean 0 and with {at(k), t≧1} independent of {at(j), t≧1} for j ≠ k. Necessary and sufficient conditions on the constants {φ (i, k)} are given for the stationarity of the process. Least squares estimators of the model parameters are derived and, under mild regularity conditions, are shown to be strongly consistent and as

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2

Tai‐Leung Chong, Terence. "The polynomial aggregated AR(1) model*." Econometrics Journal 9, no. 1 (March 1, 2006): 98–122. http://dx.doi.org/10.1111/j.1368-423x.2006.00178.x.

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3

Chan, K. S., Joseph D. Petruccelli, H. Tong, and Samuel W. Woolford. "A multiple-threshold AR(1) model." Journal of Applied Probability 22, no. 02 (June 1985): 267–79. http://dx.doi.org/10.1017/s0021900200037748.

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We consider the model Zt = φ (0, k)+ φ(1, k)Zt –1 + at (k) whenever r k−1&lt;Z t−1≦r k , 1≦k≦l, with r 0 = –∞ and rl =∞. Here {φ (i, k); i = 0, 1; 1≦k≦l} is a sequence of real constants, not necessarily equal, and, for 1≦k≦l, {at (k), t≧1} is a sequence of i.i.d. random variables with mean 0 and with {at (k), t≧1} independent of {at (j), t≧1} for j ≠ k. Necessary and sufficient conditions on the constants {φ (i, k)} are given for the stationarity of the process. Least squares estimators of the model parameters are derived and, under mild regularity conditions, are shown to be strongly cons

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4

Vrbik, Jan. "Moments of AR(1)-Model Estimators." Communications in Statistics - Simulation and Computation 34, no. 3 (July 2005): 595–600. http://dx.doi.org/10.1081/sac-200068447.

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5

Sharafi, M., and A. R. Nematollahi. "AR(1) model with skew-normal innovations." Metrika 79, no. 8 (June 29, 2016): 1011–29. http://dx.doi.org/10.1007/s00184-016-0587-7.

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6

Li, M., Q. J. Wang, J. C. Bennett, and D. E. Robertson. "A strategy to overcome adverse effects of autoregressive updating of streamflow forecasts." Hydrology and Earth System Sciences 19, no. 1 (January 6, 2015): 1–15. http://dx.doi.org/10.5194/hess-19-1-2015.

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Abstract. For streamflow forecasting, rainfall–runoff models are often augmented with updating procedures that correct forecasts based on the latest available streamflow observations of streamflow. A popular approach for updating forecasts is autoregressive (AR) models, which exploit the "memory" in hydrological model simulation errors. AR models may be applied to raw errors directly or to normalised errors. In this study, we demonstrate that AR models applied in either way can sometimes cause over-correction of forecasts. In using an AR model applied to raw errors, the over-correction usually

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7

ZHENG, Wei, Da-wu GU, and Hai-ning LU. "Application of improved AR(1) model in DNS." Journal of Computer Applications 30, no. 3 (April 6, 2010): 736–39. http://dx.doi.org/10.3724/sp.j.1087.2010.00736.

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8

Bakouch, Hassan S., and Miroslav M. Ristić. "Zero truncated Poisson integer-valued AR(1) model." Metrika 72, no. 2 (March 24, 2009): 265–80. http://dx.doi.org/10.1007/s00184-009-0252-5.

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9

El-Sayed, Sayed Mesheal, Ahmed Amin El-Sheikh, Mohamed Khalifa Ahmed Issa, and Hadia Faried Mohamed Ahmed Azmy. "A CLOSED FORM OF BIASED AR(1) MODEL." Advances and Applications in Statistics 50, no. 3 (March 10, 2017): 191–99. http://dx.doi.org/10.17654/as050030191.

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10

Franses, Philip Hans. "A model selection test for an AR (1) versus an MA (1) model." Statistics & Probability Letters 15, no. 4 (November 1992): 281–84. http://dx.doi.org/10.1016/0167-7152(92)90163-y.

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11

Chan, Ngai Hang, Deyuan Li, Liang Peng, and Rongmao Zhang. "TAIL INDEX OF AN AR(1) MODEL WITH ARCH(1) ERRORS." Econometric Theory 29, no. 5 (February 21, 2013): 920–40. http://dx.doi.org/10.1017/s0266466612000801.

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Relevant sample quantities such as the sample autocorrelation function and extremes contain useful information about autoregressive time series with heteroskedastic errors. As these quantities usually depend on the tail index of the underlying heteroskedastic time series, estimating the tail index becomes an important task. Since the tail index of such a model is determined by a moment equation, one can estimate the underlying tail index by solving the sample moment equation with the unknown parameters being replaced by their quasi-maximum likelihood estimates. To construct a confidence interv

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12

Sari, Nelfa, Maiyastri ., and Hazmira Yozza. "PENDUGAAN PARAME TER MODEL AUTOREGRESSIVE PADA DERET WAKTU." Jurnal Matematika UNAND 3, no. 4 (December 1, 2014): 28. http://dx.doi.org/10.25077/jmu.3.4.28-37.2014.

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Model deret waktu stokastik dikenal dengan model ARIMA. Model ARIMAterdiri dari model AR, MA dan ARMA. Model AR adalah bentuk regresi yangmenghubungkan suatu nilai pengamatan dengan nilai pengamatan masa lalunya padaselang waktu tertentu. Dari hubungan tersebut, terdapat parameter model AR yangakan diduga. Untuk pendugaan parameter dikhususkan untuk AR orde satu yang dinotasikan dengan AR(1) dan AR orde dua yang dinotasikan dengan AR(2). Pendugaanparameter model AR(1) dan model AR(2) ini menggunakan metode momen, metodekuadrat terkecil dan metode kemungkinan maksimum. Dari uraian ketiga metode

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13

Chong, Terence Tai-Leung. "STRUCTURAL CHANGE IN AR(1) MODELS." Econometric Theory 17, no. 1 (February 2001): 87–155. http://dx.doi.org/10.1017/s0266466601171045.

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This paper investigates the consistency of the least squares estimators and derives their limiting distributions in an AR(1) model with a single structural break of unknown timing. Let β1 and β2 be the preshift and postshift AR parameter, respectively. Three cases are considered: (i) |β1| < 1 and |β2| < 1; (ii) |β1| < 1 and β2 = 1; and (iii) β1 = 1 and |β2| < 1. Cases (ii) and (iii) are of particular interest but are rarely discussed in the literature. Surprising results are that, in both cases, regardless of the location of the change-point estimate, the unit root can always be co

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14

van Giersbergen, Noud P. A. "BARTLETT CORRECTION IN THE STABLE AR(1) MODEL WITH INTERCEPT AND TREND." Econometric Theory 25, no. 3 (June 2009): 857–72. http://dx.doi.org/10.1017/s0266466609090690.

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Bartlett corrections are derived for testing hypotheses about the autoregressive parameter ρ in the stable (a) AR(1) model, (b) AR(1) model with intercept, (c) AR(1) model with intercept and linear trend. The correction is found explicitly as a function of ρ. In the models with deterministic terms, the correction factor is asymmetric in ρ. Furthermore, the Bartlett correction is monotonically increasing in ρ and tends to infinity when ρ approaches the stability boundary of + 1. Simulation results indicate that the Bartlett corrections are useful in controlling the size of the likelihood ratio

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15

Ahmed Issa, Mohamed Khalifa. "New Estimator for AR (1) Model with Missing Observations." Journal of University of Shanghai for Science and Technology 23, no. 09 (September 6, 2021): 147–59. http://dx.doi.org/10.51201/jusst/21/09521.

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In this paper, new form of the parameters of AR(1) with constant term with missing observations has been derived by using Ordinary Least Squares (OLS) method, Also, the properties of OLS estimator are discussed, moreover, an extension of Youssef [18]has been suggested for AR(1) with constant with missing observations. A comparative study between (OLS), Yule-Walker (YW) and modification of the ordinary least squares (MOLS) is considered in the case of stationary and near unit root time series, using Monte Carlo simulation.

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16

Hamilton, David, and Ka Ho Wu. "CONFIDENCE REGIONS FOR PARAMETERS IN THE AR(1) MODEL." Journal of Time Series Analysis 16, no. 3 (May 1995): 249–65. http://dx.doi.org/10.1111/j.1467-9892.1995.tb00233.x.

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17

Griffiths, William E. "Ba yesian predictors for an ar(1) error model." Communications in Statistics - Theory and Methods 26, no. 2 (January 1997): 441–48. http://dx.doi.org/10.1080/03610929708831926.

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18

Akkaya, Ayşen D., and Moti L. Tiku. "Time series AR(1) model for short-tailed distributions." Statistics 39, no. 2 (April 2005): 117–32. http://dx.doi.org/10.1080/02331880512331344036.

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19

Hasegawa, Hikaru, Anoop Chaturvedi, and Tran van Hoa. "Bayesian Unit Root Test in Nonnormal AR(1) Model." Journal of Time Series Analysis 21, no. 3 (May 2000): 261–80. http://dx.doi.org/10.1111/1467-9892.00185.

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20

Paparoditis, Efstathios, and Dimitris N. Politis. "Large-sample inference in the general AR(1) model." Test 9, no. 2 (December 2000): 487–509. http://dx.doi.org/10.1007/bf02595747.

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21

Ling, S., and D. Li. "Asymptotic inference for a nonstationary double AR(1) model." Biometrika 95, no. 1 (January 31, 2008): 257–63. http://dx.doi.org/10.1093/biomet/asm084.

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22

Leipus, Remigijus, Anne Philippe, Vytautė Pilipauskaitė, and Donatas Surgailis. "Sample covariances of random-coefficient AR(1) panel model." Electronic Journal of Statistics 13, no. 2 (2019): 4527–72. http://dx.doi.org/10.1214/19-ejs1632.

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23

Hajrajabi, Arezo, and Afshin Fallah. "Nonlinear semiparametric AR(1) model with skew-symmetric innovations." Communications in Statistics - Simulation and Computation 47, no. 5 (June 28, 2017): 1453–62. http://dx.doi.org/10.1080/03610918.2017.1315772.

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24

Li, Lu, Chong-Yu Xu, Jun Xia, Kolbjørn Engeland, and Paolo Reggiani. "Uncertainty estimates by Bayesian method with likelihood of AR (1) plus Normal model and AR (1) plus Multi-Normal model in different time-scales hydrological models." Journal of Hydrology 406, no. 1-2 (August 2011): 54–65. http://dx.doi.org/10.1016/j.jhydrol.2011.05.052.

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25

Yang, Fu Xin, Bai Lan Zhang, and Zhi Yuan Su. "Analysis and Verification of Bullwhip Effect Based on System Dynamics." Applied Mechanics and Materials 340 (July 2013): 312–19. http://dx.doi.org/10.4028/www.scientific.net/amm.340.312.

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To study the bullwhip effect (BWE) in supply chain (SC), this paper built two system dynamics (SD) models strictly referring to the AR(1) (autoregressive process) model constructed by Frank Chen. Using Vensim simulation software, it analyzed the impact of the correlation coefficient of demand, lead time, smoothing time of demand and information to BWE, and then put forward some proposals on how to reduce BWE. By contrasting the simulation results of SD models with the AR(1) models', it verifies the validity of the AR(1) model of Frank Chen from a simulation perspective. It also shows SD model

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26

Baltagi, Badi H., and Qi Li. "Testing AR(1) against MA(1) disturbances in an error component model." Journal of Econometrics 68, no. 1 (July 1995): 133–51. http://dx.doi.org/10.1016/0304-4076(94)01646-h.

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27

Pang, Tianxiao, Terence Tai-Leung Chong, Danna Zhang, and Yanling Liang. "STRUCTURAL CHANGE IN NONSTATIONARY AR(1) MODELS." Econometric Theory 34, no. 5 (July 24, 2017): 985–1017. http://dx.doi.org/10.1017/s0266466617000317.

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This article revisits the asymptotic inference for nonstationary AR(1) models of Phillips and Magdalinos (2007a) by incorporating a structural change in the AR parameter at an unknown time k0. Consider the model ${y_t} = {\beta _1}{y_{t - 1}}I\{ t \le {k_0}\} + {\beta _2}{y_{t - 1}}I\{ t > {k_0}\} + {\varepsilon _t},t = 1,2, \ldots ,T$, where I{·} denotes the indicator function, one of ${\beta _1}$ and ${\beta _2}$ depends on the sample size T, and the other is equal to one. We examine four cases: Case (I): ${\beta _1} = {\beta _{1T}} = 1 - c/{k_T}$, ${\beta _2} = 1$; (II): ${\beta _1} = 1$

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28

Chang, Fang, Augustine C. M. Wong, and Yanyan Wu. "Asymptotic Inference for the Weak Stationary Double AR(1) Model." Open Journal of Statistics 02, no. 02 (2012): 141–52. http://dx.doi.org/10.4236/ojs.2012.22016.

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29

Lee, Sung Duck, Sun Woo Kim, and Na Rae Jo. "Comparison between homogeneity test statistics for panel AR(1) model." Korean Journal of Applied Statistics 29, no. 1 (February 29, 2016): 123–32. http://dx.doi.org/10.5351/kjas.2016.29.1.123.

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30

Garbar, Sergey. "Using AR(1) model to simulate strictly stationary random sequences." IOP Conference Series: Materials Science and Engineering 441 (November 2, 2018): 012018. http://dx.doi.org/10.1088/1757-899x/441/1/012018.

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31

Berger, James O., and Ruo-Yong Yang. "Noninformative Priors and Bayesian Testing for the AR(1) Model." Econometric Theory 10, no. 3-4 (August 1994): 461–82. http://dx.doi.org/10.1017/s026646660000863x.

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Various approaches to the development of a noninformative prior for the AR(1) model are considered and compared. Particular attention is given to the reference prior approach, which seems to work well for the stationary case but encounters difficulties in the explosive case. A symmetrized (proper) version of the stationary reference prior is ultimately recommended for the problem. Bayesian testing of the unit root, stationary, and explosive hypotheses is considered also. Bounds on the Bayes factors are developed and shown to yield answers that appear to conflict with classical tests.

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32

Anderson, T. W., R. A. Lockhart, and M. A. Stephens. "An omnibus test for the time series model AR(1)." Journal of Econometrics 118, no. 1-2 (January 2004): 111–27. http://dx.doi.org/10.1016/s0304-4076(03)00137-4.

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33

Onth, Kazuyuki, and Kenji Nakagawa. "Approximation of video cell traffic by AR(1) + IPP-model." Electronics and Communications in Japan (Part I: Communications) 78, no. 8 (August 1995): 1–9. http://dx.doi.org/10.1002/ecja.4410780801.

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34

Abramov, Oleg, Alexandr Bystrov, and Maksim Krivov. "INTEGRATION OF A COMPUTER MODEL WITH VR/AR-TECHNOLOGY." Modern Technologies and Scientific and Technological Progress 2022, no. 1 (May 16, 2022): 95–96. http://dx.doi.org/10.36629/2686-9896-2022-1-95-96.

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35

Hamza, Dhaker, Papa Ngom, Pierre Mendy, and El Hadji Deme. "GENERALIZED DIVERGENCE CRITERIA FOR MODEL SELECTION BETWEEN RANDOM WALK AND AR(1) MODEL." Journal of Statistics: Advances in Theory and Applications 17, no. 2 (May 20, 2017): 83–109. http://dx.doi.org/10.18642/jsata_7100121830.

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36

Huang, Dawei, and N. M. Spencer. "On a random vibration model." Journal of Applied Probability 33, no. 4 (December 1996): 1141–58. http://dx.doi.org/10.2307/3214992.

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A random vibration model is investigated in this paper. The model is formulated as a cosine function with a constant frequency and a random walk phase. We show that this model is second-order stationary and can be rewritten as a vector-valued AR(1) model as well as a scalar ARMA(2, 1) model. The linear innovation sequence of the AR(1) model is shown to be a martingale difference sequence while the linear innovation sequence of the ARMA(2, 1) model is only an uncorrelated sequence. A non-linear predictor is derived from the AR(1) model while a linear predictor is derived from the ARMA(2, 1) mod

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37

Huang, Dawei, and N. M. Spencer. "On a random vibration model." Journal of Applied Probability 33, no. 04 (December 1996): 1141–58. http://dx.doi.org/10.1017/s0021900200100543.

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A random vibration model is investigated in this paper. The model is formulated as a cosine function with a constant frequency and a random walk phase. We show that this model is second-order stationary and can be rewritten as a vector-valued AR(1) model as well as a scalar ARMA(2, 1) model. The linear innovation sequence of the AR(1) model is shown to be a martingale difference sequence while the linear innovation sequence of the ARMA(2, 1) model is only an uncorrelated sequence. A non-linear predictor is derived from the AR(1) model while a linear predictor is derived from the ARMA(2, 1) mod

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38

Zheng, Yanling, Xueliang Zhang, Xijiang Wang, Kai Wang, and Yan Cui. "Predictive study of tuberculosis incidence by time series method and Elman neural network in Kashgar, China." BMJ Open 11, no. 1 (January 2021): e041040. http://dx.doi.org/10.1136/bmjopen-2020-041040.

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ObjectivesKashgar, located in Xinjiang, China has a high incidence of tuberculosis (TB) making prevention and control extremely difficult. In addition, there have been very few prediction studies on TB incidence here. We; therefore, considered it a high priority to do prediction analysis of TB incidence in Kashgar, and so provide a scientific reference for eventual prevention and control.DesignTime series study.Setting Kashgar, ChinaKashgar, China.MethodsWe used a single Box-Jenkins method and a Box-Jenkins and Elman neural network (ElmanNN) hybrid method to do prediction analysis of TB incide

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39

Geng, Jin-Jun, Bing Zhang, and Yong-Feng Huang. "A MODEL OF WHITE DWARF PULSAR AR SCORPII." Astrophysical Journal 831, no. 1 (October 31, 2016): L10. http://dx.doi.org/10.3847/2041-8205/831/1/l10.

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40

Kim, Hee-Young, Christian H. Weiß, and Tobias A. Möller. "Models for autoregressive processes of bounded counts: How different are they?" Computational Statistics 35, no. 4 (March 27, 2020): 1715–36. http://dx.doi.org/10.1007/s00180-020-00980-6.

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Abstract We focus on purely autoregressive (AR)-type models defined on the bounded range $$\{0,1,\ldots , n\}$$ { 0 , 1 , … , n } with a fixed upper limit $$n \in \mathbb {N}$$ n ∈ N . These include the binomial AR model, binomial AR conditional heteroscedasticity (ARCH) model, binomial-variation AR model with their linear conditional mean, nonlinear max-binomial AR model, and binomial logit-ARCH model. We consider the key problem of identifying which of these AR-type models is the true data-generating process. Despite the volume of the literature on model selection, little is known about this

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41

Popovic, Bozidar. "AR(1) time series with approximated Beta marginal." Publications de l'Institut Math?matique (Belgrade) 88, no. 102 (2010): 87–98. http://dx.doi.org/10.2298/pim1002087p.

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We consider the AR(1) time series model Xt ? ?Xt?1 = ?t, ??p ? N \ {1}, when Xt has Beta distribution B(p, q), p ? (0, 1], q > 1. Special attention is given to the case p = 1 when the marginal distribution is approximated by the power law distribution closely connected with the Kumaraswamy distribution Kum(p, q), p ? (0, 1], q > 1. Using the Laplace transform technique, we prove that for p = 1 the distribution of the innovation process is uniform discrete. For p ? (0, 1), the innovation process has a continuous distribution. We also consider estimation issues of the model.

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42

Charbonneau, Noe L., Elise C. Manalo, Sara F. Tufa, Eric J. Carlson, Valerie M. Carlberg, Douglas R. Keene, and Lynn Y. Sakai. "Fibrillin‐1 in the Vasculature: In Vivo Accumulation of eGFP‐Tagged Fibrillin‐1 in a Knockin Mouse Model." Anatomical Record 303, no. 6 (July 13, 2019): 1590–603. http://dx.doi.org/10.1002/ar.24217.

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43

Yang, Wenqi, and Jingkun Ma. "Implied Volatility Prediction Based on Different Term Structures: An Empirical Study of the SSE 50 ETF Options Market from High-Frequency Data." E3S Web of Conferences 235 (2021): 02043. http://dx.doi.org/10.1051/e3sconf/202123502043.

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This article focuses on the implied volatility forecast of the SSE 50 ETF options market from June 1, 2017, to August 30, 2019, and constructs AR (1) model and ARMA-GARCH model based on liquidity characteristics to compare and analyze the prediction effect of implied volatility on different option types and term structures. The results show that, during the sample period of the SSE 50 ETF options market, the effect of model fitting of the ARMA-GARCH model is significantly better than the AR (1) model; the fitting sequences predicted by the two models have typical time-varying and synchronizati

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44

Francq, Christian, Lajos Horvath, and Jean-Michel Zakoïan. "SUP-TESTS FOR LINEARITY IN A GENERAL NONLINEAR AR(1) MODEL." Econometric Theory 26, no. 4 (November 4, 2009): 965–93. http://dx.doi.org/10.1017/s0266466609990430.

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We consider linearity testing in a general class of nonlinear time series models of order one, involving a nonnegative nuisance parameter that (a) is not identified under the null hypothesis and (b) gives the linear model when equal to zero. This paper studies the asymptotic distribution of the likelihood ratio test and asymptotically equivalent supremum tests. The asymptotic distribution is described as a functional of chi-square processes and is obtained without imposing a positive lower bound for the nuisance parameter. The finite-sample properties of the sup-tests are studied by simulation

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45

Amato, Rosario, Francesco Pisani, Emiliano Laudadio, Maurizio Cammalleri, Martina Lucchesi, Silvia Marracci, Luca Filippi та ін. "HIF-1-Dependent Induction of β3 Adrenoceptor: Evidence from the Mouse Retina". Cells 11, № 8 (8 квітня 2022): 1271. http://dx.doi.org/10.3390/cells11081271.

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A major player in the homeostatic response to hypoxia is the hypoxia-inducible factor (HIF)-1 that transactivates a number of genes involved in neovessel proliferation in response to low oxygen tension. In the retina, hypoxia overstimulates β-adrenoceptors (β-ARs) which play a key role in the formation of pathogenic blood vessels. Among β-ARs, β3-AR expression is increased in proliferating vessels in concomitance with increased levels of HIF-1α and vascular endothelial growth factor (VEGF). Whether, similarly to VEGF, hypoxia-induced β3-AR upregulation is driven by HIF-1 is still unknown. We u

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Kumar, Jitendra, Varun Varun, Dhirendra Kumar, and Anoop Chaturvedi. "Bayesian Unit Root Test for AR(1) Model with Trend Approximated." Statistics, Optimization & Information Computing 8, no. 2 (May 27, 2020): 425–61. http://dx.doi.org/10.19139/soic-2310-5070-786.

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The objective of present study is to develop a time series model for handling the non-linear trend process using a spline function. Spline function is a piecewise polynomial segment concerning the time component. The main advantage of spline function is the approximation, non linear time trend, but linear time trend between the consecutive join points. A unit root hypothesis is projected to test the non stationarity due to presence of unit root in the proposed model. In the autoregressive model with linear trend, the time trend vanishes under the unit root case. However, when non-linear trend

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Issa, Mohamed Khalifa Ahmed. "Weighted Least Squares Estimation for AR(1) Model With Incomplete Data." Mathematics and Statistics 10, no. 2 (March 2022): 342–57. http://dx.doi.org/10.13189/ms.2022.100209.

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Anděl, Jiři, and Tomáŝ Bartoň. "A NOTE ON THE THRESHOLD AR(1) MODEL WITH CAUCHY INNOVATIONS." Journal of Time Series Analysis 7, no. 1 (January 1986): 1–5. http://dx.doi.org/10.1111/j.1467-9892.1986.tb00481.x.

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Gazola, L., C. Fernandes, A. Pizzinga, and R. Riera. "The log-periodic-AR(1)-GARCH(1,1) model for financial crashes." European Physical Journal B 61, no. 3 (February 2008): 355–62. http://dx.doi.org/10.1140/epjb/e2008-00085-1.

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Kemp, Gordon C. R. "The Joint Distribution of Forecast Errors in the AR(1) Model." Econometric Theory 7, no. 4 (December 1991): 497–518. http://dx.doi.org/10.1017/s0266466600004734.

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Second-order asymptotic expansion approximations to the joint distributions of dynamic forecast errors and of static forecast errors in the stationary Gaussian pure AR(1) model are derived. The approximation to the dynamic forecast errors distribution can be expressed as a multivariate normal distribution with modified mean vector and covariance matrix, thus generalizing the results of Phillips [12]. However, the approximation to the static forecast errors distribution includes skewness and kurtosis terms. Thus the class of multivariate normal distributions does not provide as good approximati

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