International Journal of Research in Engineering and Science (IJRES) ISSN (Online): 2320-9364, ISSN (Print): 2320-9356 www.ijres.org Volume 5 Issue 1 Ç January. 2017 Ç PP. 27-31
Auto Regressive Process (1) with Change Point: Bayesian Approch Dr. Mayuri Pandya, Paras Sheth Department of Statistics, Maharaja Krishnkumarsinhji Bhavnagar University, Bhavnagar, Gujarat, India.
Abstract : Here we consider first order autoregressive process with changing autoregressive coefficient at some point of time m. This is called change point inference problem. For Bayes estimation of m and autoregressive coefficient we used MHRW (Metropolis Hasting Random Walk) algorithm and Gibbs sampling. The effects of prior information on the Bayes estimates are also studied. Keywords: Auto Regressive Model (1), Bayes Estimator, Change Point, Gibbs Sampling and MHRW Algorithm. I. INTRODUCTION Mayuri Pandya (2013) had studied the Bayesian analysis of the autoregressive model Xt = β1 Xt-1 + Îľt , t=1,2,â&#x20AC;Ś,m and Xt = β2 Xt-1 + Îľt , t=m+1,â&#x20AC;Ś,n where 0 < β1, β2 < 1, and Îľt was independent random variable with an exponential distribution with mean θ1 and is reflected in the sequence after Îľ m is changed in mean θ2. M. Pandya, K. Bhatt, H. Pandya, C. Thakar (2012) had studied the Bayes estimators of m, β1 and β2 under Asymmetric loss functions namely Linex loss & General Entropy loss functions of changing auto regression process with normal error. Tsurumi (1987) and Zacks (1983) are useful references on structural changes.
II.
PROPOSED AR (1) MODEL:
Let our AR(1) model be given by, đ?&#x203A;˝1 đ?&#x2018;&#x2039;đ?&#x2018;&#x2013;â&#x2C6;&#x2019;1 + đ?&#x153;&#x2013;đ?&#x2018;&#x2013; , đ?&#x2018;&#x2013; = 1,2, â&#x20AC;Ś , đ?&#x2018;&#x161;. đ?&#x2018;&#x2039;đ?&#x2018;&#x2013; = (1) đ?&#x203A;˝2 đ?&#x2018;&#x2039;đ?&#x2018;&#x2013;â&#x2C6;&#x2019;1 + đ?&#x153;&#x2013;đ?&#x2018;&#x2013; , đ?&#x2018;&#x2013; = đ?&#x2018;&#x161; + 1, â&#x20AC;Ś , đ?&#x2018;&#x203A;. where, đ?&#x203A;˝1 and đ?&#x203A;˝2 are unknown autocorrelation coefficients, đ?&#x2018;Ľđ?&#x2018;&#x2013; is the ith observation of the dependent variable, the error terms đ?&#x153;&#x2013;đ?&#x2018;&#x2013; are independent random variables and follow a N( 0, đ?&#x153;&#x17D;12 ) for i=1,2,â&#x20AC;Ś,m and a N( 0, đ?&#x153;&#x17D;22 ) for i= m+1, â&#x20AC;Ś..,n and đ?&#x153;&#x17D;12 and đ?&#x153;&#x17D;22 both are known. m is the unknown change point and đ?&#x2018;Ľ0 is the initial quantity.
III.
BAYES ESTIMATION
The Bayes procedure is based on a posterior density, say, g đ?&#x203A;˝1 , đ?&#x203A;˝2 , đ?&#x2018;&#x161; | đ?&#x2018;? , which is proportional to the product of the likelihood function L đ?&#x203A;˝1 , đ?&#x203A;˝2 , đ?&#x2018;&#x161; | đ?&#x2018;? , with a joint prior density, say, g đ?&#x203A;˝1 , đ?&#x203A;˝2 , đ?&#x2018;&#x161; representing uncertainty on the parameters values. The likelihood function of β1, β2 and m, given the sample information Zt = (xt-1, xt), t = 1, 2,..., m, m+1,..., n. is, 1 đ?&#x2018;&#x2020;đ?&#x2018;&#x161;1 đ?&#x2018;&#x2020;đ?&#x2018;&#x161;2 đ??´1đ?&#x2018;&#x161; 1 đ?&#x2018;&#x2020;đ?&#x2018;&#x203A;1 â&#x2C6;&#x2019; đ?&#x2018;&#x2020;đ?&#x2018;&#x161;1 đ?&#x2018;&#x2020;đ?&#x2018;&#x203A;2 â&#x2C6;&#x2019; đ?&#x2018;&#x2020;đ?&#x2018;&#x161;2 đ??ż đ?&#x203A;˝1 , đ?&#x203A;˝2 , đ?&#x2018;&#x161;|đ?&#x2018;? = đ??ž1 . đ?&#x2018;&#x2019;đ?&#x2018;Ľđ?&#x2018;? â&#x2C6;&#x2019; đ?&#x203A;˝1 2 + đ?&#x203A;˝1 â&#x2C6;&#x2019; . đ?&#x2018;&#x2019;đ?&#x2018;Ľđ?&#x2018;? â&#x2C6;&#x2019; đ?&#x203A;˝2 2 + đ?&#x203A;˝2 2 đ?&#x153;&#x17D;1 2 đ?&#x153;&#x17D;1 2 2đ?&#x153;&#x17D;1 2 2 đ?&#x153;&#x17D;2 2 đ?&#x153;&#x17D;2 2 đ??´2đ?&#x2018;&#x161; â&#x2C6;&#x2019;(đ?&#x2018;&#x203A;â&#x2C6;&#x2019;đ?&#x2018;&#x161; ) â&#x2C6;&#x2019; đ?&#x153;&#x17D; â&#x2C6;&#x2019;đ?&#x2018;&#x161; đ?&#x153;&#x17D;2 2đ?&#x153;&#x17D;2 2 1 (2) Where, k
k
đ?&#x2018;&#x2020;đ?&#x2018;&#x2DC;1 =
xiâ&#x2C6;&#x2019;1
2
đ?&#x2018;&#x2020;đ?&#x2018;&#x2DC;2 =
i=1 m
đ??´1đ?&#x2018;&#x161; =
xi xiâ&#x2C6;&#x2019;1 i=1 n
xi
2
i=1
xi 2
đ??´2đ?&#x2018;&#x161; = i=m+1
đ?&#x2018;&#x203A;
đ?&#x2018;&#x2DC;1 = (2đ?&#x153;&#x2039;)â&#x2C6;&#x2019;2 (3) www.ijres.org
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