Auto Regressive Process (1) with Change Point: Bayesian Approch

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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,‌,m and Xt = β2 Xt-1 + Îľt , t=m+1,‌,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?›˝1 đ?‘‹đ?‘–−1 + đ?œ–đ?‘– , đ?‘– = 1,2, ‌ , đ?‘š. đ?‘‹đ?‘– = (1) đ?›˝2 đ?‘‹đ?‘–−1 + đ?œ–đ?‘– , đ?‘– = đ?‘š + 1, ‌ , đ?‘›. where, đ?›˝1 and đ?›˝2 are unknown autocorrelation coefficients, đ?‘Ľđ?‘– is the ith observation of the dependent variable, the error terms đ?œ–đ?‘– are independent random variables and follow a N( 0, đ?œŽ12 ) for i=1,2,‌,m and a N( 0, đ?œŽ22 ) for i= m+1, ‌..,n and đ?œŽ12 and đ?œŽ22 both are known. m is the unknown change point and đ?‘Ľ0 is the initial quantity.

III.

BAYES ESTIMATION

The Bayes procedure is based on a posterior density, say, g đ?›˝1 , đ?›˝2 , đ?‘š | đ?‘? , which is proportional to the product of the likelihood function L đ?›˝1 , đ?›˝2 , đ?‘š | đ?‘? , with a joint prior density, say, g đ?›˝1 , đ?›˝2 , đ?‘š 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?‘†đ?‘š1 đ?‘†đ?‘š2 đ??´1đ?‘š 1 đ?‘†đ?‘›1 − đ?‘†đ?‘š1 đ?‘†đ?‘›2 − đ?‘†đ?‘š2 đ??ż đ?›˝1 , đ?›˝2 , đ?‘š|đ?‘? = đ??ž1 . đ?‘’đ?‘Ľđ?‘? − đ?›˝1 2 + đ?›˝1 − . đ?‘’đ?‘Ľđ?‘? − đ?›˝2 2 + đ?›˝2 2 đ?œŽ1 2 đ?œŽ1 2 2đ?œŽ1 2 2 đ?œŽ2 2 đ?œŽ2 2 đ??´2đ?‘š −(đ?‘›âˆ’đ?‘š ) − đ?œŽ −đ?‘š đ?œŽ2 2đ?œŽ2 2 1 (2) Where, k

k

đ?‘†đ?‘˜1 =

xi−1

2

đ?‘†đ?‘˜2 =

i=1 m

đ??´1đ?‘š =

xi xi−1 i=1 n

xi

2

i=1

xi 2

đ??´2đ?‘š = i=m+1

đ?‘›

đ?‘˜1 = (2đ?œ‹)−2 (3) www.ijres.org

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