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NON‐NEGATIVE AUTOREGRESSIVE PROCESSES
Author(s) -
Anděl Jiří
Publication year - 1989
Publication title -
journal of time series analysis
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.576
H-Index - 54
eISSN - 1467-9892
pISSN - 0143-9782
DOI - 10.1111/j.1467-9892.1989.tb00011.x
Subject(s) - mathematics , autoregressive model , combinatorics , independent and identically distributed random variables , estimator , statistics , random variable
. Consider a stationary autoregressive process given by X t = b 1 X t ‐1 +…+ b p X t‐p + Y t , where the Y t are independent identically distributed positive variables and b 1 ,…, b p are non‐negative parameters. Let the variables X 1 ,…, X n be given. If p = 1 then it is known that b 1 *= min( X t / X t ‐1 ) is a strongly consistent estimator for b 1 under very general conditions. In this paper the case p = 2 is analysed in detail. It is proved that min( X t / X t ‐1 )→ b 1 almost surely (a.s.) and min( X t / X t ‐2 )→ b 2 + b 1 2 a.s. as n → 8. The convergence is very slow. Denote by b 1 * and b 2 * values of b 1 and b 2 respectively which maximize b 2 + b 2 under the conditions X t ‐ b 1 X t ‐1 ‐ b 2 X t ‐2 ≥ 0 for t = 3,…, n . We prove that b 1 * b 1 and b 2 * b 2 a.s. Simulations show that b 1 * and b 2 * are better than the least‐squares estimators of the autoregressive coefficients when the distribution of Y t is exponential.