A MEDIAN UNBIASED ESTIMATOR OF THE AR(1) COEFFICIENT

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1 A MEDIAN UNBIASED ESTIMATOR OF THE AR) COEFFICIENT Ryszard Zielińsi Inst. Math. Polish Acad. Sc., Poland P.O.Box 37 Warszawa, Poland Key words: autoregressive model, median-unbiasedness ABSTRACT A proof is given that the median of the ratios of the consecutive observations of a stationary first order autoregressive process X t αx t + Y t is a median unbiased estimator of α.. INTRODUCTION The paper is concerned with the median unbiased estimation of the stationary first order autoregressive process ) X t αx t + Y t, t...,,,,... with independent innovations Y t. To the best of my nowledge there exist only two papers in the subject which contain some constructive results, namely Hurwicz 95) and Andrews 993). Both are concerned with the case that Y t are i.i.d. normal N, ) random variables. Hurwicz 95) observed that every ratio X t /X t, t, 3,..., n, is a median-unbiased estimator of α. In the basic version of the process ), where Y t are normally distributed, the ratio X t /X t has a Cauchy distribution, so that neither any ratio X t /X t nor the mean n ) n t X t/x t can be efficient. However one might conjecture that the median of the ratios X t /X t, t, 3,..., n, would be a more efficient estimate of α and perhaps an unbiased one Hurwicz95), p. 368). Andrews 993) constructed an exactly median unbiased estimator of α however his proposal suffers from two disadvantages: ) it heavy depends on the assumption of normality of innovations and ) to apply, it needs numerical

2 tables, separately for each number of observations used, or an appropriate computer procedure. An advantage of his approach was that the models he discussed were more general than our model ). The aim of this note is to prove that the Hurwicz conjecture concerning median-unbiasedness is really true. What is more, it appears that the median of the ratios is a median-unbiased estimator of α not only in the Gaussian case but whenever the medians of independent not necessary identically distributed) innovations Y, Y,..., Y n are equal to zero. It follows that the Hurwicz estimator is median-bias robust against heavy tails of innovations as well as against ε-contamination with contaminants symmetric around zero. The problem of efficiency is more difficult first of all due to the fact that it is not as clearly stated as that of unbiasedness, and will be considered elsewhere.. THE HURWICZ ESTIMATOR Our basic assumptions concerning the distributions of the innovations are that the innovations are independent, their medians are equal to zero, and they are continuous in the sense that P {Y t } P {Y t } / and P {X t } for all t,,..., n ; otherwise the Hurwicz estimator might not be defined. For a given segment ) X, X,..., X n, n fixed, of the process ) consider the ratios X /X, X 3 /X,..., X n /X n. To avoid too many technicalities we assume that n is even so that the median of the ratios is uniquely determined. As an estimator of α we tae { X 3) ˆα HUR Med, X } 3 X n,..., X X X n where Medξ, ξ,..., ξ N ) denotes the sample median of the observations, i.e. if ξ :N ξ :N... ξ N:N and N then Medξ, ξ,..., ξ N ) ξ :N. 3. RESULTS In the proof of the main result the following Lemma plays the central role. Lemma. Let ξ, ξ,..., ξ N, N odd, be random variables and let c be a constant such that

3 C) P {ξ j c} for all j,,..., N; C) for every m,,..., N, for every choice i, i,..., i m i < i <... < i m N) of integers, and for every x,..., x m P {ξ im c ξ i x,..., ξ im x m } Then P {Medξ, ξ,..., ξ N ) c}. Proof. First of all observe that for every m,,..., N and for every choice i, i,..., i m of different integers,,..., N 3) P {ξ i c, ξ i c,..., ξ im c} ) m That is a simple consequence of the following calculations P {ξ i c, ξ i c,..., ξ im c} c... c c c... P {ξ im c ξ i x,...,ξ im x m }P ξi...ξ im dx...dx m ) P ξi...ξ im dx... dx m ) P {ξ i c, ξ i c,..., ξ im c} where P ξi...ξ im is the joint distribution of ξ i... ξ im. Now we shall mae use of the following formula for the distribution function of the sample median of dependent observations David 98), Sec. 5.6) P {Medξ, ξ,..., ξ N ) c} N m ) m m ) S m where S m is the sum of n m) probabilities P {ξi c, ξ i c,..., ξ im c}. By 3), S m N m m) ) and hence 3

4 P {Medξ,ξ,..., ξ N ) c} N m ) m m N ) m )! ] )! ] )! ] / )! ] / which ends the proof of the Lemma. ) ) ) ) t ) m + ) / t t) dt ) + + t dt ) t) dt Theorem. If the innovations Y, Y,..., Y n are independent random variables such that P {Y t } P {Y t } for all t,,..., n, and P {X t } for all t,,..., n, then the Hurwicz estimator ˆα HUR is median unbiased: P α {ˆα HUR α} for all α, ) Proof. For the sequence of observations X, X,..., X n, n even, denote N n and apply the Lemma with For ξ j we have ξ X X, ξ X 3 X,..., ξ N ξ j α + Y j+ X j X n X n where Y j+ and X j are independent random variables. Now for every α 4

5 P α {ξ j α} P α { Y j+ X j } P α {Y j+, X j > } + P α {Y j+, X j < } P α{x j > } + P α{x j < } and the hypothesis C) of the Lemma holds. Similarly, for every m, 3,..., N, for every choice of integers i, i,..., i m i < i <... < i m N), and for every x, x,..., x m, taing into account that Y im + is independent of X i,..., X im, obtains P α {ξ im α ξ i x,..., ξ im x m } P α { Y i m + X im ξ i x,..., ξ im x m } P α {Y im +, X im > ξ i x,..., ξ im x m }+ + P α {Y im +, X im < ξ i x,..., ξ im x m } P α{x im > ξ i x,..., ξ im x m }+ + P α{x im < ξ i x,..., ξ im x m } so that the second hypothesis C) of the Lemma is satisfied and the Theorem follows. ACKNOWLEDGEMENTS The research has been partially supported by the grant KBN PO3D. The author is deeply indebted to Hocine Fellag Tizi-Ouzu, Algeria) for inspiring discussions. REFERENCES Andrews, D.W.K. 993), Exactly Median Unbiased Estimation of First Order Auto Regressive/Unit Root Models. Econometrica, 6, David, H.A. 98), Order Statistics, nd ed., Wiley Hurwicz, L. 95), Least Squares Bias in Time Series. In Statistical Inference in Dynamic Economic Models, ed. by T.C.Koopmans. Wiley, New Yor 5