2 ANDREW L. RUKHIN We accept here the usual in the change-point analysis convention that, when the minimizer in ) is not determined uniquely, the smal

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1 THE RATES OF CONVERGENCE OF BAYES ESTIMATORS IN CHANGE-OINT ANALYSIS ANDREW L. RUKHIN Department of Mathematics and Statistics UMBC Baltimore, MD 2228 USA Abstract In the asymptotic setting of the change-point estimation problem the limiting behavior of Bayes procedures for the zero-one loss function is studied. The limiting distribution of the dierence between the Bayes estimator and the parameter is derived. An explicit formula for the limit of the minimum Bayes ris for the geometric prior distribution is obtained from Spitzer's formula, and the rates of convergence in these limiting relations are determined. Key words and phrases Bayes ris, change-point problem, convergence rate, geometric distribution, maximum lielihood estimator, Spitzer's formula, zero-one loss function.. Asymptotic Behavior of the Bayes Estimator Under ero-one Loss Function Assume that the observed data is formed by the random subsample x ; ; x ), which is observed rst and is coming from distribution F, and by x + ; ; x n+ ) from distribution G; G 6= F. In other terms x = x ; x ; ; x ; x + ; ; x n ; x n+ ), with denoting the change-point. Denote by f and g densities of mutually absolutely continuous) probability distributions F and G, and by KF; G) and KG; F ) positive and nite) information numbers. Also let gx + j); = ; ; n; and notice that the minimal sucient statistic p x) = Q fx j) Q n+ f for parameter has the form g x ); ; f g x n) see lachy and Ruhin, 994). Thus, when distributions F and G are nown, the rst and the last observations are ancillary statistics. If is the estimate of ; = ; ; ; n, based on x ; ; x n, then its performance will be measured by the error probability n x) 6= In other terms we use the zero-one loss function L; ) = ; d 6= ; = ; d =. Assume that the prior distribution of the parameter is given by positive prior probabilities n) j = j j = ; ; n. Then the Bayes estimator n is dened by the formula " # n x) = arg max p x) = arg max log f n n g x j) + log ) Research supported by NSA Grant MD94-93-H-35.

2 2 ANDREW L. RUKHIN We accept here the usual in the change-point analysis convention that, when the minimizer in ) is not determined uniquely, the smallest value of d is taen. Then n is uniquely dened with probability. Notice that for the uniform prior distribution n is merely the maximum lielihood estimator. When is the true parametric value, the distribution of n? has the form = +d m +d+ + n x)? = d log f g x j) + log +d > ; = ; ; + d? ; log f g x j) + log m +d ; m = + d + ; ; n For any j = ; 2 let ~ j denote a random variable distributed as log f g x), when x has distribution F, and let random variable j be distributed as log f g x) for x with distribution G. For d; d n? d^ = j + d+ n x) = + d ~ j + log +d +d? > ; = ; ; + d j + log +d+ +d ; = ; ; n?? d ; assuming that ~ j and j ; j = ; 2, are mutually independent, and a similar formula holds for negative d. ut and so that 2) d^ A ) = f B ) = f j + d+ +d \ n x) = + d = Assume that for xed positive as j! ~ j + log +d +d? > g; j + log +d+ +d g; j+ j " q ; n??d \ A ) B ) with positive q. Then q q` = q +`; and q = q for some positive q. We assume that with = log q 3)? KF; G) < < KG; F ); or?e ~ j < <?E j ; j = ; 2.

3 For positive integer dene RATES OF CONVERGENCE 3 4) and 5) S = + + ~S = ~ + ~ + ; so that according to 3) ES < < E ~ S. Because of 2) for any xed, events A j); j = ; 2 ; form an increasing sequence, A j) A j + ), so that for a xed as j! A j)! A = [ i= A i) = fs d^ + ~ S?d > g Also for any xed, events B j) form a decreasing sequence, B j) B j + ), and Thus, as!, and as n?! B j)! B = +d \ n??d \ [ i= A )! B )! B i) = fs g Therefore for xed positive d, when n!, so that ^ n? )! ; 6) n x) = + d! \ \ \ and a similar formula holds for negative d. Denote Then f d = min S > ; min d f = min A B A ~S > \ B ; ~S + S d > For any xed d,f d forms a non-increasing sequence and we put If 7) f d = lim! f d ~ = inffn ~ Sn g; then according to Spitzer's formula see, Section 7, Spitzer, 966 or Corollary 2.4 in Woodroofe, 982) f = min S ~ > = ~ = ) = exp?? ~ S ) )

4 4 ANDREW L. RUKHIN Similarly we put g md = max d ~S ; max S + Sd ~ ; m which for any xed d also is a non-increasing sequence in m with and Spitzer's formula also implies that g = max g m = S max S m g d = lim m! g md = exp?? S > ) By using this notation we can reformulate 6). When ^n?)!, n x)? converges in distribution to the random variable V with f d g d ; 8) V = d = f g jdj d < In particular, 9) V = ) = min ~S > max ) S = f g When, the distribution of V was derived by Hinley 97). According to 8), n? = O ). This rate for ^ n? )! is nown to hold in a more general nonparametric setting. See Dumbgen 99), Ferger and Stute 992), Yao, Huang and Davis 994) or the monograph by Brodsy and Darhovsy 993) for various versions of this result. In our situation much more detailed statements about the limiting distributions are possible. +d Also observe that since the sums log f + g x j) + log +d ; = ; ; + d? ; and m log f +d+ g x j) + log +d ; m = + d + ; ; n; do not converge in probability, only the statement about the convergence in distribution of n? can be made. When n? = m is xed and!, under condition 2) n? converges in distribution to a random variable W m such that ) W m = d = f d g m?d d f g mjdj d < If is xed and n!, then the asymptotic distribution of n? is that of the random variable U, ) We formulate these results. f d g d U = d = f +d g jdj d <

5 RATES OF CONVERGENCE 5 Theorem. Let n be the Bayes estimator ) of the change-point parameter. Under assumption 2) when ^ n? )!, the asymptotic distribution of n? is given by random variable V in 8) with lim n = ) = f g from 9). If is xed, for n!, n? converges in distribution to the random variable specied in ), and, when n? = m is xed and n!, n? converges in distribution to ). Motivated by Theorem from now on, we assume the truncated) geometric prior distribution with probabilities j =?q?q q j j = ; ; ; n, = log q. Values of q larger than n+ are of interest when the change-point is more liely to happen at the end of the observation period. This situation occurs in many quality control type problems. Under the geometric prior distribution the sets A j) and B j) do not depend on j and T+d for d, A = g n??d). Thus 2) = f d and Tn??d B ) n x)? = d = f d g n??d) d ; f +d) g n?)jdj d < ; Our goal is to derive the rate of convergence in Theorem. Lemma. Let sums S and S ~ be dened by 4) and 5) respectively and let d be a non negative integer. Then for any f d? f d ~S > f d ; and for any m g md? g d S > g md >m roof One has f d? f d = min S > ; min d ~S + S d > ; min j> ~S j + S d ~Sj + S d ; min S > ; min ~S + S d > d j> ~Sj ; min S > ; min ~S + S d > d j> ~Sj j> min S > ; min d ~S + S d > The last inequality here follows from the fact that the sums S ~ ;, and Sj ~ ; j >, are positively dependent see Robbins 954), and the second inequality is proved similarlyy. 2 To estimate the probabilities in the right-hand sides of Lemma we need the following notation. Let m u) = inf s m 2 u) = inf s log log g s f?s? su f s g?s? su ;

6 6 ANDREW L. RUKHIN Clearly both these functions are concave, non-increasing and non-positive; m u) vanishes if and only if u?kf; G), and m 2 u) vanishes when u?kg; F ). If u < KG; F ), then m u) = and when u < KF; G), one has m 2 u) = inf <s< log inf log <s< g s f?s? su f s g?s? su R For example, if u < KG; F ), the minimum of the convex function, log f?s g s? su, of s; s, is attained in the interval [; ). Indeed its derivative at s =, equal to KG; F )?u, is positive. Therefore m 2 u) = = inf log <s< inf log <s< f s x)g?s x) dx? su f?s x)g s x) dx?? s)u This argument shows that under condition 3) ; m?u)? u 3) m 2?) = m ) + < Theorem 2. One has 4) 8 < n x) = + d? V = d f d g n??d) h expf+)m )g?expfm )g + expfn??d+)m2?)g?expfm 2?)g?expfm )g + expfn?+)m2?)g?expfm 2?)g f +d) g n?)jdj h expf+d+)m )g i i d d < roof By Cherno's inequality Lemma 9.. in Cherno, 972) ~S inf f?s x)g s x) dxe?s = e m) ; s> and similarly S > ) inf s> f s x)g?s x) dxe s = e m2?) According to 2) for d one needs an estimate for the dierence f d g n??d)? f d g = [f d? f d ]g n??d) + f d [g n??d)? g ] Lemma shows that for example, 5) expfn?? d + )m 2?)g g n??d)? g g n??d)? expfm 2?)g Inequality 4) follows now from 5), a similar inequality for f d, and the fact that both sequences f d and g md are non-increasing. 2

7 RATES OF CONVERGENCE 7 According to Theorem 2 the rate of convergence of n? to V is exponential and is determined by numbers m ) and m 2?) = m ) +. When =n!, lim n! n log j n x) = + d? V = d j m ) _? )m 2?) The same result holds for the convergence of n? to random variables W m and U. For example, as! n x) = + d expf+)m fd g )g m?d)?expfm? W m = d )g d expf+d+)m f +d) g )g n?)jdj?expfm )g d < Theorems and 2 give the asymptotic results about the behavior of the corresponding ris function, or the probability of the correct decision. When, say, = and n! n x) = = max mn? S m = g n?)! g f g ; and a similar formula holds for = n. Thus for the uniform prior distribution the limiting frequentist ris of the Bayes estimator,? V = ) = lim n 6= ), which is asymptotically constant over the \central" part of the parameter space, exceeds the limiting values at the end-points = and = n. 2. Asymptotic Behavior of the Bayes Minimum Ris In this Section we loo at the limiting minimum Bayes ris, which is just the average error probability, for geometric prior distributions. When q = it is shown in Ruhin 994) that B n = n + n = n = )! inf n B n = V = ) = f g For q 6= the average probability of the correct decision B n in the change-point problem also converges to B = inf n B n Ruhin, 995). First of all we show here that 6) B = 8 >< > exp exp Indeed for < q < B n = n? n?? q? q n+ h S ) + q ~S < io ; < q ; h io ~S < + q S ) ; < q = =q < n q n = ) =? q? q n+ and as n! and is xed, g n?)! g. Therefore where ~ is dened in 7). B n! B =? q)g q f =? q)g n q f g n?) ; q ~ > ) ;

8 8 ANDREW L. RUKHIN One has? q) =?? q) q ~ > ) =?? q) ~ = ) q =? q ~ ) q ~ = ) According to Spitzer's formula see Woodroofe, 987, Corollary 2.3 with s = q, t = ) for any < q < q ~ = ) =? exp which proves 6) in this case. When q > one has with q = =q B n = qn q? ) q n+? n m=? q ) ~S < ; q m n?m n = n? m) As n! for any xed m, n?m n = n? m) = f g m! f g m f. Therefore Let = inffn S n > g. Then q m g m =? q? B n! B =? q )f q m g m q m m) =? q "? Another application of Spitzer's formula shows that? q ) q m g m = exp? q S > ) q = ) and this concludes the proof of 6). According to 6) the limiting Bayes gain, B, is a continuous function of q; < q <. Indeed as q!, B! f g. This corresponds to the fact that the Cesaro mean, corresponding to q =, must coincide with the limiting version of Abel's means dening B for q 6=. Now we use Lemma and Theorem 2 to get the rates of convergence in these limiting relations. When < q < B n? B =? q) h n q f g n?)? g )? Inequality 5) with d =, and 3) show that B n? B? q) n n+ ) qn+ q f g +? q n+ q f g n? expfn? + )m 2?)g? expfm 2?)g ; n # q f g n?) i + q n+

9 ? q) expfn + )m 2?)g? expfm 2?)g RATES OF CONVERGENCE 9 n expf?m 2?)? )g + q n+ =? q) expfn + )[m 2?)? m )]g? expfn + )m 2?)g? expfm 2?)g)expf?m )g? ) + q n+ 7) h? q) expfm q n+ )g i? expfm )g)? expfm 2?)g) + A similar estimate with q and m 2?) replaced by q and m )) holds when q >. Thus in both of these cases the rate of convergence is exponential, but the situation changes for q =. Then B n? B = n + n f g n?? f g ) n + " n 8) = n + n? f? f ) g n?) + g n?)? g f f g n?) expf + )m )g? expfm )g + n # expfn? + )m 2?)g f g n?)? expfm 2?)g gn expfm )g n + [? expfm )g] + f n expfm 2?)g 2 [? expfm 2?)g] 2 We sum up results obtained in this Section. Theorem 3. For the geometric prior distribution, the average probability of the correct decision, B n converges, so that 6) holds. For q 6= the rate of convergence is exponential as in 7). When q =, this rate is given by 8). Thus, when < q <, the minimum Bayes ris is formed by weighted sum of the asymptotic values of the ris function adjacent to the origin, and for q > the minimum Bayes ris is formed by the sum of riss close to = n. In these case the convergence of B n to B is exponential. For the uniform prior distribution the limiting ris of the corresponding Bayes estimator,? B, coincides with the minimum Bayes ris, but the convergence rate is much slower. Yao 987) obtained an inequality for the probability n 6= ), where is the estimator of based on doubly innite sample ; x? ; x ; x ;, and n is based on observations x [n?)=2] ; ; x? ; x ; x ; ; x [n=2]. Interestingly, this bound is not an exponential one either, but has the order O? [ ^ n? )]?. Notice that most convergence rates in the renewal theory are of exponential order see Stone, 965).

10 ANDREW L. RUKHIN 3. Change-oint Estimation for Exponential Distributions In this section we loo at an example of the change-point estimation when F and G are two exponential distributions with densities fx) = e?x and gx) = e?x for x >. When q =, V = ) = f g can be found explicitely 9) f g =? + log < ;? =? log = > see Woodroofe 982, p 34). Figure shows the graphs of the probability of the correct decision of the Bayes estimator against the uniform prior distribution when n = 5 and = 2. The constant function in this and the following gures represents the Bayes gain, the average probability of the correct decision), whose exact value 7 is to be compared with the value 53 from 9). The bound 8) gives the value of 9, which demonstrates that it is rather sharp. In Figures 2 and 3 similar graphs are given for q = 2 and q = 8. The Bayes gain, when q = 2, equals to 98, and for q = 8 this quantity is equal to 24. A simple numerical calculation shows that m log 2) =?25 and m log 8) =?34, so that according to Theorem 3 for q = 2, B 5? B 7, and for q = 8, B 5? B 28. These bounds are also pretty accurate Figure. The graphs of n = when n = 5 and q = and the Bayes ris

11 RATES OF CONVERGENCE Figure 2. The graph of n = ) when q = Figure 3. The graph of n = ) when q = 8 References. B. E. Brodsy and B. S. Darhovsy. Nonparametric Methods in Change{oint roblems. Kluwer, Dordrecht, H. Cherno. Sequential Analysis and Optimal Design. Regional Conferences Series in Applied Mathematics. SIAM, hiladelphia, A, D. Ferger and W. Stute. Convergence of changepoint estimators. Stochastic roc. Appl., 42345{35, D. V. Hinley. Inference about the change-point in a sequence of random variables. Biometria, 57{7, 97.

12 2 ANDREW L. RUKHIN 5. D. lachy and A. L. Ruhin. Unbiasedness and suciency in the change-point problem. In. Lachout and J. A. Vise, editors, roceedings of 2th rague Conference on Information Theory, Statistical Decision Functions and Random rocesses. Academy of Sciences of Czech Republic, rague, H. Robbins. A remar on the joint distribution of cumulative sums. Ann. Math. Statist., 25964{66, A. L. Ruhin. Asymptotic minimaxity in the change-point problem. In D. Siegmund, E. Carlstein, and H. Mueller, editors, roceedings on Change-oint roblems. Institute of Mathematical Statistics, Hayward, CA, A. L. Ruhin. Asymptotic behavior of Bayes estimatiors in the change-point problem. Math. Methods Statist, 4, F. Spitzer. rinciples of Random Wal. Van Nostrand, New Yor, C. J. Stone. On moment generating function and renewal theory. Ann. Math. Statist., 36298{3, M. Woodroofe. Nonlinear Renewal Theory in Sequential Analysis. Regional Conferences Series in Applied Mathematics. SIAM, hiladelphia, A, Y. Yao, D. Huang, and R. A. Davis. On almost sure behavior of change-point estimators. In D. Siegmund, E. Carlstein, and H. Mueller, editors, roceedings on Change-oint roblems. Institute of Mathematical Statistics, Hayward, CA, Y. C. Yao. Approximating the distribution of the maximum lielihood estimate of the change-point in a sequence of independent random variables. Ann. Statist., 532{ 328, 987.