2 S. S. DRAGOMIR, N. S. BARNETT, AND I. S. GOMM Theorem. Let V :(d d)! R be a twce derentable varogram havng the second dervatve V :(d d)! R whch s bo

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1 J. KSIAM Vol.4, No., -7, 2 FURTHER BOUNDS FOR THE ESTIMATION ERROR VARIANCE OF A CONTINUOUS STREAM WITH STATIONARY VARIOGRAM S. S. DRAGOMIR, N. S. BARNETT, AND I. S. GOMM Abstract. In ths paper we establsh an upper bound for the estmaton error varance of a contnuous stream wth a statonary varogram V whch s assumed to be of the r-holder type (Lpschtzan) on [d d] : Functonal propertes for the mappng (t) :E X 2 X (t) t 2 [ d] are also gven.. Introducton In [], the authors consdered X (t) as denng the qualty of a product at tme t where X (t) s a contnuous tme stochastc process whch may be non-statonary. Typcally, X (t) represents a contnuous stream ndustral process such as s common n many areas of the chemcal ndustry. The paper [] was concerned wth ssues related to samplng the stream wth a vew to estmatng the mean qualty characterstc of the ow, X over the nterval [ d]: Speccally, focus was on obtanng the samplng h locaton, sad to be optmal, whch mnmzes the estmaton error varance, E X 2 X (t) t d: Gven that t s as speced, h the problem s to nd the value of t (the samplng locaton) that mnmzes E X 2 X (t) : It s shown that for constant streamows, the optmal samplng pont s the mdpont of [ d] for stuatons where the process varogram, V (u) 2 E h X X (t) 2 where V () V (u) V (u) s statonary (note that varogram statonarty s not equvalent to process statonarty). The paper [] contnues to consder optmal samplng locatons for stuatons where the stream ow ratevares. The optmal samplng locaton s seen to depend on both the ow rate functon and the form of the process varogram - some examples are gven. In [2], rather than focussng on the optmal samplng pont, the authors have focussed on the actual value of the estmaton error varance tself. They obtaned the followng result by employng an nequalty of the Ostrowsk type for double ntegrals. 99 Mathematcs Subject Classcaton. Prmary 62 X xx Secondary 26 D 5. Key words and phrases. Error varance, Contnuous stream wth statonary varogram.

2 2 S. S. DRAGOMIR, N. S. BARNETT, AND I. S. GOMM Theorem. Let V :(d d)! R be a twce derentable varogram havng the second dervatve V :(d d)! R whch s bounded. If kv k : sup t2(d d) jv (t)j < then (.) E X 2 X (t) for all t 2 [ d] : " 4 + t d V Note that the best nequalty we can get from (:) s that one for whch t t gvng the bound h X X (t ) 2 d2 E V : 6 It should be noted that the above result requres double derentabltyofv n (d d) and that ths condton does not hold for the case of a lnear varogram. That s, V (u) a juj u 2 R. For other results on Ostrowsk's nequalty we refer to the recent papers [3]- [7] and the book [8]. In ths note we pont out another bound for the estmaton error varance whch does not requre the derentablty of V: Some functonal propertes are also gven. 2. The Results Frstly, let us recall the concept of the r-holder type mappngs. Denton. The mappng f :[a b] R! R s sad to be of the r-holder type wth r 2 ( ] f (2.) jf (x) f (y)j H jx yj r for all x y 2 [a b] wth a certan H>: If r we get the classcal concept of Lpschtzan mappngs. Example. If r 2 ( ] then the mappng f (x) x r satses the condton (2.2) jf (x) f (y)j jx r y r jjx yj r for all x y 2 [ ) whch shows that f s of the r-holder type wth the constant H on every closed nterval [a b] : Example 2. Any derentable mappng f :[a b]! R havng the dervatve bounded n (a b) s Lpschtzan on (a b) : The followng result holds. Theorem 2. Assume that the varogram V : [d d]! R s of the r-holder type on [d d] wth the constant H>: Then we have the nequalty " (2.3) E X 2 X (t) 2H d for all t 2 [ d] : t r+ +(d t) r+ r + 2Hd r +

3 FURTHER BOUNDS FOR THE ESTIMATION ERROR VARIANCE 3 Proof. From [], usng an dentty gven n [9], t can be shown that E X 2 X (t) 2 d Also, observe that (see []) t d2 V (v u) dudv: V (v t) dv t and V (t u) du t and then we get the dentty (2.4) E X 2 X (t) d V (v t) dv + d d V (v t) dv + d V (v u) dvdu V (t u) du V (t u) du [V (v t)+v (t u) V (v u)] dvdu: Usng the fact that V s of the r-holder type, we can wrte that (2.5) jv (v t) V (v u)j H jv t v + uj r H ju tj r for all u v t 2 [ d]and (2.6) jv (t u)j jv (t u) V ()j H jt uj r

4 4 S. S. DRAGOMIR, N. S. BARNETT, AND I. S. GOMM for all t x 2 [ d]: Now, usng (2:4) (2:6) we get E X 2 X (t) 2H d 2H d 2H d " [V (v t)+v (t u) V (v u)] dvdu jv (v t) V (v u)+v (t u)j dvdu jv (v t) V (v u)j + jv (t u)j dvdu [H jt uj r + H jt uj r ] dvdu jt uj r du (t u) r du + t r+ +(d t) r+ r + t (u t) r du and the rst nequalty n(2:3) s proved. The second part s obvous. Corollary. If V s Lpschtzan wth the constant L>, then we have the nequalty: " h (2.7) E X 2 t X (t) Ld: Proof. Choose r to get n the rght hand sde of the nequalty (2:3) : t 2 +(d t) 2 " t : Then, by (2:3), we deduce (2:7) : Remark. It s easy to see that the mappng g :[ d]! R, g (t) :t r+ +(d t) r+ has the propertes and nf g (t) g t2[ d] d dr+ 2 2 r sup g (t) g () g (d) d r+ t2[ d] whch shows that the best nequalty we can get from (2:3) s that one for whch t t gettng (2.8) E h X X (t ) 2 2r Hd r r + :

5 FURTHER BOUNDS FOR THE ESTIMATION ERROR VARIANCE 5 For the Lpschtzan case, we get (2.9) E h X X (t ) 2 2 Ld: Dene the mappng :[ d]! R gven by h (t) E X 2 X (t) : The followng property of contnuty for holds. Theorem 3. If V s of the r-holder type wth the constant H> on the nterval [ d] then s of the r-holder type wth the constant 2H: Proof. Let t t 2 2 [ d] : Then we have j (t 2 ) (t )j 2H jt 2 t j r 2H jt 2 t j r and the theorem thus proved. [V (v t 2 )+V (t 2 u) V (v u)] dudv [V (v t )+V (t u) V (v u)] dudv [(V (v t 2 ) V (v t )) + (V (t 2 u) V (t u))] dudv [jv (v t 2 ) V (v t )j + jv (t 2 u) V (t u)j] dudv [H jt 2 t j r + H jt 2 t j r ] dudv Corollary 2. If V s L-Lpschtzan on [ d] then s 2L-Lpschtzan on [ d] : The followng result concernng the convexty property of the mappng dened above on[ d] holds. Theorem 4. If the varogram V :[d d] s monotonc nondecreasng on the nterval [ d] then () s convex on [ d] : Proof. We know that for all t 2 [ d] Then (t) 2 d t d2 (t) 2 [V (t) V (d t)] : d V (v u) dudv:

6 6 S. S. DRAGOMIR, N. S. BARNETT, AND I. S. GOMM Now, let t t 2 2 [ d] wth t 2 >t : Then 2 d 2 d (t 2 ) (t ) (t 2 t ) (t ) 2 V (u) du + t2 V (u) du 2 d [V (t ) V (d t )] (t 2 t ) 2 t t V (u) du dt2 As V s nondecreasng on [ d] then 2 d Z dt V (u) du (t 2 t ) V (t )+(t 2 t ) V (d t ) : 2 t V (u) du (t 2 t ) V (t ) and whch mples that t dt2 V (u) du (t 2 t ) V (d t ) (t 2 ) (t ) (t 2 t ) (t ) for all t 2 >t 2 [ d] whch shows that the mappng () sconvex on [ d] : References [] N.S. Barnett, I.S. Gomm, and L. Armour: Locaton of the optmal samplng pont for the qualty assessment of contnuous streams, Australan J. Statstcs, 37(2), 995, [2] N.S. Barnett and S.S. Dragomr, A note on bounds for the estmaton error varance of a contnuous stream wth statonary varogram, J. KSIAM, Vol. 2 (2)(998), [3] N.S. Barnett and S.S. Dragomr: An Ostrowsk's type nequalty for double ntegrals and applcatons to cubature formulae, submtted. [4] S.S. Dragomr and S. Wang, A new nequalty of Ostrowsk's type n L norm and applcatons to some specal means and to some numercal quadrature rules, Tamkang J. of Math., 28 (997), [5] S.S. Dragomr and S. Wang, An nequalty of Ostrowsk-Gruss' type and ts applcatons to the estmaton of error bounds for some specal means and for some numercal quadrature rules, Computers Math. Applc., 33(997), 5-2. [6] S.S. Dragomr and S. Wang, Applcatons of Ostrowsk's nequalty to the estmaton of error bounds for some specal means and some numercal quadrature rules, Appl. Math. Lett., (998), 5-9. [7] S.S. Dragomr and S. Wang, A new nequalty of Ostrowsk's type n L p norm and applcatons to some specal means and to some numercal quadrature rules, submtted. [8] D.S. MITRINOVI C, J.E. PE CARI C and A.M. FINK: Inequaltes for Functons and Ther Integrals and Dervatves, Kluwer Academc Publshers, 994. [9] I.W. Saunders, G.K. Robnson, T. Lwn and R.J. Holmes, A smpled varogram method for the estmaton error varance n samplng from contnuous stream, Internat. J. Mneral Processng, 25(989),

7 FURTHER BOUNDS FOR THE ESTIMATION ERROR VARIANCE 7 School of Communcatons and Informatcs Vctora Unversty of Technology, PO Box 4428, MC melbourne Cty, 8 Vctora, Australa. emalfnel, sever, sgg@matlda.vu.edu.au