FURTHER BOUNDS FOR THE ESTIMATION ERROR VARIANCE OF A CONTINUOUS STREAM WITH STATIONARY VARIOGRAM

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1 FURTHER BOUNDS FOR THE ESTIMATION ERROR VARIANCE OF A CONTINUOUS STREAM WITH STATIONARY VARIOGRAM N. S. BARNETT, S. S. DRAGOMIR, AND I. S. GOMM Abstract. In this paper we establish an upper boun for the estimation error variance of a continuous stream with a stationary variogram V which is assume to be of r-höler type (Lipschitzian) on,.. Introuction In, the authors consiere X (t) as efining the quality of a prouct at time t where X (t) is a continuous time stochastic process which may be non-stationary. Typically, X (t) represents a continuous stream inustrial process such as is common in many areas of the chemical inustry. The paper was concerne with issues relate to sampling the stream with a view to estimating the mean quality characteristic of the flow, X, over the interval,. Specifically, focus was on obtaining the sampling location, sai to be optimal, ( ) 2 which minimizes the estimation error variance, E X X (t), t. Given that t is as specifie, the problem is to fin the value of t (the sampling ( ) 2 location) that minimizes E X X (t). It is shown that for constant stream flows, the optimal sampling point is the mipoint of, for situations where the process variogram, V (u) = 2 E ( X X (t) ) 2, where V () =, V ( u) = V (u) is stationary (note that variogram stationarity is not equivalent to process stationarity). The paper continues to consier optimal sampling locations for situations where the stream flow rate varies. The optimal sampling location is seen to epen on both the flow rate function an the form of the process variogram - some examples are given. In 2, rather than focussing on the optimal sampling point, the authors have focusse on the actual value of the estimation error variance itself. They obtaine the following result by employing an inequality of the Ostrowski type for ouble integrals. Date: March 3, Mathematics Subject Classification. Primary 62 X xx; Seconary 26 D 5. Key wors an phrases. Error variance, Continuous stream with stationary variogram.

2 2 N. S. BARNETT, S. S. DRAGOMIR, AND I. S. GOMM Theorem. Let V : (, ) R be a twice ifferentiable variogram having the secon erivative V : (, ) R which is boune. If V := sup t (,) V (t) <, then (.) for all t,. ( ) 2 E X X (t) 4 + ( t 2 2 ) 2 2 V Note that the best inequality we can get from (.) is that one for which t = t = 2 giving the boun ( E X X (t ) ) V. It shoul be note that the above result requires ouble ifferentiability of V in (, ) an that this conition oes not hol for the case of a linear variogram. That is, V (u) = a u, u R. For other results on Ostrowski s inequality we refer to the recent papers 3-7 an the book 8. In this note we point out another boun for the estimation error variance which oes not require the ifferentiability of V. Some functional properties are also given. 2. The Results Firstly, let us recall the concept of r-höler type mappings. Definition. The mapping f : a, b R R is calle of r-höler type with r (, if (2.) f (x) f (y) H x y r for all x, y a, b with a certain H >. If r =, we get the classical concept of Lipschitzian mappings. Example. If r (,, then the mapping f (x) = x r satisfies the conition (2.2) f (x) f (y) = x r y r x y r for all x, y, ), which shows that f is of r-höler type with the constant H =, on every close interval a, b. Example 2. Any ifferentiable mapping f : a, b R having the erivative boune in (a, b) is Lipschitzian on (a, b). The following result hols. Theorem 2. Assume that the variogram V :, R is of r-höler type on, with the constant H >. Then we have the inequality ( ) 2 E X X (t) 2H t r+ + ( t) r+ (2.3) 2H r + r + for all t,.

3 FURTHER BOUNDS FOR THE ESTIMATION ERROR VARIANCE 3 Proof. From, using an ientity given in 9, it can be shown that ( ) 2 E X X (t) = 2 t V (u) u + t V (u) u } 2 b V (v u) uv. Also, observe that (see ) V (v t) v = t V (u) u + t V (u) u an V (t u) u = t V (u) u + t V (u) u an then we get the ientity (2.4) ( ) 2 E X X (t) = = 2 V (v t) v + b = 2 b V (v t) v + V (v u) vu V (t u) u V (t u) u V (v t) + V (t u) V (v u) vu. Using the fact that V is of r Höler type, we can write that (2.5) V (v t) V (v u) H v t v + u r = H u t r for all u, v, t, an (2.6) V (t u) = V (t u) V () H t u r

4 4 N. S. BARNETT, S. S. DRAGOMIR, AND I. S. GOMM for all t, x,. Now, using (2.4) (2.6), we get ( ) 2 E X X (t) = 2 = 2H = 2H = 2H V (v t) + V (t u) V (v u) vu V (v t) V (v u) + V (t u) vu V (v t) V (v u) + V (t u) vu H t u r + H t u r vu t u r u t (t u) r u + t r+ + ( t) r+ r + t (u t) r u an the first inequality in (2.3) is prove. The secon part is obvious. Corollary. If V is Lipschitzian with the constant L >, then we have the inequality: ( ) ( ) 2 t 2 E X X (t) (2.7) 2 L. Proof. Choose r = to get in the right han sie of the inequality t 2 + ( t) 2 ( ) t 2 = Then, by (2.3), we euce (2.7). Remark. It is easy to see that the mapping g :, R, g (t) := t r+ + ( t) r+ has the properties ( ) inf g (t) = g = r+ t, 2 2 r an sup g (t) = g () = g () = r+ t, which shows that the best inequality we can get from (2.3) is that one for which t = t = 2, getting ( E X X (t ) ) 2 2 r H r (2.8) r +. For the Lipschitzian case, we get (2.9) E ( X X (t ) ) 2 2 L.

5 FURTHER BOUNDS FOR THE ESTIMATION ERROR VARIANCE 5 Define the mapping ξ :, R given by ( ) 2 ξ (t) = E X X (t). The following property of continuity for ξ hols. Theorem 3. If V is of r-höler type with the constant H > on the interval,, then ξ is of r-höler type with the constant 2H. Proof. Let t, t 2,. Then we have = = ξ (t 2 ) ξ (t ) 2 V (v t 2 ) + V (t 2 u) V (v u) uv 2 V (v t ) + V (t u) V (v u) uv (V (v t 2 ) V (v t )) + (V (t 2 u) V (t u)) uv = 2H t 2 t r 2 2 = 2H t 2 t r an the theorem thus prove. V (v t 2 ) V (v t ) + V (t 2 u) V (t u) uv H t 2 t r + H t 2 t r uv Corollary 2. If V is L-Lipschitzian on,, then ξ is 2L-Lipschitzian on,. The following result concerning the convexity property of the mapping ξ efine above on, hols. Theorem 4. If the variogram V :, is monotonic nonecreasing on the interval,, then ξ ( ) is convex on,. Proof. We know that for all t, Then ξ (t) = 2 t V (u) u + t V (u) u } 2 ξ (t) = 2 V (t) V ( t). V (v u) uv.

6 6 N. S. BARNETT, S. S. DRAGOMIR, AND I. S. GOMM Now, let t, t 2, with t 2 > t. Then ξ (t 2 ) ξ (t ) (t 2 t ) ξ (t ) = 2 t2 } { t2 V (u) u + V (u) u 2 t } t V (u) u + V (u) u 2 V (t ) V ( t ) (t 2 t ) = 2 t2 } t V (u) u V (u) u (t 2 t ) V (t ) + (t 2 t ) V ( t ). t 2 t As V is nonecreasing on,, then t2 an which implies that t t V (u) u (t 2 t ) V (t ) t 2 V (u) u (t 2 t ) V ( t ) ξ (t 2 ) ξ (t ) (t 2 t ) ξ (t ) for all t 2 > t,, which shows that the mapping ξ ( ) is convex on,. References N.S. Barnett, I.S. Gomm, an L. Armour: Location of the optimal sampling point for the quality assessment of continuous streams, Australian J. Statistics, 37(2), 995, N.S. Barnett an S.S. Dragomir, A note on bouns for the estimation error variance of a continuous stream with stationary variogram, J. KSIAM, Vol. 2 (2)(998), N.S. Barnett an S.S. Dragomir: An Ostrowski s type inequality for ouble integrals an applications to cubature formulae, submitte. 4 S.S. Dragomir an S. Wang, A new inequality of Ostrowski s type in L norm an applications to some special means an to some numerical quarature rules, Tamkang J. of Math., 28 (997), S.S. Dragomir an S. Wang, An inequality of Ostrowski-Grüss type an its applications to the estimation of error bouns for some special means an for some numerical quarature rules, Computers Math. Applic., 33(997), S.S. Dragomir an S. Wang, Applications of Ostrowski s inequality to the estimation of error bouns for some special means an some numerical quarature rules, Appl. Math. Lett., (998), S.S. Dragomir an S. Wang, A new inequality of Ostrowski s type in L p norm an applications to some special means an to some numerical quarature rules, submitte. 8 D.S. MITRINOVIĆ, J.E. PE CARIĆ an A.M. FINK: Inequalities for Functions an Their Integrals an Derivatives, Kluwer Acaemic Publishers, I.W. Sauners, G.K. Robinson, T. Lwin an R.J. Holmes, A simplifie variogram metho for the estimation error variance in sampling from continuous stream, Internat. J. Mineral Processing, 25(989), School of Communications an Informatics, Victoria University of Technology, PO Box 4428, MC melbourne City, 8 Victoria, Australia. aress: {neil, sever, isg}@matila.vu.eu.au URL: aress: sever@matila.vu.eu.au