Fenstad, Grete Usterud

NORA - Norwegian Open Research Archives

Statistical Research Report Institute of Mathematics University of Oslo A NOTE ON A THEOREM OF BARANKIN AND GURLAND by Grete Usterud Fenstad ;~-.. '-No. 2 July 1965 A NOTE ON A THEOREM OF BARANKDJ AND G ·oRLAND In their paper "On asymptotically normal, efficient estimators: 1 11 , [1], Barankin and Gurland have left a·gap in their proof of Theorem 4.1 ([1; p.9~). We shall in this note restate the theorem and give a complete proof. For convenience we shall repeat the necessary definitions and notations used in [1]. DEFINITION 1 • r-. in the sample space ~L <(· A family c~.J of probability distributions is said to belong to the class n·o if (i) there is a 1-1 correspondence between the elements (ii) (iii) (iv) (v) (vi) Jr.-.., ru; of ~- and the points of ~ , an open set in a k-dimensional Euclidean space; t7\ elements of "--( .... ~ corresponding to distinct points in ~~ are distinct; there is a non-negative function p on il x dil and a measure )-1 responding to . :"\ ln _ .. -". 7' such that the element of 0'.) cor-has the density function with respect to l/ I / for each X E f.2. the function p (x' • ) is continuously differentiable in !:@; the differentiations ·• d ( i p(• g)dt• * 1 ' k 0 g~( = . . . ' .._/ ' /"'"'\ .. 0... / can be taken under the integral sign; the integrals (r.~ y 1 ( u ln E p(-• 'g)~ 'ot = 1 ' k ~ d g* . . . ' l. I are finite for all gc: ® -2--. (vii) for each G E @: the matrix o ln p p(. ,G)d;~ a G..\ I il' . I, :,.,, '),{ ' /) = 1 k 't ' •.•.• ' H is non-singular. 0r DEFINITION 2. The family of distributions v- = l_ p{ · , G) d 1t-t , I G E- @) E lT 0 is said to belong to the class TI 1 if there ,is a finite set of s ~ k f1 -measurable, real-valued functions on _fl \ ¢~ = ( 4> 1 , ••• , cPs) 1 such that (i) the moments Ai (9) ~ l (j)ip{ • ,9)df' and a .. (G) = j' ( ¢ . - A. (G)) ( ¢ . - AJ. ( G ~p ( • , G) d,P. lJ ..... l l J ~L I are finite for all i, j = 1 , ••• , s and all G E ® ; and are twice continuously differentiable functions af G; (ii) the matrix I' 11 :1 i!a· .(G)jlt . . 1 il lJ lt l' J = ' ••• ' s is non-singular for each G €:. (:rl (iii) the matrix ' ,: I I '0 Ai 'I ,, i 1 ' I ?J Gy;. ll = ... ' s I ~ .I ot= 1 , k I • I ' is of rank k for each GE ®; (iv) the differentiations '(j J cp i p ( • ' g ') 'Y~ i = 1 ' ... ' s -cg ~ 1 ' k ~ ..:,")_ = ... ' can be taken under the sign of integration (v) if j' s denotes an s-dimensional Euclidean space, then the mapping A, of ® into J'S defined by is homeomorphic. -.3-The above definitions reproduces the conditions of [1], how-ever, not every regularity assumption stated in the definitions are needed for the proof below (e,g. we do not have to assume any regularity conditions on the probability function p(• ,G) except for the existence of a set of functions cP such that their first moments A(G) are finite and satisfy Definition 2 (iii) and (v)). Following Barankin and Gurland we shall need the following definitions. DEFINITION 3. E and F are two spaces. An (E,F)-separator: D, is a real-valued function on E x F, having the properties (i) for each point e E: E, inf D(e,f)) -oo, and fEF (ii) this infimum is attained in F. DEFINITION 4. A ( r s, @ )-separator, D, is said to be ( ~, cP) -regular if (i) for each Gc ® and each G' ={= G we have D(A(9) ,G 1 )) D(A(G) ,I (ii) (iii) for each g 1 E ® , there is a neighborhood NA( g, )C I' s A ( G 1 ) , and a neighborhood Ng' C ® of Q 1 , such that in NA(G 1 ) x Ng 1 , the partial derivatives tR ' ,, = 1 ' ••• , .k are continuous a 2D(z,G) 0 zi aG~ i = 1, ••• , s for each 6 1 £ QP , there exist spherical neighborhoods of N~(G') C NA(G') , of A(G 1 ) and Ng~ C Ng 1 of G' such that ,~ in i 1~2 I. u .D(z,G) (.lQ ~G-1 ~ ~ '-" /l ! . *,;·\ = 1, ••• ,k and for each inf D(z,G) ~ inf D(z,G) ~ E N~l G£@ -Ngl =!= 0 -4-. (iv) there exists at least one Borel-measurable function H0 on fl s. · to @ such that D ( z , H0 ( z ) ) = inf D ( z , G ) , z 6 f' s G~@ We shall also need the tubular neighborhood theorem (see [2; 5.5 Theorem!) which essentially states: ...1 Let f be a homeomorhic continuousl~ differentiable mapping from an open set M to r m. The Jacobian of f .has maximal. rank in every point of M. Then there is a neighborhood W of f(M) and a continuously differentiable mapping r of W to f(M) such that r ( y) = y for every y £ f ( M) • We now state Theorem 4.1 of [1] in a slightly changed form: THEOREM. Let D be a ( S:), ¢)-regular ( r s, @ )-separator. Then there is an open neighborhood S of R and a mapping G of S to ® having the following properties: (a) G is continuousl~ differBntiable on s (b) G is the inverse maE:Qing of A on R (c) inf GE ill' D(z,G) = D(z,G(z)) when z E S. Finally, G is the unique ma:J2J2ing with the :12roperties (a), (b) and (c) RE1UlliK. Where· Barankin and Gurland are sound we sketch the argument for completeness and convenience. Where Barankin and Gurland are incomplete we shall be quite explicit, hoping that we are not having any gap. PROOF. 1) Local inverse. Let G' be a fixed but arbitrary point in @ • For every z E NA( G,) there is a G0E N~, such that inf D(z,G) = D(z,G0 ) (Definition 4 (iii)). Define then GE@ -5-Gg 1 is shown to be unique because of the existence of continuous derivatives (ii) and because on No No A( Q I ) X g!• , ::::;= 0 I Then Definition 4 (i) gives Gg,(A(G'))= G' • At last, the implicit function theorem might be used to prove that Gg, is continuously differentiable on N~(Q')" Summing up~ we have for every G' E Of; found a mapping Gg, which locally has the properties (a), (b), (c) and is unique. 2) Global inverse. We shall here be more explicit. One easily verifies that the mapping A (Definition 2 (v)) has the propertie~ of the mapping f in the tubular neighborhood theorem. Thus there is a neighborhood S0 of R = A( 4P ) and a continuously differentiable mapping r of S0 to R, such that r(z) = z for every z E R. Let for every Q the neighborhood N~ be replaced by a reduced neighborhood N~ with radius equal to 1/3 of the radius of Ng. The neighborhoods N~ have the property that: implies is either contained in or 0 NQ" The neighborhoods N~(Q) will be replaced by neighborhoods constructed in the following way: (i) for every Q E @ , let N~(g)C N~(G) be a neighborhood of A(Q) such that Gg maps Gg is continuous.) " ' NA(Q) into Ng • (This is possible since {ii) set 11 J . ll Nl(Q) = NA(Q) (1 l z e: S0 ) r(z) E. NA(Q) n One may verify that NA(Q) really are that (i) also applies for N~(G) as well as for neighborhoods, and ll NA(G). The neighborhoods NA(Q) now have the nice property that: Nl(G'{l N~(G") 9= 0 implies that there is a QE ® such that I (\ I A ( Q ) f_ N, ( {) I '\ I I N_A( Q " ) • -6-I ("' I ( ) 1 I For, if z E NA(G 1 )' t NA(Gn), then r z E NA(G 1 ) n NA(G') and there is a G E ® such that A(G) = r(z). Define S = U Nl(G) Gc® . We are going to show that if I n I () () z E. N A( G 1 ) 1 NA( G") then GG 1 z = GG" z • Let zENJi.(G 1 )nNl(G")' the~ there is a G0 E ® such that-· ( 0)- I () , 1 ( ( 0)) ( ( 0)) I n'lll A G t:- NA(G') NA(G") . Hence, G0 = GG 1 A G = GG" A G E NG 1 ' NG" because of property ( i) of the neighborhoods Nl( G). Thus N~ InN~" =)= 0 I I 0 and NG 1 U NG" is contained in for example NG'' Therefore, since both GG,(z) and GG 11 (z) minimize D(z,G) and both lie in N~, , they have to b'e equal because of the locally uniqueness of GG. Define G to be G(z) = GG(z) for every ZE: N~(G) G is a mapping of S to aD and G has the properties (a), (b) and (c) in the theorem. It remains to prJve that 3) G is the unique mapping wiht these properties. .Assume that also G1 has the properties (a), (b) and (c), but that there is a point z 0 E S of the locally such that G(z 0 ) =I= G1 (z 0 ). 1 Let z 0 tS Nl(Go). Because uniqueness of G, G1 (z 0 ) ¢ NG • Let C be the straight I 0 line between z 0 and A( G 0 ), then C c NA( G . ) • Since G and G 1 are 0 continuous and N~ is open, there is a point G1 £N~ on G1 (C) (the image of C by G9) which is not on G(C). But then there must also exist a point G"EN~. on G(C) such that G1 and G11 are the images I 0 of the same Z 1 t::.N1~(Go)• But this is imposs;Lble, since D(z 1 ,G) cannot I II pave two minimizing points in NG • I! ·• 0 We have in part 2) of the proof explicitly constructed the I neighborhoods NA(G) by means of the tubulnr neighborhood theorem. In Barankin and Gurland it is not evident how they construct the neighbor-hoods such that they have the property: .;:7-I I rA I I _NA(G')n NA(G") =J= VJ implies that RrJ(NA(G')UNA(G")) is connected. If two neighborhoods NA(G') and NA(G") intersect without the intersection having common points with R, it is not difficult to replace these neighborhoods by others and having an empty intersection as might be seen from the following picture: ,. R \ .,..,--, '\ / ' I I \ I ---+--. __ ._ __ __~___ -- ---j------\A(G") / I "- '-- _../ / ......___ ___ _/ / The difficulty arises when given an A(G') there is an infinite number of points A(G") such that NA(G') (1 NA(G") =J= ¢ but NA( G, /1 NA( G" /! R == rj) If one carries through a similar reduction of the neighborhoods as above for each A(G") , the reduced NA(G') may in the end turn out to be empty! REFERENCES [1] E.W. Barankin and J. Gurland (1951): "On asymptotically normal, efficient estimators: I", University of California Publications in Statistics, Vol.1, No.6, pp.89-130. [2J J.R. Munkres (1963): "Elementary differential topology", Annals of Mathematical Studies, Number 54, Princeton University Press. **** ;~1~~--}~ ** 

A NOTE ON A THEOREM OF BARANKIN AND GURLAND

Abstract

Similar works

Full text

Available Versions

NORA - Norwegian Open Research Archives