- Email: [email protected]
Sufficiency and ancillarity in characterization problems
Journal of Statistical Planning and Inference 102 (2002) 223–228
www.elsevier.com/locate/jspi
Su"ciency and ancillarity in characterization problems Abram Kagan ∗ Department of Mathematics, University of Maryland, College Park, MD 20742, USA Received 28 July 2000; received in revised form 13 December 2000; accepted 27 March 2001
Abstract It is shown that in some setups of interest, independence of two statistics is equivalent to su"ciency of one of them and ancillarity of the other for a properly constructed parametric family of distributions. This observation reveals the origin of some characterization results. c 2002 Elsevier Science B.V. All rights reserved. MSC: primary 62E10; secondary 62B05 Keywords: Ancillarity; Characterization; Su"ciency; Uniform distribution
1. Introduction This section contains a brief review of basic statistical concepts. For details see Lehmann (1986, Chapter 4). Let P = {P ; ∈ } be a family of probability distributions on a measurable space (X; A) parametrized by a general parameter : A subalgebra B ⊂ A is su#cient for P (or, as they say sometimes, for ) if for any A ∈ A there exists a statistic A such that P (A|B) =
A;
∈ :
(1)
A su"cient subalgebra B is complete if E ( ) = 0 ∀ ∈ ⇒ P { = 0} = 1
∀ ∈ :
(2)
A subalgebra C ⊂ A is similar for P if for any C ∈ C P (C) = const;
∈ :
(3)
We say that subalgebras A1 ; A2 are P-independent if for any A1 ∈ A1 ; A2 ∈ A2 P (A1 ∩ A2 ) = P (A1 )P (A2 );
∈ :
∗ Tel.: +1-301-405-5456; fax: +1-301-314-0827. E-mail address: [email protected] (A. Kagan).
c 2002 Elsevier Science B.V. All rights reserved. 0378-3758/02/$ - see front matter PII: S 0 3 7 8 - 3 7 5 8 ( 0 1 ) 0 0 1 0 9 - 4
224
A. Kagan / Journal of Statistical Planning and Inference 102 (2002) 223–228
For a statistic T; T : (X; A) → (T; F), we shall denote by (T ) the subalgebra of A generated by T , i.e., A ∈ (T ) ⇔ A = T −1 F
for an F ∈ F:
We say that P is linked if for any pair ; ∈ there exists a sequence 0 ; 1 ; : : : ; n ; n+1 of points in with 0 = ; n+1 = such that Pj ; Pj+1 are not mutually singular, i.e., Pj (A) = 1 ⇒ Pj+1 (A) ¿ 0;
j = 0; 1; : : : ; n:
A well known theorem due to Basu (1955, 1958) claims that if B is a su#cient subalgebra for a linked family P then a subalgebra C which is P-independent of B; is similar. The linkage condition of P may not be omitted as the following example demonstrates. Take a bivariate random vector (X; Y ) with P {(X; Y ) = (; )} = 1: The subalgebra B = (X ) is su"cient for P = {P ; ∈ R} since knowing X means knowing . The subalgebra C = (Y ) is P-independent of B while P (Y = y) = 1 if y = and =0 if y = so that C is not similar. If B is a complete su"cient subalgebra then any similar subalgebra C is P-independent of B (this result does not require the linkage condition). The converse of Basu’s theorem is false, in general. A subalgebra B which is P-independent of a similar subalgebra C may not be su"cient, as simple examples show. However, if a subalgebra which is P-independent of a similar subalgebra is big enough, it must be su"cient. To state the result, let A1 ∨ A2 be the smallest subalgebra of A that contains both A1 and A2 . Lemma 1. A subalgebra B which (i) is P-independent of a similar subalgebra C and (ii) complements C to the whole -algebra A (i.e.; B ∨ C = A); is su#cient. Proof. See Kagan (1966) or Barra (1971). Fisher called a similar subalgebra C ancillary for P if it has a P-independent complement. It is likely that Fisher knew that the complement must be su"cient for P. If this is the case, Lemma 1 has been known since long ago. Basu’s theorem can be used for proving independence of some statistics; see Ferguson (1962) for this matter. Similarly, Lemma 1 can be used for characterization of distributions by independence of special statistics. The general scheme is as follows. Suppose that two statistics, T1 = T1 (X ); T2 = T2 (X ) are independent when the distribution of a random element X is P. We associate with P a family P = {P ; ∈ } in such a way that (i) T1 and T2 remain independent when the distribution of X is P , i.e., T1 and T2 are P-independent, (ii) one of the statistics, say T2 is similar for P, and (iii) (T1 ) ∨ (T2 ) = (X ), i.e., T2 is ancillary. Then, by virtue of Lemma 1, T1 is su"cient for P and this opens the door for applying techniques of su"ciency for characterizing P.
A. Kagan / Journal of Statistical Planning and Inference 102 (2002) 223–228
225
James (1979, Lemma 4.1) proved a special case of Lemma 1 in connection to a characterization of the distribution of independent positive random variables X1 ; : : : ; Xn by independence of the vector (X2 ; =X1 ; : : : ; Xn =X1 ) and a homogeneous statistic G(X1 ; : : : ; Xn ). For related earlier results see Ferguson (1964, 1967), Klebanov (1973). An expository paper Gather (1996) contains recent references. In the next section, we illustrate (Theorem 1) the method based on Lemma 1 in a setup involving the maximum and minimum order statistics and the vector of Studentized residuals. A referee noticed that the statement of Theorem 1 is some kind of statistical folklore. It deals with su"ciency in the so called nonregular case when the family under study is dominated but its members are not mutually absolutely continuous. Functional equations arising in the theory of su"cient statistics in independent samples have been studied since long ago, mainly in the regular case; see Darmois (1935), Koopman (1936), Pitman (1936), Dynkin (1951), and a monograph Linnik (1968). The regularity conditions leading to exponential families were not always explicitly and correctly stated as noticed in Brown (1964). The nonregular case is more complicated since minimal su"cient statistics may include, besides summational statistics appearing in the regular case, order and, possibly, other statistics. Theorem 1 explicitly states regularity conditions characterizing the family of uniform distributions, within the class of families with location and scale parameters, by su"ciency of the maximum and minimum order statistics. The method based on Lemma 1 has limitations. The most serious is that it requires the complementary character of independent statistics and does not work, for example, in studying independence of the sample mean and variance. As shown in Pe˜na et al. (1992), there is a class of families not admitting ancillary statistics.
2. Characterization Theorem that follows is an illustration of approaching to characterization via su"ciency. Theorem 1. Let X =(X1 ; : : : ; Xn ) be a sample of size n ¿ 3 from a population with an absolutely continuous distribution function F(x). Assume the density f(x) continuous and positive in an interval (a; b) for some a; b; −∞ ¡ a ¡ b ¡ + ∞ and zero for outside of the interval (a; b). If Xn : 1 ¡ Xn : 2 ¡ · · · ¡ Xn : n are the order statistics of the sample; then the pair T1 = (Xn : 1 ; Xn : n ) of the extreme statistics is independent of the vector T2 = (Xn : 2 − Xn : 1 )=(Xn : n − Xn : 1 ); : : : ; (Xn : n−1 − Xn : 1 )=(Xn : n − Xn : 1 ) (equivalent to the vector of the Studentized residuals) i6 the population is uniform; i.e.; f(x) =
1 ; b−a
a ¡ x ¡ b; =0 elsewhere:
(4)
226
A. Kagan / Journal of Statistical Planning and Inference 102 (2002) 223–228
There is no f(x) continuous and positive on (−∞; a) nor on (b; ∞) nor on (−∞; +∞) for which T1 and T2 are independent. Proof. Introduce a location=scale parameter family P = {P ; ∈ }; = ("; ) ∈ = R × R+ , P (A) = −n f((x1 − ")= ) : : : f((x n − ")= ) d x1 : : : d x n : (5) A
Lemma 2. Independence of T1 ; T2 when " = 0; = 1 implies their P-independence. Proof. One has for A1 ; A2 from the -algebras of Borel sets in R2 and Rn−1 , respectively, P"; (T1 ∈ A1 ; T2 ∈ A2 ) = P0; 1 (T1 ∈ A1 + "; T2 ∈ A2 ) = P0; 1 (T1 ∈ A1 + ")P0; 1 (T2 ∈ A2 ) = P"; (T1 ∈ A1 )P"; (T2 ∈ A2 ): Su"ce to consider the case of a = 0; b = 1: The general case reduces to this by introducing new random variables Xj = (Xj − a)=b; j = 1; : : : ; n. From P {t ¡ Xn : 1 ¡ Xn : n ¡ s} = {F((s − ")= ) − F((t − ")= )}n ;
"¡t ¡s¡" +
one gets the density function g(t; s; ) of T1 , g(t; s; ) = n(n − 1) −2 f((t − ")= )f((s − ")= ) ×{F((s − ")= ) − F((t − ")= )}n−2 ;
(6)
if " ¡ t ¡ s ¡ " + , and g(t; s; ) = 0 elsewhere. Due to su"ciency of T1 for P, the following decomposition holds for " ¡ min(x1 ; : : : ; x n ) = t ¡ s = max(x1 ; : : : ; x n ) ¡ " + : n −n f((xj − ")= ) = n(n − 1) −2 f((t − ")= )f((s − ")= ) 1
×{F((s − ")= ) − F((t − ")= )}n−2 h(x1 ; : : : ; x n )
so that for 0 ¡ min(x1 ; : : : ; x n ) = t ¡ s = max(x1 ; : : : ; x n ) ¡ 1 n f(xj ) = n(n − 1) n−2 f(t)f(s) {F(s) − F(t)}n−2 h( x1 + "; : : : ; x n + "): 1
(7)
Fixing t; s; 0 ¡ t ¡ s ¡ 1 and choosing in (7) x1 = · · · = x n−1 = t; results in {f(t)}n−2 = n(n − 1)
x n = s;
" = −t=(s − t);
F(s) − F(t) s−t
= 1=(s − t)
n−2 h(0; : : : ; 0; 1):
(8)
A. Kagan / Journal of Statistical Planning and Inference 102 (2002) 223–228
227
Letting t → 0+ one gets from (8) F(s) = s;
0¡s¡1
implying (4). Notice that due to the assumed continuity and positivity of f(x) in (a; b), h(0; 0; : : : ; 0; 1) ¿ 0: Now, from (8) follows that if limx→a+ = 0 there is no F such that T1 ; T2 are independent. Also, if f(x) is continuous and positive on (−∞; a), then F must be linear on an unbounded interval and this is impossible for a distribution function. This proves the necessity part of Theorem. To prove su"ciency, note that if (4) holds, T1 is a su"cient statistic for the family P of distributions (5), the latter being the family of all uniform distributions on ("1 ; "2 ); "1 ¡ "2 : Actually, T1 (a two-dimensional statistic) is a complete su"cient statistic for P (a family parametrized by a two-dimensional parameter). Here is a proof of this known fact. Let (t; s) be a non-negative function; one may assume (t; s) = 0 for t ¿ s. For any "1 ; "2 ; "1 ¡ "2 choose " and in such a way that " = "1 , " + = "2 : Let 1{t6s} denote the indicator of the set {(t; s): t 6 s}: The condition E { (Xn : 1 ; Xn : n )} = 0
for all ∈ R × R+
is equivalent to that for all "1 ; "2 ; "1 ¡ "2 "2 "2 (t; s) (s − t)n−2 1{t6s} dt ds = 0: "1
"1
(9)
Since mes{(t; s): (s − t)1{t6s} = 0} = 0, (9) shows that mes{(t; s): (t; s) = 0} = 0 and, thus, P { (Xn : 1 ; Xn : n ) = 0} = 1;
∀ ∈ R × R+ :
The statistic T2 , being similar for P, is independent of T1 . Acknowledgements I am grateful to two anonymous referees for their constructive comments. References Barra, J.R., 1971. Fondamentales de Statistique MathKematique (English translation (1981): Mathematical Basis of Statistics. Academic Press, New York) Dunod, Paris. Basu, D., 1955. On statistics independent of a complete su"cient statistic. Sankhya 15, 377–380. Basu, D., 1958. On statistics independent of a su"cient statistic. Sankhya 20, 223–226. Brown, L., 1964. Su"cient statistics in the case of independent random variables. Ann. Math. Statist. 35, 1456–1474. Darmois, G., 1935. Sur les lois de probabilitKe a estimation exhaustive. C. R. Acad. Sci. Paris 260, 1265–1266.
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A. Kagan / Journal of Statistical Planning and Inference 102 (2002) 223–228
Dynkin, E.B., 1951. Necessary and su"cient statistics for a family of probability distributions. Uspehi Mathem. Nauk, VI, 66 –90 (English translation (1961): Select. Trans. Math. Statist. Prob. 1 23– 41) (in Russian). Ferguson, T.S., 1962. Location and scale parameters in exponential families of distributions. Ann. Math. Statist. 33, 986–1001. Ferguson, T.S., 1964. A characterization of the exponential distribution. Ann. Math. Statist. 35, 1199–1207. Ferguson, T.S., 1967. On characterizing distributions by properties of order statistics. Sankhya A29, 265–277. Gather, U., 1996. Characterizing distributions by properties of order statistics—a partial review. In: Nagaraja, H.N., Sen, P.K., Morrison, D.F. (Eds.), Statistical Theory and Applications. Springer, Berlin, pp. 89–103. James, I.R., 1979. Characterization of a family of distributions by the independence of size and shape variables. Ann. Statist. 7, 869–881. Kagan, A.M., 1966. Two remarks on characterization of su"ciency. In: Sirazhdinov, S.Kh. (Ed.), Limit Theorems and Statistical Inference. Izdat. Fan, Tashkent, pp. 60 – 66 (in Russian). Klebanov, L.B., 1973. On the characterization of a family of distributions by the property of independence of statistics. Theory Prob. Appl. 18, 608–611. Koopman, B.O., 1936. On distributions admitting a su"cient statistic. Trans. Amer. Math. Soc. 39, 399–409. Lehmann, E.L., 1986. Testing Statistical Hypotheses., 2nd Edition. Wiley, New York. Linnik, Yu.V., 1968. Statistical Problems with Nuisance Parameters. Amer. Math. Soc. (Translation Series), Vol. 20. American Mathematical Society, Providence, RI. Pe˜na, E.A., Rohatgi, V.K., SzKekely, G.J., 1992. On the non-existence of ancillary statistics. Statist. Probab. Lett. 15, 357–360. Pitman, E.J.G., 1936. Su"cient statistics and intrinsic accuracy. Proc. Camb. Phil. Soc. 32, 567–579.
www.elsevier.com/locate/jspi
Su"ciency and ancillarity in characterization problems Abram Kagan ∗ Department of Mathematics, University of Maryland, College Park, MD 20742, USA Received 28 July 2000; received in revised form 13 December 2000; accepted 27 March 2001
Abstract It is shown that in some setups of interest, independence of two statistics is equivalent to su"ciency of one of them and ancillarity of the other for a properly constructed parametric family of distributions. This observation reveals the origin of some characterization results. c 2002 Elsevier Science B.V. All rights reserved. MSC: primary 62E10; secondary 62B05 Keywords: Ancillarity; Characterization; Su"ciency; Uniform distribution
1. Introduction This section contains a brief review of basic statistical concepts. For details see Lehmann (1986, Chapter 4). Let P = {P ; ∈ } be a family of probability distributions on a measurable space (X; A) parametrized by a general parameter : A subalgebra B ⊂ A is su#cient for P (or, as they say sometimes, for ) if for any A ∈ A there exists a statistic A such that P (A|B) =
A;
∈ :
(1)
A su"cient subalgebra B is complete if E ( ) = 0 ∀ ∈ ⇒ P { = 0} = 1
∀ ∈ :
(2)
A subalgebra C ⊂ A is similar for P if for any C ∈ C P (C) = const;
∈ :
(3)
We say that subalgebras A1 ; A2 are P-independent if for any A1 ∈ A1 ; A2 ∈ A2 P (A1 ∩ A2 ) = P (A1 )P (A2 );
∈ :
∗ Tel.: +1-301-405-5456; fax: +1-301-314-0827. E-mail address: [email protected] (A. Kagan).
c 2002 Elsevier Science B.V. All rights reserved. 0378-3758/02/$ - see front matter PII: S 0 3 7 8 - 3 7 5 8 ( 0 1 ) 0 0 1 0 9 - 4
224
A. Kagan / Journal of Statistical Planning and Inference 102 (2002) 223–228
For a statistic T; T : (X; A) → (T; F), we shall denote by (T ) the subalgebra of A generated by T , i.e., A ∈ (T ) ⇔ A = T −1 F
for an F ∈ F:
We say that P is linked if for any pair ; ∈ there exists a sequence 0 ; 1 ; : : : ; n ; n+1 of points in with 0 = ; n+1 = such that Pj ; Pj+1 are not mutually singular, i.e., Pj (A) = 1 ⇒ Pj+1 (A) ¿ 0;
j = 0; 1; : : : ; n:
A well known theorem due to Basu (1955, 1958) claims that if B is a su#cient subalgebra for a linked family P then a subalgebra C which is P-independent of B; is similar. The linkage condition of P may not be omitted as the following example demonstrates. Take a bivariate random vector (X; Y ) with P {(X; Y ) = (; )} = 1: The subalgebra B = (X ) is su"cient for P = {P ; ∈ R} since knowing X means knowing . The subalgebra C = (Y ) is P-independent of B while P (Y = y) = 1 if y = and =0 if y = so that C is not similar. If B is a complete su"cient subalgebra then any similar subalgebra C is P-independent of B (this result does not require the linkage condition). The converse of Basu’s theorem is false, in general. A subalgebra B which is P-independent of a similar subalgebra C may not be su"cient, as simple examples show. However, if a subalgebra which is P-independent of a similar subalgebra is big enough, it must be su"cient. To state the result, let A1 ∨ A2 be the smallest subalgebra of A that contains both A1 and A2 . Lemma 1. A subalgebra B which (i) is P-independent of a similar subalgebra C and (ii) complements C to the whole -algebra A (i.e.; B ∨ C = A); is su#cient. Proof. See Kagan (1966) or Barra (1971). Fisher called a similar subalgebra C ancillary for P if it has a P-independent complement. It is likely that Fisher knew that the complement must be su"cient for P. If this is the case, Lemma 1 has been known since long ago. Basu’s theorem can be used for proving independence of some statistics; see Ferguson (1962) for this matter. Similarly, Lemma 1 can be used for characterization of distributions by independence of special statistics. The general scheme is as follows. Suppose that two statistics, T1 = T1 (X ); T2 = T2 (X ) are independent when the distribution of a random element X is P. We associate with P a family P = {P ; ∈ } in such a way that (i) T1 and T2 remain independent when the distribution of X is P , i.e., T1 and T2 are P-independent, (ii) one of the statistics, say T2 is similar for P, and (iii) (T1 ) ∨ (T2 ) = (X ), i.e., T2 is ancillary. Then, by virtue of Lemma 1, T1 is su"cient for P and this opens the door for applying techniques of su"ciency for characterizing P.
A. Kagan / Journal of Statistical Planning and Inference 102 (2002) 223–228
225
James (1979, Lemma 4.1) proved a special case of Lemma 1 in connection to a characterization of the distribution of independent positive random variables X1 ; : : : ; Xn by independence of the vector (X2 ; =X1 ; : : : ; Xn =X1 ) and a homogeneous statistic G(X1 ; : : : ; Xn ). For related earlier results see Ferguson (1964, 1967), Klebanov (1973). An expository paper Gather (1996) contains recent references. In the next section, we illustrate (Theorem 1) the method based on Lemma 1 in a setup involving the maximum and minimum order statistics and the vector of Studentized residuals. A referee noticed that the statement of Theorem 1 is some kind of statistical folklore. It deals with su"ciency in the so called nonregular case when the family under study is dominated but its members are not mutually absolutely continuous. Functional equations arising in the theory of su"cient statistics in independent samples have been studied since long ago, mainly in the regular case; see Darmois (1935), Koopman (1936), Pitman (1936), Dynkin (1951), and a monograph Linnik (1968). The regularity conditions leading to exponential families were not always explicitly and correctly stated as noticed in Brown (1964). The nonregular case is more complicated since minimal su"cient statistics may include, besides summational statistics appearing in the regular case, order and, possibly, other statistics. Theorem 1 explicitly states regularity conditions characterizing the family of uniform distributions, within the class of families with location and scale parameters, by su"ciency of the maximum and minimum order statistics. The method based on Lemma 1 has limitations. The most serious is that it requires the complementary character of independent statistics and does not work, for example, in studying independence of the sample mean and variance. As shown in Pe˜na et al. (1992), there is a class of families not admitting ancillary statistics.
2. Characterization Theorem that follows is an illustration of approaching to characterization via su"ciency. Theorem 1. Let X =(X1 ; : : : ; Xn ) be a sample of size n ¿ 3 from a population with an absolutely continuous distribution function F(x). Assume the density f(x) continuous and positive in an interval (a; b) for some a; b; −∞ ¡ a ¡ b ¡ + ∞ and zero for outside of the interval (a; b). If Xn : 1 ¡ Xn : 2 ¡ · · · ¡ Xn : n are the order statistics of the sample; then the pair T1 = (Xn : 1 ; Xn : n ) of the extreme statistics is independent of the vector T2 = (Xn : 2 − Xn : 1 )=(Xn : n − Xn : 1 ); : : : ; (Xn : n−1 − Xn : 1 )=(Xn : n − Xn : 1 ) (equivalent to the vector of the Studentized residuals) i6 the population is uniform; i.e.; f(x) =
1 ; b−a
a ¡ x ¡ b; =0 elsewhere:
(4)
226
A. Kagan / Journal of Statistical Planning and Inference 102 (2002) 223–228
There is no f(x) continuous and positive on (−∞; a) nor on (b; ∞) nor on (−∞; +∞) for which T1 and T2 are independent. Proof. Introduce a location=scale parameter family P = {P ; ∈ }; = ("; ) ∈ = R × R+ , P (A) = −n f((x1 − ")= ) : : : f((x n − ")= ) d x1 : : : d x n : (5) A
Lemma 2. Independence of T1 ; T2 when " = 0; = 1 implies their P-independence. Proof. One has for A1 ; A2 from the -algebras of Borel sets in R2 and Rn−1 , respectively, P"; (T1 ∈ A1 ; T2 ∈ A2 ) = P0; 1 (T1 ∈ A1 + "; T2 ∈ A2 ) = P0; 1 (T1 ∈ A1 + ")P0; 1 (T2 ∈ A2 ) = P"; (T1 ∈ A1 )P"; (T2 ∈ A2 ): Su"ce to consider the case of a = 0; b = 1: The general case reduces to this by introducing new random variables Xj = (Xj − a)=b; j = 1; : : : ; n. From P {t ¡ Xn : 1 ¡ Xn : n ¡ s} = {F((s − ")= ) − F((t − ")= )}n ;
"¡t ¡s¡" +
one gets the density function g(t; s; ) of T1 , g(t; s; ) = n(n − 1) −2 f((t − ")= )f((s − ")= ) ×{F((s − ")= ) − F((t − ")= )}n−2 ;
(6)
if " ¡ t ¡ s ¡ " + , and g(t; s; ) = 0 elsewhere. Due to su"ciency of T1 for P, the following decomposition holds for " ¡ min(x1 ; : : : ; x n ) = t ¡ s = max(x1 ; : : : ; x n ) ¡ " + : n −n f((xj − ")= ) = n(n − 1) −2 f((t − ")= )f((s − ")= ) 1
×{F((s − ")= ) − F((t − ")= )}n−2 h(x1 ; : : : ; x n )
so that for 0 ¡ min(x1 ; : : : ; x n ) = t ¡ s = max(x1 ; : : : ; x n ) ¡ 1 n f(xj ) = n(n − 1) n−2 f(t)f(s) {F(s) − F(t)}n−2 h( x1 + "; : : : ; x n + "): 1
(7)
Fixing t; s; 0 ¡ t ¡ s ¡ 1 and choosing in (7) x1 = · · · = x n−1 = t; results in {f(t)}n−2 = n(n − 1)
x n = s;
" = −t=(s − t);
F(s) − F(t) s−t
= 1=(s − t)
n−2 h(0; : : : ; 0; 1):
(8)
A. Kagan / Journal of Statistical Planning and Inference 102 (2002) 223–228
227
Letting t → 0+ one gets from (8) F(s) = s;
0¡s¡1
implying (4). Notice that due to the assumed continuity and positivity of f(x) in (a; b), h(0; 0; : : : ; 0; 1) ¿ 0: Now, from (8) follows that if limx→a+ = 0 there is no F such that T1 ; T2 are independent. Also, if f(x) is continuous and positive on (−∞; a), then F must be linear on an unbounded interval and this is impossible for a distribution function. This proves the necessity part of Theorem. To prove su"ciency, note that if (4) holds, T1 is a su"cient statistic for the family P of distributions (5), the latter being the family of all uniform distributions on ("1 ; "2 ); "1 ¡ "2 : Actually, T1 (a two-dimensional statistic) is a complete su"cient statistic for P (a family parametrized by a two-dimensional parameter). Here is a proof of this known fact. Let (t; s) be a non-negative function; one may assume (t; s) = 0 for t ¿ s. For any "1 ; "2 ; "1 ¡ "2 choose " and in such a way that " = "1 , " + = "2 : Let 1{t6s} denote the indicator of the set {(t; s): t 6 s}: The condition E { (Xn : 1 ; Xn : n )} = 0
for all ∈ R × R+
is equivalent to that for all "1 ; "2 ; "1 ¡ "2 "2 "2 (t; s) (s − t)n−2 1{t6s} dt ds = 0: "1
"1
(9)
Since mes{(t; s): (s − t)1{t6s} = 0} = 0, (9) shows that mes{(t; s): (t; s) = 0} = 0 and, thus, P { (Xn : 1 ; Xn : n ) = 0} = 1;
∀ ∈ R × R+ :
The statistic T2 , being similar for P, is independent of T1 . Acknowledgements I am grateful to two anonymous referees for their constructive comments. References Barra, J.R., 1971. Fondamentales de Statistique MathKematique (English translation (1981): Mathematical Basis of Statistics. Academic Press, New York) Dunod, Paris. Basu, D., 1955. On statistics independent of a complete su"cient statistic. Sankhya 15, 377–380. Basu, D., 1958. On statistics independent of a su"cient statistic. Sankhya 20, 223–226. Brown, L., 1964. Su"cient statistics in the case of independent random variables. Ann. Math. Statist. 35, 1456–1474. Darmois, G., 1935. Sur les lois de probabilitKe a estimation exhaustive. C. R. Acad. Sci. Paris 260, 1265–1266.
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Dynkin, E.B., 1951. Necessary and su"cient statistics for a family of probability distributions. Uspehi Mathem. Nauk, VI, 66 –90 (English translation (1961): Select. Trans. Math. Statist. Prob. 1 23– 41) (in Russian). Ferguson, T.S., 1962. Location and scale parameters in exponential families of distributions. Ann. Math. Statist. 33, 986–1001. Ferguson, T.S., 1964. A characterization of the exponential distribution. Ann. Math. Statist. 35, 1199–1207. Ferguson, T.S., 1967. On characterizing distributions by properties of order statistics. Sankhya A29, 265–277. Gather, U., 1996. Characterizing distributions by properties of order statistics—a partial review. In: Nagaraja, H.N., Sen, P.K., Morrison, D.F. (Eds.), Statistical Theory and Applications. Springer, Berlin, pp. 89–103. James, I.R., 1979. Characterization of a family of distributions by the independence of size and shape variables. Ann. Statist. 7, 869–881. Kagan, A.M., 1966. Two remarks on characterization of su"ciency. In: Sirazhdinov, S.Kh. (Ed.), Limit Theorems and Statistical Inference. Izdat. Fan, Tashkent, pp. 60 – 66 (in Russian). Klebanov, L.B., 1973. On the characterization of a family of distributions by the property of independence of statistics. Theory Prob. Appl. 18, 608–611. Koopman, B.O., 1936. On distributions admitting a su"cient statistic. Trans. Amer. Math. Soc. 39, 399–409. Lehmann, E.L., 1986. Testing Statistical Hypotheses., 2nd Edition. Wiley, New York. Linnik, Yu.V., 1968. Statistical Problems with Nuisance Parameters. Amer. Math. Soc. (Translation Series), Vol. 20. American Mathematical Society, Providence, RI. Pe˜na, E.A., Rohatgi, V.K., SzKekely, G.J., 1992. On the non-existence of ancillary statistics. Statist. Probab. Lett. 15, 357–360. Pitman, E.J.G., 1936. Su"cient statistics and intrinsic accuracy. Proc. Camb. Phil. Soc. 32, 567–579.