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On fractional uniform order statistics
Statistics & Probability Letters 58 (2002) 93–96
On fractional uniform order statistics M.C. Jones ∗ Department of Statistics, The Open University, Walton Hall, Milton Keynes MK7 6AA, UK Received June 2001
Abstract The fractional uniform order statistics of Stigler [J. Amer. Statist. Assoc. 72 (1977) 544] are shown to correspond to a randomised version of the alternative fractional uniform order statistics de1ned by taking c 2002 Elsevier Science B.V. All rights reserved. convex combinations of consecutive uniform order statistics. Keywords: Beta distribution; Generalised arc-sine distribution; Linear combinations of order statistics
1. Introduction There are, it would seem, two principal de1nitions of “fractional order statistics” in the literature. The 1rst, most obvious and most ubiquitous would seem to be the de1nition of Xa : n , 1 ¡ a ¡ n, say, as a convex combination of Xj : n and Xj+1 : n , namely Xa : n ≡ Xj+c : n = (1 − c)Xj; n + cXj+1 : n :
(1)
Here, j = a denotes the largest integer less than or equal to a, c = a − j, and Xj : n is the jth order statistic from an independent and identically distributed sample of size n from distribution F. A quite diCerent de1nition of fractional order statistic is due to Stigler (1977). Let Xa : n = F −1 (Ua : n ) where Ua : n is a fractional order statistic associated with the uniform distribution (on [0; 1]) de1ned to be a quantity following the beta distribution (on [0; 1]) with parameters a and n + 1 − a, written Beta(a; n + 1 − a). (This de1nition is appropriate for 0 ¡ a ¡ n + 1.) Notice that this distribution is the appropriate one for an ordinary order statistic in the uniform case. With asymptotic arguments, Stigler (1977) shows that the distribution associated with his de1nition of fractional order statistics — which is admitted not to be a practical one — is a good approximation to the distribution of the corresponding convex combination type of fractional order statistic. ∗
Fax: +44-1908-652140. E-mail address: [email protected] (M.C. Jones).
c 2002 Elsevier Science B.V. All rights reserved. 0167-7152/02/$ - see front matter PII: S 0 1 6 7 - 7 1 5 2 ( 0 2 ) 0 0 1 1 9 - 0
94
M.C. Jones / Statistics & Probability Letters 58 (2002) 93–96
Hutson (1999) makes practical use of Stigler’s observation in providing high-quality nonparametric con1dence intervals for quantiles. The main result of this paper brings Stigler’s fractional order statistics a little closer to the convex combination type of fractional order statistics. It pertains to the uniform case, that is Ua : n ≡ Uj+c : n = (1 − C)Uj; n + CUj+1 : n ;
(2)
where C ∼ Beta(c; 1 − c);
0 ¡ c ¡ 1;
independent of the U ’s:
(We also de1ne U0 : n ≡ 0, Un+1 : n ≡ 1 and C ≡ 0 when c = 0.) This claim is proved in Section 2. The distribution of C is called the generalised arc-sine distribution. It is, perhaps, surprising that the appropriate distribution for C is a U-shaped distribution; this is the case since all generalised arc-sine distributions are uni-antimodal with antimode at 1 − c. More generally, Stigler (1977) introduced his fractional order statistics via a Dirichlet process de1nition, a major element of which is the speci1cation of the joint distribution of a set of M fractional order statistics. In the uniform case, and in our own notation, this joint density is M +1 (uk − uk −1 )ak −1 fa1 ; :::; aM +1 (u1 ; : : : ; uM ) = P(n + 1) (3) P(ak ) k=1
on 0 ≡ u0 ¡ u1 ¡ · · · ¡ uM ¡ uM +1 ≡ 1. The key to “fractionality” is that the parameters are arbitrary positive real numbers except for the constraint that a1 + · · · + aM +1 = n + 1. If M 6 n, (3) reduces to the joint distribution of M ordinary uniform order statistics Ui1 : n ; : : : ; UiM : n whenever aj = ij − ij−1 , j = 1; : : : ; M + 1, having set i0 = 0 and iM +1 = n + 1 (David, 1981, Section 2.2). We extend construction (2) to the joint distribution of several fractional uniform order statistics in Section 3. Some brief closing remarks are oCered in Section 4.
2. Proof of distribution of construction (2) It is trivially true that Ua : n ∼ Beta(a; n−a+1) for a=1; 2; : : : ; n. For 1 ¡ a ¡ n, a = 2; 3; : : : ; n−1, the joint distribution of Uj : n ; Uj+c : n and Uj+1 : n is fj; j+c; j+1 (u; v; w) = fj; j+1 (u; w)fj+c|j; j+1 (v|u; w) =
nu j−1 (1 − w)n−j−1 (v − u)c−1 (w − v)−c ; B(j; n − j) B(c; 1 − c)
(4)
0 ¡ u ¡ v ¡ w ¡ 1. The 1rst term in (4) is the joint distribution of consecutive order statistics (David, 1981, Section 2.2); the second term is the appropriately linearly translated Beta(c; 1 − c) density. Now, v u j−1 (v − u)c−1 du = B(j; c)v j+c−1 0
M.C. Jones / Statistics & Probability Letters 58 (2002) 93–96
and
v
1
95
(1 − w)n−j−1 (w − v)−c dw = B(n − j; 1 − c)(1 − v)n−j−c ;
so the marginal density of Ua : n is nB(j; c)B(n − j; 1 − c) j+c−1 v (1 − v)n−j−c = {B(a; n − a + 1)}−1 v a−1 (1 − v)n−a B(j; n − j)B(c; 1 − c) as required. Appropriate modi1cations of the above argument can be made to show that the claim is true for 0 ¡ a ¡ 1 and n ¡ a ¡ n + 1 also.
3. Joint distribution Wherever two or more fractional order statistics, Uj+c1 ; n ; : : : ; Uj+cm ; n are required that all fall between Uj and Uj+1 , i.e. 0 ≡ c0 ¡ c1 ¡ · · · ¡ cm ¡ cm+1 ≡ 1, write Uai : n ≡ Uj+ci : n = (1 − Ci )Uj; n + Ci Uj+1 : n ;
i = 1; : : : ; m;
(5)
but now de1ne the joint distribution of C1 ; : : : ; Cm by the generalised arc-sine version of the joint distribution given at (3), i.e. (3) with ak = ck − ck −1 , k = 1; : : : ; m + 1 and n = 1. Then, linearly transforming this density to its equivalent conditional version on (uj : n ; uj+1 : n ) yields the following very similar form fj+c1 ; ::: ; j+cm (u1 ; : : : ; um ) =
m+1 k=1
(uk − uk −1 )ck −ck −1 −1 P(ck − ck −1 )
;
(6)
uj : n ≡ u0 ¡u1 ¡· · ·¡um ¡um+1 ≡ uj+1 : n . This retains its simplicity because (um+1 −u0 ) (ck −ck −1 −1) =1. With this de1nition for the joint distribution of two or more random fractional uniform order statistics with indices between consecutive order statistics, we can proceed to the full joint distribution of any number of random fractional uniform order statistics. Let 0 ≡ c‘; 0 ¡ c‘; 1 ¡ · · · ¡ c‘; m‘ ¡ c‘; m‘ +1 ≡ 1, ‘ = 0; : : : ; n + 1. The joint density of the n + n+1 ‘=0 m‘ order statistics Uc0; 1 ; n ; : : : ; Uc0; m0 : n ; U1 : n ; U1+c1; 1 ; n ; : : : ; U1+c1; m1 : n ; U2 : n ; : : : ; Un : n ; Un+cn+1; 1 ; n ; : : : ; Un+cn+1; mn+1 : n is readily calculated by multiplying the joint density of U1 : n ; :::; Un : n , which is the constant P(n + 1) on 0 ¡ u1 ¡ · · · ¡ um ¡ 1, by the conditional densities of the fractional order statistics. These, thanks to the Markov property of ordinary order statistics (David, 1981, Section 2.7), yield n m ‘ +1 (u‘+c‘; k − u‘+c‘; k −1 )c‘; k −c‘; k −1 −1 ; P(n + 1) P(c‘; k − c‘; k −1 )
(7)
‘=0 k=1
0 ≡ uc0; 0 ¡ uc0; 1 ¡ · · · ¡ uc0; m0 ¡ uc0; m0 +1 ≡ u1 ≡ u1+c1; 0 ¡ u1+c1; 1 ¡ · · · ¡ un−1+cn−1; mn−1 +1 ≡ un ≡ un+cn; 0 ¡ un+cn+1; 1 ¡ · · · ¡ un+cn+1; mn ¡ un+cn+1; mn +1 ≡ 1.
96
M.C. Jones / Statistics & Probability Letters 58 (2002) 93–96
Of course, there is no special role for U1 : n ; : : : ; Un : n in (7), and the same form holds for the density function without (any subset of) the ordinary order statistics too. The integration concerned with marginalising over any Ui : n (or indeed any Ua : n ) is exactly the same as in the ordinary order statistics case since the integration depends in no way on any of the powers being integer. We have thus established the desired product form for the joint distribution of random fractional uniform order statistics de1ned by (2) and (5). This holds whatever the manner in which the fractional order statistics are con1gured relative to the ordinary order statistics. 4. Closing remarks Stigler (1977) described his de1nition of fractional order statistics as a “purely technical creation”. The connection between two forms of fractional uniform order statistics established in this paper would make Stigler’s de1nition operational in the uniform case if practitioners were happy with the random element introduced via the generalised arc-sine distribution : : : which, in general, they will not be. An added diQculty in the nonuniform case is that one has to know F to transform to and from the uniform case, unless content to approximate further by employing an estimate of F. Perhaps Stigler’s fractional order statistic construction remains a largely technical creation. References David, H.A., 1981. Order Statistics, 2nd Edition. Wiley, New York. Hutson, A.D., 1999. Calculating nonparametric con1dence intervals for quantiles using fractional order statistics. J. Appl. Statist. 26, 343–353. Stigler, S.M., 1977. Fractional order statistics, with applications. J. Amer. Statist. Assoc. 72, 544–550.
On fractional uniform order statistics M.C. Jones ∗ Department of Statistics, The Open University, Walton Hall, Milton Keynes MK7 6AA, UK Received June 2001
Abstract The fractional uniform order statistics of Stigler [J. Amer. Statist. Assoc. 72 (1977) 544] are shown to correspond to a randomised version of the alternative fractional uniform order statistics de1ned by taking c 2002 Elsevier Science B.V. All rights reserved. convex combinations of consecutive uniform order statistics. Keywords: Beta distribution; Generalised arc-sine distribution; Linear combinations of order statistics
1. Introduction There are, it would seem, two principal de1nitions of “fractional order statistics” in the literature. The 1rst, most obvious and most ubiquitous would seem to be the de1nition of Xa : n , 1 ¡ a ¡ n, say, as a convex combination of Xj : n and Xj+1 : n , namely Xa : n ≡ Xj+c : n = (1 − c)Xj; n + cXj+1 : n :
(1)
Here, j = a denotes the largest integer less than or equal to a, c = a − j, and Xj : n is the jth order statistic from an independent and identically distributed sample of size n from distribution F. A quite diCerent de1nition of fractional order statistic is due to Stigler (1977). Let Xa : n = F −1 (Ua : n ) where Ua : n is a fractional order statistic associated with the uniform distribution (on [0; 1]) de1ned to be a quantity following the beta distribution (on [0; 1]) with parameters a and n + 1 − a, written Beta(a; n + 1 − a). (This de1nition is appropriate for 0 ¡ a ¡ n + 1.) Notice that this distribution is the appropriate one for an ordinary order statistic in the uniform case. With asymptotic arguments, Stigler (1977) shows that the distribution associated with his de1nition of fractional order statistics — which is admitted not to be a practical one — is a good approximation to the distribution of the corresponding convex combination type of fractional order statistic. ∗
Fax: +44-1908-652140. E-mail address: [email protected] (M.C. Jones).
c 2002 Elsevier Science B.V. All rights reserved. 0167-7152/02/$ - see front matter PII: S 0 1 6 7 - 7 1 5 2 ( 0 2 ) 0 0 1 1 9 - 0
94
M.C. Jones / Statistics & Probability Letters 58 (2002) 93–96
Hutson (1999) makes practical use of Stigler’s observation in providing high-quality nonparametric con1dence intervals for quantiles. The main result of this paper brings Stigler’s fractional order statistics a little closer to the convex combination type of fractional order statistics. It pertains to the uniform case, that is Ua : n ≡ Uj+c : n = (1 − C)Uj; n + CUj+1 : n ;
(2)
where C ∼ Beta(c; 1 − c);
0 ¡ c ¡ 1;
independent of the U ’s:
(We also de1ne U0 : n ≡ 0, Un+1 : n ≡ 1 and C ≡ 0 when c = 0.) This claim is proved in Section 2. The distribution of C is called the generalised arc-sine distribution. It is, perhaps, surprising that the appropriate distribution for C is a U-shaped distribution; this is the case since all generalised arc-sine distributions are uni-antimodal with antimode at 1 − c. More generally, Stigler (1977) introduced his fractional order statistics via a Dirichlet process de1nition, a major element of which is the speci1cation of the joint distribution of a set of M fractional order statistics. In the uniform case, and in our own notation, this joint density is M +1 (uk − uk −1 )ak −1 fa1 ; :::; aM +1 (u1 ; : : : ; uM ) = P(n + 1) (3) P(ak ) k=1
on 0 ≡ u0 ¡ u1 ¡ · · · ¡ uM ¡ uM +1 ≡ 1. The key to “fractionality” is that the parameters are arbitrary positive real numbers except for the constraint that a1 + · · · + aM +1 = n + 1. If M 6 n, (3) reduces to the joint distribution of M ordinary uniform order statistics Ui1 : n ; : : : ; UiM : n whenever aj = ij − ij−1 , j = 1; : : : ; M + 1, having set i0 = 0 and iM +1 = n + 1 (David, 1981, Section 2.2). We extend construction (2) to the joint distribution of several fractional uniform order statistics in Section 3. Some brief closing remarks are oCered in Section 4.
2. Proof of distribution of construction (2) It is trivially true that Ua : n ∼ Beta(a; n−a+1) for a=1; 2; : : : ; n. For 1 ¡ a ¡ n, a = 2; 3; : : : ; n−1, the joint distribution of Uj : n ; Uj+c : n and Uj+1 : n is fj; j+c; j+1 (u; v; w) = fj; j+1 (u; w)fj+c|j; j+1 (v|u; w) =
nu j−1 (1 − w)n−j−1 (v − u)c−1 (w − v)−c ; B(j; n − j) B(c; 1 − c)
(4)
0 ¡ u ¡ v ¡ w ¡ 1. The 1rst term in (4) is the joint distribution of consecutive order statistics (David, 1981, Section 2.2); the second term is the appropriately linearly translated Beta(c; 1 − c) density. Now, v u j−1 (v − u)c−1 du = B(j; c)v j+c−1 0
M.C. Jones / Statistics & Probability Letters 58 (2002) 93–96
and
v
1
95
(1 − w)n−j−1 (w − v)−c dw = B(n − j; 1 − c)(1 − v)n−j−c ;
so the marginal density of Ua : n is nB(j; c)B(n − j; 1 − c) j+c−1 v (1 − v)n−j−c = {B(a; n − a + 1)}−1 v a−1 (1 − v)n−a B(j; n − j)B(c; 1 − c) as required. Appropriate modi1cations of the above argument can be made to show that the claim is true for 0 ¡ a ¡ 1 and n ¡ a ¡ n + 1 also.
3. Joint distribution Wherever two or more fractional order statistics, Uj+c1 ; n ; : : : ; Uj+cm ; n are required that all fall between Uj and Uj+1 , i.e. 0 ≡ c0 ¡ c1 ¡ · · · ¡ cm ¡ cm+1 ≡ 1, write Uai : n ≡ Uj+ci : n = (1 − Ci )Uj; n + Ci Uj+1 : n ;
i = 1; : : : ; m;
(5)
but now de1ne the joint distribution of C1 ; : : : ; Cm by the generalised arc-sine version of the joint distribution given at (3), i.e. (3) with ak = ck − ck −1 , k = 1; : : : ; m + 1 and n = 1. Then, linearly transforming this density to its equivalent conditional version on (uj : n ; uj+1 : n ) yields the following very similar form fj+c1 ; ::: ; j+cm (u1 ; : : : ; um ) =
m+1 k=1
(uk − uk −1 )ck −ck −1 −1 P(ck − ck −1 )
;
(6)
uj : n ≡ u0 ¡u1 ¡· · ·¡um ¡um+1 ≡ uj+1 : n . This retains its simplicity because (um+1 −u0 ) (ck −ck −1 −1) =1. With this de1nition for the joint distribution of two or more random fractional uniform order statistics with indices between consecutive order statistics, we can proceed to the full joint distribution of any number of random fractional uniform order statistics. Let 0 ≡ c‘; 0 ¡ c‘; 1 ¡ · · · ¡ c‘; m‘ ¡ c‘; m‘ +1 ≡ 1, ‘ = 0; : : : ; n + 1. The joint density of the n + n+1 ‘=0 m‘ order statistics Uc0; 1 ; n ; : : : ; Uc0; m0 : n ; U1 : n ; U1+c1; 1 ; n ; : : : ; U1+c1; m1 : n ; U2 : n ; : : : ; Un : n ; Un+cn+1; 1 ; n ; : : : ; Un+cn+1; mn+1 : n is readily calculated by multiplying the joint density of U1 : n ; :::; Un : n , which is the constant P(n + 1) on 0 ¡ u1 ¡ · · · ¡ um ¡ 1, by the conditional densities of the fractional order statistics. These, thanks to the Markov property of ordinary order statistics (David, 1981, Section 2.7), yield n m ‘ +1 (u‘+c‘; k − u‘+c‘; k −1 )c‘; k −c‘; k −1 −1 ; P(n + 1) P(c‘; k − c‘; k −1 )
(7)
‘=0 k=1
0 ≡ uc0; 0 ¡ uc0; 1 ¡ · · · ¡ uc0; m0 ¡ uc0; m0 +1 ≡ u1 ≡ u1+c1; 0 ¡ u1+c1; 1 ¡ · · · ¡ un−1+cn−1; mn−1 +1 ≡ un ≡ un+cn; 0 ¡ un+cn+1; 1 ¡ · · · ¡ un+cn+1; mn ¡ un+cn+1; mn +1 ≡ 1.
96
M.C. Jones / Statistics & Probability Letters 58 (2002) 93–96
Of course, there is no special role for U1 : n ; : : : ; Un : n in (7), and the same form holds for the density function without (any subset of) the ordinary order statistics too. The integration concerned with marginalising over any Ui : n (or indeed any Ua : n ) is exactly the same as in the ordinary order statistics case since the integration depends in no way on any of the powers being integer. We have thus established the desired product form for the joint distribution of random fractional uniform order statistics de1ned by (2) and (5). This holds whatever the manner in which the fractional order statistics are con1gured relative to the ordinary order statistics. 4. Closing remarks Stigler (1977) described his de1nition of fractional order statistics as a “purely technical creation”. The connection between two forms of fractional uniform order statistics established in this paper would make Stigler’s de1nition operational in the uniform case if practitioners were happy with the random element introduced via the generalised arc-sine distribution : : : which, in general, they will not be. An added diQculty in the nonuniform case is that one has to know F to transform to and from the uniform case, unless content to approximate further by employing an estimate of F. Perhaps Stigler’s fractional order statistic construction remains a largely technical creation. References David, H.A., 1981. Order Statistics, 2nd Edition. Wiley, New York. Hutson, A.D., 1999. Calculating nonparametric con1dence intervals for quantiles using fractional order statistics. J. Appl. Statist. 26, 343–353. Stigler, S.M., 1977. Fractional order statistics, with applications. J. Amer. Statist. Assoc. 72, 544–550.