The Canadian Journal of Statistics Vol. 8. No. I. 1980, Pages 79-85 La Revue Canadienne de Statistique

79

y-confidence tolerance regions M. EVANS'

University of Toronto

Key w o r k and phrases: Tolerance regions, multivariate regression, regression with

AMS 1980 subject classifications: Primary 62F25, 62899. normal error.

ABSTRACT

Tolerance regions are discussed that have the property of covering an event of interest with prescribed confidence y. An exact expression is derived for the evaluation of confidence levels for tolerance regions associated with the multivariate regression model with normal error.

1. INTRODUCTION

The theory of tolerance regions is concerned with making inferences, based upon observations from a first random system, about events for a second random system. The inferences take the form of elements of the event space for the second system, together with a statistical property.

Primarily, two types of tolerance region have been considered, namely ,B-expecta- tion and Bantent tolerance regions. The theory of P-expectation tolerance regions, where we require the average probability content of the tolerance region to be at least P, is discussed for example, in Guttman (1970a) and Evans ( 1977). A discussion of p-content tolerance regions, where we require that the tolerance region have probability content at least /3 with some prescribed confidence, can be found in Guttman (1970% 1970b).

In an application, suppose the investigator has some event of interest and wants to select his tolerance region based upon criteria related to this event. Then a third type of tolerance region might be of interest. In this case we require that the tolerance region cover the event of interest with confidence y. Tolerance regions satisfying this property are discussed in this paper. This type of tolerance region has been considered previously by Wilks (1962), Owen (1964, 1965) and John (1968). In this paper we present an expression for the exact evaluation, via numerical integration, of such tolerance regions for the multivariate regression model with normal error.

2. TOLERANCE REGIONS

Suppose we have a statistical model Al for an observable response variable X from a random system and a statistical model A 2 for a concealed response value Y from a related random system. Let { P, : w E Q} and { Qu : w E Q} be the classes of probability measures in YUI and A*, respectively, and assume that the indexing set Q is the same and that the actual parameter values are the same. This establishes the connection between the two random systems.

Research supported in part by the Natural Sciences and Engineering Research Council Canada.

80 EVANS Vol. 8, No. 1

If ;X is the sample space in 41 and .43 is the event space in .& then a tolerance region T is a measurable mapping T: % - .43. For a &expectation tolerance region we require that

for every w E 0 and for a ,&content tolerance region at confidence level y we require that

Pw[Qw(T(X)) = P I = Y (2)

for every w E 52. A tolerance region T is usually associated with an event A ( w ) dependent upon the distribution: i.e., A : Q + a. Accordingly we would want to find T satisfying ( I ) or (2) and such that T is positioned in the sample space, perhaps in some optimal way, relative to A.

In this paper we ask that T possess a confidence property with respect to the event A.

DEFINITION. A tolerance region T is a y-confidence tolerance region for A if Pw[A(w) E T(X)I = y

for every w E Q. Clearly if QW(A(u)) L 8 for every w E Q then T is also a 8-content tolerance

region at confidence level y. Suppose that T is a y-confidence tolerance region for A ( w ) ; then C: 9” + P(Q)

given by C( X ) = ( w : A ( w ) c T( X ) ) is a y-confidence region for w. Further if C is a y-confidence region for w then T : X+ D given by T ( X ) = UwEc(x) A ( w ) is a y- confidence tolerance region for A(w) . If we call two exact y-confidence regions C1, Cz for w equivalent if C: 3 4 P(Q) defined by C ( X ) - C l ( X ) f l C Z ( X ) is an exact y-confidence region for w, and similarly for exact y-confidence tolerance regions for A(@), then we have the following result. (That the equivalence relation is well- defined is easily seen by noting that if CI and Cz are equivalent exact y-confidence regions for w then Pw(w E c l ( X ) n C l ( X ) ) = 0 for every w.)

LEMMA 1. The equivalence classes of exact y-confidence regions C for w, satisfving A ( 8 ) s A ( w ) implies 8 E C ( X ) are in 1-1 correspondence with the equivalence classes of exact y-confidence tolerance regions for A ( @ ) .

Proof. Suppose that TI and TZ are equivalent exact y-confidence tolerance regions for A ( o ) and CI, C2 are the corresponding exact y-confidence regions as defined above. Then w E Cl( X ) n CZ( X ) if and only if A( 0) c TI ( X ) i l Tz( X ) and thus CI and Cz are equivalent. Similarly if CI and Cz are equivalent exact y-confidence regions for w then TI and Tz, derived from CI and CZ resptctively are equivalent exact y-confidence tolerance regions for A ( w ) . Finally we note that if C ( X ) = (a: A ( w ) 5; T ( X ) ) and T ’ ( X ) = UwEc(x) A(o) , then T and T’ are equivalent exact y- confidence tolerance regions for A ( w ) and this establishes the lemma. Q.E.D.

This correspondence can sometimes be useful in the construction of y-confidence tolerance regions.

U wEC(X)

1980 yCONFlDENCE TOLERANCE REGIONS 81

3. NORMAL THEORY

In this section we consider the construction of y-confidence tolerance regions for the multivariate regression model with normal error. In the first system with observ- able variable X we take the model described by

X = RV + E,

where E - .Npxn(O; 1 1 8 Z), V = ( v l vectors and

- - v,) is an r x n full-rank matrix of input

Q = (( R, Z) : R E RPXr, Z is p X p positive definite} . For the second system we take the model described by

y = Rw + e,

where e - .,.Cr,(O; C) and w E R‘ is a possibly new input vector. For the multivariate normal distribution the events of primary interest are those

whose boundaries are the ellipsoidal contours of the density function; namely for probability content /3 the event

A,(Bw, Z) = ( y : ( y - Rw)’C-‘(y - Rw) 5 X;(l - P ) } .

We will restrict ourselves, on purely intuitive grounds, to tolerance regions of the form

T(X) = T(Bxw, Sx) (y: (y - Bxw)’Sxl(y - BXW) 5 kP}

where Bx = XV’(VV’)-’ is the matrix of regression coefficients, SX = (X - BxV)(X - BxV)’ is the inner product matrix of residual vectors; we will then determine k so as to obtain y-confidence for A ( w ) . Our actual calculations, however, will produce the confidence y as a function of k. For a discussion of the use of this form of tolerance region in the context of p-content tolerance regions see Guttman (1970a).

For convenience, in the algebraic manipulations to come, we introduce a positive affine group Y acting on RpXn x RP. The group is given by 48 - {[B, C] : B E Rpxr, C isp x p positive definite and [BI, CI][BP, CP] = [BI + CIBZ, CICP], [B, C]-’ = [-C-IB, C-’1 and i = [O, I]. The action of 48 on Rpxn X RP is given by [B, C].(X, y ) = (BV + CX, Bw + Cy). This induces the action [B, C].(R, Z) = (B + CR, CZC’) on the parameter space Q.

We make use of the following factorization of a p x p symmetric positive definite matrix S; namely S = QDQ’, where D = diag(d:, . . . , d j ) is the matrix of ordered eigenvalues of S (df 2 &+I ) and Q is p X p orthogonal. We obtain a positive square root factorization of S by writing S* = QD’Q’, where D* = diag(d1, . . . , d,) and d, is the positive square root of df. We note that the tolerance region T can be written as T(X) = T(Bxw, S X ) = [Bx, S$-]T(O, I) where T(0, I) = {y:y’y I k’}. Further we can write Aa(Rw, Z) = (6, Z*]AB(O, I) where AB(O, I) = {y :y’y = x i ( 1 - P ) ) . Now putting

[B, S’] = [6,Z*]I-1[Bx, S i ] = [Z-’(Bx - R), Z - ’ S i ]

82 EVANS Vol. 8, No. 1

we see that Aa(6w, Z) 5; T(X) if and only if

Aa(O, I) E [B, S']T(O, I) = Bw + S'T(0, I) = Bw + QD'Q'T(0, I) - Q(Q'Bw + Di T(0, I)) - Q(b + D'T(0, I)),

where b - Q'Bw and we have used the fact that a sphere maps 1-1 onto itself under an orthogonal transformation. Using this fact again we have that A,, is covered by T if and only if

(1)

Applying the orthogonal transformation P = -I to both sides of (1) we have that A,, is covered by T if and only if

Ap(0, I) Z; b + D'T(0, I).

1) E T(0, D); (2)

i.e., if and only if the ball with centre b and radius ko = is contained in the region bounded by the ellipsoid with centre 0 and principal semi-axes of lengths dl k, . . . , d,k along the first through pth coordinate axes of Rp respectively.

The confidence y is the probability that (2) holds and is obtained by the probability integral over ((b, D):AB(b, I) G T(0, D)}. It is clear that if (2) holds then A,,(O, I) s T(0, D) and thus ko s d,k. For fixed D with ko 5 d,k we then let I(D) be the set of values b for which (2) holds. As a preliminary step we obtain the probability content P(D) of I(D).

To obtain the region I(D) we effectively roll a sphere of radius ko on the insides of the ellipsoids dT(0, s2D) where s E (ko/d,k, 11. The union of all the surfaces traced by the centre of the sphere as it rolls gives the region I(D). We start with the centre of the sphere at (sd,k - ko)e,, where e, = (0, . . . , 0, I)', and roll on the inside of dT(0, s2D) along the ellipses formed by cutting dT(0, s'D) with 2-dimensional planes containing the %,-axis. Due to the symmetry of T(0, D) we need only calculate the probability content of I+(D) = I(D) tl (R+) , and I+(D) is obtained by rolling the sphere on the insides of d'T(0, s2D) = U ( 0 , s'D) f l ( R+), for s E (ko/d,k, I]. In general, when rolling on a given ellipse, the sphere will jam; i.e., touch the ellipsoid at more than one point, before we have completed rolling along the portion of the ellipse in ( I?+),. Thus the sphere is rolled until it jams. We show that the sphere with centre at (sd,k - ko)e, is contained in T(0, s2D) for s E (ko/d,k, 11 and calculate when the sphere jams as we roll.

The quation of the ellipsoid dT(0, s'D) is given by x'D-'x - s2kz. Thus the equation of the tangent plane to dT(0, s2D) at xo is xLD-'(x - XO) - 0. Therefore D-'XO/(~ D-'xo 11 is the normal vector to the ellipsoid at XO. Now consider the sphere with centre at

uo xo - ~oD-'xo/II D-'xo 11 (11 D-'xo 11 I - koD-')~~/ll D-'xo 11 (4)

and radius ko. We have the following result:

LEMMA 2. A sphere in Rp of radius ko and centre at (4) is contained in T(0, sZD) ifand only if11 D-'xo 11 L ko/d:.

Proof. We make use of the identity (y - x)'D-'(y + x) - 0 for x and y in V ( 0 .

1980 y-CONFIDENCE TOLERANCE REGIONS 83

s2D). The sphere is contained in T(0, s'D) if and only if for any y in dT(0, sZD) we have ki 5 (y - UO)'(Y - UO) = ( y - XO)'(Y - XO) + 2(y - ~o)'koD-'~o/I/ D-'XO(~ + k ? ~

= ( y - XO)'(Y - XO) - (~O/IID-'XOII)((Y - XO)'D-'(-ZXO)

+ ( Y - xo)'D-'(y + a)] + &$

(y - XO)'(Y - XO) - (ko/II D - ' x o ~ ~ ) ( Y - ~o)'D-l(y - XO) + k; = (y - XO)'[I - (ko/l(D-'~oII)D-'](y - XO) + ki.

Since y - xo assumes all directions in RP as y ranges in dT(0, s2D) we see that the inequality holds if and only if the matrix I - (ko/II D-Ixo(l)D-' is positive semi- definite which is true if and only if 11 D-'xo 11 ? ko/dg. Q.E.D.

An immediate consequence of Lemma 2 is that the sphere of radius ko 5 sdpk and centre at (sd,k - ko)ep is contained in T(0, s2D) since IID-'(sd,ke,,)ll = sk/dp ? (b/dpk)(k/dp) = ko/dg. Further we note that for x E dT(0, s2D) as 11 x 11 increases 11 D-'x 11 decreases since the points D-'x lie on the ellipsoid dT(0, s2D-'). This fact, Lemma 2 and (4) show that the sphere can be rolled, as previously discussed, until the pth coordinate of its centre is 0 and it then jams.

We now calculate P(D). We hBve that b - .,y^p(O; c;1), where co" = 1 + w'(VV')-'w so that

P(D) = 2p I (2a~t)-~/'exp{-b'b/2c~} db. ( 5 ) P ( D I

For b E P ( D ) we have that

b = (11 D-'xII I - koD-')~/lI D - ' x ~ (

for a unique x E T+(O, D). The existence of x is obtained by noting that there is a smallest ellipsoid of the form dT(0, sZD) where s E (ko/d,k, 13 which contains the sphere of radius k ~ , centered at b. The uniqueness of x follows by noting that Lemma 2 is true for this unique smallest ellipsoid and b,, # 0 implies 11 D-lx 11 >&old; which implies that the inequality in the proof is strict.

We calculate P(D) by rolling the sphere on the inside of the ellipsoids dT(0, s2D) for s E (b /d ,k , I ] and summing the probability density on the surfaces generated by the centre of the sphere as it rolls. We first make the transformation b + x which has Jacobian determinant

J(b+x)= 11 - ( ~ o / ~ [ D - ' x ~ ~ ) D - ' + (ko/IID-'xll)(D-'~/ll D-'X[I)(D-'X/I~ D-'~ll)'l

= II - ( b / ~ ~ D - l ~ ~ ~ ) D - ' 1 1 I + ( I - (ko/II D-'xII)D-')-'

.(ko/ll D - ~ X I I ) ( D - W I I D - ~ X ~ I ) ( D - W I I D-IXIIY I P

=I n (1 - ko/d?llD-'XII) ( 1 + (ko/llD-'xll) 1-1

P * C (xT/d: 11 D-'X 112)(1 - ko/dfll D-'X ll)-').

;-1

Next we make the transformation x + y, where y = D-*x which has Jacobian

84 EVANS Vol. 8, No. 1

determinant I D I *. Finally we make the transformation y -., (s, 8) where

yl = sk sin d1 . -. sin t9p-l

p = sk sin d1 . sin dp-ecos dP-'

yp = sk cos 81

for 8, E [0, 2v), i = 2, , . . , p - 1, and d1 E [0, v] . This transformation has Jacobian H G ' (sin d,)p-i- ' . Then utting J(s , 8 ) = J ( b determinant J(y -P s, 8 ) = sP-'kP-'

+ 8. s) = J(b+ x)J(x+ y)J(y+ s, 8) if 11 D-'x(s, 8)II rko/dp and 0 otherwise and P

we have that

From Srivastava and Khatri (1979) we have the joint density function of (dl, . . . , dp), namely

P

g(dt, . . . , d,) - C ( p , n ) n dYPexp{-d :/2) n (da - dj) 1-1 I-?

for 0 < dp < dP-' < - 0 < dl c co, where

This gives the probability y( k) that (2) is true; namely

y(k) = f f - [ P(D)g(dl, . . . , dp) ddl . 8 dd,. w dp

This expression shows how to calculate k, using numerical integration, to obtain some prescribed confidence for the tolerance region.

ACKNOWLEDGEMENTS

a referee for suggestions that led to substantial improvements over an earlier version. The author would like to thank John Wilker for the method of proof used in Lemma 2 and

RESUME

Dans cet article nous etudions des regions de confiance qui ont la proprikte de couvrir un evenement choisi a un niveau de confiance prkdetennine y . On obtient une expression exacte pour evaluer les niveaux des regions de confiance associees au modble de regression multivariee avec des errcurs normales.

1980 y-CONFIDENCE TOLERANCE REGIONS 85

REFERENCES Ellison, R.E. (1964). On two-sided tolerance intervals for a normal distribution. Ann. Math. Statist., 35 ,

Evans, M. (1977). Sampling and Structural Tolerance Regions. Ph.D. Thesis, Dept. of Mathematics,

Guttman, 1. (1970a). Construction of /?-content tolerance regions at confidence level y for large samples

Guttman. 1. ( 1970b). Statistical Tolerance Regions, Classical and Bayesian. Griffin. London. John, S. (1968). A central tolerance region for the multivariate normal distribution. J. Roy. Statist. Soc.

Owen. D.B. (1964). Control of percentages in both tails of the normal distribution. Technomrtrics. 6,377-

Owen, D.B. (1965). A special case of a bivariate non-central t-distribution. Biometrika. 52. 437-446. Srivastava, M.S., and Khatri. C.G. ( 1979). An Introduction to Multivariate Statistics. North Holland.

Wilks, S.S. (1962). Mathematical Statistics. Wiley. New York.

762-772.

University of Toronto.

from the k-variate normal distribution. Ann. Math. Statist.. 41. 376400.

Ser. B, 30. 599-601.

387.

Amsterdam.

Received I May 1979 Revised 7 December 1979

Department of Statistics University of Toronto

Toronto, Ontario MSS I A l