journal of Statistical Planning and Inference 12 (1985) 203-212

North-Holland

203

O N E F F I C I E N T U S E OF A U X I L I A R Y I N F O R M A T I O N

Ashok SAHAI*

University of Roorkee, Roorkee, UP 247667, India

Ajit SAHAI**

University of Lucknow, Badshah Bagh, Lucknow, UP 226007, India

Received 15 August 1980; revised manuscript received 22 March 1985 Recommended by M.N. Murthy

Abstract: Srivastava (1967), Reddy (1974) and Sahai (1979) proposed ratio-cure-product estimators to improve the ratio and product estimators. Three new ratio-cure-product estimators are proposed. The relative efficiencies of all these estimators are studied comparatively, through

an empirical study, for the case of a fairly large sample from a large bivariate population.

AMS Subject Classification: 62D05.

Key words: First and second orders of large sample approximation; Mean square error; Relative efficiency; Empirical study.

1. Introduction and notations

Suppose a study involves a variable, say Y, for a large population. Let C(Y) be the coefficient of variation for the variable Y. In some cases a close guess for the value of C(Y) might be available (see Gleser and Healy (1976), Lee (1981)). Further, let X be an auxiliary variable with its coefficient of variation C(X). Introduce Q for the correlation of X and Y, C(Z) for C(Y)/C(X) and G for O- C(Z). We assume the availability of a good guess for the value of C(Z). Also, the past association with the experimental material might provide a close guess for the value of Q. Thus, we would have a good guess, say g, for the value of G.

This paper addresses the problem of efficiently estimating the population mean, ~', using a fairly large sample, of size n, from the large bivariate population characterized as above.

Let the sample (population) means for the two variables be denoted by y(~') and • (X), respectively. Set

*Presently at Dept. Statistics, P.O. Box 35047, University of Dar es Salaam, Tanzania. **Presently at ICDS, Human Nutrition Unit, All-India Inst. Med. Sci., New Delhi, India 110029.

0378-3758/85/$3.30 © 1985, Elsevier Science Publishers B.V. (North-Holland)

204 A. Sahai, A. Sahai / On efficient use of auxiliary information

(Y/~")- 1 =e and ( X / X ) - 1 =e'. (1.1)

One easily checks that

E(e)=E(e')=O, Var(e)=Cz(Y)/n,

Var(e') = C2(X)/n, Cov(e', e) = G. C2(X)/n. (1.2)

Hence both e and e' are of order n-t/2.

In some applications the sampled large bivariate population might happen to be nearly normal. The joint distribution of e and e' could be taken to be normal to the extent of normality of the sampled population and /o r to the extent of largeness of n (by the central limit theorem). This leads us to the applicability of the results in Sukhatme and Sukhatme (1970) (4.7, p. 145):

M(0, 4)= 12F. R(p), M(1, 3) = 12F. G. R0v),

M(1, 2) = M(2, 1)=0, M(2, 2)= 4F. H . R0v),

where M(i, j) = E((e)i(e') j) with i, j positive integers,

H = (1 + 2~o2) • C2(Z) , F = cZ(x)/(4n. C2(Z))

and R0v) = C2(y)/n.

(1.3)

Let the symbol M 1 ( . ) ( M 2 ( . ) ) denote the first (second) order approximation, up to the terms of order n - i (n-E), to the mse, M(. ), of the estimator obtained by re- taining terms up to the second (fourth) degree in e and/or e'. Introduce R(. ) for the relative mse of the estimator of ~', i.e., R(- ) = M ( - )/~'2. Also, let Ri(.)= Mi( . ) /~-2 i-- 1,2.

Let B stand for the regression coefficient of Y on X, i.e., B=~o. S(Z) where S(Z) = S(Y)/S(X), S(Y)= C(Y). f" and S(X)= CO(). X. With a bivariate sample and known X the following classical estimators of F are well known:

~'(R) = ty/x). X:

~'(P) = (y. x)/X:

F(L) =y + b. (X-.e):

Ratio estimator,

Product estimator,

Linear regression estimator,

where b is the sample counterpart (estimate) of the population regression coefficient B, and

Y(D) = y + bo" ( X - ~ ) : Difference estimator,

where b0 is the guessed value of B. It may be noted that, to the extent of C(Z) being a more stable population para-

meter than S(Z), G would be guessable more closely than B. Hence the estimators exploiting the guess g, such as those given in the following section, have an edge over the difference estimator using the guess b 0. Also, if the sample is so large as to justify RI(. ) for the estimators, the linear regression estimator might be better

A. Sahai, A. Sahai / On efficient use of auxiliary information 205

motivated to the extent that it would, in that case, be a more reasonable proposition to estimate the population parameters rather than to guess them.

In view of the above, we study the estimators in the following section for the case of a sample which is not very large but only fairly large. Consequently, the study of the estimators in terms of R2(. ) might be appropriate. Therefore, let

RE(. )=(R(.P)/R2(. )). 100°7o (1.4)

define the relative efficiency of these estimators with respect to the usual unbiased

estimator Y.

2. The proposed estimators

Murthy (1964) recommended the use of ~'(R) when G>½, that of F(P) when G < -½ and that of)7 otherwise. In implementing this recommendation the guessed value g of G would have to be used whereas none of the estimators used g explicitly. The possibility of more efficient estimation through an explicit use of g motivated the proposition of the ratio-cum-product estimators:

Y(1) = y- (3~'/£) g (Srivastava (1967)),

~'(2) =y- 3(/(3~'+ g ( £ - X ) ) (Reddy (1974)),

~'(3)=y. ((1 +g) . f f+(1-g) .~) / ( (1-g) . f f+(1 +g).o?) (Sahai (1979)).

We further propose the following estimators on Sahai (1979)'s lines:

Y'(4) =.P. ((2 + g)- ,s~_ g. £)/((2 - g)- 37" + g. ~),

Y'(5) =.P- ( (2 -g ) - ~ + g. X)/((2 + g) -£-g- , I~) .

Interestingly, these two estimators are dual to each other in the sense that the expres- sion for one is obtainable from that of the other through an interchange of 9~ and X" followed by the inversion of the coefficient of.P. In this set-up of duality the ratio and product estimators are self-dual and so are Srivastava (1967)'s and Sahai (1979)'s estimators. However, Reddy (1974)'s estimator is not self-dual and its dual, as follows, is also included in our study:

~F(6) = (1 - g)..P + g- ~'(R).

Using (1.1) and (1.2) we easily check the following for these estimators Y(i), i= 1,2, ..., 6:

RI(Yr(i)) = Q. R(y) where Q= 1 +g. (g-2G)/C2(Z).

Hence, for g - G , Q - ( 1 - 0 2) and R1(~'(i))-(1 _~2). R(y). Whereas the sample is supposed to be only fairly large we proceed to determine the RE(IT(i))'s, RE(~'(R)) and RE(Y(P)) by using (1.3) further. Hence we obtain, using (1.4), the following

206 A. Sahai, A. Sahai / On efficient use of auxiliary information

RE(. )'s for these estimators:

(RE(~'(I)))-~ = Q + F . (4g-(2g + 1). H - 4 g . (7g + 2)-(g + 1). G

+g2. (7g+ 11). (g+ 1)), (2.1)

(RE(Y(2))) - 1 = Q + 4F. (3 g 2. H - 18 g 3. G + 9g 4), (2.2)

(RE(~r(3)))-I=Q+F.(4g.(Eg+ 1) .H-6g . (5g+ 1)- (g+ 1)-G

+ 9g 2- (g+ 1)2), (2.3)

(RE(Y'(4)))- 1 = Q + F" (8g 2" H - 30g 3. G + 9g4), (2.4)

(RE(~'(5))) -1 = Q + F-(8g(g + 1)- H - 6 g . (5g + 2)-(g + 2)- G

+ 9g 2. (g + 2)2), (2.5)

(RE(~'(6))) -~=Q+4F.(g. (g+2) .H-6g.(2g+I) .G+9g2) , (2.6)

(RE(~'(R)))-1 = 1 + (1 - 2 G ) / C E ( z ) + 12F- ( H - 6 G + 3), (2.7)

(RE(~'(P)))- 1 = 1 +(1 + 2G)/CE(Z)+4F • H. (2.8)

3. The comparison - An empirical study

The performance of a ratio-cum-product estimator would depend on the propor- tionate error in guessing G, say E(G) ( = ( g - G)/G). However, it would depend on the values of the population parameters too. Consequently the comparisons of (2.1) to (2.8) are complex. To demonstrate this and to bring out the sensitivity of the estimators to the error E(G), an empirical study is carried out with thirty values of n: 10, 20, ..., 300; thirty values each of C(Z) and C(X): 0.10, 0.20, . . . , 3.00; six values of Q: + / -0 .9 , + / - 0 . 6 , + / - 0 . 3 ; five values of E(G): + / - 0 . 4 , + / -0 .2 ,0 .0 . The RE(.) 's , as per (2.1) to (2.8), were computerized for these 810000 value- combinations. A comparison of RE(-) ' s for each value-combination led to the ascertainment of the RF(. )'s for the estimators where RF(. ) stands for the relative frequency, for the estimator, of having the highest RE(. ), i.e., that of being the best estimator. It turned out that

RF(~'(4)) = 0.406,

RF(~'(R)) =0.086,

RF(Y'(1)) = 0.054,

RF(Y'(6)) = 0.124,

RF(~'(P)) =0.085,

RF(Y'(3)) =0.042.

RF(Y'(2)) = 0.106,

RF(Y'(5)) = 0.073,

Hence the empirical probability that y would be left unimproved by these estimators comes to be as low as 0.024.

A finer description of the empirical study is afforded through the first three tables in the appendix. From Table 1 we infer (empirically) that: for negative and low cor- relation, i~(6) is quite probably the best estimator; for negative and moderately high

A. Sahai, A. Sahai / On efficient use o f auxiliary information 207

(very high) correlation, Y(4) (Y(2)) is most probably the best estimator; for positive and not-very-high correlation, Y(4) is quite probably the best estimator; for positive and very high correlation no estimator stands out well. Also, as from Table 2, Y(4) turns out to be the most robust (least sensitive) against the error E(G) except for the case of the error E(G) being rather highly negative when :Y(6) gets to be the best most often. Moreover, it may be noted that the estimator ~'(4) is more sensitive to a negative error than it is to a positive error. Table 3 brings forth the effect of the values of C(Z) and C(X) on the relative performances of the estimators. Whereas F(4) happens to be uniformly the best more often, a lower value of C(Z) is more favourable to it.

The last six tables in the appendix illustrate, very briefly, the possible gains through the proposed estimators; the trend of RE(- ) values with increasing values of n and the effect of E(G) values on the relative performances of the estimators. The corresponding six value-combinations of the parameters G, Q, C(Z) and C(X) were selected in order to represent, briefly, some of the possibilities: Q = + / - 0 . 9 , + / -0 .6 , + / - 0 . 3 ; C(X)= 1.00, 1.25,2.00,2.75,3.00; C(Z)=0.50, 1.00,2.50. In this context we note that, as observed by Murthy (1967), mostly C(Z)>_ 1 in practice. Tables 4, 6, 7 and 8 illustrate that, even though G>0.5 or G < - 0 . 5 , Y(R) or Y(P) might be worse than y unless n > 3 0 or so. In particular, Table 8 illustrates the typical possibility of all estimators turning out to be worse than 37 unless n is at least 20. The entries of the six tables also illustrate the conclusion from the Tables 1 and 3, e.g., Y(6) is the most efficient estimator when p = - 0 . 3 (in Table 9).

Hence, we conclude from the empirical study that, other things remaining the same, the proposed estimator Y(4) happens to be the most potential estimator for being recommended.

Lastly, we note that the empirical results presented in this section are well founded and appropriate only for bivariate data which is not in deviation from what would lead us to the bivariate normal model assumed for the joint distribution of e and e'.

Appendix

The first three tables enlist the values of the relative frequency RF(. ) for the estimators according to p-value, E(G)-value, and C(Z)- and C(X)-value range, respectively. In each row of the tables the boldface value is that of the estimator, with highest RF(. ) value, which is the best most probably. The last six tables in- dicate some relative efficiencies RF(- ) for the estimators with respect to 37 for the hypothetical population characterized at the top of the table. In each row of the tables the relative efficiency printed boldface is that of the most efficient estimator for the respective values of n and E(G).

208 A. Sahai, A. Sahai / On efficient use o f auxiliary information

Table 1 Relative frequency RF(-) values for the estimators

Estimators

O Y(1) ~'(2) Y(3) Y(4) Y(5) F(6) Y(R) ~'(P) y

- 0 . 9 0.000 0.489 0.000 0.328 0.000 0.040 0.000 0.130 0.013 - 0 . 6 0.063 0.000 0.046 0 . 4 0 0 0.000 0.141 0.000 0.330 0.020 - 0.3 0.020 0.000 0.177 0.073 0.127 0.533 0.000 0.053 0.016

0.3 0.007 0.000 0.000 0 . 8 9 5 0.000 0.000 0.037 0.000 0.061 0.6 0.007 0.000 0.000 0.627 0.027 0.011 0.299 0.000 0.029 0.9 0.225 0.145 0.027 0.112 0.284 0.018 0.180 0.000 0.009

Table 2 Relative frequency RF(-) values for the estimators

Estimators

E(G) ~'(1) ~(2) Y(3) Y(4) Y(5) ~F(6) Y(R) Y'(P) p

- 0.4 0.011 0.087 0.000 0.225 0.137 0.308 0.119 0.102 0.011 - 0.2 0.101 0.183 0.039 0.341 0.084 0.149 0.041 0.046 0.016

0.0 0.000 0.259 0.055 0 . 4 5 1 0.065 0.105 0.015 0.029 0.021

0.2 0.087 0.000 0.050 0.562 0.044 0.056 0.088 0.082 0.031 0.4 0.068 0.000 0.065 0.450 0.035 0.002 0.167 0.169 0.044

Table 3 Relative frequency RF(. ) values for the estimators

Value ranges Estimators

C(Z) C(X) Y-(I) ~(2) ~'(3) Y(4) ~'(5) t'(6) Y(R) Y(P) y

0.1-1.0

1 . 1 - 2 . 0

2.1-3.0

0.1-1.0 0.047 0.108 0.057 0.583 0.048 0.110 0.024 0.023 0.000 1 . 1 - 2 . 0 0.047 0.114 0.057 0.590 0.050 0.110 0.019 0.013 0.000 2.1-3.0 0.047 0.117 0.056 0.591 0.051 0.111 0.017 0.007 0.003

0.1-1.0 0.040 0.082 0.047 0.315 0.070 0.106 0.170 0.170 0.000 1.1-2.0 0.047 0.092 0.046 0.342 0.073 0.113 0.146 0.131 0.010 2.1-3.0 0.055 0.102 0.044 0.347 0.076 0.124 0.118 0.087 0.047

0.1-1.0 0.070 0.117 0.032 0.344 0.098 0.131 0.100 0.106 0.002 1 . 1 - 2 . 0 0.066 0.114 0.026 0.307 0.098 0.154 0.090 0.108 0.036 2.1-3.0 0.064 0.107 0.012 0.233 0.092 0.157 0.090 0.123 0.122

A. Sahai, A. Sahai / On efficient use o f auxiliary information 209

Table 4 Relative efficiencies (in %) of the estimators for hypothetical population I: G = - 0 . 9 0 , p = - 0 . 9 0 ,

c(X) =3.00, C(Z)= 1.00

Estimators

n E(G) ~'(1) ~'(2) ~'(3) ~'(4) Y(5) ~'(6) ~'(R) ~'(P)

10 - 0.4 100.2 128.9 94.4 109.8 92.8 96.9 3.0 39. l

10 - 0.2 60.2 160.2 59.2 105.9 35.9 26.4 3.0 39.1

l0 0.0 43.7 165.1 43.7 109.9 21.0 12.6 3.0 39.1

10 0.2 36.8 88.7 36.6 121.2 14.9 7.5 3.0 39.1

10 0.4 35.4 35.7 33.9 131.9 11.8 4.9 3.0 39.1

20 - 0 . 4 151.8 182.7 145.0 162.6 143.2 147.9 5.4 72.5

20 - 0.2 106.2 236.2 104.7 171.5 66.5 49.9 5.4 72.5

20 0.0 80.7 251.4 80.7 181.8 40.4 24.7 5.4 72.5

20 0.2 68.1 148.2 67.7 190.9 28.8 14.7 5.4 72.5

20 0.4 63.6 64.1 61.2 185.5 22.7 9.7 5.4 72.5

30 - 0 . 4 183.3 212.1 176.6 193.6 174.8 179.5 7.3 101.4

30 - 0.2 142.5 280.6 140.6 216.0 92.9 70.9 7.3 101.4

30 0.0 112.5 304.4 112.5 232.6 58.4 36.2 7.3 101.4

30 0.2 94.9 190.9 94.5 236.2 41.8 21.6 7.3 101.4

30 0.4 86.6 87.2 83.7 214.7 32.8 14.3 7.3 101.4

Table 5

Relative efficiencies (in 070) of the estimators for hypothetical population II: G=0 .45 , Q=0.90, C(X) = 2.75, C(Z) = 0.50

Estimators

n E(G) Y'(1) ~'(2) Y(3) ~'(4) F(5) Y(6) Y'(R) ~'(P).

l0 - 0.4 434.4 240.8 335.2 225.4 2338.5 815.4 9.9 9.5

l0 - 0.2 418.5 325.9 355.1 267.4 206.9 265.4 9.9 9.5

10 0.0 221.3 360.6 221.3 293.1 67.5 110.6 9.9 9.5

l0 0.2 108.2 242.5 114.7 286.5 31.7 58.6 9.9 9.5

l0 0.4 58.1 118.9 61.9 242.8 17.7 36.0 9.9 9.5

20 - 0.4 363.8 272.1 323.6 262.0 551.9 452.2 17.4 10.4

20 - 0.2 433.5 377.9 396.8 335.4 283.4 333.8 17.4 10.4

20 0.0 311.6 428.0 311.6 376.5 119.7 182.8 17.4 10.4

20 0.2 174.4 315.1 182.7 350.0 59.1 103.8 17.4 10.4

20 0.4 98.1 172.4 103.3 273.5 33.5 64.5 17.4 10.4

30 - 0.4 345.1 284.5 320.0 277.0 439.9 393.8 23.3 10.8

30 - 0.2 438.7 399.1 413.0 366.4 323.2 365.1 23.3 10.8

30 0.0 360.7 456.4 360.7 416.0 161.2 233.6 23.3 10.8

30 0.2 219.1 349.9 227.8 377.9 83.2 139.5 23.3 10.8

30 0.4 127.2 202.7 133.0 285.4 47.7 87.8 23.3 10.8

210 A. Sahai, A. Sahai / On efficient use o f auxiliary information

Table 6

Relative efficiencies (in %) of the estimators for hypothetical population III: G=0.60, ~=0.60, C(X)=2.00, C(Z)= 1.00

Estimators

n E(G) ~'(1) ~'(2) Y-(3) ~'(4) ~'(5) Y-(6) Y-(R) y-(p)

10 - 0 . 4 120.4 121.4 118.2 123.9 116.5 116.7 46.6 25.7

10 - 0 . 2 113.9 118.6 112.8 121.9 97.9 105.5 46.6 25.7

10 0.0 100.5 109.1 100.5 115.8 74.4 90.2 46.6 25.7 10 0.2 83.3 92.6 83.9 106.6 53.7 74.7 46.6 25.7

10 0.4 66.0 72.2 66.5 95.3 38.1 61.1 46.6 25.7

20 - 0 . 4 130.9 131.4 129.6 132.9 128.6 128.6 67.9 28.2

20 - 0.2 130.6 133.5 129.8 135.6 119.3 124.8 67.9 28.2 20 0.0 122.3 128.5 122.3 133.0 100.8 114.4 67.9 28.2

20 0.2 107.8 115.4 108.3 125.6 79.5 100.4 67.9 28.2

20 0.4 90.4 96.0 90.9 114.5 60.3 85.7 67.9 28.2

30 - 0 . 4 134.8 135.2 133.9 136.2 133.1 133.2 80.1 29.2

30 - 0 . 2 137.2 139.4 136.7 140.9 128.7 132.9 80.1 29.2

30 0.0 131.9 136.6 131.9 140.0 114.4 125.6 80.1 29.2 30 0.2 119.5 125.6 119.9 133.5 94.6 133.3 80.1 29.2 30 0.4 103.1 107.9 103.5 122.7 74.7 98.9 80.1 29.2

Table 7

Relative efficiencies (in %) of the estimators for hypothetical population IV! G = - 0 . 6 0 , Q=-0 .60 , C(X) =2.75, C(Z) = 1.00

Estimators

n E(G) Y-(l) Y'(2) Y'(3) Y-(4) ~'(5) Y'(6) $'(R) Y'(P)

10 - 0.4 132.7 106.8 127.8 110.6 163.7 186.1 4.5 47.6

10 - 0 . 2 104.5 98.9 102.7 103.3 90.1 81.9 4.5 47.6

10 0.0 81.9 86.0 81.9 94.2 55.9 43.0 4.5 47.6

10 0.2 66.3 68.6 67.0 84.0 38.8 26.0 4.5 47.6 10 0.4 56.0 50.0 56.7 73.3 29.2 17.3 4.5 47.6

20 - 0 . 4 137.8 122.4 135.1 124.8 152.8 162.0 7.9 68.9 20 - 0 . 2 124.2 120.1 122.9 123.3 113.4 106.6 7.9 68.9

20 0.0 107.5 110.9 107.5 117.5 82.4 67.5 7.9 68.9 20 0.2 92.5 94.7 93.2 108.4 61.9 44.4 7.9 68.9 20 0.4 80.5 74.2 81.2 97.0 48.5 30.8 7.9 68.9

30 - 0 . 4 139.6 128.7 137.8 130.4 149.5 155.2 10.5 81.1

30 - 0 . 2 132.4 129.3 131.5 131.8 124.1 118.6 10.5 81.1 30 0.0 120.0 122.8 120.0 128.1 97.8 83.2 10.5 81.1 30 0.2 106.5 108.4 107.1 120.1 77.2 58.2 10.5 81.1 30 0.4 94.3 88.4 95.0 108.8 62.3 41.8 10.5 81.1

A. SahaL A. Sahai / On efficient use o f auxiliary information 211

Table 8 Relative efficiencies (in °70) of the estimators for hypothetical population V: G=0.75, Q=0.30,

cO()= 1.25, C(Z) =2.5

Estimators

n E(G) Y(1) 17(2) Y(3) 17(4) 17(5) ~'(6) Y(R) ~'(P)

i0 - 0.4 95.7 98.5 95.6 101.2 90.8 92.8 73.5 63.1

10 - 0.2 90.9 93.5 90.8 97.7 84.4 88.2 73.5 63.1

10 0.0 84.9 87.0 84.9 92.9 76.9 82.9 73.5 63.1

10 0.2 78.2 79.2 78.2 87.2 69.0 77.3 73.5 63.1

10 0.4 71.1 70.6 71.1 80.9 60.9 71.6 73.5 63.1

20 - 0.4 101.6 103. l 101.5 104.5 98.7 99.9 87.7 67.0 20 - 0.2 99.3 100.9 99.3 103.3 95.3 97.7 87.7 67.0

20 0.0 95.8 97.1 95.8 100.7 90.5 94.5 87.7 67.0 20 0.2 91.2 91.9 91.2 97.1 84.6 90.6 87.7 67.0

20 0.4 85.8 85.4 85.8 92.6 77.9 86.2 87.7 67.0

30 - 0.4 103.7 104.8 103.6 105.7 101.7 102.5 93.7 68.4

30 - 0.2 102.5 103.6 102.5 105.2 99.6 101.3 93.7 68.4

30 0.0 100.1 101.0 100.1 103.6 96.2 99.1 93.7 68.4

30 0.2 96.6 97.1 96.6 100.9 91.5 96.1 93.7 68.4 30 0.4 92.2 91.9 92.1 97.2 85.9 92.4 93.7 68.4

Table 9 Relative efficiencies (in °70) of the estimators for hypothetical population VI: G = - 0 . 7 5 , ~ = - 0 . 3 0 ,

C(X)= 1.00, C(Z) =2.5

Estimators

n E(G) 17(1) i7(2) 17(3) 17(4) 17(5) 17(6) 17(R) 17(P)

l0 - 0.4 108.0 101.8 107.9 103.6 112.8 115.2 47.3 96.3

10 - 0.2 106.6 98.7 106.6 101.6 111.4 114.3 47.3 96.3

10 0.0 103.6 94.0 103.6 98.4 107.9 110.2 47.3 96.3

10 0.2 99.5 87.9 99.5 94.1 102.9 103.6 47.3 96.3 10 0.4 94.7 80.7 94.6 88.9 97.0 95.4 47.3 96.3

20 - 0.4 108.1 104.9 108.0 105.8 110.4 111.6 56.9 102.1

20 - 0.2 108.0 103.8 108.0 105.4 110.4 111.8 56.9 102.1 20 0.0 106.7 101.3 106.7 103.8 108.9 110.0 56.9 102.1 20 0.2 104.3 97.5 104.3 101.2 106.1 106.4 56.9 102.1

20 0.4 101.0 92.4 100.9 97.6 102.3 101.4 56.9 102.1

30 - 0.4 108.1 106.0 108.1 106.6 109.7 110.4 61.1 104.2

30 - 0.2 108.5 105.6 108.5 160.7 110.1 111.0 61.1 104.2 30 0.0 107.7 104.0 107.7 150.8 109.2 110.0 61.1 104.2 30 0.2 105.9 101.2 105.9 120.8 107.2 107.4 61.1 104.2 30 0.4 103.3 97.1 103.2 100.9 104.3 103.5 61.1 104.2

212 A. Sahai, A. Sahai / On efficient use o f auxiliary information

Acknowledgement

The authors are grateful to the referees for pointing out mistakes in the original draft and for suggesting many improvements in the presentation of the paper.

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