On two methods of bias reduction in the estimation of ratios

Transcription

1 Biometriha (1966), 53, 3 and 4, p Printed in Great Britain On two methods of bias reduction in the estimation of ratios BY J. N. K. RAO AOT) J. T. WEBSTER Texas A and M University and Southern Methodist University SUMMARY Quenouille's method of bias reduction, based on splitting the sample at random into groups, is applied to estimation of ratios and the optimum choice of the number of groups, g, is investigated assuming a model. It is shown that both the bias and variance of the estimator are decreasing functions of g. The estimator is compared with a modified ratio estimator assuming the same model. 1. INTRODUCTION Durbin (1959) has applied Quenouille's (1956) method of bias reduction to the estimation of ratios. Suppose r denotes the oustomary ratio estimator of the form r = y/x based on n observations whose bias is en (n~ ) where c is a constant and y and x are unbiased estimators of the population mean of the characters ' y' and ' x' respectively. Let the sample be divided at random into g groups, each of size p, where n = pg. Let r\ denote the ratio estimator calculated from the sample after omitting the jth group (i.e. based on p(g 1) observations). Then the estimator where r Qi = gr (g l)r'j, has bias of order n~ at most. Durbin has shown that if the regression of y on x is linear (i.e. y = a+fix + u, where u is the error term) and x is normally distributed, r Q with g = has a smaller asymptotic variance than r. Recently, one of us (Rao, 1965) has derived the asymptotic bias and asymptotic variance of r Q for general g, to 0{n~*), assuming the above model. It was shown that both the asymptotic bias and asymptotic variance of r Q are in fact decreasing functions of g so that g = n would be the optimum choice. Durbin has also considered the case where the regression of y on x is linear but xjn has the gamma-distribution with parameter h. He has shown that, although the variance of r Q with g = compared to the variance of r is slightly increased, the reduction in bias is such that the mean square error (MSE) of r Q is reduced. In this paper, we shall derive the bias and variance of r Q for general g assuming Durbin's second model. We shall show that the bias of r Q is a decreasing function oig. The variance of r Q is derived explicitly for the case of integer m( = nh). However, since the resulting expression is not in a closed form, we make a numerical study of the behaviour of the variance of r Q as a function of g and show that it is a decreasing function of g. Hence, both the bias and variance are decreasing functions of g for the case of integer m at least, so that g = n would be the optimum choice. We also show numerically that r Q has a smaller variance than r for g >. We shall compare r Q with the modified ratio estimator r M (Tin, 1965) assuming the above model. We show that, although the bias of r M is slightly smaller than the bias of r 0 (with g = n), r u has a slightly larger MSE than r Q for m > and/or a = 0 (i.e. when the bias of

2 57 J. N. K. RAO AND J. T. WBBSTEB TQ and r is zero). But the difference is so small that it is difficult to choose between r M and r Q on the basis of mean square error only. However, r 0 has an advantage over r u with regard to the variance estimator. Following Tukey (195) we can use the simple estimator «(r 0 ) = g~\g- I)" 1 (r Qj -r Q f () as the variance estimator of TQ since the g estimators r Qi (called pseudo-values) may be treated as approximately independent and since r Q is the mean of the r Qi. We are, at the present time, making a detailed investigation of the bias and MSE of v(r Q ) assuming various models, and the results will be reported later. It may be noted that all our results are exact for any sample size, n. We confine ourselves to simple random sampling and assume the population size is infinite, to simplify the discussion.. BIAS OV r AND r Q Let (y it xj, i = 1,...,», denote a simple random sample of n observations on the characters 'y'and ' a;' and let y and x denote the sample means. Let y f = a+/3x i + u i where E(u t \x) = 0 and F(«i x) = ns where i is a constant of 0(n -1 ). Let the variates xjn have the gamma distribution with parameter h, so that x = xjn has the gamma distribution with parameter m = nh. The customary ratio estimator of p => E(y)/E(x) based on the n observations in the sample is r = y/x. Therefore, using the model, we have r =fi + (a + u)/x and the bias of r is given by B(r) = E(r)-p = a (!-r--) =, * T, (3) r ' \m-l m] m{m 1) since E{\jx) = l/(w-l). By splitting the sample at random into g groups, each of size p, we get the ratio estimator r 'i ~ y'ife'i ^roin the sample after omitting the jth. group, where y\ = (ny py~])l(n p) and x'j {rix px f )l{n p) where y t and Xj are the sample means for the jth. group. Then Quenouille's estimator r Q is given by where u'j = (nu puf)l(n p) and u } is the sample mean of the U'B for the jth group. Now since {(g 1 )jg} xj has the gamma distribution with parameter (n p) h = {m(g 1 )}jg, we have and the bias of r Q is <g-l)m-g m(m-l){m(g-l)-gy It is easily seen that -B(r o ) is a monotonically decreasing function of g (> ) and, hence, the smallest value of -B(r o ) occurs at g = n. (6)

3 Two methods of bias reduction VARTA-wmi! OF r AND From Durbin (1959), the variance of r is given by >W»M _ 1\ (<m 9V ' ' Turning to the variance of r Q, we get from (4) where E{r Q fi)is given by (5). We now evaluate and (a; We have g /l\ 1 /1\ (y-l)«it remains, therefore, to evaluate E{\j{xx'j)} and J?{l/(^Xy)}. It may be noted that these expectations are the inverse raw product moments of correlated gamma variates. We were able to evaluate explicitly -E7{l/(zzJ)} for all values of m, but E{\\(x\ xj)} for only integer m and that too not in a closed form. Theorems 1 and below enable us to evaluate these expectations. THEOREM: 1. Let X x and X t be independent gamma variates urith parameters a and b respectively. Then.. E (*!(*!+ X a )J = (a-l)(a + b-) Proof. Let X 3 = X x + X t ; then the joint density of X t and X 3 is Therefore /(X^Z,) = {r(a)r(6)}-mx,-x 1 ) 6-1 Xrierp(-X,). where Ei^Xi 1 ; a 1,6) denotes the expected value of 1/Jf when X x and X a have a gamma distribution with parameters a 1 and 6 respectively, i.e. when X 3 has the gamma distribution with parameter a Hence we get ().

4 574 J. N. K. RAO AND J. T. WEBSTEB THEOBHM. Let X v X and X 3 be independent gamma variates with parameters a, a and b respectively. Then, provided a and b are integers, m[ 1 ) _ r(a + 6-)r -» Ux 1+ x )(x a +x )j~ r»r(6) U-o l } { a+6-3 (13) Proof. We first make the transformations where J^ = X 1 + X andw^ = X a + X. Since the Jacobian of the transformation is equal to TT, we get, after integrating over W a, Now, for a >, r(a+br(a+6-) O(a,6) = i?(o-l,6)-c(o-1,6 + 1) a where )(o,6) = f' P (l-s 1 ) a -i(l->s )«Jo Ji-^r, It can be easily shown that Further, for any integer d, l), () Hence, from (15), (), (17) and (1) we get (13) and (14). It may be noted that Theorem can be easily generalized to the case where X x and X have different parameters. However, this generalization is not needed for our present purpose. Now, replacing o and b by {m(g l)}/g and mjg respectively in (), we get g-1 \xxjj (m-){g(m-l)-m}' x'j)] is given by the right-hand side of (13) with a ~ m/g and 6 = {m(g )jg) if mjg >, and by the right-hand side of (14) with a = 1 and b = m if m-g. Therefore, using (11), (13), (14) and (19) in (), we get V(r Q ) explicitly as a function

5 Two methods of bias reduction 575 of g and m. However, since the resulting expression would not be in a closed form, it is difficult to investigate analytically the behaviour of V{r Q ) as a function g for fixed m. Therefore, we have made a numerical investigation using an IBM 7094; the results are given in Table 1. We find from Table 1 that V(r Q ) decreases monotonically as g increases so that the smallest values of V(r Q ) occur at g = n. Hence, both the bias and variance are decreasing functions of g for integer m at least, so that the optimum choice would be g = n. Also, we find from Table 1 that, although V(r Q ) for g = can be slightly larger than V{r) (as pointed out by Durbin, 1959), F(r<j) for g > is smaller than V(r). Table 1. Coefficients C lt...,c e in the variance formulae V(r Q ) = V(r) = C s op + C t and V(r M ) = CsP? + G g for selected values of m, g, and n m n C t C t TO n Cf C t COMPABISON OF r<j WITH THE MODIFIED RATIO ETIMATOB r M Tin (1965) has proposed the modified ratio estimator where «= (n-1) <-x -y). He has made asymptotic comparisons between r M, r Q with g = and Beale's (196) ratio estimator - IV (1) (0) for our present model and for two other situations. His comparisons lead to the conclusion that r u has the smallest asymptotic variance with r B and r 0 next in that order. (There seems to be an error in the coefficients of a and a? in Tin's formula for E(r M ) and V(r M ) under our model.) Since we have shown that g = n is the optimum choice, one should compare r M with rq for g = n. We shall now derive the exact bias and variance of r M under our model and then compare with r Q for g = n.

6 576 J. N. K. RAO AND J. T. WEBSTEB Substitution of t/ 4 = a+px t + u t in (0) leads to Hence, -fl -_;«(! ) (3) and the bias of r M is Also = E(r M -P)>-EHr M -P), (5) E{s%lx^) and E(*/x?). Theorem 3 below enables- Therefore, it remains to evaluate E^x*), us to evaluate these expectations. (6) THEOBEM 3. Let Z lt...,z n be independent gamma variates withparameter h, where Z i = xjn. Then, for i 4= j, T(a + h)t(b + h) 1 E II (m + a + b-t) t-i where m = nh, c is an integer greater than zero and a and b are integers greater than or equal to zero. Proof. (sz/l-r^)j 0 -J exp(-sz t) " (a + h)t(b + h) where the expectation on the right-hand side denotes the expectation of ll(lz,) c when Z t (f+t+j), Z t and Z t have the gamma distribution with parameters h, a + h and b + h respectively. Hence, we get (7). Using Theorem 3, we have E (I) = orhj [ n * E n (m-l)(m-) (7) () (9) and (n -1) (TO - 1) (m - ) (m +1) (m + ) (m + 3)" (30)

7 Substitution of () in (4) leads to Two methods of bias reduction 577 which is slightly smaller than jb( r g) at g = n. Also, substituting (9) and (30) in (3), (5) and (6), we get after simplification v(r M )-a m a (m' + 3ro + ) mn ( m*(m + 4) (m* + 5m + 6) + mn \ + (m -4)(m -l)(m + 3) (n-l)(wi -4)(m a -l)(m + 3)J* We have tabulated V(r M ) in Table 1 for selected values of m and n; for m >, V(r M ) = MSE(r Jf ) and V(r Q ) = MSE(r Q ), 9 = n, to six decimal plaoes. Hence, from Table 1 it follows that, although the bias of r M is slightly smaller than the bias of r Q (with g = n), r M has a slightly larger mean square error than r Q for m > and/or a = 0 (i.e. when the bias of r M and r Q is zero). But the difference is so small that it is difficult to choose between r Q and r M on the basis of mean square error only. However, as mentioned earlier in 1, the estimator r Q has an advantage over r M with regard to the variance estimator. REFERENCES JJKAIE, E. M. L. (196). Some use of computers in operational research. IndustrieUe Organization 31, 7-. DUBBIN, J. (1969). A note on the application of Quenouille's method of bias reduction to estimation of ratios. Biometrika 46, QuENOUiLiJffi, M. H. (1956). Notes on bias in estimation. Biometrika 43, RAO, J. N. K. (1965). A note on estimation of ratios by Quenouille's method. Biometrika 5, TIN, M. (1966). Comparison of some ratio estimators. J. Amer. Statist. Ass. 60, TUXBY, J. W. (195). Bias and confidence in not-quite large samples. Ann. Math. Statist. (Abstract), 9, 614. K '