Pergamon Microelectron. Reliab., Vol. 36, No. 9. pp. 1249-1252. 1996 Copyright ~) 1996 ElsevierScience Ltd Printed in Great Britain. All rights reserved 0026-2714/96 $15.00+.00 0026-27 ! 4(95) 00158- !

PROPERTIES OF A CONSISTENT ESTIMATION PROCEDURE IN ULTRASTRUCTURAL MODEL WHEN RELIABILITY RATIO IS KNOWN

ANIL K. SRIVASTAVA and SHALABH Department of Statistics, University of Lucknow, Lucknow, India

(Received Jbr publication 1 August 1995)

Abstract--To overcome the problem of inconsistency of the ordinary least squares (OLS) estimators for regression coefficients in a linear measurement error model, we assume the reliability ratio to be known a priori and utilize this information to form consistent estimators in an ultrastructural model. Assuming the distributions of measurement errors and the random error component to be not necessarily normal, the efficiency properties of the estimator for slope parameter have been analysed and the effect of departure from normality is examined. Copyright © 1996 Elsevier Science Ltd.

1. INTRODUCTION 2. MODEL SPECIFICATION The least squares procedure is well known to yield Let us consider the following linear ultrastructural the minimum variance estimator in the class of linear model and unbiased estimators for the coefficients in a linear Y~=~+flX, (i= 1,2 ..... n) (1) regression model. This optimal property implicitly assumes that the observations on variables are Yi = Y, + u~ (2) correctly measured. Such a specification is often not xi=Xi+vi o (3) tenable in many scientific investigations and the observations are invariably subject to measurement X i = m i + wi, (4) errors. When these errors are negligibly small they may not appreciably alter the inferences deduced from where ~ is the intercept term and fl is the slope the application of least squares. On the other hand, parameter. Here, Yi and x i are the observed values when the measurement errors are not negligibly small, corresponding to true values of the study variable Y, the inferences are significantly altered and in fact and explanatory variable Xi, respectively; u~ and v~ are become invalid. The main reason is that the esti- the measurement errors associated with the study mators arising from least squares procedure are variable and explanatory variable, respectively; and wi not even consistent in the presence of measurement is the random error component associated with the errors. explanatory variable. The quest for consistent estimation has received Let us make some assumptions about the error considerable attention in the literature and has led to terms, u~, u2,. •., u, are assumed to be identically and the evaluation of several estimators but little is independently distributed with mean 0 and variance known about their properties, particularly when the ~r2 along with third and fourth central moments ytua 3 errors are not necessarily normally distributed; see and (72u + 3) a4, respectively. Similarly, v~, rE,..., v, e.g. Fuller [1]. The plan of this paper is as follows. are also assumed to be identically and independently In Section 2, we consider an ultrastructural formula- distributed with mean 0 and variance a 2 along tion of the linear regression model with measure- with third and fourth central moments ~,tva 3 and ment errors; see Dolby [2]. In Section 3, we describe (72v + 3)a~4, respectively. The random error compo- a consistent estimation procedure when reliability nents wl, w2 ..... w, are also assumed to be identically ratio is known, while in Section 4 we analyse the and independently distributed with mean 0, variance 3 large sample asymptotic properties of this pro- O'w,2 third central moment as 71wa,~ and the fourth cedure. Finally, Section 5 draws some conclusions. central moment as (3'2w + 3)a~. Here 71 and 72 are In the end, some expectations are derived in the the Pearson measure of excess of skewness and Appendix. kurtosis, respectively in the error distribution of the variable in the suffix. Correspondence: Dr. Anil K. Srivastava, 15, Shastri Nagar, It is further assumed that u~, u2 ..... u.; v~, v 2..... v. Department of Statistics, University of Lucknow, Lucknow- and wt, w 2..... w. are mutually independent of each 226 004, India. other.

1249 1250 A.K. Srivastava and Shalabh

It may be noticed here that no specific form of the which is indeed the ratio of variance of X to variance error distribution has been assumed. All that is of its observed counterpart x. assumed is the existence of first four moments of the In many scientific investigations, the value of p is error distributions. correctly known. Such a knowledge may arise from To study the efficiency behaviour of estimators in some theoretical considerations or from empirical a comprehensive manner, we introduce the following experience. Or a reasonably accurate estimate of p quantities: may be available from some other independent studies. Moreover, the reliability ratio is not difficult d : Smm + 2; 0 --.< d < 1, (5) to find in many applications and there are various procedures to do it; see e.g. Ashley and Vanghan [4], where Lord and Novick [5] and Marquis et al. [6]. 1" 1~ Smm = ~ 2 (mi -- th)2 with rh = m i. (6) Assuming p to be known, a correction for removing /1 /1 the contamination of measurement errors is applied It is observed that in case of functional form of the to the least squares estimator; see Fuller ([1], p. 5). measurement error model, the Xis are all fixed, so This yields the following consistent estimator of/3: that w~ = 0 for all i = 1,2,...,n and consequently a,2, = 0. This makes d equal to 0. Similarly, d is equal b- s~. (11) pSxx to 1 in case of structural form of the measurement error model because all means of X~s are the same which is termed as attenuated or adjusted or im- implying that Smm = 0. Thus, d serves as a measure of maculate estimator. departure of ultrastructural model from its two forms, namely the functional and the structural. 4. PROPERTIES OF ESTIMATOR In order to study asymptotic properties, the ml m 2..... m, are assumed to be asymptotically The asymptotic properties of the estimator b have cooperative in the sense that rh = n -~ y'" m~ and been studied by Fuller and Hidiroglou [7] for the Sm,. = n- ~ ~" (m i -- m)2 tend to finite limits as/1 grows structural variant of ultrastructural model under the large. This specification is required to avoid the assumption of normality. We, however, assume that presence of any trend; see Schneeweiss [3]. errors are not necessarily normally distributed, and consider a general model. In order to study the large sample asymptotic 3. A CONSISTENT ESTIMATION PROCEDURE properties of the consistent estimator b, let Combining (1), (2) and (3), we get Zvt, = n- U2v'Av - nu2a 2

Yi = O: + fix i + (U i -- fit;i). (7) Zww = n- l/2w'Aw -- nt/2a 2 Writing 1" l" Zmu = n ~/2m'Au; Z,, v = n- U2m'Av Sxx = 2 (Xi -- .~)2; ~ : 2 Xi l1 n Z,,w = n-l/2m'Aw; Z,v = n l/2u'Av

1" 1 Z,,~, = n- l/2v'Aw; Z,w = n- l/2u'Aw, (12) s~,. = .- y (x, - ~)(y, - ;): ; = - ~ y, (8) n /1 where u = Col.(u 1, u 2..... u,) the least squares estimator of/3 is given by v = Col.(v1, v2..... v.) # _ ~x, (9) w = Col.(w~, w2 ..... w,) Sxx e = Col.(l, 1..... 1) which is a consistent estimator and possesses minimum variance in the class of linear and unbiased estimators 1 A =l,--ee'. of/3 provided that the observations on explanatory n variable are free from any measurement error. When these observations are contaminated by measurement Further, we introduce the following notation: errors, it is well known that D is neither consistent nor unbiased. r= - ' 0

2 + Z~ + Z~w) + p)ZI~ Smm + a,,, + (1 - 2p)(Zm~ 2(1 - p - 2 a~ (101 Smm -4- aw JI- + /3- ~(Z,.. + Z,~ + Z,~)]. (15) Properties of a consistent estimation procedure 1251

Employing eqns (1)-(4), we can express (16) as Equation (23) is also obtained when the distribu- follows: tions are not mesokurtic but satisfy the constraint: 72v P d2 [b;fl] n_1/2 (1--P)[1 +n-l/Zh~]-lh - (25) 72w 1 - p

=(1-P)[n-1/2h).-n lhxh~,] +Op(n >2). Further, from (23), we notice that the relative bias P is always positive. This bias increases when (16) 72,. Pde . > (26) Thus, the relative bias to order O(n- ~) is given by 72w 1 -- p while it may decrease when the inequality (26) holds true with a reversed sign. Similarly, we observe from (20) and (24) that the = (1 7_ p) [n- ~/2E(h,) - n- ~E(h~h,)]. (17) relative mean squared error to the order of our P approximation may decline making b more efficient Similarly, the relative mean squared error to order than in the mesokurtic or normal case when both the O(n- I) is distributions of v and w are platykurtic. If both the distributions are leptokurtic, the efficiency declines. When one is platykurtic and the other is leptojurtic, the net effect on the efficiency depends upon the value /1 ,[-I - Pl2~-- 2 of the factor (72~, + d272w) • If this factor is negative, = LIT-j u8) the precision increases while it decreases when this factor is positive. Substituting the expected values of hy, hxhy and h 2 The above discussion brings out the effect of from the Appendix, we obtain the following results. departure from normality of the error distribu- tions on the efficiency properties of the consistent Theorem estimator b. Next, let us consider the two popular variants of the The large sample asymptotic approximations for ultrastructural model, namely the structural and the relative bias and relative mean squared error of functional which are the two extremes of the ultra- b up to order O(n-1) are given by structural model. Setting d = 0 and d = 1, we find

RB(b) - (1 - p) [B + (1 - P)Y2v + d272w] (19) (1 - p) n RB(b)f - [1 + 2p + 72,~] (27) /1

RM(b) = (1- p)2 [M + )'2~, + d272,a.1, (20) /1 RB(b)s = {1 -p) [(1 - P)72~, - P)'2,~,] (28) where n B = (1 - d) + 2p(1 - d) 2 (21) R B(b) f = (-1 ~ P-)2--[p~l-~._ p) -]- 72 t,] (29) l+r M - 2(1 - d) 2. (22) p(1 - p) /1 fil 2 )) + 72° + ~'2~ • (30) An interesting observation emerging from the expressions (19) and (20) is that only the relative It is thus seen that the estimator b is generally biased peakedness of the error distributions associated with for both the structural and functional models. It the explanatory variable influences the efficiency vanishes for the structural model in two cases. One is properties of the estimator; the skewness has no role when the error distributions are mesokurtic and the to play. Further, the skewness and kurtosis of the other is when the error distributions are not meso- error distribution associated with study variable kurtic but satisfy the condition (25) with d = 1. exhibits no influence at all on the properties of Comparing the relative mean squared errors and estimator, at least to the order of our approximation. observing that (2 + 7>~) cannot be negative, we find When the distributions of v and w are mesokurtic that the estimator b is more efficient when used for (or normal) so that 72, = ",'2w = 0, the expressions (19) functional model in comparison to structural model. and (20) reduce to the following:

RB(b) - (1 - p) B (23) 5. CONCLUSIONS tl We have considered the estimation of slope RM(b) = !1_- p)~ M. (24) coefficient in a linear regression model when both /1 the variables are subject to measurement errors. 1252 A.K. Srivastava and Shalabh

Employing an ultrastructural framework and assuming 5. F. M. Lord and M. R. Novick, Statistical Theories of the reliability ratio to be known, a consistent Mental Test Scores. Addison-Wesley, Reading, MA estimation procedure is described and its large sample (1968). 6. K. H. Marquis, M. S. Marquis and J. M. Polich, Response asymptotic properties are studied when errors are not bias and reliability in sensitive topic surveys, J. Am. Star. necessarily normally distributed. Our analysis has Ass. 81, 381-389 (1986). revealed some interesting results related to effect of 7. W.A. Fuller and M. A. Higiroglou, Regression estimation departure from the normality of measurement errors after correcting for attenuation, J. Am. Stat. Ass. 33, 99-104 (1978). on the efficiency properties of the consistent estimator. 8. T. D. Stanley, Forecasting from fallible data: correcting The performance of this estimator in two popular prediction bias with Stein-rule least squares, J. Forecasting models, namely functional and structural models, is 7, 103-113 (1988). also studied. Our investigations can be easily extended to the case of a model containing more than one explanatory APPENDIX variable on the lines of Fuller ([1], Section 2.2) and Up to order O(1), we have Fuller and Hidiroglou [7]. Further, it would be n- U2E(hy) = p(l - d) (A1) interesting to conduct some Monte Carlo studies and to apply bootstrap techniques to some real world data. E(hxhy) = p2[d2y2w - 2(1 - d) 2] - p(l - P)Y2~, (A2) These may shed some additional light on the perfor- p(l + r) mance properties of estimator. Employing the present E(h 2) - + p2[(y2v - 2) + d2(~,2w - 2) + 4d]. (1 - p) consistent estimator, one can formulate predictors for (A3) the actual and average values of the study variable and explanatory variables. Analysing the properties Proof. It may be recalled that the three sets (ul, u 2..... u,) @1, v2..... v,) and (wl, w 2..... w,) are of predictors will be an interesting exercise. stochastically independent. Further, the elements in each We have assumed that the true value of reliability set are identically and independently distributed. Utilizing ratio is known. Often this value may not be available. these properties, it is easy to see that In such circumstances, it may be imperative to n- 1/:E(Zww) = - ~, estimate it, for instance, using repeated observations or utilizing some independent sample or obtaining n- 1/~E(Zoo) = -~ some guess. Every one of these propositions will alter while the expected values of Zm., Z,.v, Z,.w, Z.L,, Zvw, and the properties of estimator. To what extent will be an Z.w are all zero. interesting question for investigation; see Stanley [8]. Employing these results we obtain from eqn (15) the result (A1). Similarly, dropping the terms with zero expected values, AcknowledtJement--The second author gratefully acknow- we find from eqns (14) and (15) the following result: ledges the financial support from the Council of Scientific and Industrial Research (CSIR) of India in the form of a E(hxhr) = (1 - p)_ E[-(1 - p)Z2~, - pZL2,~, Junior Research Fellowship.

+ 2(1 - p)(Z~v + Z~) + 4(1 - p)Z~w]. REFERENCES It can be easily verfied that Measurement Error Models. 1. W. A. Fuller, John Wiley, E(Z~.) = ~t(2 + ~,) New York (1987). 2. G. R. Dolby, The ultrastructural relation: a synthesis of E(Z~) = at(2 + ~) the functional and structural relations, Biometrika 63, 39-50 (1976). E(zL) = "~S~m 3. H. Schneeweiss, Note on a linear model with errors in E(zL,) = ~ts.. the variables and with trend, Stat. Papers, 32, 261-264 (1991). 4. R. Ashley and D. Vaughan, Measuring measurement error up to order 0(1). in economic times series, J. Business Econ. Stat. 4, 95-103 Using these expressions, we find the result (A2). (1986). Equation (A3) can be derived in a similar manner.