ESTIMATION OF THE RECIPROCAL OF THE MEAN OF THE INVERSE GAUSSIAN DISTRIBUTION WITH PRIOR INFORMATION

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1 STATISTICA, anno LXVIII, n., 008 ESTIMATION OF THE RECIPROCAL OF THE MEAN OF THE INVERSE GAUSSIAN DISTRIBUTION WITH PRIOR INFORMATION 1. INTRODUCTION The Inverse Gaussian distribution was first introduced by (Schrodinger, 1915). Stemming from the earlier work by (Tweedie, 1947), the Inverse Gaussian (IG) distribution has received considerable attention in the statistical literature during the last two decades. Over the years it has found alications in different areas ranging from reliability and life testing to meteorology, biology, economics, medicine, market surveys and remote sensing. See (Seshadri, 1999) for an extensive list of alications (Sen and Khattree, 000). The review aer by (Folks and Chhikara, 1978) has resented various interesting roerties of the IG distribution. There are various alternative forms of a IG distribution available in the literature. Out of these we choose to work with the most familiar one having robability density function (.d.f). 1 λ λ ( x µ ) f( x, µλ, ) = ex 3 π x µ x, x > 0, λ > 0, µ > 0 (1) Here, the arameters µ and λ are called location and shae arameters resectively. The mean and the variance of the IG distribution corresonding to (1) 3 are resectively given by E( X ) = µ and V( X ) = µ λ. In cases, where the distribution has arisen from the inverse Gaussian rocess, one may be interested in 1 estimating the recirocal of the mean ie.. µ. Let x1, x,..., x n be a random samle of size n from the Inverse Gaussian distribution (1). It is well known that the maximum likelihood estimators for µ and λ are ˆ µ x n i = x = and ˆ n λ = i = 1 n V n 1 1, where V = i = 1 xi x.

2 0 The samle mean x follows Inverse Gaussian distribution with arameters µ and nλ. The samle mean x is unbiased for µ where as ˆλ is a biased estimator λ. The statistics ( xv, ) are jointly comlete sufficient statistics for ( µ, λ ) if both µ and λ are unknown. When λ is known, the uniformly minimum variance unbiased estimator (UMVUE) of the recirocal mean (1/ µ ) is ˆ 1 1 θ1 = x nλ () with the variance ˆ 1 1 Var( θ1) = + nλ µ nλ (3) We note that in many ractical situations the value of λ is not known, in such a case the maximum likelihood estimator (mle), uniformly minimum variance unbiased estimator (UMVUE) and minimum mean squared error (MMSE) estimators of (1/ µ ) are resectively given by and ˆ 1 θ ml =, (MLE) (4) x ˆ 1 V θ UMVUE =, (UMVUE) (5) x n( n 1) ˆ 1 V θ MMSE =, (MMSE) (6) x n( n+ 1) It is ointed out in (Sen and Khattree, 000) that the estimators ˆml θ ˆMMSE θ are articular members of the following class of estimators { V ; 0}, ˆUMVUE θ and ˆ 1 θ = α α (7) x The class ˆ θ is a convex subsace of the real line. But it does suffer from certain undesirable features. For instance, this class does not ensure the non-negativity of the estimators and unless α = 0, with ositive robability, any estimator includ-

3 Estimation of the recirocal of the mean of the Inverse Gaussian distribution etc. 03 ing the UMVUE and MMSE in ˆ θ can take negative values and hence be out of the arameter sace. Following (Lehmann, 1983,. 114) one may comment on ˆUMVUE θ as follows ˆUMVUE θ can take negative values although the estimate is known to be non-negative. Excet when n is small, the robability of such values is not large, but when they do occur they cause an embarrassment. This difficulty can be avoided by relacing it by zero whenever it leads to a negative value, the resulting estimator of course, will no larger be unbiased. This roblem has been further discussed by (Pandey and Malik, 1989). It is to be mentioned that the estimator ˆ θ 1 in () can be used only when the value of the arameter λ is known. The value of λ is not known in most of the ractical situations. However, in many ractical situations the exerimenter has some rior estimate regarding the value of the arameter, either due to his exerience or due to his acquaintance with the behavior of the system. Thus the exerimenter may have evidence that the value of λ is in the neighborhood of λ 0, a known value. We call λ 0 the exerimenter s rior guess see (Pandey and Malik, 1988). The study of the estimators based on rior oint estimate (or guess value) λ 0 revealed that these are better (in terms of mean squared error) than the usual estimators when the guess value is in vicinity of true value. This roerty necessitated the use of reliminary test of hyothesis to decide whether guess value is in vicinity of true value or not. The intention behind reliminary test was that if guess value is in vicinity of the true value, the estimator based on rior oint estimate or guess value λ 0 should be used otherwise usual estimators. In this aer, we define the estimator of the recirocal of the mean (1/ µ ) by incororating the additional information λ 0 of λ. The mean square error criterion will be used to judge the merits of the suggested estimator. Numerical illustrations are given in the suort of resent study. The relevant references in this connection are (Thomson, 1968), (Singh and Saxena, 001), (Singh and Shukla, 000, 003), (Tweedie, 1945, 56, 57a, 57b), (Travadi and Ratani, 1990), (Jani, 1991), (Kourouklis,1994), (Iwase and Seto, 1983, 85), (Iwase, 1987), (Korwar, 1980), (Srivastava et al. 1980), and (Singh and Pandit, 007, 008).. SUGGESTED ESTIMATOR AND ITS PROPERTIES If samle size is sufficiently large the maximum likelihood estimators (MLE) are consistent and efficient in terms of mean squared error. If samle size is small, there is relatively little information about the arameter available from the samle and if there is any rior estimate for the arameter, the shrinkage method can be useful, see (Pandey, 1983). Let the rior oint estimate λ 0 of the arameter λ be available. Then we define the following estimator for (1/ µ ) as

4 04 ˆ 1 θ x δ 1 n 0 = + ˆ δ W δ0, (8) where δ ˆ ˆ 0 = 1/ λ0, δ = 1/ λ = V/( n 1) is the UMVE of δ = 1/ λ ; W is a constant such that MSE of ˆ θ is least and is a non-zero real number. Taking exectation both sides of (8) we have ˆ E( V ) E( θ) E δ0 Wδ = + 0 x n ( n 1) (9) We know for I.G. distribution (1) that 1 1 µ E = 1+ x µ nλ (10) and n+ 1 Γ E( V ) = λ n 1 Γ (11) Using (10) and (11) in (9) we have n+ 1 ˆ Γ 1 E( θ) = + δ0 W + δ0 µ nλ n n 1 λ Γ ( n 1) Thus we get the bias of ˆ θ as n+ 1 ˆ 1 1 Γ 1 B( θ ) 0 W 0 nλ n δ δ = + n 1 λ Γ (1) The mean squared error of ˆ θ is given by

5 Estimation of the recirocal of the mean of the Inverse Gaussian distribution etc. 05 n+ 4 1 Γ 1 (1 ) δ MSE = MSE η W η. + + x n ( n 1) n 1 Γ n+ 1 Γ + Wη ( n 1) n 1 Γ n+ 1 Γ δ 1 η Wη + n ( n 1) n 1 Γ, (13) 1 where δ = λ, δ0 λ η = δ = λ and δ 1 3δ = + = + n MSE x n λµ λ n µ n (14) The MSE at (13) is minimized for δ0 δ δ0 W = W( ), (15) δ δ where n+ 1 Γ n 1 W( ) = n+ 4 1 Γ Since δ in (15) is unknown, therefore, relacing δ by its estimate ˆ δ, we get an estimate of W as ˆ δ0 δ δ 0 Ŵ = W( ) ˆ δ δ 0 (16)

6 06 Thus relacing W by Ŵ in (8), we get a class of shrinkage estimators of (1/ ) µ as ˆ θ 1 1 [ ( )( ˆ )] ( ) = 0 W 0 x n δ + δ δ = + W( ) x n λo λ λ 0 (17) As ointed out earlier regarding UMVUE ˆUMVUE θ, we note that the resulting family of shrinkage estimators ˆ θ ( ) can also take negative values for smaller values of n. However, this difficulty can be avoided by relacing it by zero whenever it yields negative values. The bias of ˆ θ ( ) is given by (17) ˆ δ B( θ( )) = [{1 W( )}{1 η}] (18) n and the MSE of ˆ θ ( ) is given by ˆ 1 δ ( W( )) MSE( θ( )) = MSE + ( η 1) (1 W( )) 1 + x n ( n 1) (19) which is less than 1 MSE x if 1 G < η < 1+ where G ( W( )) G1 = 1 (1 W( )) ( n 1) Thus we have the following roosition. Proosition.1 The class of shrinkage estimators ˆ θ ( ) has smaller relative mean squared error (RMSE) than that of ˆ 1 θ ml = for η (1 G1,1 + G1). x The MSE exression at (19) can be re-exressed as

7 Estimation of the recirocal of the mean of the Inverse Gaussian distribution etc. 07 ˆ ˆ δ (1 + W( )) MSE( θ( ) ) = MSE( θumvue) + ( η 1) (1 W( )) (1 W( )), n ( n 1) (0) or MSE( θ ) MSE( θ ) ( η 1) (1 ( )) n ( n 1) ( n+ 1) ˆ ˆ δ ( W( )) ( ) = MMSE + W + where ˆ 1 MSE( θumvue ) = + nµλ nn ( 1) λ and ˆ 1 ( n + ) MSE( θmmse ) = + nµλ n ( n+ 1) λ, (1) () (3) Now we state the following roositions which can be easily roved from (0) and (1). Proosition. The class of shrinkage estimators ˆ θ ( ) has smaller relative mean squared error (RMSE) than that of UMVUE ˆ 1 V θumvue =, for η (1 G, 1 + G ). x n( n 1) where G 1 (1 ( W( )) ) = ( n 1)(1 W( )). Proosition.3 The class of shrinkage estimators ˆ θ ( ) has smaller RMSE than that of MMSE estimator ˆ 1 V θmmse = x n( n+ 1) for η (1 G3, 1 + G3),

8 08 where G 1 ( W( )) = (1 W( )) ( n+ 1) ( n 1) SPECIAL CASE In this section we discuss the roerties of an estimator ˆ θ (1) (say), which is a articular member of the roosed class of shrinkage estimators ˆ θ ( ) defined at (17). For =1, the estimator θ ( ) reduces to the estimator ˆ θ (1), thus ˆ ˆ 1 1 n 1 ˆ θ(1) = δ0 + ( δ δ0 ) x n n+ 1 ˆ δ 0 = θmmse nn ( + 1) (4) Putting =1 in (18) and (19) we get the bias and MSE of ˆ θ (1) resectively as ˆ (1 η ) δ B( θ(1) ) = nn ( + 1) and (5) ˆ ˆ 4δ MSE( θ(1) ) = MSE( θmmse ) + η( η ), (6) n ( n+ 1) where MSE( ˆ θ MMSE ) is given by (3). In order to comare the bias of ˆMMSE θ with B( ˆ θ (1)), we write the bias of ˆMMSE θ as ˆ δ B( θ MMSE ) = nn ( + 1) (7) It follows from (5) and (7) that B( ˆ θ ) < B( ˆ θ ) if (1) MMSE 0< η < (8)

9 Estimation of the recirocal of the mean of the Inverse Gaussian distribution etc. 09 Further from (6) we note that MSE( ˆ ) < MSE( ˆ ) if θ(1) θ MMSE 0< η < (9) Thus we state the following theorem. Theorem 3.1 The estimator ˆ θ (1) is less biased as well more efficient than MMSE estimator ˆMMSE θ iff 0< η <. Remark 3.1 It is to be noted that the estimator ˆMMSE θ is the minimum mean squared error estimator so it has less MSE than that of usual estimator ˆ θ ml = 1/ x and UMVUE ˆUMVUE θ. Thus it is interesting to note from (9) that the estimator ˆ θ (1) is more efficient than the estimators ˆ θ, ˆ ml θ UMVUE and ˆMMSE θ under the condition 0 < η <. 4. NUMERICAL ILLUSTRATION AND CONCLUSIONS To get tangible idea about the erformance of the roosed class of shrinkage estimators ˆ θ ( ) over usual estimator ˆ 1 θ ml =, UMVUE x ˆ 1 V θumvue = and MMSE estimator ˆ 1 V θ MMSE =, we x n( n 1) x n( n+ 1) have comuted the ercent relative efficiencies (PRE s) of ˆ θ ( ) with resect to ˆ θ, ˆ ml θ UMVUE and ˆMMSE θ resectively using the following formulae: ˆ ˆ A PRE( θ( ), θ ml ) = 100 ( A+ A ) ˆ ˆ PRE( θ( ), θ UMVUE) = 100 ( A+ A1 ) ˆ ˆ 1 na na 3 PRE( θ( ), θ MMSE) = 100 ( A+ A1 ) (30) (31) (3) n where A = 3 + C, (33)

10 10 ( W( )) A1 = ( η 1) (1 W( )) + 1, (34) ( n 1) A A 1 = + C n 1, (35) 1 ( n + ) = +, (36) C nn ( + 1) 3 µ C = λ 1, is the coefficient of variation. The PRE s have been comuted for different values of = ± 1, ±. n = 5, 10, 15; η = 0.5(0.5)1.75 ; and C = 1,5. The comuted values are dislayed in tables 4.1(a). 4.(a) and 4.3(a). We have also comuted the range of η for different values of, n, η and C; and resented in tables 4.1(b), 4.(b) and 4.3(b). We note that ( n 7)( n 9) W ( ) = ( n 1) W ( 1) = ( n 5) ( n 1) W (1) = ( n 1) ( n + 1) ( n 1) W () = ( n+ 5)( n+ 3) It is observed from tables 4.1(a), 4.(a) and 4.3(a) that the ercent relative efficiencies of ˆ θ ( ), =±, ± 1; with resect to ˆml θ, ˆUMVUE θ and ˆMMSE θ resectively (.. i e PRE( ˆ θ, ˆ θ ), PRE( ˆ θ, ˆ θ ) and PRE( ˆ θ, ˆ θ )): ( ) ml ( ) UMVUE ( ) MMSE (i) attain their maximum at η = 1 (i.e. when λ coincide with λ 0 ), (ii) decrease as n increases, (iii) increase as the value of coefficient of variation (C) increases, (iv) is more than 100% for the largest range of dominance of η when =1, (i.e. ˆ θ (1)), but the gain in efficiency is smaller comared to other estimators ˆ θ ( ),

11 Estimation of the recirocal of the mean of the Inverse Gaussian distribution etc. 11 ˆ θ( 1) and ˆ θ (). However, the estimator ˆ θ () seems to be the good intermediate choice between the estimators ˆ θ( ), ˆ θ( 1) and ˆ θ (1). Comaring the results of tables 4.1(a), 4.(a) and 4.3(a) it is seen that the gain in efficiency by using ˆ θ ( ) over ˆml θ is the largest followed by ˆUMVUE θ and then ˆMMSE θ. Also the roosed family of shrinkage estimators ˆ ( ) θ is more efficient than ˆml θ for widest range of η followed by ˆUMVUE θ and ˆMMSE θ see tables 4.1(b), 4.(b) and 4.3(b). It is exected too. Thus we conclude that the suggested family of shrinkage estimators ˆ θ ( ) is to be recommended for its use in ractice when (i) the guessed value λ 0 moves in the vicinity of the true value λ. (ii) samle size n is small (i.e. in the situations where the samling is costly) and (iii) the oulation is heterogeneous. TABLE 4.1(a) Percent relative efficiency of ˆ θ ( ) over ˆml θ C P 1 η n

12 1 TABLE 4.(a) Percent relative efficiency of ˆ θ ( ) over ˆUMVUE θ C P η n

13 Estimation of the recirocal of the mean of the Inverse Gaussian distribution etc. 13 TABLE 4.3(a) Percent relative efficiency of ˆ θ ( ) over ˆMMSE θ C P η n

14 14 n Estimator TABLE 4.1(b) Range of η for different values of n ˆ θ( ) (0.0000,.8708) (0.0000,.0383) (0.0000,.3186) ˆ θ( 1) (0.0000,.0000) (0.0000, ) (0.0000, ) ˆ θ (1) (0.0000, ) (0.0000, ) (0.0000, ) ˆ θ () (0.0000,.374) (0.0000,.6774) (0.0000, ) n Estimator TABLE 4.(b) Range of η for different values of n ˆ θ( ) (0.0000,.1067) (0.3006, ) (0.3033, ) ˆ θ( 1) (0.1591, ) (0.0609, ) (0.3780, 1.96) ˆ θ (1) (0.0000,.574) (0.0000,.09) (0.0000,.099) ˆ θ () (0.0693, ) (0.1436, ) (0.1658, 1.834) n Estimator TABLE 4.3(b) Range of η for different values of n ˆ θ( ) (0.0446,1.9554) (0.3348,1.6651) (0.378,1.67) ˆ θ( 1) (0.40,1.7598) (0.199,1.8701) (0.1061,1.8939) ˆ θ (1) (0.0000,.0000) (0.0000,.0000) (0.0000,.0000) ˆ θ () (0.1635,1.8365) (0.1951,1.8049) (0.056,1.7944) ACKNOWLEDGEMENT The authors are thankful to the executive editor Professor ssa. Marilena Pillati and an anonymous referee for their constructive suggestions regarding imrovement of the aer. School of Studies in Statistics Vikram University, Ujjain, India HOUSILA P. SINGH SHEETAL PANDIT

15 Estimation of the recirocal of the mean of the Inverse Gaussian distribution etc. 15 REFERENCES J.L. FOLKS, R.S. CHHIKARA, (1978), The Inverse Gaussian distribution and its alication, a review, Journal of Royal Statistical Society, B, 40, R.V. HOGG, (1974), Adotive robust rocedures, A artial review and some suggestions for future alications and theory, Journal of the American Statistical Association, 69, KOSEI IWASE, (1987), UMVU estimation for the Inverse Gaussian distribution I ( µ, c µ ) with known C, Communication Statistics, 16, 5, KOSEI IWASE, NORAKI SETO, (1983), UMVUE s for the Inverse Gaussian distribution, Journal of the American Statistical Association, 78, KOSEI IWASE, NORAKI SETO, (1985), UMVU estimators of the mode and limits of an interval for the Inverse Gaussian distribution, Communication in Statistics, Theory and Methods, 14, P.N. JANI, (1991), A class of shrinkage estimators for the scale arameter of the exonential distribution, IEEE Transaction on Reliability, 40, R.M. KORWAR, (1980), On the UMVUE s of the variation and its recirocal of an Inverse Gaussian distribution, Journal of the American Statistical Association, 75, S. KOUROUKLIS, (1994), Estimation in the two-arameter exonential distribution with rior information, IEEE Transaction on Reliability, 43, 3, E.L. LEHMANN, (1983), Theory of oint estimation, Wiley, New York, B.N. PANDEY, (1983), Shrinkage estimation of the exonential scale arameter, IEEE Transaction on Reliability, 3,, B.N. PANDEY, H.J. MALIK, (1988), Some imroved estimators for a measure of disersion of an Inverse Gaussian distribution, Communication in Statistics, Theory and Methods, 17, 11, B.N. PANDEY, H.J. MALIK, (1989), Estimation of the mean and the recirocal of the mean of the Inverse Gaussian distribution, Communication Statistics, Simulation, 18(3), E. SCHRONDINGER, (1915), Zur theories der fall-und Steigvursuche on Tailchenmit Brownsher Bewegung, Physikaliche Zeitschrift, 16, SEN, R. KHATTREE, (000), Revisiting the roblem of estimating the Inverse Gaussian arameters, IAPQR Transactions, 5,, R. SESHADRI, (1999), The Inverse Gaussian distribution, Statistical Theory and Alication, Sringer Verlag, New York. H.P. SINGH, S.K. SHUKLA, (000), Estimation in the two-arameter Weibull distribution with rior information, IAPQR. Transactions, 5,, H.P. SINGH, S. SAXENA, (001), Imroved estimation in one-arameter Exonential distribution with rior information, Gujarat Statistical Review, 8, 1-, H.P. SINGH, S.K. SHUKLA, (003), A family of shrinkage estimators of the square of mean in normal distribution, Statistical Paers, Verlag, 44, H.P. SINGH, S. PANDIT, (007), Imroved estimation if Inverse Gaussian shae arameter and measure of disersion with rior information, Journal of statistical Theory and Practice, 1,, H.P. SINGH, S. PANDIT, (008), Estimation of shae arameter and measure of disersion of Inverse Gaussian distribution using rior information, Bulletin of statistics and economics,, S08, V.K. SRIVASTAVA, T.D. DWIVEDI, S. BHATNAGAR, (1980), Estimation of the square of mean in normal oulation, Statistica, XL, 4, J.R. THOMPSON, (1968), Some shrinkage technique for estimating the mean, Journal of the American Statistical Association, 63,

16 16 R.J. TRAVEDI, R.T. RATANI, (1990), On estimation of reliability function for Inverse Gaussian distribution with known coefficient of variation, IAPQR, Transactions, 5,, M.C.K. TWEEDIE, (1945), Inverse Statistical Variate, Nature, M.C.K. TWEEDIE, (1947), Functions of statistical variate with given means, with secial inference to lalacian distributions, Proceedings of the Cambridge Philosohical Society, 43, M.C.K. TWEEDIE, (1956), Some statistical roerties of Inverse Gaussian distribution. Virginia Journal Science, 7, M.C.K. TWEEDIE, (1957 a), Statistical roerties of Inverse Gaussian distribution, I, Annals of Mathematical Statistics, 8, M.C.K. TWEEDIE, (1957 b), Statistical roerties of Inverse Gaussian distribution, II, Annals of Mathematical Statistics, 8, SUMMARY Estimation of the recirocal of the mean of the Inverse Gaussian distribution with rior information This aer considers the roblem of estimating the recirocal of the mean (1/ µ ) of the Inverse Gaussian distribution when a rior estimate or guessed value λ 0 of the shae arameter λ is available. We have roosed a class of estimators ˆ θ ( ), say, for (1/ µ ) with its mean squared error formula. Realistic conditions are obtained in which the estimator ˆ θ ( ) is better than usual estimator, uniformly minimum variance unbiased estimator (UMVUE) and the minimum mean squared error estimator (MMSE). Numerical illustrations are given in suort of the resent study.