Malaysian Jurnal f Mathematical Sciences 4(): 7-4 () On Huntsberger Type Shrinkage Estimatr fr the Mean f Nrmal Distributin Department f Mathematical and Physical Sciences, University f Nizwa, Sultanate f Oman E-mail: alhemyari@unizwa.edu.m; drzuhair@yah.cm ABSTRACT The present article deals with an imprved estimatr f nrmal mean which is btained by cnsidering single stage prcedure with apprpriate shrinkage weight functin. Indeed tw shrunken testimatrs f Huntsberger-type are prpsed fr the mean µ f nrmal distributin when a prir estimate µ f the mean µ is available. The gdness f this prcedure as a means f maximizing the relative efficiency is explained in this paper. The expressins fr the bias mean squared errr and relative efficiency f the prpsed testimatrs are derived. The perfrmances f the prpsed testimatrs are cmpared with classical and existing testimatrs based n the criteria f biased rati and relative efficiency t search fr a better estimatr. Keywrds: Nrmal distributin; Huntsberger-type shrinkage testimatr; Preliminary test; Bias rati; Relative efficiency. The mdel INTRODUCTION The nrmal distributin plays an imprtant rle in bth the applicatin and inferential statistics. In mdeling applicatins, the nrmal curve is an excellent apprximatin t the frequency distributins f bservatins taken n a variety f variables and as a limiting frm f varius ther distributins (Davisn, (3)). Many psychlgical measurements and physical phenmena can be apprximated well by the nrmal distributin. In additin, there are many applicatins f the nrmal distributin in engineering. One applicatin deals with analysis f items which exhibit failure due t wear, such as mechanical devices. Other applicatins are, the analysis f the variatin f cmpnent dimensins in manufacturing, mdeling glbal irradiatin data, and the intensity f laser light, and s n. Indeed the wide applicatin and ccurrence f the nrmal distributin in life testing and reliability prblems are a wnder. In the cntext f reliability prblems and life testing, a number f failure time data have been examined (Bain and Engelharadt, (99)) and it was shwn that the nrmal distributin give quite a gd fit fr the mst cases.
Incrprating a guess value, and PSE In many prblems, the experimenter has sme prir infrmatin regarding the value f µ either due t past experiences r t his familiarity with the behavir f the ppulatin. Hwever, in certain situatins the prir infrmatin is available nly in the frm f an initial guess value (natural rigin) µ f µ. In such a situatin it is natural t start with an estimatr X f µ and mdify it by mving it clser t µ, s that the resulting estimatr, thugh perhaps biased, has smaller mean squared errr than that f X in sme interval arund µ. This methd f cnstructing an estimatr f µ that incrprates the prir infrmatin µ leads t what is knwn as a shrunken estimatr. Cnsider the Huntsberger, (955) type shrinkage estimatr ɶ µ = { ψ ( X )( X µ ) + µ }, () ϕ where ψ ( X )( ψ ( X ) ), represents a weighting functin specifying the degree f belief in µ. A number f authrs (Gdman, (953), Thmpsn, (968a), Arnld, (969) and Mehta and Srinivasan, (97)) have tried t develp new s shrinkage estimatrs f the frm () fr special ppulatins by chsing different weight functins. The relevance f such types f shrinkage estimatrs lies in the fact that, thugh perhaps they are biased, have smaller MSE than X in sme interval arund µ. Shrinkage estimatrs f the frm () have the disadvantage that it necessarily uses the prir value in the cnstructin f final estimatrs. Hwever, it is nt necessary that the prir value is clse t true value. T emply this idea in estimatin f the mean µ, a preliminary test is first cnducted t check the clseness f µ t µ befre using it in a shrinkage estimatr. Define ϕ ( X), if H : µ = µ, ψ( X) =, if H : µ µ, () where ϕ( X ). Thus, the preliminary shrinkage estimatr (PSE) f µ crrespnding t ψ ( X ) is given by, ˆ µ = {[ ϕ ( X )( X µ ) + µ ] I + [ X ] I }, (3) R R 8 Malaysian Jurnal f Mathematical Sciences
where I R and On Huntsberger Type Shrinkage Estimatr fr the Mean f Nrmal Distributin I are respectively the indicatr functins f the acceptance R regin R and the rejectin regin R. A number f ther authrs (Thmpsn, (968a, 968b), Davis and Arnld, (97), Hiran, (977), Kamb et al., (99), Singh et al., (4), Singh and Saxena, (5), Lemmer, (6), Al-Hemyari et al., (9b), Al-Hemyari, (9a, )) have tried t develp new shrinkage estimatrs f the frm (3) fr special ppulatins by chsing different weight functins and R. In this paper tw preliminary shrinkage estimatrs f Huntsberger-type fr the mean µ f nrmal distributin N ( µ, σ ) when σ is knwn r unknwn are prpsed, and the expressins fr the bias, mean squared errr, and relative efficiency are derived and studied numerically. The discussin regarding the cmparisns with the earlier knwn results are made and the usefulness f these testimatrs under different situatins is prvided as cnclusins frm varius numerical tables btained frm simulatin results. TESTIMATOR µɶ AND ITS PROPERTIES WITH KNOWN σ Let X be nrmally distributed with unknwn µ and knwn variance σ. Assume that a prir estimate past. The first prpsed testimatr is, ɶ µ = {[ X ae n b( X ) / µ σ ( X µ )] I R +[ X ] µ abut µ is available frm the I R }. (4) Fr this testimatr we cnsidered R as the cmmnly used acceptance regin f the hypthesis H : θ = θ against the alternative H : θ θ. If α is the level f significance f the test, then the preliminary test regin R is given by R = { X : T ( X ) [ L, U ]}, (5) α / α / where l α / and U α / are the lwer and upper (α /) percentile pints f the statistic T( X ) used fr testing the abve hypthesis. If the standard nrmal statistic T( X ) is used, the regin R is given by: σ σ [ ( n b X µ ) / σ R } = [ µ zα /, µ + zα / ], ϕ( X ) = [ a e ], (6) n n Malaysian Jurnal f Mathematical Sciences 9
b, a, and z α / is the upper ( α / ) percentile pint f the standard nrmal distributin. The bias expressin f ~µ is defined by, B( ɶ µ µ ) = E( ɶ µ ) µ R nb( X µ ) / σ µ µ µ σ = (( ae )( X ) + ) f ( X /, ) d X + X f ( X / µ, σ ) d X µ, R (7) where R is the rejectin regin and f ( X µ, σ ) is the p.d.f. f X. Simple calculatins lead t bλ /(b+ ) σ ae B( ɶ µ µ, R ) = ( ) [ b + J 3/ ( a, b ) + λ J ( a, b )]. (8) n (b + ) The mean squared errr expressin f ~µ is defined by, MSE( ɶ µ µ, R ) E( ɶ µ µ ) (( ae )( X µ ) µ µ ) nb( X µ ) / σ = = + R f X d X X f X d X ( / µ, σ ) + ( µ ) ( / µ, σ ). R (9) After sme algebraic manipulatins, it can be easily shwn that σ 5 / bλ /(b+ ) MSE ( ɶ µ µ, R ) = a(b + ) e [ (b + ) J ( a, b ) n + λ ( b) b + J ( a, b ) bλ J ( a, b ) + a (4b + ) e 5 / ] bλ /(4b+ ) x (4b + ) J ( a3, b3 ) + λ 4b + J ( a3, b 3) + λ J ( a3, b3 )], () where a = λ z, b = λ + z, a = ( λ z )/ b +, b = ( λ + z )/ b +, α / α / α / α / a = ( λ z )/ 4b, b = ( λ + z )/ 4b, λ = n( µ µ )/ σ, 3 α / 3 α / 3 Malaysian Jurnal f Mathematical Sciences
and On Huntsberger Type Shrinkage Estimatr fr the Mean f Nrmal Distributin b j i y / Ji( aj, bj) = y e dy, i =,,, j =,. π aj Fr numerical cmputatins we may use the relatins: ( ) J( a, b) = exp( a ) exp( b ) π, () and ( ) J ( a, b) = J ( a, b) + exp( a ) exp( b ) π. () Remark : Chice f a It seems reasnable t select a that minimizes the MSE ( ɶ µ µ ). Setting ( / a) MSE ( ɶ µ µ ) ) t zer, we get, ( ( ) 5 / bλ /(b+ )(4b+ ) a = a = (4b + ) /(b + ) e {[(b + ) J ( a, b ) + λ ( b) b + J ( a, b ) bλ J ( a, b )]/[(4b + ) J ( a, b ) + λ 4b + J ( a, b ) 3 3 3 3 3 3 + λ J ( a, b )]} (3) Since MSE ( ɶ µ µ ) = a σ 5/ bλ /(4b+ ) (4b + ) e (4b + ) J ( a3, b3 ) + λ 4b + J ( a3, b3) + λ J ( a3, b3 ) ], n it fllws that the minimizing value f a [,] is given by: if a, aɶ = a if a, (4) if a. The efficiency f ~µ relative t X is defined by, Eff ( ɶ µ µ ) = σ / MSE ( ɶ µ µ ). (5) Malaysian Jurnal f Mathematical Sciences 3
Remark : Sme prperties. It is easily seen that B ( ɶ µ µ ) is an dd functin f λ, whereas E( n µɶ ), MSE ( ɶ µ µ ) and Eff ( ɶ µ µ ) are all even functins f λ. Als Eff ( ɶ µ µ ) = as λ ±. Since lim B ( ɶ µ µ ) =, and lim MSE ( ɶ µ µ ) =, µɶ is a asympttically n n unbiased and cnsistent estimatr f µ. Als µɶ dminates X in large n and n in the sense that lim [ MSE ( ɶ µ µ ) MSE ( X)]. n Remark 3: Special cases. It may be nted here, when ϕ ( X ) = c, c is cnstant the equatins (8), () & (5) agree with the result f Thmpsn, (968a); when the hypthesis H : µ = µ is accepted with prbability ne the same expressin aggress with the result f Mehta and Srinivasan, (97); als when a =, the same expressin aggress with the result f Hiran, (977), and when ϕ ( X ) = c, c, if X R and ϕ ( X ) = c, if X R, the result agrees with the result f Al-Hemyari et al., (9b). When TESTIMATOR µɶ WITH σ UNKNOWN σ is unknwn, it is estimated by n i i= s = ( X X ) /( n ). Again taking regin R as the pretest regin f size α fr testing H : µ = µ against H : µ µ in the testimatr µɶ defined in (3) and denting the resulting estimatr as µɶ. The testimatr µɶ emplys the interval R given by, R = µ tα /, n s / n, µ + tα /, n s / n, (6) where t α /, n is the upper ( α / ) percentile pint f the t distributin with n degrees f freedm. The expressins fr bias and MSE and are given respectively by: 3 Malaysian Jurnal f Mathematical Sciences
On Huntsberger Type Shrinkage Estimatr fr the Mean f Nrmal Distributin bλ /(b ) σ + ae * * * * B( ɶ µ µ ) = [ b + J 3/ ( a, b ) + λ J( a, b )] f ( s σ ) ds (7) n (b + ) MSE ( ɶ µ µ ) = σ n { a(b + ) e [(b + ) J ( a, b ) + λ ( b) b + J ( a, b ) 5/ bλ /(b+ ) * * * * 5/ bλ /(4b+ ) * * * * 3 3 λ 3 3 bλ J ( a, b )] + a (4b + ) e [(4b + ) J ( a, b ) + 4b + J ( a, b ) + λ J ( a, b )]} f ( s σ ) ds, * * 3 3 where * s * s * a3 = ( λ tα /, n )/ 4b +, b3 = ( λ + tα /, n )/ 4b +, a = ( λ tα /, n x σ σ s * s * s * x )/ b +, b = ( λ + tα /, n )/ b +, a = λ tα /, n, b = λ + tα /, n, σ σ σ and f ( s σ ) is the p. d.f. f s. If µ = µ, the abve expressins reduce t: B ( ɶ µ µ ) =, (9) [ σ a a MSE( ɶ µ µ, R ) = + ( α)[ ] 4 a t 3/ 3/ α /, n Γ( n / ) / n (b + ) (4b + ) t α /, n (4b + ) n / n [ n π ( n ) Γ(( n ) / ) x x( + ) ] + a tα /, n Γ / n n / n tα /, n (4b + ) π ( n ) Γ (4b + ) + }. n Remark 4: Chice f a Prceeding in the manner as in last sectin, we get the minimizing value f a as fllws, ( α ) a a t b + [ 3 α n ( ) n = =, Γ ( + t α, n ( b + ) x x( n ) n ) ]] n [ π ( n ) Γ ( b + ) ( α ) ( 4b + ) [ ( n ) Γ ( b + ) x x + t ( b + ) n + t 3 α, n Γ ( α, n ( n )) / n [( π 4 4 n ]] () Malaysian Jurnal f Mathematical Sciences 33
Since σ ( α) n n MSE ( ɶ µ, Y µ R ) = + 4 t 3/ α /, n Γ /[ π( n ) Γ (4b + ) a n (4b + ) n / tα /, n (4b + ) + } >, () n it fllws that the minimizing value f a [,] is given by: if a aɶ = a if a () if a The relative efficiency f µɶ defined by, Eff ( ɶ µ µ ) = σ / MSE( ɶ µ µ ). (3) SIMULATION AND NUMERICAL RESULTS T bserve the behavir f the prpsed testimatrs, and t give useful cmparisn between the prpsed, classical, and existing estimatrs, the cmputatins f relative efficiency, bias rati, were dne fr the testimatrs µɶ and µɶ. Specifically, fr testimatr ~µ these cmputatins were dne fr α =.,.,.5,.,.5, b =.,.,., and λ =.(.) 4, whereas fr µɶ this was dne fr α =.,.,.5, b =.,., n = 4(4) and in rder t cmpare ur testimatrs µɶ and µɶ with thse f Al-Hemyari et al.,(9), Saxena and Singh, (6), Kamb et al., (99), Hiran, (977), Davis and Arnld, (97), Arnld, (969) and Thmpsn, (968a). Sme f these cmputatins are given in Tables t 3. We make the fllwing bservatins frm tables f cmputatins presented in this paper: i) The testimatr µɶ is biased (see Table ). The bias rati is reasnably small if the prir pint estimate µ desn t deviate t much frm the true value µ. 34 Malaysian Jurnal f Mathematical Sciences
On Huntsberger Type Shrinkage Estimatr fr the Mean f Nrmal Distributin ii) It is bserved that, fr λ 3, µɶ has smaller mean squared errr than the classical estimatr X. Thus µɶ may be used t imprve the efficiency if λ belng t the regin λ 3. iii) Fr fixed n and α, the relative efficiency f µɶ is increases with decreasing value f b. iv) As expected ur cmputatins (see Table ) shwed that the relative efficiency f µɶ decreases with size α f the pretest regin, i.e. α =. gives higher relative efficiency than fr ther values f α. v) Fr fixed b, and α, the relative efficiency f ~µ is maximum when λ, and decreases with increasing value f λ. v) Fr any fixed α and b, the relative efficiency is a decreasing functin f n when λ. xi) The behaviural pattern f testimatr ~µ is similar t that f ~µ as fr relative efficiency, and bias rati are cncerned (see Table 3 ). TABLE : Shwing Eff ( ɶ µ µ ) when b =.,., α =.,.5,.,.35, and λ =.(.)()3. b α....3.4.5.6. 8.643 7.35 6.644 6.35 5.7 5.377 4.67..5 6.554 5.399 4.937 4.39 3.93 3.663 3.. 4.893 4.53 3.847 3.893 3.85.964.7.35 3.776 3.783.655.444.3.96.9. 6.53 6.87 5.74 5.8 4.693 4.35 3.83..5 5.57 5.4 4.77 4.63 3.864 3.5.93. 3.973 3.6 3.64.953.75.43.66.35.95.794.75.635.49.97.5 Malaysian Jurnal f Mathematical Sciences 35
TABLE (cntinued): Shwing Eff ( ɶ µ µ ) when b =.,., α =.,.5,.,.35, and λ =.(.)()3. b α.7.8.9..5..5 3.. 4. 3.856 3.46.937.99.37.99.5..5.83.63.37.84.74.36.4.5..74.573.985.765.499.4.9..35.93.893.8.73.5.53.97.. 3.53 3.64.735.595.8.5.83.3..5.7.494..975.645.4....73.99.854.673.588.63.6..35.98.796.633.43.3.48.3.957 TABLE : Shwing B ( ɶ µ µ ) when b =.,., α =.,.5,.,.35, and λ =.(.)()3. b α....3.4.5.6.. -.49 -.5 -.66 -. -. -.8..5. -.36 -. -.6 -.94 -.9 -.4.. -. -.85 -.43 -.7 -.85 -.95.35. -.5 -.537 -.5 -.48 -.6 -.73.. -.4 -.4 -.6 -. -. -.7..5. -.35 -. -.58 -.9 -. -.4.. -. -.83 -.4 -.7 -.85 -.94.35. -.4 -.5 -.4 -.4 -.6 -.73 36 Malaysian Jurnal f Mathematical Sciences
On Huntsberger Type Shrinkage Estimatr fr the Mean f Nrmal Distributin TABLE (cntinued): Shwing B ( ɶ µ µ ) when b =.,., α =.,.5,.,.35, and λ =.(.)()3. b α.7.8.9..5..5 3.. -.3 -.33 -.45 -.5 -.94 -.36 -.385 -.43..5 -.6 -.9 -.35 -.38 -.63 -.38 -.36 -.4. -.98 -.4 -.7 -.3 -.56 -.94 -.333 -.395.35 -.88 -.9 -.97 -.98 -.36 -.77 -.3 -.376. -.3 -.3 -.43 -.45 -.9 -.34 -.383 -.49..5 -. -.4 -.53 -.36 -.6 -.33 -.36 -.47. -.97 -.3 -.6 -. -.55 -.93 -.33 -.394.35 -.87 -.9 -.95 -.97 -.35 -.76 -.3 -.374 TABLE 3: Shwing Eff ( ɶ µ µ ) when b =.,., α =.,.5,.,.35, and λ =.. b n α..5..35 4 4.78.34 7.56 5.35 8 7.34.79.838 7.83. 7.93 4.48.78 8.935 6 8.3 5.6 4.464 9.673 8.58 5.855 3.45.38 4 4.643.45 7.534 5.5 8 7.7.6.769 7.773. 7.7 4.78.85 8.86 6 8.85 5. 3.347 9.5 8.349 5.69 4.34.8 Malaysian Jurnal f Mathematical Sciences 37
CONCLUSION Testimatr µɶ is better than that f, Saxena and Singh, (6), Kamb et al., (99), Hiran, (977), Davis and Arnld, (97), Arnld, (969) and Thmpsn, (968a) bth in terms f higher relative efficiency and barder range f λ fr which efficiency is greater than unity and better than Al-Hemyari et al., (9b) if λ.4. Als, n cmparing Table 3 (fr unknwn σ case) with the crrespnding results f Saxena and Singh, (6), Hiran, (977), Davis and Arnld, (97), Arnld, (969) and Thmpsn, (968a), it has been bserved that µɶ is als much better in terms f higher relative efficiency than the existing testimatrs with unknwn σ. ACKNOWLEDGEMENTS I am pleased t thankful t the referees, whse suggestins imprved the paper cnsiderably. REFERENCES Al-Hemyari, Z A.. Sme testimatrs fr the scale parameter and Reliability Functin f Weibull Lifetime Data. J. f Risk and Reliability, 4:-9. Al-Hemyari, Z A. 9a. Reliability Functin Estimatr with Expnential Failure Mdel fr Engineering Data. Prceedings f the 9 Internatinal Cnference f Cmputatinal Statistics and Data Engineering, IAENG, Lndn, II:38-34. Al-Hemyari, Z A. Khurshid, A. Al-Jbry, A. 9b. On Thmpsn type estimatrs fr the mean f nrmal distributin. Revista Investigacin Operacinl, 3():9-6. Arnld, J. C. 969. A mdified technique fr imprving an estimate f the mean. Bimetrics, 5:45-44. Bain, L. J. and Engelharadt, M. 99. Statistical Analysis f Reliability and Life Testing Mdels. New Yrk: Marcel Dekker. Davisn, A. C. 3. Statistical mdels. Cambridge University Press. 38 Malaysian Jurnal f Mathematical Sciences
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