Estimation Approach to Ratio of Two Inventory Population Means in Stratified Random Sampling

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1 American Journal of Operational Researc 05, 5(4): 96-0 DOI: 0.593/j.ajor Estimation Approac to Ratio of Two Inventor Population Means in Stratified Random Sampling Subas Kumar Yadav, S. S. Misra,*, Alok Kumar Sukla Department of Matematics and Statistics (A Centre of Excellence), Dr. RM Avad Universit, Faizabad, U.P., India Department of Statistics, D.A-V, College, Kanpur, U.P., India Abstract Te present paper proposes a new approac of estimation to te inventor sstem wic undergoes te process of stratification b using te tecnique of stratified random sampling. Tis seeks to develop te estimation of ratio of two population means using auxiliar variable under stratified random sampling troug generalized ratio tpe estimator. Te statistical analses of existing estimators and proposed estimator ave been discussed and condition of efficient estimator as been attained. Wit te elp of computing algoritm, a numerical illustration of te problem as been also presented to meet te scientific footings of teor of estimation. Kewords Stratified Inventor, Ratio Estimator, Auxiliar Variable and Computing Algoritm. Introduction Usable and idle resources referring to man, material, macine and mone are called inventor items; vide Ackoff and Sasieni (993). Heterogeneous inventor items are stratified for optimal control of inventor sstem. Stratification of inventor sstem attempts to provide te mecanism to optimize te inventor control for eac stratum wic is believed to be ver efficient and easil minimizes te costs. Tis also makes andling of inventor eas for eac stratum and accordingl te stratum domain experts are supposed to supervise te field works of te inventor andling and management, vide for example Alex and Benn (009). In tis situation, stratified random sampling is used to develop te efficient estimator to estimate te population parameters of stratified inventor items for caracteristics under stud is not omogeneous, vide for example Cocran (940, 977). A few researces are available in tis field and terefore demands deep exploration in tis area of interdisciplinar researc. Under tis sampling sceme, te wole population of inventor items is divided into relativel omogeneous groups. Ten te required sample is drawn b taking appropriate subsamples from tese stratums using simple random sampling tecnique. Te fres objective of tis paper is to develop estimation approac to te ratio of two population means in stratified inventor sstem b evolving te efficient estimator using * Corresponding autor: sant_x003@aoo.co.in (S. S. Misra) Publised online at ttp://journal.sapub.org/ajor Coprigt 05 Scientific & Academic Publising. All Rigts Reserved stratified random sampling tecnique for inventor populations under consideration. Tis proposes estimation of ratio of two population means using auxiliar variable under stratified random sampling troug generalized ratio tpe estimator. Te expressions for te bias and mean square error (MSE) ave been obtained up to te first order of approximation. Te minimum MSE of proposed estimator is also obtained. An empirical stud is also carried out to meet te teoretical findings.. Statistical Analsis Te estimation of te ratio of two population means as been studied b man autors in te literature including Murt and Sing (965), Rao and Pareira (968), Sa and Sa (978), Ra and Sing (985), Upadaa and Sing (985), Upadaa et.al (985), Sing and Rani (005, 006) and Sindu et. al (009) etc. et us consider te population of inventor items under stud consisting of units and tis population of size is divided into stratums eac of size (,,..., ) and te required sample of size n is drawn b taking n (,,..., ) units from corresponding strata. Tus = and n= n. et 0 and be te two main variables under stud and x be te auxiliar variable. 0i, i and x i (,,...,, i =,,..., ) are te observations on te i t unit of te t stratum for te variables 0, and x respectivel.

2 American Journal of Operational Researc 05, 5(4): ow we ave te following notations wic ave been used trougout tis paper. Y0 = 0i, te t strata mean for te main i = variable under stud 0. Y = i, te t strata mean for te main i = variable under stud. X = xi, te t strata mean for te auxiliar i = variable under stud x. Y0 = Y 0 = WY 0, te inventor = = population mean for te main variable under stud 0. Y = Y = WY, te inventor = = population mean for te main variable under stud. X= X = WX, te inventor = = population mean for te auxiliar variable under stud x. n 0 = 0i n, te sample mean of 0 for t i = strata. n = i n, te sample mean of for t strata. i = n x = xi n, te sample mean of x for t strata. i = W =, te weigt of t strata. Y0 R =, te ratio of two population means of stud Y variables. It is well known tat te appropriate estimators for te estimation of inventor population parameters are te corresponding statistics. Tus appropriate estimators for population means Y 0, Y and X are te usual unbiased estimators, te sample means in stratified random sampling = W, st = W and 0st 0 x = Wx respectivel. st Te conventional estimator for te ratio of two population means R in stratified random sampling is defined as, ˆ 0st Rst = (.) st As it is well known tat te use of auxiliar information supplied b te auxiliar variable enances te efficienc of te estimator of an parameter, so Sing (965) proposed te traditional ratio tpe estimator of ratio of two population means using auxiliar information in stratified random sampling as, 0st X G st = (.) st xst Te bias and mean square error of G st, up to te first order of approximations respectivel are, S Sx S0 + Y X YY 0 st = γ (.3) S0x S x BG ( ) R W + + YX 0 YX S0 S Sx + + Y0 Y X st = γ S0 S0x Sx MSE( G ) R W + YY Y X YX 0 0 (.4) Jan et.al (04) proposed te following modified estimator using median (M d ) of te auxiliar information as, G st X + M = 0st d st xst (.5) Te bias and mean square error of G st, up to te first order of approximation is, S S x + Y ( X + Md ) S0 S 0x BG ( st ) = R W γ + (.6) YY 0 Y0( X+ Md ) S x + Y ( X + Md )

3 98 Subas Kumar Yadav et al.: Estimation Approac to Ratio of Two Inventor Population Means in Stratified Random Sampling S0 S Sx + + Y0 Y θ st = γ S0 S0x Sx MSE( G ) R W were, θ = ( X + M d ), 0 = 0i 0 = i x = ( i ) ( ) S, ( ) S, S x x, + YY Yθ Yθ 0 0i 0 i 0 0 (.7) S = ( )( ), S = ( )( x x ), 0x 0i 0 i S = x x ( )( ) x i i 3. Statistical Analsis of Proposed Estimator Motivated b Jan et.al (04), we propose te following generalized estimator for te ratio of two inventor population means as, α 0st X + Md τ st = (3.) st xst + Md Were α is a suitable constant be determined suc tat te mean square error of τ st is minimum. In order to stud te large sample properties of te proposed estimator we ave assumed tat, 0 = Y0( + e0), = Y( + e) and x = X( + e) suc tat Ee ( 0) = Ee ( ) = Ee ( ) And S0 Ee ( 0) Wγ Y0 S Ee ( ) Wγ Y =, =, Sx Ee ( ) = Wγ, X S0 Ee ( 0e) = Wγ, YY 0 S0x Ee ( 0e) = Wγ, YX 0 Sx Ee ( e) = Wγ. YX Using above expressions, te bias and te mean square error of te proposed estimator, up to te first order of approximation respectivel are, S α( + αθ ) Sx + Y X st = γ S0 αθs0 x αθs x B( τ ) R W + YY 0 Y0X YX (3.) S0 S αθ S x + + Y0 Y X S ( ) 0 S MSE τ 0x st = R W γ + αθ YY 0 Y0X Sx αθ YX (3.3) X were, θ =. X + Md Tis mean square error is minimum for, S0x Sx Wγ YX 0 YX α = Sx θ Wγ X And te minimum mean square error is, S0 S S0 Wγ + Y0 Y YY 0 MSEmin ( τ st ) = R S0x Sx Wγ YX 0 YX Sx Wγ X (3.4) (3.5)

4 American Journal of Operational Researc 05, 5(4): Efficienc Comparison From (.4) and (3.5) we ave tat te proposed estimator τ st is better tan te estimator G st, if MSE( G st ) MSEmin ( τ st ) > 0, if S0x Sx W γ YX 0 YX Sx R Wγ 0 > (4.) X Sx S0x Sx + Wγ X YX 0 YX Under te above laid down condition, te proposed estimator is believed to perform more efficientl tan te estimator G st considered in (.) of ratio of two population means for it will ave lesser mean squared error as compared to G st. From (.7) and (3.5) we ave tat te proposed estimator τ st is better tan te estimator G st, if MSE( Gst ) MSEmin ( τ st ) > 0, if S0x Sx W γ YX 0 YX Sx R Wγ 0 > (4.) X Sx S0x Sx + Wγ θ Y0θ Yθ Te proposed estimator τ st given in (3.) is supposed to provide to lesser mean squared error as compared to G st under te laid down condition in (4.). In brief, te psical significance of expressions given in (4.) and (4.) attempts to seek exceedingl better estimator aving least mean square error as compared to previousl existing ones. 5. Computing Algoritm and umerical Illustration Te following algoritm as been developed to compute te estimator and its efficienc. i. Begin ii. Data input iii. Compute sample mean of first inventor population iv. Compute sample mean of second inventor population v. Compute sample and population means of auxiliar inventor population vi. Compute estimator for ratio of two inventor populations vii. Compute biases for all estimators viii. Compute MSE ix. Compute efficienc (Percentage Relative Efficienc-PRE) x. If PRE is greater tan previous ones xi. Find efficient estimator xii. Data output xiii. End o Start Input Data Define Means Input Sample Data Compute MSE Compute Diff Is diff greater tan zero Yes Condition Attained Find Estimator Input for PRE Compute PRE Output PRE End Figure 5.. Tabular form of te algoritm (Computing flow cart)

5 00 Subas Kumar Yadav et al.: Estimation Approac to Ratio of Two Inventor Population Means in Stratified Random Sampling Table 5.. Computed Data Statistics Y 0 = 9.40 Y 0 = 09.0 Y = Y = X = 3757 X = S 0 ( ) = S 0 ( ) = S ( ) = S ( ) = S x = S x = 6.79 S 0 = S 0 = 45.0 S 0x = S 0x = S x = S x = Md ( x ) = Table 5.. Computed Bias, MSE and PRE of different estimators w.r.t. G st Estimators BIAS M S E PRE G st.4e e G st 8.4E E τ st -9.94E E To meet out te teoretical findings, we ave considered te data in Murt (967) were te main variables 0 and are te number of workers and te fixed capital respectivel along wit te output as te auxiliar variable. Te size of te population is 0 and is divided into two stratums eac of size 5. Te sample size is 4 b taking from eac stratum. Following are te parameters computed. 6. Observations and Conclusions Wen inventor items are eterogeneous and uge, te estimation approac is onl panacea for estimating te caracteristics of inventor populations under consideration wic oterwise seems difficult to control and manage for an organization. So for te estimation of ratio of two inventor population means ave been discussed in paper, one is man-inventor population and anoter is mone-inventor population as defined b Ackoff and Sasieni (993). Under te section numerical illustration, two eterogeneous inventor populations are given wose estimators are given tereafter final estimator as ratio as been developed wose bias and mean square errors are lesser tan all previous estimators and percentage relative efficienc is given wic is iger tan previous ones, vide table 5., row 3. Tus proposed estimator is most efficient estimator of ratio of two inventor population means using auxiliar information among suc estimators as it as lesser mean square error. Tus te proposed estimator sould be preferred for te estimation of ratio of two inventor population means. REFERECES [] Ackoff R and Sasieni M W (993), Fundamentals of Operations Researc, Wile Eastern td. [] Cocran W G (940), te estimation of ields of cereal experiments b sampling for te ratio of grain to total produce, Jour. Agri. Sci., 59, 5-6. [3] Cocran W G (977), Sampling Tecniques, Wile Eastern td, tird edition. [4] Gerskov Alex and Moldovanu Benn (009), Dnamic Revenue Maximization wit Heterogeneous Objects: A Mecanism Design Approac, American Economic Journal: Microeconomics, Volume, umber, pp (3). [5] Jan, R Maqbool, S Amad, A and azir, A (04), Modified Ratio Tpe Estimator of Two Population Means in Stratified Sampling, Indian Streams Researc Journal, 4,5, -6. [6] Murti M and Sing M P (965), On te estimation of ratio and product of te population parameters, Sanka, B, 7, [7] Rao J..K. & Pereira.P. (968), On double ratio estimators, Sanka, A, 30, [8] Ra S.K. & Sing R.K. (985), Some estimators for te ratio and product of population parameters. Journal of te Indian Societ of Agricultural Statistics, 37 (), -0. [9] Sa S.M. & Sa D.. (978), Ratio-cum-product estimator for estimating ratio (product) of two population parameters, Sanka, C, 40 (), [0] Sindu S.S., Tailor R. & Sing S. (009), On te estimation of population proportion, Applied Matematical Science, 3(35),

6 American Journal of Operational Researc 05, 5(4): [] Sing G.. and Rani R. (005, 006), Some linear transformations on auxiliar variable for estimating te ratio of two population means in sample surves, Model Assisted Statistics and Applications, (), IOS Press, -5. [] Upadaa.. & Sing H.P. (985), A class of estimators using auxiliar information for estimating ratio of two finite means. Gujarat Statistical Review, (), 7-6. [3] Upadaa.. Sing H.P. & Vos, J.W.E. (985), On te estimation of population means and ratios using supplementar, Statistica. eerlandica, 39(3),