# A Novel and Improved Logarithmic Ratio-Product Type Estimator of Mean in Stratified Random Sampling

Authors: Soofia Iftikhar; Alamgir Khalil; Amjad Ali
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

Population mean is an important characteristic from many perspectives in statistical methodology. In this connection, many separate ratio type estimators of population mean have been developed by the researchers to gain more precision and efficiency available in the literature. This study suggests a new class of logarithmic separate ratio-product type estimators of a finite population mean in stratified random sampling using one auxiliary variable to achieve enhanced precision and efficiency. The mean square error expression for the proposed class of estimators and some selected estimates are derived till first order of approximation. Three real data sets and a simulation study reveal that the proposed estimators outperform other estimators used in this analysis.


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## Body

## 1. Introduction

Researchers developed many estimators for mean in different scenarios. This study focuses on mean estimation in stratified random sampling. In the literature, there are many ratios, products, regressions, and exponential estimators for the population mean in stratified random sampling. Haq and Shabbir [1] introduced two improved family of estimators for population mean in simple as well as in stratified random sampling. For population mean in stratified random sampling, Kadilar and Cingi [2, 3] suggested some ratio estimators. Koyuncu and Kadilar [4, 5] proposed mean estimators using single as well as two auxiliary variables in stratified random sampling. In the presense of nonresponse, Kumar and Bhougal [6] proposed estimator of the population mean. In stratified random sampling using two auxiliary variables, Malik and Singh [7] proposed the estimator of population mean. Using information on sensitive variable in stratified double phase sampling, Mushtaq et al. [8] suggested estimator of the population mean. Pandey [9] introduced product-cum-power estimators of the population mean. In the presence of nonresponse under stratified two-phase random sampling, Sanaullah et al. [10] proposed a generalized exponential type ratio-cum-ratio and product-cum-product estimators for population mean. In simple as well as in stratified random sampling, Shabbir and Gupta [11, 12] proposed some new estimators of population mean, and Singh et al. [13] proposed exponential type estimator of population mean [14].Till date many ratios, products, exponential, and regression type estimators have been developed. Developing logarithmic type estimators is relatively new in the field. Izunobi and Onyeka [15] utilized the concept of logarithm and developed logarithmic type ratio and product estimators of the population mean in simple random sampling technique. They compared these logarithmic type estimates with the customary ratio and exponential type estimates. Their newly developed estimators turned out to be the most efficient among all three. A separate ln-type type estimator of variance in stratified random sampling was introduced by Cekim and Kadilar [16] as an alternate to another type of estimators used in their study. They concluded that their proposed estimator was the best alternative. In the same fashion, Cekim and Kadilar [17] introduced a family of logarithmic type estimators of variance in simple random sampling. Their developed estimators turned out to be the efficient ones as compared to the ratio, regression, and exponential type estimators. They also concluded that the use of the “ln function” in the proposed estimators, improved the efficiency of the estimators.However, less work has been done in the literature deriving logarithmic type estimators for the mean. The aim of this paper is to develop more precise and efficient logarithmic ratio-product type estimator for the first time for the population mean in stratified random sampling.In order to describe our proposed estimator, we proceed as follows:Let the finite populationψ = {ψ1, ψ2, …, ψN} consisting of η units which are divided in to “d” strata. Let the dth (d = 1, 2, 3, …, Z) stratum contains ηd units such that ∑d=1Zηd=η A sample of size цd from each stratum is drawn without replacement as well as independently under simple random sampling such that ∑d=1Zцd = ц. Let the sample mean estimates of response and auxiliary variables be y¯d = ∑i=1ndWdydi/цd and x¯d = ∑i=1ndWdxdi/цd respectively. The corresponding population characteristics will be Ψ¯ = ∑d=1ZWdΨ¯d and X¯ = ∑d=1ZWdψ¯d, respectively. The variances of response and auxiliary variables are hyd2 = ∑d=1Zydi−Ψ¯d2/ηd−1 and hxd2 = ∑d=1Zxdi−ψ¯d2/ηd−1 and hxdyd=∑d=1Zydi−Ψ¯dxdi−ψ¯d/ηd−1 is the covariance between X and Y, respectively. Furthermore, we assume Δ0d = y¯d−Ψ¯d/Y¯d, Δ1d = x¯d−ψ¯d/ψ¯d, Δ0dΔ1d = x¯d−ψ¯dy¯d−Ψ¯d/ψ¯dΨ¯d where E (Δ0d) = E(Δ1d) = 0, E(Δ0d2) = ∑d=1Zwd21−fd/цdhyd2/Ψ¯d2, E (Δ1d2) = ∑d=1Zwd21−fd/цdhxd2/ψ¯d2, E (Δ0dΔ1d) = ∑d=1Zwd21−fd/цdhxdyd/X¯dΨ¯d. ρxyd represents the correlation between the values of X and Y variables of the dth stratum, Cxd represents the coefficient of variation of the auxiliary variable in the dth stratum. β2xd is the coefficient of kurtosis of x variable in the dth stratum.


## 2. Some Selected Estimators of Variance from the Literature

(i)
The separate ratio estimator is as follows:(1)y¯RS=∑d=1Zwdy¯dψ¯dx¯d.Expressingy¯RS in terms of Δi’s(2)y¯RS=∑d=1ZwdΨ¯d1+Δ0d1+Δ1d−1=∑d=1ZwdΨ¯d1+Δ0d1−Δ1d+Δ1d2.Ignoring higher order terms ofΔi’s, we get(3)=∑d=1ZwdΨ¯d1−Δ1d+Δ1d2+Δ0d−Δ0dΔ1dy¯RS−Y¯=∑d=1ZwdΨ¯d−Δ1d+Δ1d2+Δ0d−Δ0dΔ1d.Upon squaring the above equation, we get MSE value as below:(4)Ey¯RS−Y¯=∑d=1Zwd2Ψ¯d2E−Δ1d+Δ1d2+Δ0d−Δ0dΔ1d2=∑d=1Zwd2Ψ¯d2EΔ1h2+Δ0h2−Δ0hΔ1h,MSEy¯RS=∑d=1Zwd21−fdцdhyd2+Rd2ηxd2−2Rdηxdηydρxyd,wherefd = цd/ηd and Rd = Ψ¯/X¯.(ii)
A stratified version of Bahl and Tuteja [18] exponential is as follows:(5)y¯BT=∑d=1Zwdy¯dexpψ¯d−x¯dψ¯d+x¯d.Expressingy¯BT in terms of Δi’s(6)y¯BT=∑d=1ZwdΨ¯d1+Δ0dexp−Δ1d2+Δ1d,y¯BT=∑d=1ZwdΨ¯d1+Δ0dexp−Δ1d21+Δ1d2,y¯BT=∑d=1ZwdΨ¯d1+Δ0dexp−Δ1d2+Δ1d22,y¯BT=∑d=1ZwdΨ¯d1+Δ0d1−Δ1d2+Δ1d22,y¯BT=∑d=1ZwdΨ¯d1+Δ0d−Δ1d2−Δ1dΔ0d2+Δ1d22,y¯BT−Ψ¯=∑d=1ZwdΨ¯dΔ0d−Δ1d2−Δ1dΔ0d2+Δ1d22.Squaring both sides and then applying expectation we get(7)MSEy¯BT=∑d=1Zwd21−fdцdηyd2+Rd2ηxd24−Rdηxdηydρxyd.(iii)
Motivated by Sisodia and Dwivedi [19] and Kadilar and Cingi [3], Tailor and Lone [20] proposed the following estimators:(8)y¯TRS1=∑d=1Zwdy¯dψ¯d+Cxdx¯d+Cxd,y¯TRS2=∑d=1Zwdy¯dψ¯d+β2xdx¯d+β2xd,y¯TRS3=∑d=1Zwdy¯dCxd∗ψ¯d+β2xdCxd∗x¯d+β2xd,y¯TRS4=∑d=1Zwdy¯dβ2xd∗ψ¯d+Cxdβ2xd∗x¯d+Cxd.Expressingy¯TRS1 it in terms of Δi’s(9)y¯TRS1=∑d=1ZwdΨ¯d1+Δ0d1+ψ¯dΔ1dx¯d+Cxd−1,⊕y¯TRS1=∑d=1ZwdΨ¯d1+Δ0d1−ψ¯dΔ1dψ¯d+Cxd+ψ¯dΔ1dψ¯d+Cxd2,y¯TRS1−Ψ¯=∑d=1ZwdΨ¯dΔ0d−ψ¯dψ¯d+CxdΔ1d+ψ¯dψ¯d+Cxd2Δ1d2ψ¯dψ¯d+CxdΔ0dΔ1d.Squaring both sides and applying expectation, we get MSE(10)MSEy¯TRS1=∑d=1Zwd21−fdцdhyd2+Rd2ψ¯dψ¯d+Cxd2hxd2−2Rdψ¯dψ¯d+β2xdhxdhydρxyd.In the same fashion, the MSE expressions for the rest of the three estimators are obtained.(11)MSEy¯TRS2=∑d=1Zwd21−fdцdhyd2+Rd2ψ¯dψ¯d+β2xd2hxd2−2Rdψ¯dψ¯d+β2xdhxdhydρxyd,MSEy¯TRS3=∑d=1Zwd21−fdцdhyd2+Rd2Cxd∗ψ¯dCxd∗ψ¯d+β2xd2hxd2−2RdCxd∗ψ¯dCxd∗ψ¯d+β2xdhxdhydρxyd,MSEy¯TRS4=∑d=1Zwd21−fdцdhyd2+Rd2β2xd∗ψ¯dβ2xd∗ψ¯d+Cxd2hxd2−2Rdβ2xd∗ψ¯dβ2xd∗ψ¯d+Cxdhxdhydρxyd..


## 3. The Proposed Estimator

Inspired by Iznobi and Onyeka [15] and Kumar and Boughal [6], we suggest a general class of logarithmic ratio-product type estimator:(12)y¯SSPi=∑d=1Zwdy¯dλa∗ψ¯d+ba∗x¯d+b+1−λlnx¯dlnψ¯d,wherei=1,2,3,4,5,λ where, a and b takes the values Cd, β2xd and ρxyd respectively. Furthermore, when λ = 1, y¯SSPi reduces to a customary ratio type estimator, i.e., y¯SSPi  =  ∑d=1Zwdy¯da∗ψ¯d+b/a∗x¯d+b.Also, whenλ  = 0, y¯SSPi reduces to a stratified version of product type estimator by Iznobi and Onyeka [15], i.e., y¯SSPi=∑d=1Zwdy¯dlnx¯d/lnψ¯d.The class of proposed estimators is as follows:(13)y¯SSP1=∑d=1Zwdy¯dλψ¯dx¯d+1−λlnx¯dlnψ¯d,wherea=1,b=0,y¯SSP2=∑d=1Zwdy¯dλψ¯d+Cxdx¯d+Cxd+1−λlnx¯dlnψ¯d,wherea=1,b=Cxd,y¯SSP3=∑d=1Zwdy¯dλβ2xd∗ψ¯d+Cxdβ2xd∗x¯d+Cxd+1−λlnx¯dlnψ¯d,wherea=β2xd,b=Cxd,y¯SSP4=∑d=1Zwdy¯dλβ2xd∗ψ¯d+ρxydβ2xd∗x¯d+ρxyd+1−λlnx¯dlnψ¯d,wherea=β2xd,b=ρxyd,y¯SSP5=∑d=1Zwdy¯dλρxyd∗ψ¯d+Cxdρxyd∗x¯d+Cxd+1−λlnx¯dlnψ¯d,wherea=ρxyd,b=Cxd.In order to find the MSE expressions, we writey¯SSPi in terms of Δi’s as follows:(14)y¯SSPi=∑d=1ZwdΨ¯d1+Δ0dλ1+a∗ψ¯da∗ψ¯d+bΔ1d−1+1−λ1+lnψ¯d1−Δ1dlnX¯d,y¯SSPi=∑d=1ZwdΨ¯d1+Δ0d1−λΔ1da∗ψ¯da∗ψ¯d+b+Δ1dlnψ¯d−Δ1d22lnX¯d−λΔ1dlnX¯d−λΔ1d22lnX¯d,y¯SSPi=∑d=1ZwdΨ¯d1+Δ0d−λΔ1da∗ψ¯da∗ψ¯d+b+Δ1dlnX¯d−Δ1d22lnX¯d−λΔ1dlnX¯d−λΔ1d22lnX¯d−λΔ1dΔ0da∗ψ¯da∗ψ¯d+b+Δ1dΔ0dlnψ¯d−−λΔ1dΔ0dlnψ¯d,y¯SSPi−Y¯=∑d=1ZwdΨ¯dΔ0d+Δ1d1lnψ¯d−λ1lnψ¯d+a∗ψ¯da∗ψ¯d+b2+Δ1dΔ0d1lnψ¯d−λ1lnψ¯d+a∗ψ¯da∗ψ¯d+b−Δ1d22lnψ¯d1+λ.Squaring both sides and applying expectation also ignoring higher powers ofΔ's we get MSE expression as follows:(15)MSEy¯SSPi=∑d=1Zwd21−fdndhyd2+Rd2λ1lnψ¯d+a∗ψ¯da∗ψ¯d+b−1lnψ¯d2hxd2−2Rdλ1lnψ¯d+a∗ψ¯da∗ψ¯d+b−1lnψ¯dhxdhydρxyd.For finding the optimum value of MSE, we differentiate MSE(y¯SSPi) w.r.t. λ and equating it to zero, i.e., d/dλ MSE(y¯SSPi) = 0.On simplification, we get(16)λ∗=∑d=1Zwd21−fd/nd1/lnψ¯d+a∗ψ¯d/a∗ψ¯d+bRdhydhxdρxyd−Rd2hxd2/lnψ¯d∑d=1Zwd21−fd/ndRd2hxd21/lnψ¯d+a∗ψ¯d/a∗ψ¯d+b2.Putting this value ofλ∗ in the expression for MSE(y¯SSPi), we get(17)MSEy¯SSPiopt=∑d=1Zwd21−fdndhyd2+Rd2λ∗1lnψ¯d+a∗ψ¯da∗ψ¯d+b−1lnψ¯d2hxd2−2Rdλ∗1lnψ¯d+a∗ψ¯da∗ψ¯d+b−1lnψ¯dhxdhydρxyd.This is the optimum value of MSE(y¯SSPi) for i = 1,2,3,4,5.


## 4. Numerical Results

To evaluate the performance of the suggested proposed estimator, we consider three data sets.Dataset 1.
Source: Singh and Manghat [21], page 208.
Y: expected sale for the current summer, X: the number of refrigerators sold during last summer.(18)η=1334,η1=400,η2=216,η3=364,η4=364,hx1=14.5256,hx2=14.0394,hx3=13.8851hx4=12.1559,hy1=12.9112,hy2=13.2015,hy3=15.0532,hy4=13.0618,ρxy1=0.79,ρxy2=0.87,ρxy3=0.92ρxy4=0.91,ц1=14,ц2=9,ц3=12,ц4=7,Z=4,W1=0.298,W2=0.161,W3=0.271,W4=0.271,ψ¯1=76.21,ψ¯2=58.11,ψ¯3=69.08,ψ¯4=63.71,Ψ¯1=79.35,Ψ¯2=59.44,Ψ¯3=76.66,Ψ¯4=64.57,Cx1=0.1906,Cx2=0.2416,Cx3=0.201,Cx4=0.1908,β2x1=2.22,β2x2=2.29,β2x3=1.96,β2x4=2.47.Dataset 2.
Source: Murthy [22], Page 228.
Y: output for the factories in the region, X: fixed capital.(19)η=10,η1=5,η2=5,hx1=74.87,hx2=66.346,hy1=615.92,hy2=340.38,ρxy1=0.82,ρxy2=0.98,ц1=2,ц2=2,Z=2,W1=0.5,W2=0.5,ψ¯1=214.4,ψ¯2=333.8,Ψ¯1=1925.8,Ψ¯2=3115.6,Cx1=0.34,Cx2=0.19,β2x1=1.88,β2x1=2.32.Dataset 3.
Source: Singh and Manghat [21], page 218.
Y: juice quantity per cane (grams), X: weight of cane (grams).(20)η=25,η1=6,η2=12,η3=7,hx1=52.0257,hx2=43.3712,hx3=53.7631,hy1=8.9443,hy2=15.0506,hy3=10.9654,ρxy1=0.9456,ρxy2=0.9482,ρxy3=0.7532ц1=3,ц2=6,ц3=3,Z=3,W1=0.24,W2=0.48,W3=0.28,ψ¯1=366.67,ψ¯2=310.83,ψ¯3=317.14,Ψ¯1=135,Ψ¯2=99.17,Ψ¯3=80.71,Cx1=0.1418881,Cx2=0.139532,Cx3=0.169523,β2x1=2.2866,β2x2=3.2690,β2x3=3.1306.
The corresponding MSE values of the proposed estimators along with estimators used in the analysis are presented in Table1 below.
Table1 shows the mean square error value of the class of estimators, i.e., y¯SSPi where i = 1, 2, 3, 4, and 5 is higher as compared to the y¯RS, y¯BT, y¯TRS1, y¯TRS2, y¯TRS3, and y¯TRS4.
The percentage relative efficiency is given by PRE =MSEy¯RS/MSEi×100 where i = y¯RS, y¯BT, y¯TRS1, y¯TRS2, y¯TRS3, y¯TRS4, y¯SSP1, y¯SSP2, y¯SSP3, y¯SSP4, and y¯SSP5.
Using the above formula, the PRE of all the estimators are calculated and presented in Table2.
Table2 shows that the PRE of y¯SSPi where i = 1, 2, 3, 4, and 5. is higher as compared to y¯RS, y¯BT, y¯TRS1, y¯TRS2, y¯TRS3, and y¯TRS4.Table 1
Mean square error (MSE) values of all the estimators.
MSEDataset 1Dataset 2Dataset 3MSE(y¯RS)1.19585228816025.030812.838428161MSE(y¯BT)1.66286912910913.025852.377039268MSE(y¯TRS1)1.19152446815986.235742.835827829MSE(y¯TRS2)1.15463475915715.174222.795581801MSE(y¯TRS3)1.09109954914768.005702.599766403MSE(y¯TRS4)1.19386736516005.882202.837416046MSE(y¯SSP1)1.0769181689150.7668441.963066923MSE(y¯SSP2)1.0770560189156.0606981.963114059MSE(y¯SSP3)1.0769611879153.8847371.963009877MSE(y¯SSP4)1.0774670039155.1272181.962873250MSE(y¯SSP5)1.0768845709157.4637301.963055715The bold values represent the MSE values of the proposed estimator.Table 2
The PRE of all the estimators on three data sets.
EstimatorsDataset1Dataset 2Dataset 3(y¯RS)100100100(y¯BT)71.9150146.8431119.4102(y¯TRS1)100.3632100.2427100.0917(y¯TRS2)103.5697101.9717101.5326(y¯TRS3)109.6006108.5118109.1801(y¯TRS4)100.1663100.1196100.0357(y¯SSP1)111.0439175.1223144.5915(y¯SSP2)111.0297175.0210144.5880(y¯SSP3)111.0394175.0626144.5957(y¯SSP4)110.9874175.0389144.6058(y¯SSP5)111.0474174.9942144.5923The bold values represent the MSE values of the proposed estimator.


## 5. Efficiency Comparisons

The MSE’s of the proposed estimators are smaller under the following conditions:MSE (y¯SSPi) < MSE (y¯RS) iff(21)∑d=1Zwd21−fdndhyd2+Rd2λ∗1lnψ¯d+a∗ψ¯da∗ψ¯d+b−1lnψ¯d2hxd2−2Rdλ∗1lnψ¯d+a∗ψ¯da∗ψ¯d+b−1lnψ¯dhxdhydρxyd<∑d=1Zwd21−fdцdhyd2+Rd2hxd2−2Rdhxdhydρxyd.On simplification, we get(22)∑d=1Zwd21−fdndRd2hxd2∑d=1Zwd21−fd/nd1/lnψ¯d+a∗ψ¯d/a∗ψ¯d+bRdhydhxdρxyd−Rd2hxd2/lnψ¯d∑d=1Zwd21−fd/ndRd2hxd21/lnψ¯d+a∗ψ¯d/a∗ψ¯d+b21lnψ¯d+a∗ψ¯da∗ψ¯d+b−1lnψ¯d2−1−2Rdhxdhydρxyd∑d=1Zwd21−fd/nd1/lnψ¯d+a∗ψ¯d/a∗ψ¯d+bRdhydhxdρxyd−Rd2hxd2/lnψ¯d∑d=1Zwd21−fd/ndRd2hxd21/lnψ¯d+a∗ψ¯d/a∗ψ¯d+b21lnψ¯d+a∗ψ¯da∗ψ¯d+b−1lnψ¯d−1<0.Equation (22) holds for all i = 1, 2, 3, 4, and 5 as follows:(i)
y¯SSP1 vs y¯RS(23)∑d=1Zwd21−fd/ndRd2hxd2∑d=1Zwd21−fd/nd1/lnψ¯d+1Rdhydhxdρxyd−Rd2hxd2/lnψ¯d∑d=1Zwd21−fd/ndRd2hxd21/lnψ¯d+1211/lnψ¯d+1−1lnψ¯d2−1−2Rdhxdhydρxyd∑d=1Zwd21−fd/nd1/lnψ¯d+1Rdhydhxdρxyd−Rd2hxd2/lnψ¯d∑d=1Zwd21−fd/ndRd2hxd21/lnψ¯d+1211/lnψ¯d+1−1lnψ¯d−1<0.(ii)
y¯SSP2 vs y¯RS(24)∑d=1Zwd21−fd/ndRd2hxd2∑d=1Zwd21−fd/nd1/lnψ¯d+ψ¯d/ψ¯d+CxdRdhydhxdρxyd−Rd2hxd2/lnψ¯d∑d=1Zwd21−fd/ndRd2hxd21/lnψ¯d+ψ¯d/ψ¯d+Cxd21lnψ¯d+ψ¯dψ¯d+Cxd−1lnψ¯d2−1−2Rdhxdhydρxyd∑d=1Zwd21−fd/nd1/lnψ¯d+ψ¯d/ψ¯d+CxdRdhydhxdρxyd−Rd2hxd2/lnψ¯d∑d=1Zwd21−fd/ndRd2hxd21/lnψ¯d+ψ¯d/ψ¯d+Cxd21lnψ¯d+ψ¯dψ¯d+Cxd−1lnψ¯d−1<0.(iii)
y¯SSP3 vs y¯RS(25)∑d=1Zwd21−fd/ndRd2hxd2∑d=1Zwd21−fd/nd1/lnψ¯d+β2xd∗ψ¯d/β2xd∗ψ¯d+CxdRdhydhxdρxyd−Rd2hxd2/lnψ¯d∑d=1Zwd21−fd/ndRd2hxd21/lnψ¯d+a∗ψ¯d/a∗ψ¯d+b21lnψ¯d+β2xd∗ψ¯dβ2xd∗ψ¯d+Cxd1lnψ¯d2−1−2Rdhxdhydρxyd∑d=1Zwd21−fd/nd1/lnψ¯d+β2xd∗ψ¯d/β2xd∗ψ¯d+CxdRdhydhxdρxyd−Rd2hxd2/lnψ¯d∑d=1Zwd21−fd/ndRd2hxd221/lnψ¯d+β2xd∗ψ¯d/β2xd∗ψ¯d+Cxd21lnψ¯d+β2xd∗ψ¯dβ2xd∗ψ¯d+Cxd−1lnψ¯d−1<0.(iv)
y¯SSP4 vs y¯RS(26)∑d=1Zwd21−fdndRd2hxd2∑d=1Zwd21−fd/nd1/lnψ¯d+β2xd∗ψ¯d/β2xd∗ψ¯d+ρxydRdhydhxdρxyd−Rd2hxd2/lnψ¯d∑d=1Zwd21−fd/ndRd2hxd21/lnψ¯d+β2xd∗ψ¯d/β2xd∗ψ¯d+ρxyd21/lnψ¯d+β2xd∗ψ¯d/β2xd∗ψ¯d+ρxyd−1/lnψ¯d2−1−2Rdhxdhydρxyd∑d=1Zwd21−fd/nd1/lnψ¯d+β2xd∗ψ¯d/β2xd∗ψ¯d+ρxydRdhydhxdρxyd−Rd2hxd2/lnψ¯d∑d=1Zwd21−fd/ndRd2hxd21/lnψ¯d+β2xd∗ψ¯d/β2xd∗ψ¯d+ρxyd21/lnψ¯d+β2xd∗ψ¯d/β2xd∗ψ¯d+ρxyd−1/lnψ¯d−1<0.(v)
y¯SSP5 vs y¯RS(27)∑d=1Zwd21−fdndRd2hxd2∑d=1Zwd21−fd/nd1/lnψ¯d+ρxyd∗ψ¯d/ρxyd∗ψ¯d+CxdRdhydhxdρxyd−Rd2hxd2/lnψ¯d∑d=1Zwd21−fd/ndRd2hxd21/lnψ¯d+ρxyd∗ψ¯d/ρxyd∗ψ¯d+Cxd21/lnψ¯d+ρxyd∗ψ¯d/ρxyd∗ψ¯d+Cxd−1/lnψ¯d2−1−2Rdhxdhydρxyd∑d=1Zwd21−fd/nd1/lnψ¯d+ρxyd∗ψ¯d/ρxyd∗ψ¯d+CxdRdhydhxdρxyd−Rd2hxd2/lnψ¯d∑d=1Zwd21−fd/ndRd2hxd21/lnψ¯d+ρxyd∗ψ¯d/ρxyd∗ψ¯d+Cxd21/lnψ¯d+ρxyd∗ψ¯d/ρxyd∗ψ¯d+Cxd−1/lnψ¯d−1<0.(1)
MSE (y¯SSPi) < MSE (y¯BT) iff(28)∑d=1Zwd21−fdndhyd2+Rd2λ∗1lnψ¯d+a∗ψ¯da∗ψ¯d+b−1lnψ¯d2hxd2−2Rdλ∗1lnψ¯d+a∗ψ¯da∗ψ¯d+b−1lnψ¯dhxdhydρxyd<∑d=1Zwd21−fdцdhyd2+Rd2hxd24−Rdhxdhydρxyd.On simplification, we get(29)∑d=1Zwd21−fdndRd2hxd2λ∗1lnψ¯d+a∗ψ¯da∗ψ¯d+b−1lnψ¯d2−14−2Rdhxdhydρxydλ∗1lnψ¯d+a∗ψ¯da∗ψ¯d+b−1lnψ¯d−12<0.Equation (29) holds for all i = 1, 2, 3, 4, 5 as follows:(vi)
y¯SSP1 vs y¯BT(30)∑d=1Zwd21−fdndRd2hxd2∑d=1Zwd21−fd/nd1/lnψ¯d+1Rdhydhxdρxyd−Rd2hxd2/lnψ¯d∑d=1Zwd21−fd/ndRd2hxd21/lnψ¯d+121lnψ¯d+1−1lnψ¯d2−14−2Rdhxdhydρxyd∑d=1Zwd21−fd/nd1/lnψ¯d+1Rdhydhxdρxyd−Rd2hxd2/lnψ¯d∑d=12wd21−fd/ndRd2hxd21/lnψ¯d+121lnψ¯d+1−1lnψ¯d−12<0.(vii)
y¯SSP2 vs y¯BT(31)∑d=1Zwd21−fdndRd2hxd2∑d=1Zwd21−fd/nd1/lnψ¯d+ψ¯d/ψ¯d+CxdRdhydhxdρxyd−Rd2hxd2/lnψ¯d∑d=1Zwd21−fd/ndRd2hxd21/lnψ¯d+ψ¯d/ψ¯d+Cxd21lnψ¯d+ψ¯dψ¯d+Cxd−1lnψ¯d2−14−2Rdhxdhydρxyd∑d=1Zwd21−fd/nd1/lnψ¯d+ψ¯d/ψ¯d+CxdRdhydhxdρxyd−Rd2hxd2/lnψ¯d∑d=1Zwd21−fd/ndRd2hxd21/lnψ¯d+ψ¯d/ψ¯d+Cxd21lnψ¯d+ψ¯dψ¯d+Cxd−1lnψ¯d−12<0.(viii)
y¯SSP3 vs y¯BT(32)∑d=1Zwd21−fdndRd2hxd2∑d=1Zwd21−fd/nd1/lnψ¯d+β2xd∗ψ¯d/β2xd∗ψ¯d+CxdRdhydhxdρxyd−Rd2hxd2/lnψ¯d∑d=1Zwd21−fd/ndRd2hxd21/lnψ¯d+β2xd∗ψ¯d/β2xd∗ψ¯d+Cxd21lnψ¯d+β2xd∗ψ¯dβ2xd∗ψ¯d+Cxd−1lnψ¯d2−14−2Rdhxdhydρxyd∑d=1Zwd21−fd/nd1/lnψ¯d+β2xd∗ψ¯d/β2xd∗ψ¯d+CxdRdhydhxdρxyd−Rd2hxd2/lnψ¯d∑d=1Zwd21−fd/ndRd2hxd21/lnψ¯d+β2xd∗ψ¯d/β2xd∗ψ¯d+Cxd21lnψ¯d+β2xd∗ψ¯dβ2xd∗ψ¯d+Cxd−1lnψ¯d−12<0.(ix)
y¯SSP4 vs y¯BT(33)∑d=1Zwd21−fdndRd2hxd2∑d=1Zwd21−fd/nd1/lnψ¯d+β2xd∗ψ¯d/β2xd∗ψ¯d+ρxydRdhydhxdρxyd−Rd2hxd2/lnψ¯d∑d=1Zwd21−fd/ndRd2hxd21/lnψ¯d+β2xd∗ψ¯d/β2xd∗ψ¯d+ρxyd21lnψ¯d+β2xd∗ψ¯dβ2xd∗ψ¯d+ρxyd−1lnψ¯d2−14−2Rdhxdhydρxyd∑d=1Zwd21−fd/nd1/lnψ¯d+β2xd∗ψ¯d/β2xd∗ψ¯d+ρxydRdhydhxdρxyd−Rd2hxd2/lnψ¯d∑d=1Zwd21−fd/ndRd2hxd21/lnψ¯d+β2xd∗ψ¯d/β2xd∗ψ¯d+ρxyd21lnψ¯d+β2xd∗ψ¯dβ2xd∗ψ¯d+ρxyd−1lnψ¯d−12<0.(x)
y¯SSP5 vs y¯BT(34)∑d=1Zwd21−fdndRd2hxd2∑d=1Zwd21−fd/nd1/lnψ¯d+ρxyd∗ψ¯d/ρxyd∗ψ¯d+CxdRdhydhxdρxyd−Rd2hxd2/lnψ¯d∑d=1Zwd21−fd/ndRd2hxd21/lnψ¯d+ρxyd∗ψ¯d/ρxyd∗ψ¯d+Cxd21lnψ¯d+ρxyd∗ψ¯dρxyd∗ψ¯d+Cxd−1lnψ¯d2−14−2Rdhxdhydρxyd∑d=1Zwd21−fd/nd1/lnψ¯d+ρxyd∗ψ¯d/ρxyd∗ψ¯d+CxdRdhydhxdρxyd−Rd2hxd2/lnψ¯d∑d=1Zwd21−fd/ndRd2hxd21/lnψ¯d+ρxyd∗ψ¯d/ρxyd∗ψ¯d+Cxd21lnψ¯d+ρxyd∗ψ¯dρxyd∗ψ¯d+Cxd−1lnψ¯d−12<0.(1)
MSE (y¯SSPi) < MSE (y¯TRS1) iff(35)∑d=1Zwd21−fdndhyd2Rd2λ∗1lnψ¯d+a∗ψ¯da∗ψ¯d+b−1lnψ¯d2hxd2−2Rdλ∗1lnψ¯d+a∗ψ¯da∗ψ¯d+b−1lnψ¯dhxdhydρxyd<∑d=1Zwd21−fdцdhyd2+Rd2ψ¯dψ¯d+Cxd2hxd2−2Rdψ¯dψ¯d+Cxdhxdhydρxyd.On simplification, we get(36)∑d=1Zwd21−fdndRd2hxd2λ∗1lnψ¯d+a∗ψ¯da∗ψ¯d+b−1lnψ¯d2−ψ¯dψ¯d+Cxd2−2Rdhxdhydρxydλ∗1lnψ¯d+a∗ψ¯da∗ψ¯d+b−1lnψ¯d−ψ¯dψ¯d+Cxd<0.Equation (36) holds for all i = 1, 2, 3, 4, and 5 as follows:(xi)
y¯SSP1 vs y¯TRS1(37)∑d=1Zwd21−fdndRd2hxd2∑d=1Zwd21−fd/nd1/lnψ¯d+1Rdhydhxdρxyd−Rd2hxd2/lnψ¯d∑d=1Zwd21−fd/ndRd2hxd21/lnψ¯d+121lnψ¯d+1−1lnψ¯d2−ψ¯dψ¯d+Cxd2−2Rdhxdhydρxyd∑d=1Zwd21−fd/nd1/lnψ¯d+1Rdhydhxdρxyd−Rd2hxd2/lnψ¯d∑d=1Zwd21−fd/ndRd2hxd21/lnψ¯d+121lnψ¯d+1−1lnψ¯d−ψ¯dψ¯d+Cxd<0.(xii)
y¯SSP2 vs y¯TRS1(38)∑d=1Zwd21−fdndRd2hxd2∑d=1Zwd21−fd/nd1/lnψ¯d+ψ¯d/ψ¯d+CxdRdhydhxdρxyd−Rd2hxd2/lnψ¯d∑d=1Zwd21−fd/ndRd2hxd21/lnψ¯d+ψ¯d/ψ¯d+Cxd21lnψ¯d+ψ¯dψ¯d+Cxd−1lnψ¯d2−ψ¯dψ¯d+Cxd2−2Rdhxdhydρxyd∑d=1Zwd21−fd/nd1/lnψ¯d+ψ¯d/ψ¯d+CxdRdhydhxdρxyd−Rd2hxd2/lnψ¯d∑d=1Zwd21−fd/ndRd2hxd21/lnψ¯d+ψ¯d/ψ¯d+Cxd21lnψ¯d+ψ¯dψ¯d+Cxd−1lnψ¯d−ψ¯dψ¯d+Cxd<0.(xiii)
y¯SSP3 vs y¯TRS1(39)∑d=1Zwd21−fdndRd2hxd2∑d=1Zwd21−fd/nd1/lnψ¯d+β2xd∗ψ¯d/β2xd∗ψ¯d+CxdRdhydhxdρxyd−Rd2hxd2/lnψ¯d∑d=1Zwd21−fd/ndRd2hxd21/lnψ¯d+β2xd∗ψ¯d/β2xd∗ψ¯d+Cxd21lnψ¯d+β2xd∗ψ¯dβ2xd∗ψ¯d+Cxd−1lnψ¯d2−ψ¯dψ¯d+Cxd2−2Rdhxdhydρxyd∑d=1Zwd21−fd/nd1/lnψ¯d+β2xd∗ψ¯d/β2xd∗ψ¯d+CxdRdhydhxdρxyd−Rd2hxd2/lnψ¯d∑d=1Zwd21−fd/ndRd2hxd21/lnψ¯d+β2xd∗ψ¯d/β2xd∗ψ¯d+Cxd21lnψ¯d+β2xd∗ψ¯dβ2xd∗ψ¯d+Cxd−1lnψ¯d−ψ¯dψ¯d+Cxd<0.(xiv)
y¯SSP4 vs y¯TRS1(40)∑d=1Zwd21−fdndRd2hxd2∑d=1Zwd21−fd/nd1/lnψ¯d+β2xd∗ψ¯d/β2xd∗ψ¯d+ρxydRdhydhxdρxyd−Rd2hxd2/lnψ¯d∑d=1Zwd21−fd/ndRd2hxd21/lnψ¯d+β2xd∗ψ¯d/β2xd∗ψ¯d+ρxyd21lnψ¯d+β2xd∗ψ¯dβ2xd∗ψ¯d+ρxyd−1lnψ¯d2−ψ¯dψ¯d+Cxd2−2Rdhxdhydρxyd∑d=1Zwd21−fd/nd1/lnψ¯d+β2xd∗ψ¯d/β2xd∗ψ¯d+ρxydRdhydhxdρxyd−Rd2hxd2/lnψ¯d∑d=1Zwd21−fd/ndRd2hxd21/lnψ¯d+β2xd∗ψ¯d/β2xd∗ψ¯d+ρxyd21lnψ¯d+β2xd∗ψ¯dβ2xd∗ψ¯d+ρxyd−1lnψ¯d−ψ¯dψ¯d+Cxd<0.(xv)
y¯SSP5 vs y¯TRS1(41)∑d=1Zwd21−fdndRd2hxd2∑d=1Zwd21−fd/nd1/lnψ¯d+ρxyd∗ψ¯d/ρxyd∗ψ¯d+CxdRdhydhxdρxyd−Rd2hxd2/lnψ¯d∑d=1Zwd21−fd/ndRd2hxd21/lnψ¯d+ρxyd∗ψ¯d/ρxyd∗ψ¯d+Cxd21lnψ¯d+ρxyd∗ψ¯dρxyd∗ψ¯d+Cxd−1lnψ¯d2−ψ¯dψ¯d+Cxd2−2Rdhxdhydρxyd∑d=1Zwd21−fd/nd1/lnψ¯d+ρxyd∗ψ¯d/ρxyd∗ψ¯d+CxdRdhydhxdρxyd−Rd2hxd2/lnψ¯d∑d=1Zwd21−fd/ndRd2hxd21/lnψ¯d+ρxyd∗ψ¯d/ρxyd∗ψ¯d+Cxd21lnψ¯d+ρxyd∗ψ¯dρxyd∗ψ¯d+Cxd−1lnψ¯d−ψ¯dψ¯d+Cxd<0.(1)
MSE (y¯SSPi) < MSE (y¯TRS2) iff(42)∑d=1Zwd21−fdndhyd2+Rd2λ∗1lnψ¯d+a∗ψ¯da∗ψ¯d+b−1lnψ¯d2hxd2−2Rdλ∗1lnψ¯d+a∗ψ¯da∗ψ¯d+b−1lnψ¯dhxdhydρxyd<∑d=1Zwd21−fdцdhyd2+Rd2ψ¯dψ¯d+β2xd2hxd2−2Rdψ¯dψ¯d+β2xdhxdhydρxyd.On simplification, we get(43)∑d=1Zwd21−fdndRd2hxd2λ∗1lnψ¯d+a∗ψ¯da∗ψ¯d+b−1lnψ¯d2−ψ¯dψ¯d+β2xd2−2Rdhxdhydρxydλ∗1lnψ¯d+a∗ψ¯da∗ψ¯d+b−1lnψ¯d−ψ¯dψ¯d+β2xd<0.Equation (43) holds for all i = 1, 2, 3, 4, and 5 as follows:(xvi)
y¯SSP1 vs y¯TRS2(44)∑d=1Zwd21−fdndRd2hxd2∑d=1Zwd21−fd/nd1/lnψ¯d+1Rdhydhxdρxyd−Rd2hxd2/lnψ¯d∑d=1Zwd21−fd/ndRd2hxd21/lnψ¯d+121lnψ¯d+1−1lnψ¯d2−ψ¯dψ¯d+β2xd2−2Rdhxdhydρxyd∑d=1Zwd21−fd/nd1/lnψ¯d+1Rdhydhxdρxyd−Rd2hxd2/lnψ¯d∑d=1Zwd21−fd/ndRd2hxd21/lnψ¯d+121lnψ¯d+1−1lnψ¯d−ψ¯dψ¯d+β2xd<0.(xvii)
y¯SSP2 vs y¯TRS2(45)∑d=1Zwd21−fdndRd2hxd2∑d=1Zwd21−fd/nd1/lnψ¯d+ψ¯d/ψ¯d+CxdRdhydhxdρxyd−Rd2hxd2/lnψ¯d∑d=1Zwd21−fd/ndRd2hxd21/lnψ¯d+ψ¯d/ψ¯d+Cxd21lnψ¯d+ψ¯dψ¯d+Cxd−1lnψ¯d2−ψ¯dψ¯d+β2xd2−2Rdhxdhydρxyd∑d=1Zwd21−fd/nd1/lnψ¯d+ψ¯d/ψ¯d+CxdRdhydhxdρxyd−Rd2hxd2/lnψ¯d∑d=1Zwd21−fd/ndRd2hxd21/lnψ¯d+ψ¯d/ψ¯d+Cxd21lnψ¯d+ψ¯dψ¯d+Cxd−1lnψ¯d−ψ¯dψ¯d+β2xd<0(xviii)
y¯SSP3 vs y¯TRS2(46)∑d=1Zwd21−fdndRd2hxd2∑d=1Zwd21−fd/nd1/lnψ¯d+β2xd∗ψ¯d/β2xd∗ψ¯d+CxdRdhydhxdρxyd−Rd2hxd2/lnψ¯d∑d=1Zwd21−fd/ndRd2hxd21/lnψ¯d+β2xd∗ψ¯d/β2xd∗ψ¯d+Cxd21lnψ¯d+β2xd∗ψ¯dβ2xd∗ψ¯d+Cxd−1lnψ¯d2−ψ¯dψ¯d+β2xd2−2Rdhxdhydρxyd∑d=1Zwd21−fd/nd1/lnψ¯d+β2xd∗ψ¯d/β2xd∗ψ¯d+CxdRdhydhxdρxyd−Rd2hxd2/lnψ¯d∑d=1Zwd21−fd/ndRd2hxd21/lnψ¯d+β2xd∗ψ¯d/β2xd∗ψ¯d+Cxd21lnψ¯d+β2xd∗ψ¯dβ2xd∗ψ¯d+Cxd−1lnψ¯d−ψ¯dψ¯d+β2xd<0.(xix)
y¯SSP4 vs y¯TRS2(47)∑d=1Zwd21−fdndRd2hxd2∑d=1Zwd21−fd/nd1/lnψ¯d+β2xd∗ψ¯d/β2xd∗ψ¯d+ρxydRdhydhxdρxyd−Rd2hxd2/lnψ¯d∑d=1Zwd21−fd/ndRd2hxd21/lnψ¯d+β2xd∗ψ¯d/β2xd∗ψ¯d+ρxyd21lnψ¯d+β2xd∗ψ¯dβ2xd∗ψ¯d+ρxyd−1lnψ¯d2−ψ¯dψ¯d+β2xd2−2Rdhxdhydρxyd∑d=1Zwd21−fd/nd1/lnψ¯d+β2xd∗ψ¯d/β2xd∗ψ¯d+ρxydRdhydhxdρxyd−Rd2hxd2/lnψ¯d∑d=1Zwd21−fd/ndRd2hxd21/lnψ¯d+β2xd∗ψ¯d/β2xd∗ψ¯d+ρxyd21lnψ¯d+β2xd∗ψ¯dβ2xd∗ψ¯d+ρxyd−1lnψ¯d−ψ¯dψ¯d+β2xd<0.(xx)
y¯SSP5 vs y¯TRS2(48)∑d=1Zwd21−fdndRd2hxd2∑d=1Zwd21−fd/nd1/lnψ¯d+ρxyd∗ψ¯d/ρxyd∗ψ¯d+CxdRdhydhxdρxyd−Rd2hxd2/lnψ¯d∑d=1Zwd21−fd/ndRd2hxd21/lnψ¯d+ρxyd∗ψ¯d/ρxyd∗ψ¯d+Cxd21lnψ¯d+ρxyd∗ψ¯dρxyd∗ψ¯d+Cxd−1lnψ¯d2−ψ¯dψ¯d+β2xd2−2Rdhxdhydρxyd∑d=1Zwd21−fd/nd1/lnψ¯d+ρxyd∗ψ¯d/ρxyd∗ψ¯d+CxdRdhydhxdρxyd−Rd2hxd2/lnψ¯d∑d=1Zwd21−fd/ndRd2hxd21/lnψ¯d+ρxyd∗ψ¯d/ρxyd∗ψ¯d+Cxd21lnψ¯d+ρxyd∗ψ¯dρxyd∗ψ¯d+Cxd−1lnψ¯d−ψ¯dψ¯d+β2xd<0.(1)
MSE (y¯SSPi)< MSE (y¯TRS3) iff(49)∑d=1Zwd21−fd/ndhyd2+Rd2λ∗1lnψ¯d+a∗ψ¯da∗ψ¯d+b−1lnψ¯d2hxd2−2Rdλ∗1lnψ¯d+a∗ψ¯da∗ψ¯d+b−1lnψ¯dhxdhydρxyd<∑d=1Zwd21−fd/ndhyd2+Rd2Cxd∗ψ¯dCxd∗ψ¯d+β2xd2hxd2−2RdCd∗ψ¯dCd∗ψ¯d+β2xdhxdhydρxyd.On simplification, we get(50)∑d=1Zwd21−fdndRd2hxd2λ∗1lnψ¯d+a∗ψ¯da∗ψ¯d+b−1lnψ¯d2−Cxd∗ψ¯dCxd∗ψ¯d+β2xd2−2Rdhxdhydρxydλ∗1lnψ¯d+a∗ψ¯da∗ψ¯d+b−1lnψ¯d−Cxd∗ψ¯dCxd∗ψ¯d+β2xd<0.Equation (50) holds for all i = 1, 2, 3, 4, and 5 as follows(xxi)
y¯SSP1 vs y¯TRS3(51)∑d=1Zwd21−fdndRd2hxd2∑d=1Zwd21−fd/nd1/lnψ¯d+1Rdhydhxdρxyd−Rd2hxd2/lnψ¯d∑d=1Zwd21−fd/ndRd2hxd21/lnψ¯d+121lnψ¯d+1−1lnψ¯d2−Cxd∗ψ¯dCxd∗ψ¯d+β2xd2−2Rdhxdhydρxyd∑d=1Zwd21−fd/nd1/lnψ¯d+1Rdhydhxdρxyd−Rd2hxd2/lnψ¯d∑d=1Zwd21−fd/ndRd2hxd21/lnψ¯d+121lnψ¯d+1−1lnψ¯d−Cxd∗ψ¯dCxd∗ψ¯d+β2xd<0.(xxii)
y¯SSP2 vs y¯TRS3(52)∑d=1Zwd21−fdndRd2hxd2∑d=1Zwd21−fd/nd1/lnψ¯d+ψ¯d/ψ¯d+CxdRdhydhxdρxyd−Rd2hxd2/lnψ¯d∑d=1Zwd21−fd/ndRd2hxd21/lnψ¯d+ψ¯d/ψ¯d+Cxd21lnψ¯d+ψ¯dψ¯d+Cxd−1lnψ¯d2−Cxd∗ψ¯dCxd∗ψ¯d+β2xd2−2Rdhxdhydρxyd∑d=1Zwd21−fd/nd1/lnψ¯d+ψ¯d/ψ¯d+CxdRdhydhxdρxyd−Rd2hxd2/lnψ¯d∑d=1Zwd21−fd/ndRd2hxd21/lnψ¯d+ψ¯d/ψ¯d+Cxd21lnψ¯d+ψ¯dψ¯d+Cxd−1lnψ¯d−Cxd∗ψ¯dCxd∗ψ¯d+β2xd<0.(xxiii)
y¯SSP3 vs y¯TRS3(53)∑d=1Zwd21−fdndRd2hxd2∑d=1Zwd21−fd/nd1/lnψ¯d+β2xd∗ψ¯d/β2xd∗ψ¯d+CxdRdhydhxdρxyd−Rd2hxd2/lnψ¯d∑d=1Zwd21−fd/ndRd2hxd21/lnψ¯d+β2xd∗ψ¯d/β2xd∗ψ¯d+Cxd21lnψ¯d+β2xd∗ψ¯dβ2xd∗ψ¯d+Cxd−1lnψ¯d2−Cxd∗ψ¯dCxd∗ψ¯d+β2xd2−Cxd∗ψ¯dCxd∗ψ¯d+β2xd2−2Rdhxdhydρxyd∑d=1Zwd21−fd/nd1/lnψ¯d+β2xd∗ψ¯d/β2xd∗ψ¯d+CxdRdhydhxdρxyd−Rd2hxd2/lnψ¯d∑d=1Zwd21−fd/ndRd2hxd21/lnψ¯d+β2xd∗ψ¯d/β2xd∗ψ¯d+Cxd21lnψ¯d+β2xd∗ψ¯dβ2xd∗ψ¯d+Cxd−1lnψ¯d−Cxd∗ψ¯dCxd∗ψ¯d+β2xd<0.(xxiv)
y¯SSP4 vs y¯TRS3(54)∑d=1Zwd21−fdndRd2hxd2∑d=1Zwd21−fd/nd1/lnψ¯d+β2xd∗ψ¯d/β2xd∗ψ¯d+ρxydRdhydhxdρxyd−Rd2hxd2/lnψ¯d∑d=1Zwd21−fd/ndRd2hxd21/lnψ¯d+β2xd∗ψ¯d/β2xd∗ψ¯d+ρxyd21lnψ¯d+β2xd∗ψ¯dβ2xd∗ψ¯d+ρxyd−1lnψ¯d2−Cxd∗ψ¯dCxd∗ψ¯d+β2xd2−2Rdhxdhydρxyd∑d=1Zwd21−fd/nd1/lnψ¯d+β2xd∗ψ¯d/β2xd∗ψ¯d+ρxydRdhydhxdρxyd−Rd2hxd2/lnψ¯d∑d=1Zwd21−fd/ndRd2hxd21/lnψ¯d+β2xd∗ψ¯d/β2xd∗ψ¯d+ρxyd21lnψ¯d+β2xd∗ψ¯dβ2xd∗ψ¯d+ρxyd−1lnψ¯d−Cxd∗ψ¯dCxd∗ψ¯d+β2xd<0.(xxv)
y¯SSP5 vs y¯TRS3(55)∑d=1Zwd21−fdndRd2hxd2∑d=1Zwd21−fd/nd1/lnψ¯d+ρxyd∗ψ¯d/ρxyd∗ψ¯d+CxdRdhydhxdρxyd−Rd2hxd2/lnψ¯d∑d=1Zwd21−fd/ndRd2hxd21/lnψ¯d+ρxyd∗ψ¯d/ρxyd∗ψ¯d+Cxd21lnψ¯d+ρxyd∗ψ¯dρxyd∗ψ¯d+Cxd−1lnψ¯d2−Cxd∗ψ¯dCxd∗ψ¯d+β2xd2−2Rdhxdhydρxyd∑d=1Zwd21−fd/nd1/lnψ¯d+ρxyd∗ψ¯d/ρxyd∗ψ¯d+CxdRdhydhxdρxyd−Rd2hxd2/lnψ¯d∑d=1Zwd21−fd/ndRd2hxd21/lnψ¯d+ρxyd∗ψ¯d/ρxyd∗ψ¯d+Cxd21lnψ¯d+ρxyd∗ψ¯dρxyd∗ψ¯d+Cxd−1lnψ¯d−Cxd∗ψ¯dCxd∗ψ¯d+β2xd<0.(1)
MSE (y¯SSPi) < MSE (y¯TRS4) iff(56)∑d=1Zwd21−fdndhyd2+Rd2λ∗1lnψ¯d+a∗ψ¯da∗ψ¯d+b−1lnψ¯d22Rdλ∗1lnψ¯d+a∗ψ¯da∗ψ¯d+b−1lnψ¯dhxdhydρxyd<∑d=1Zwd21−fdцdhyd2+Rd2β2xd∗ψ¯dβ2xd∗ψ¯d+Cxd2hxd2−2Rdβ2xd∗ψ¯dβ2xd∗ψ¯d+Cxdhxdhydρxyd.On simplification, we get(57)∑d=1Zwd21−fdndRd2hxd2λ∗1lnψ¯d+a∗ψ¯da∗ψ¯d+b−1lnψ¯d2−β2xd∗ψ¯dβ2xd∗ψ¯d+Cxd2−2Rdhxdhydρxydλ∗1lnψ¯d+a∗ψ¯da∗ψ¯d+b−1lnψ¯d−β2xd∗ψ¯dβ2xd∗ψ¯d+Cxd<0.Equation (57) holds for all i = 1, 2, 3, 4, and 5 as follows:(xxvi)
y¯SSP1 vs y¯TRS4(58)∑d=1Zwd21−fdndRd2hxd2∑d=1Zwd21−fd/nd1/lnψ¯d+1Rdhydhxdρxyd−Rd2hxd2/lnψ¯d∑d=1Zwd21−fd/ndRd2hxd21/lnψ¯d+121lnψ¯d+1−1lnψ¯d2−β2xd∗ψ¯dβ2xd∗ψ¯d+Cxd2−2Rdhxdhydρxyd∑d=1Zwd21−fd/nd1/lnψ¯d+1Rdhydhxdρxyd−Rd2hxd2/lnψ¯d∑d=1Zwd21−fd/ndRd2hxd21/lnψ¯d+121lnψ¯d+1−1lnψ¯d−β2xd∗ψ¯dβ2xd∗ψ¯d+Cxd<0.(xxvii)
y¯SSP2 vs y¯TRS4(59)∑d=1Zwd2Rd2hxd2∑d=1Zwd21−fd/nd1/lnψ¯d+ψ¯d/ψ¯d+CxdRdhydhxdρxyd−Rd2hxd2/lnψ¯d∑d=1Zwd21−fd/ndRd2hxd21/lnψ¯d+ψ¯d/ψ¯d+Cxd21lnψ¯d+ψ¯dψ¯d+Cxd−1lnψ¯d2−β2xd∗ψ¯dβ2xd∗ψ¯d+Cxd2−2Rdhxdhydρxyd∑d=1Zwd21−fd/nd1/lnψ¯d+ψ¯d/ψ¯d+CxdRdhydhxdρxyd−Rd2hxd2/lnψ¯d∑d=1Zwd21−fd/ndRd2hxd21/lnψ¯d+ψ¯d/ψ¯d+Cxd21lnψ¯d+ψ¯dψ¯d+Cxd−1lnψ¯d−β2xd∗ψ¯dβ2xd∗ψ¯d+Cxd<0.(xxviii)
y¯SSP3 vs y¯TRS4(60)∑d=1Zwd21−fdndRd2hxd2∑d=1Zwd21−fd/nd1/lnψ¯d+β2xd∗ψ¯d/β2xd∗ψ¯d+CxdRdhydhxdρxyd−Rd2hxd2/lnψ¯d∑d=1Zwd21−fd/ndRd2hxd21/lnψ¯d+β2xd∗ψ¯d/β2xd∗ψ¯d+Cxd21lnψ¯d+β2xd∗ψ¯dβ2xd∗ψ¯d+Cxd−1lnψ¯d2−β2xd∗ψ¯dβ2xd∗ψ¯d+Cxd2−2Rdhxdhydρxyd∑d=1Zwd21−fd/nd1/lnψ¯d+β2xd∗ψ¯d/β2xd∗ψ¯d+CxdRdhydhxdρxyd−Rd2hxd2/lnψ¯d∑d=1Zwd21−fd/ndRd2hxd21/lnψ¯d+β2xd∗ψ¯d/β2xd∗ψ¯d+Cxd21lnψ¯d+β2xd∗ψ¯dβ2xd∗ψ¯d+Cxd−1lnψ¯d−β2xd∗ψ¯dβ2xd∗ψ¯d+Cxd<0.(xxix)
y¯SSP4 vs y¯TRS4(61)∑d=1Zwd21−fdndRd2hxd2∑d=1Zwd21−fd/nd1/lnψψ¯d+β2xd∗ψ¯d/β2xd∗ψ¯d+ρxydRdhydhxdρxyd−Rd2hxd2/lnψ¯d∑d=1Zwd21−fd/ndRd2hxd21/lnψ¯d+β2xd∗ψ¯d/β2xd∗ψ¯d+ρxyd21lnψ¯d+β2xd∗ψ¯dβ2xd∗ψ¯d+ρxyd−1lnψ¯d2−β2xd∗ψ¯dβ2xd∗ψ¯d+Cxd2−2Rdhxdhydρxyd∑d=1Zwd21−fd/nd1/lnψ¯d+β2xd∗ψ¯d/β2xd∗ψ¯d+ρxydRdhydhxdρxyd−Rd2hxd2/lnψ¯d∑d=1Zwd21−fd/ndRd2hxd21/lnψ¯d+β2xd∗ψ¯d/β2xd∗ψ¯d+ρxyd21lnψ¯d+β2xd∗ψ¯dβ2xd∗ψ¯d+ρxyd−1lnψ¯d−β2xd∗ψ¯dβ2xd∗ψ¯d+Cxd<0.(xxx)
y¯SSP5 vs y¯TRS4(62)∑d=1Zwd21−fdndRd2hxd2∑d=1Zwd21−fd/nd1/lnψ¯d+ρxyd∗ψ¯d/ρxyd∗ψ¯d+CxdRdhydhxdρxyd−Rd2hxd2/lnψ¯d∑d=1Zwd21−fd/ndRd2hxd21/lnψ¯d+ρxyd∗ψ¯/ρxyd∗ψ¯d+Cxd21lnψ¯d+1−1lnψ¯d2−β2xd∗ψ¯dβ2xd∗ψ¯d+Cxd2−2Rdhxdhydρxyd∑d=1Zwd21−fd/nd1/lnψ¯d+ρxyd∗ψ¯d/ρxyd∗ψ¯d+CxdRdhydhxdρxyd−Rd2hxd2/lnψ¯d∑d=1Zwd21−fd/ndRd2hxd21/lnψ¯d+ρxyd∗ψ¯/ρxyd∗ψ¯d+Cxd21lnψ¯d+1−1lnψ¯d−β2xd∗ψ¯dβ2xd∗ψ¯d+Cxd<0.The above three data sets are used for checking the efficiency conditions 1.1–6.5. Table3 represents the values of these conditions for the three data sets.Table 3
Efficiency conditions for the three data sets.
ConditionData set 1Data set 2Data set 3(1) MSE (y¯SSPi) < MSE (y¯RS) iff ∑d=1Zwd21−fd/ndRd2hxd2λ∗1/lnψ¯d+a∗ψ¯d/a∗ψ¯d+b−1/lnψ¯d−1−2Rdhxdhydρxydλ∗1/lnψ¯d+a∗ψ¯d/a∗ψ¯d+b−1/lnψ¯d−1<0y¯SSP1 vs y¯RS−0.1189 < 0−6874.2640 < 0−0.8754 < 0y¯SSP2 vs y¯RS−0.1188 < 0−6868.9701 < 0−0.8753 < 0y¯SSP3 vs y¯RS−0.1189 < 0−6871.1461 < 0−0.8754 < 0y¯SSP4 vs y¯RS−0.1184 < 0−6869.9036 < 0−0.8756 < 0y¯SSP5 vs y¯RS−0.1190 < 0−6867.5671 < 0−0.8754 < 0(2) MSE (y¯SSPi) < MSE (y¯BT) iff ∑d=1Zwd21−fd/ndRd2hxd2λ∗1/lnψ¯d+a∗ψ¯d/a∗ψ¯d+b−1/lnψ¯d2−1/4−2Rdhxdhydρxydλ∗1/lnψ¯d+a∗ψ¯d/a∗ψ¯d+b−1/lnψ¯d−1/2<0y¯SSP1 vs y¯BT−0.5860 < 0−1762.2590 < 0−0.4140 < 0y¯SSP2 vs y¯BT−0.5858 < 0−1756.9651 < 0−0.4139 < 0y¯SSP3 vs y¯BT−0.5859 < 0−1759.1411 < 0−0.4140 < 0y¯SSP4 vs y¯BT−0.5854 < 0−1757.8986 < 0−0.4142 < 0y¯SSP5 vs y¯BT−0.5860 < 0−1755.5621 < 0−0.4140 < 0MSE (y¯SSPi) < MSE (y¯TRS1) iff ∑d=1Zwd21−fd/ndRd2hxd2λ∗1/lnψ¯d+a∗ψ¯d/a∗ψ¯d+b−1/lnψ¯d2−ψ¯d/ψ¯d+Cxd2−2Rdhxdhydρxydλ∗1/lnψ¯d+a∗ψ¯d/a∗ψ¯d+b−1/lnψ¯d−ψ¯d/ψ¯d+Cxd<0y¯SSP1 vs y¯TRS1−0.1292 < 0−6905.4411 < 0−0.8773 < 0y¯SSP2 vs y¯TRS1−0.1290 < 0−6900.1473 < 0−0.8772 < 0y¯SSP3 vs y¯TRS1−0.1291 < 0−6902.3233 < 0−0.8773 < 0y¯SSP4 vs y¯TRS1−0.1286 < 0−6901.0808 < 0−0.8775 < 0y¯SSP5 vs y¯TRS1−0.1292 < 0−6898.7443 < 0−0.8773 < 0MSE (y¯SSPi) < MSE (y¯TRS2) iff ∑d=1Zwd21−fd/ndRd2hxd2λ∗1/lnψ¯d+a∗ψ¯d/a∗ψ¯d+b−1/lnψ¯d2−ψ¯d/ψ¯d+β2xd2−2Rdhxdhydρxydλ∗1/lnψ¯d+a∗ψ¯d/a∗ψ¯d+b−1/lnψ¯d−ψ¯d/ψ¯d+β2xd<0y¯SSP1 vs y¯TRS2−0.1292 < 0−6905.4411 < 0−0.8773 < 0y¯SSP2 vs y¯TRS2−0.1290 < 0−6900.1473 < 0−0.8772 < 0y¯SSP3 vs y¯TRS2−0.1291 < 0−6902.3233 < 0−0.8773 < 0y¯SSP4 vs y¯TRS2−0.1286 < 0−6901.0808 < 0−0.8775 < 0y¯SSP5 vs y¯TRS2−0.1292 < 0−6898.7443 < 0−0.8773 < 0MSE (y¯SSPi) < MSE (y¯TRS3) iff ∑d=1Zwd21−fd/ndRd2hxd2λ∗1/lnψ¯d+a∗ψ¯d/a∗ψ¯d+b−1/lnψ¯d2−Cxd∗ψ¯d/Cxd∗ψ¯d+β2xd2−2Rdhxdhydρxydλ∗1/lnψ¯d+a∗ψ¯d/a∗ψ¯d+b−1/lnψ¯d−Cxd∗ψ¯d/Cxd∗ψ¯d+β2xd<0y¯SSP1 vs y¯TRS3−0.6154 < 0−7427.7317 < 0−1.1637 < 0y¯SSP2 vs y¯TRS3−0.6152 < 0−7422.4379 < 0−1.1637 < 0y¯SSP3 vs y¯TRS3−0.6153 < 0−7424.6138 < 0−1.1638 < 0y¯SSP4 vs y¯TRS3−0.6148 < 0−7423.3713 < 0−1.1639 < 0y¯SSP5 vs y¯TRS3−0.6154 < 0−7421.0348 < 0−1.1637 < 0MSE (y¯SSPi) < MSE (y¯TRS4) iff∑d=1Zwd21−fd/ndRd2hxd2∑d=1Zwd21−fd/ndRd2hxd2λ∗1/lnψ¯d+a∗ψ¯d/a∗ψ¯d+b−1/lnψ¯d2−β2xd∗ψ¯d/β2xd∗ψ¯d+Cxd2−2Rdhxdhydρxydλ∗1/lnψ¯d+a∗ψ¯d/a∗ψ¯d+b−1/lnψ¯d−β2xd∗ψ¯d/β2xd∗ψ¯d+Cxd<0y¯SSP1 vs y¯TRS4−0.1235 < 0−6890.7319 < 0−0.8759 < 0y¯SSP2 vs y¯TRS4−0.1234 < 0−6885.4381 < 0−0.8759 < 0y¯SSP3 vs y¯TRS4−0.1235 < 0−6887.6140 < 0−0.8760 < 0y¯SSP4 vs y¯TRS4−0.1230 < 0−6886.3715 < 0−0.8761 < 0y¯SSP5 vs y¯TRS4−0.1236 < 0−6884.0350 < 0−0.8759 < 0The bold values represent the MSE values of the proposed estimator.Table3 shows that the efficiency conditions for the three datasets are satisfied.


## 6. Simulation Study

For simulation study, we considered a bivariate normal population of size 10,000. This population is then divided into three strata of sizes 3000, 3000, and 4000, respectively. We considered proportional allocation procedure to draw samples of sizes 30, 30, and 40. The samples are drawn from each stratum with parametersμ1 = (25.75417, 103.4167) and Σ1=40.1475767.4717467.47174133.9015 of stratum-I, μ2 = (28.94, 110.9231) and Σ2 =30.3335741.0341741.0341766.24359 of stratum-II, and μ3 = (25.84286, 104.2857) and Σ3=45.4458481.141481.1414154.989 of stratum-III. For the values of parameters, we considered a data set from Singh and Manghat [21], page: 213.The sample values of the estimates are generatedR = 500, 1000, 10,000, and 100,000 times and their MSE values are calculated using the relationship MSE (y¯i) = ∑i=1Ry¯i−Ψ¯2/R where y¯i is the ith sample value of the estimate among R generated sample values and Ψ¯ represents the true value of the estimator, i.e., its value calculated from the population. Table 4 represents the values of the MSE for R = 500, 1000, 10,000, and 100,000, respectively.Table 4
The MSE values of all the estimators forR = 500, 1000, 10,000, and 100,000.
MSE valuesR = 500R = 1000R = 10,000R = 100,000MSE (y¯RS)0.14440.14970.14550.1447MSE (y¯BT)0.25600.25970.25300.2502MSE (y¯TRS1)0.14470.14990.14570.1448MSE (y¯TRS2)0.14950.15450.15030.1494MSE (y¯TRS3)0.18610.19110.18590.1845MSE (y¯TRS4)0.14450.14980.14550.1447MSE (y¯SSP1)0.12530.13100.12820.1277MSE (y¯SSP2)0.12500.13080.12770.1273MSE (y¯SSP3)0.12500.13070.12790.1274MSE (y¯SSP4)0.12950.13550.13190.1318MSE (y¯SSP5)0.12500.13080.12780.1274The bold values represent the MSE values of the proposed estimator.It is evident from Table4 that the suggested estimators have minimum MSE among all estimators used in the analysis for all R = 500, 1000, 10,000, and 100,000. Moreover it is also observed that MSE’s of the proposed estimators y¯SSPi for i = 1, 2, 3, 4, and 5 approach almost the values 0.1277, 0.1273, 0.1274, 0.1318, and 0.1274, respectively. Whereas, the MSE’s of y¯RS, y¯BT, y¯TRS1, y¯TRS2 , y¯TRS3, and y¯TRS4 approach the values 0.1447, 0.2502, 0.1448, 0.1494, 0.1845, and 0.1447, respectively, as we increase the number of simulations. However, our proposed estimator outperforms all other estimators used in the study all levels of simulations.


## 7. Conclusion

A new class of logarithmic ratio-product type estimators,y¯SSPi for i = 1, 2, 3, 4, and 5, for population mean is proposed in this study. MSE expression along with efficiency conditions was derived. Based on three data sets, the MSE values of the suggested estimators have been found smaller than the MSE’s of y¯RS, y¯BT, y¯TRS1, y¯TRS2, y¯TRS3, and y¯TRS4. The efficiency conditions were derived and also satisfied for all the three data sets. The percentage relative efficiency of the new class of logarithmic ratio type estimators was found higher compared to the estimators used in the analysis. Moreover, the simulation study also supported the claim that the suggested class of logarithmic type estimators is more efficient than the ones considered in the analysis. Hence, we conclude that based on all three modes of comparisons the suggested class of estimators is more efficient than the estimators used in this study.



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*Source: 6179625-2022-06-08.xml*