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Srinivasa Rao Kolli, U.V.Adinarayana Rao ESTIMATING POPULATION AVERAGE USING INDEX-TYPE ESTIMATORS
RT&A, No 3 (74) Volume 18, September 2023
THE EFFICIENCY OF ESTIMATING A POPULATION AVERAGE USING INDEX-TYPE ESTIMATORS IN SEQUENTIAL SAMPLING
Srinivasa Rao Kolli1 ,U.V.Adinarayana Rao2, Adilakshmi Siripurapu3 Taviti Naidu Gongada4
Dept. of Operations, GITAM school of Business, GITAM (Deemed to be University), Visakhapatnam, AP, India12,4
Dept. of Basic Science and Humanities, Vignan's Institute of Information Technology (A), Duvvada, Visakhapatnam, AP, India3
srinivaskolli4@gmail.com1 auppu@gitam.edu 2 laxmimaths2008@gmail.com3 dr.tavitinaidugongada@gmail.com4
Abstract
This paper discusses the difficulty of approximating the population average of a variable y by knowledge about a supplementary variable x in the context of two successive (rotation) sampling occasions. The paper proposes a group of exponential-class estimators that includes the regular balanced estimator, produce-class estimator, and proportion-class estimator and suggests that these estimators are superior to existing estimators. The paragraph also mentions that the paper discusses optimal substitute statements and then the implementation of the recommended estimators, which may be important considerations for practical applications of the proposed methods. Finally, an empirical study is mentioned as supporting evidence for the research.
Keywords: Auxiliary variable, Study variable, Bias, Mean squared error, Successive sampling
1. Introduction
It is general towards procedure the supplementary knowledge at the approximation stand to achieve developed estimations of the population average y of the analyze variate x. Auxiliary information can be valuable in improving estimates of population parameters by incorporating additional relevant data. Ratio, Product, and Regression methods are good examples in this context. This statement is also generally true. Ratio, Product, and Regression methods are commonly used techniques for estimating auxiliary information. When the auxiliary variable x is positively (high) correlated with the study variable y, the Ratio estimation method is quite effective. The Ratio estimation method is effective when the auxiliary variable and the study
Srinivasa Rao Kolli, U.V.Adinarayana Rao
ESTIMATING POPULATION AVERAGE RT&A, No 3 (74) USING INDEX-TYPE ESTIMATORS_Volume 18, September 2023
variable have a positive correlation. When there is a negative correlation between the auxiliary variable and the study variable, the Product estimation method is commonly employed. The theory of successive sampling has been developed by various researchers, starting with [1] and followed [2], [4], [5] and several others. The mentioned researchers have indeed contributed to developing the theory of successive sampling.
Auxiliary information was utilized by [6] and [7] multiple times to estimate the current population means in successive sampling. Supplementary information implemented by [6] and [7] within the framework of subsequent selection to estimate the population mean on the current occasion. Recent research conducted by [8] where they utilized auxiliary information on both events and proposed various estimators for estimating the population mean on the current occasion within the framework of two-occasion successive (rotation) sampling. In their work on two-occasion subsequent selection, [8] employed auxiliary information to develop estimators designed explicitly for estimating the population mean on the current occasion. These researchers have successfully used supplementary information and developed various estimation methods to evaluate the population mean on the recent occurrence in the context of successive sampling.
2. Introduces the formulation and notation of the proposed estimator
Consider a set V = [v1,v2,.....vK } Consisting of K elements. Let [y,x] Represent the study and
auxiliary variables, respectively, with [yi,Xi] denoting the values on the ith unit Vi[i = 1,2,3,.....K}
of the population. Furthermore, let [y, x] Represent the population values of [y, x] Respectively. In this scenario, we assume that the population mean.? Of the auxiliary variable x is already known. Subsequently, to estimate the population mean Y Of the study variable y, a simple random sample of size k is selected from the population V without replacement. The Classical Ratio Estimator and the Ordinary Product Estimator are two methods used to estimate the population mean Y. The Classical Ratio Estimator is defined as lF = FX, where F = - x ^ 0 is the estimate of the Ratio F of
X
the population means. The unweighted sample means y and x,, represented as y = 1'Eli=1yi and x = iYJi=1xL, respectively, are utilized. The efficiency of the classical ratio estimator relies on a
strong positive correlation between the variables y and x. On the other hand, the Ordinary Product
s ~
Estimator for Y is defined as ls = ^ where S = y.x is the estimate of the Product S of the population means. The Product Estimator is commonly employed when there is an expectation of a strong negative correlation between the two variables.
The research on estimating the population mean y has led to the proposal of different estimators by various researchers. The Estimator ls, credited to [9] and revisited by [10] is one such example. Additionally, [11] suggested Ratio and Product-Type Exponential Estimators for estimating the population mean Y.
\—1
The Ratio-Type Exponential Estimator lFeis defined as y e^^, while the Product-Type
Exponential Estimator lSe is defined as y eiTx+x\ The simple Expansion Estimator l0, which is used for estimating the population mean Y When none of the previously mentioned estimators are suitable. The Expansion Estimator is defined as/0 = y = 1'Zli=1yL Which is the unweighted sample mean of y. Overall, the sentence accurately presents the contributions of different researchers in proposing various estimators for the population mean y.
Srinivasa Rao Kolli, U.V.Adinarayana Rao
ESTIMATING POPULATION AVERAGE RT&A, No 3 (74) USING INDEX-TYPE ESTIMATORS_Volume 18, September 2023
It discusses the findings of [12] regarding the variability of sample means and the
coefficients of variation. The variability of the sample mean x is usually less than that of the
sample mean y . If then introduces B[x] as the coefficient of variation of x and B[y] as the
coefficient of variation of y . The sentence further provides the relationships between the squared
coefficients of variation and the population coefficients of variation. It states that B2[x] = [1fcfl] Bj:
and B2[y] = [1-fl] By where g = ^ is the sampling fraction, Bx = -j is the population coefficient of
-
y Y is the population coefficient of variation for j, 7
variation for x, and By = -r is the population coefficient of variation for y, 7X = [^
1]-1 Sf=i[xi - and ry2 = [tf - 1]-1 Si=i[yi - ?] Represent the sums of squared deviations from the respective population means. If Dx = where b is a constant between 0 and 1, then D [x] = ¿D [y] ; 0 < ft < 1 This implies that the coefficient of variation for x is proportional to the coefficient of variation for y. [10] findings on the variability of sample means and the coefficients of variation explicitly focus on the relationship between D [x] and D [y] when D [x] = ¿D [y].
ZF ¿/ £ >0.5,
r ' b
y if - 0.5 < 1 < 0.5, 7
'f i/ 7 < -0.5 , b
7V
Where, y = and = [tf - 1]-1 - *] bi - ?]
'x'y
- > -1 and - < 1 as M < 1
& b & & If I —
3.The Suggested Estimator
Motivated by [13] based on that motivation, they derived a modified Exponential-Type Estimator for estimating the population mean Y as
f[x+0ft]-[X+0x])
W = ye{[*+0XHx+0*]}
fx[l-0]+[0-l]X)
y gfX [l+0] + [0 + l]£j
r ......
y g(.[0+l]£+[0 + l]Î) ([0-l][X-x])
= y el[0+i][x+x]j
Where 0 is a scalar used as a design parameter. for 0 = 1 ZMe = y 0 = 0, ZMe = ZPe the value of 0 (presumably a variable or parameter) is very large, the proposed estimator
([0-1][j?-x])
lim lMe = limy ef[0+1][*+;e]}
fix-xi}
s ZFe = y ef[x+y]}
3.1.Sampling bias and mean squared error of estimator
Sampling bias refers to the systematic error when the sample used for estimation does not represent the studied population. It can occur due to non-random sampling, non-response bias, or selection bias. Understanding and addressing sampling bias is crucial for obtaining reliable and accurate estimators.
Srinivasa Rao Kolli, U.V.Adinarayana Rao
ESTIMATING POPULATION AVERAGE RT&A, No 3 (74)
USING INDEX-TYPE ESTIMATORS_Volume 18, September 2023
y = Y[1 + e0], x = X[1 + e1] 3 H[h0] = H[h1]= 0 & H[h2] = 2 m2] = Hlflo2 H[hohi] = [^YDyDx
f -[0-1]hi ) lMh = Y[1 + ho]e^+1][2+*i]}
= Y[1 + h0]e{[i+® }
{«^[i+hi]-
Where H =
= Y[1 + h0]e{ 2 [1+2] } (!)
[1-0]
[1+0]
lMh^Y[1 + h0+?hl + ?h0h1 + ^h2] (2)
[lMh-Y]?Ylh0+^ + «-^ + ^h2] (3)
A[U] = Y^ + (4)
= A0[f][K+^]
Where A0=^-^ and A0 = ^
0 k 0 (1+0)
(lMe-Y)2 = Y2[hl+!?f + Re0ei] (5)
MSE(lMh) = MSE(lMh), =(1-f)Y2 [d22 + RyDyDx + ^] (6)
= U0[1+^-2f+bRY] Where U0 = (1-3k>T* = Var(y) The MSE (lMh) is minimum when R0 = —2\
MSE of (lMh) = U0(1- Y2) (7)
%= y + p(X-x) (8)
Table 1: PREs Estimator lMhis better than y, lF , ls lFh and lsh
Estimator PREs (* y)
Population
1 2 3 4 5
y 100.0000 100.0000 100.0000 100.0000 100.0000
h 66.5810 30.5860 156.3967 31.1061 56.2431
h 10.5463 7.6514 25.8171 92.9342 167.5887
781.3982 292.0779 197.7846 54.9135 74.5067
24.2836 19.0754 47.1121 133.0386 133.064
3.2. Effectiveness Relationship
Scalar R(or0)" and true optimum value:
Var ®) = MSE(y) = (1-f) T2 = (1-f-) y2C2y = U0 (9)
A(lF)=A0(A-Y) A(ls) = A0V
A(lPe) = (y) (3b - 47) (10)
A(lFe) = (£) (4y - b) MSE(lF) = U0[1 + b2-2yb]
Srinivasa Rao Kolli, U.V.Adinarayana Rao ESTIMATING POPULATION AVERAGE USING INDEX-TYPE ESTIMATORS
RT&A, No 3 (74) Volume 18, September 2023
MSE= U»[1 + ô2 +2y6] MSE(Z5) = Uo[l + (J) (6 - 4y)] MSE(is) = Uo[l + © (6 + 47)] Where ¿0 = ^^fBy
M5£(y) - min.MSE(ZMe) = U» 72 > 0 MS£(Zfi) - min. MSE(ZMe) = U» (a - 7)° > 0 MSE(Zs) - min. MS£(ZMe) = U» (a + 7)° > 0
MS£(ifle) - min.MS£(ZMe) = - k)2 > 0 MS£(ifle) - mm.MS£(ZMe) = ^>(f + r)2 > 0
(11)
(12)
(13)
(14)
(15)
• The usual Unbiased Estimator y does not require the correlation between the study variable y and the auxiliary variable x to be zero. when 7 = 0
• Usual Ratio Estimator ) and mentioning a condition when b = 7; in this condition, it states that both the Estimators i5 and ZMe are equally efficient.
• Ratio-Type Exponential Estimator [11], denoted as ifie, does not hold when b = 27 , which represents the scenario where both the estimators ZFe and ZMe exhibit equal efficiency.
• Product-Type Exponential Estimator [11], referred to as ZFe, is not applicable when b = -27 , which corresponds to the situation where both the estimators i5e and ZMe demonstrate equal efficiency.
Table 2: The proposed estimator ZMh is superior to y, ZF , Zs ZFh and Zsh when the value of R falls within a specific range.
Populati The proposed estimator iMft Outperforms other estimators when the value of R lies on within a specific range.
y
u
u
The standard range of R in which iM„is better than y, if ,
(-1.7765,0)
(-2,0.2235)
(-3.7765,2)
(-1,-0.7765)
(-2.7765,1)
(-1,-0.7765)
«5 ifft and iJft
1
2 (-1.3669,0) (-2,0.6331) (-3.3669,2) (-1,-0.3669) (-2.3669,1) (-1,-0.3669)
3 (-2.5742,0) (-2,-0.5742) (-4.5742,2) (-1.5742, -1) (-3.5742,1) (-1.5742,- 0.5740)
4 (0,1.8673) (-2,3.8673) (-0.1328,2) (-1,2.8673) (0.8673,1) (0.8673,1)
5 (0,6.3059) (-2,8.3059) (2,4.3054) (-1,7.3059) (1,5.3059) (-1,4.3054)
3.3.When the scalar R (or 0) does not align with its actual optimum value: MS£(iMe) - Variance (y) = U062 [^ +
= .M![fi2 -2flfl0]
(fl-flo)° i.e < №>1
min (0, - < fi < max. (0, -
From eq. (15) and eq. (16), we have,
Srinivasa Rao Kolli, U.V.Adinarayana Rao
ESTIMATING POPULATION AVERAGE RT&A, No 3 (74) USING INDEX-TYPE ESTIMATORS_Volume 18, September 2023
b
MSE(lMe) - MSE (lF) = U0b2[^ + ^-1 +
= U0b2[T + lt-1-R0]
= H°f[R2-2RR0-4-4R0]
= Eo^[R2-2RR0 + R2-4-4R0-R0)]
4
U0b
4
(R - R0)2 <(2+ R0)2 i.e., IR - R0I < 12 + R0I min, {-2,2(1-^)} <R < max {-2,2(1- °f)}
From eq. (16) and eq. (17), we have, MSE(lMe) - MSE (ls) = U0b2[^ + ^-1- 2f]
[(R-R0)2-(2+R0)2] (17)
: U0b2[^-^-1 + R0]
= H0L[R2-2RR0-4 + 4R0] = ^[R2-2RR0 + R20-4 + 4R0-R20]
= uob2 rm n \2 m n v
(R - R0)2 <(2- R0)2 i.e., IR - R0I <I2- R0I
V)
2
4 [(R-R0)2-(2-R0)2] (18)
min, {-2,2(1+°f)}< R < max {-2,2(1+ Ipi}
From eq. (17) and eq. (18), we have, MSE(lMe) - MSE (lFe) = U0b2[^ + ^-± +1]
0 L 4 2 4 2.
= H0L[R2-2RR0-1-2R0]
= ^[R2-2RR0 + R20-1-2R0-R20] = ^[(R-R0)2-(1 + R0)2] (19)
(R - R0)2 <(1 + R0)2 i.e., IR - R0I < I1 + R0I min {-1,(1-4i)}<R<max{-1,(1-4i)} From eq. (18) and eq. (19), we have MSE( lMe) - MSE (lFe) = U0b2[^ + ^-±- I]
4 b 4 bi
= ^[R2 - 2RR0-1 + 2R0] = !±f[R2- 2RR0 + R0-1 + 2R0 - R0] = ^[(R-R0)2-(1-R0)2] (20)
(R - R0)2 <(1- R0)2 i.e., IR - R0I <I1- R0I min, {1- (1 + 4f)} < R < max{1 - (1 + 4j)}
The usual Unbiased Estimator x is applicable when IR -fl0| < |fl0| Which can be equivalently expressed as min {(0, - 4)} < R < max { (0, - 4)}
The usual Ratio Estimator lF is valid when IR - fi0I < I2 + fi0I , which can be alternately expressed as min {2, -2 (1 - 4)} < R < max {-2,2 (1 - 4)}
The usual Product Estimator ls is valid when IR - fi0I < I2 - fi0I, which can be alternately expressed as min {2, -2 (1 + 4)} < R < max {-2,2 (1 + 4)}
Srinivasa Rao Kolli, U.V.Adinarayana Rao
ESTIMATING POPULATION AVERAGE RT&A, No 3 (74) USING INDEX-TYPE ESTIMATORS_Volume 18, September 2023
• The Ratio-Type Exponential Estimator ZFe introduced by Bah and Tuteja 1991[11] is applicable if - fl0| < |1 + fl0|, which can be equivalently expressed as min {-1, (l -
• The Product-Type Exponential Estimator ZFe introduced Bahl and Tuteja 1991[11] if
- fi0| < |1 - fi0| which can be equivalently expressed as min {1 - (1 + y)} < R < rn«{1- (1 + ?)}
Table 3: PREs Estimator ZMh concerning y, for different values of R
R 0 PREs (*>0
Population
1 2 3 4 5
-2.0000 -3.0000 66.5810 30.5860 156.3967 31.1061 56.2431
-1.7500 -3.6667 105.4984 45.4209 182.7966 35.5534 60.2317
-1.5000 -5.0000 187.1938 73.6467 202.4393 40.8775 64.5841
-1.2500 -9.0000 383.2829 135.4864 208.2654 47.2636 69.3319
-1.0000 0.0000 781.3982 292.0779 197.7846 54.9135 74.5067
-0.7500 7.0000 738.4366 585.7795 175.3442 64.0167 80.1386
-0.5000 3.0000 353.0569 448.2367 148.3074 74.6863 86.2537
-0.2500 1.6667 174.9985 200.1900 122.3232 86.8380 92.8714
0.0000 1.0000 100.0000 100.0000 100.0000 100.0000 100.0000
0.2500 0.6000 63.7373 57.9872 81.8494 113.0935 107.6315
0.5000 0.3333 43.8933 37.4100 67.4411 124.3406 115.7344
0.7500 0.1429 31.9699 26.0031 56.0835 131.5695 124.2463
1.0000 0.0000 24.2836 19.0754 47.1121 133.0386 133.0649
1.2500 0.1111 19.0533 14.5708 39.9776 128.3594 142.0399
1.5000 -0.2000 15.3392 11.4840 34.2528 118.7285 150.9669
1.7500 0.2727 12.6097 9.2794 29.6138 106.2421 159.5862
2.0000 -0.3333 10.5463 7.6514 25.8171 92.9342 167.5887
4. The precision of first-order approximations to mean squared errors (MSEs).
Once we have compared the MSEs of the Proposed Estimator and other estimators using firstorder approximations, our attention turns towards evaluating the accuracy of these approximations by deriving second-order approximations for the MSEs. In this analysis, we assume that Cx = Cx = C,, taking a value of 1, and that the sample is drawn from a large Bivariate Normal population. However, it is essential to mention that more complex expressions are derived for other scenarios, as discussed in [13]
0_n = tfn_ = 071 = 0
30 = ^03 = ^12 = ^21
004 = 040 = 3Ö4 031 = 013 = 37Ö4
022 = (1 + 2r2)fl4
-°eJ - fca
ZMe = 7(1 + 2 ( 2)
Where E(e£e/) - , i,; = 0,1,2,4 and a = 2 fori + = 4
= ?(1 + h0) (l + — + fl(fl-2)ftl | + ! fi(fi3-12fi2 + 36fi-24)h4-|
V. 2 8 48 384 J
¿i=£ , ¿2=^2
1 2 2 8
£ _ fi(fi2-6fi + 6)hl ^ _ fi(fi3-12fi2 + 36fi-24)h|
3 48 , 4 384
Then ZMe = ?(1 + ft0)[1 + + 62ft2 + M3 + ¿^4 + •••]
= ?[1 + + + ¿ft? + ¿2M2 + + M4+........] (22)
By disregarding terms of ft's with powers higher than four, we obtain the following.
-1
Srinivasa Rao Kolli, U.V.Adinarayana Rao
ESTIMATING POPULATION AVERAGE RT&A, No 3 (74) USING INDEX-TYPE ESTIMATORS_Volume 18, September 2023
result:
lMe = Y[1 + h0 + bi(hi + Khi) + b2(h\ + Kh2) + b3(hl + KhD + b^hR (23)
By computing the expectation of both sides of equation (23), we derive the bias of lMe as follows:
A(lMe) = ?■£-{(Ybi + b2) + 3-^ (Yb3 + b4)} (24)
By squaring both sides of equation (24) and disregarding terms of h's with powers higher than two, we obtain the following result:
( lMe - Y)2 = Y2[h20 + 2b1h0h1 + blhl + 2(bf + b2)h0hl + 2b1h2h1 + 2b1b2h3- + (b\ +
2b2)h20hl +2(b-+ 2bib2)hh- + (b2 + 2bib-)h4} (25)
2 2 2
MSE(lMe) = {MSE(lMe) + [(b2 + 2b2)(1 + 2y2) + 6(h- + 2bib2) + 3(a.22 + 2bib2)]}
= MSE(lMe) + (^j;-) [7R4 + 4R-(14y - 9) + 4R2(16y2 - 36y + 17) -
16R(4y2 - 3y + 2)} (26)
R « R0 = -2y(with a =1). Then putting R = -2y in eq. (25), we have,
If a reliable estimate of R0 is available, denoted as R ~ R0 = -2y (with a = 1), then substituting = -2y into equation (26), we obtain the following result:
MSE(lMe) = MSE(lMe) + —y(4 + 11y - 10y2 - 5y-) = MSE( lMe){1+!fky^f^} = --f{(1 - y2) + — y(1 - y) (4 + 15y + 5y2)} MSE(lMe) = MSE(ytf) =Df(1- Y2) (27)
MSE(lRe) =MSE(lRe){1+1D2k((l4--2^28r2))}
=D^{(5-y)+6k(143-248y+12Qy2)}
MSE(lRe) = MSE(lRe){1+^(■
' 7-40y
)}
(5+4y)
= Dl!1{(4+r)+D;27 - 40r)} (28)
r)2y2
MSE(lRe)=D-^r(5-4y)
n2y2
MSE(lRe) =D2-(5 + 4y) (29)
MSE(lRe) = MSE(lMe) = ^{(l - y)2 +-D2(143 - 312y - 48y2 + 160y- + 80y4)) (30)
{(i -y)2 (143 - 312y - 48y2 + 160y- + 80y4)} > 0 MSE(lRe) - MSE (I Me) = ^{(l + Y)2 +7^(7- 104y - 176y2 + 160y- + 80y4))
((i + Y)2 +Dk(7 - 104Y - 176y2 + 160y- + 80y4)} > 0 (31)
2
MSE(yth)=D2Y2(1-y2)(i + 1i-)
MSE(yth) - MSE(lMe) = --f{(1- Y2) -\Y(1 - Y)(4 + 15y + 5y2)} > 0 (32)
If {(1 - y2) -D2y(1 - y)(4 + 15y + 5y2)} > 0 Dx = Dx = D, where b=1 the Mean Squared Error (MSE) of the usual Ratio Estimator lF=p (J) and the Product Estimator ls=$ (J)
MSE(lp)2 = MSE(lp)i {1+^(6- 37)}
DJf(2(1-y) + 6Jf(2-3y + y2)
Srinivasa Rao Kolli, U.V.Adinarayana Rao
ESTIMATING POPULATION AVERAGE RT&A, No 3 (74) USING INDEX-TYPE ESTIMATORS_Volume 18, September 2023
MS^^MS^M^gl+f}
2(1+y)+2!(1 + 2y2)Y! (33)
k
From eq. (32) and eq. (33 ), we have MS£(Zf)2 - MS£(ZFe)2 = {(3 - 7) + £ [6(1 - y)(2 - 7)- 1 (143 - 248y + 128y2)]}
{(3 +T [6(1 - K)(2 - K)- ^ (143 - 2487 + 128K2)]} > 0 (34) from eq. (33) and eq. (34), we have,
MS£(Zs)2 - MS£(Zsh)2 = p2p) {(3 + K)+^2(57 + 1687)}
{(4+^)+6S(57 + 168^)}>0 (35)
5.Conclusion
The research findings highlight the significant impact of introducing new product-type and ratiotype estimators on the efficiency of estimating a population average by index-type estimators in the sequential random sample. The findings demonstrate that incorporating these new types of estimators increases the population average approximations' accuracy, thereby enhancing the overall efficiency of the sampling process. These findings have practical implications for researchers and practitioners in various fields, providing valuable insights to optimize their sampling strategies and obtain more reliable population mean estimates. Future research can explore additional aspects of these new estimators and their potential applications in different sampling scenarios to enhance the efficiency of sequential sampling techniques further.
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