APPROXIMATE OPTIMUM STRATA BOUNDARIES FOR PROPORTIONAL ALLOCATION USING RANKED SET SAMPLING Текст научной статьи по специальности «Математика»

APPROXIMATE OPTIMUM STRATA BOUNDARIES FOR PROPORTIONAL ALLOCATION USING RANKED SET SAMPLING

Khalid Ul Islam Rather1*, S.E.H Rizvi2, Manish Sharma3, M. Iqbal Jeelani4,

Faizan Danish5

Division of Statistics and Computer Science, SKUAST-Jammu, India1-2-3 Division of Social and Basic Sciences Faculty of Forestry, SKUAST-Kashmir, India4 Department of Mathematics, School of Advanced Science(SAS) VIT-AP India5 [email protected] [email protected] [email protected] [email protected] [email protected]

Abstract

Ranked set sampling is an approach to data collection originally combines simple random sampling with the field investigator's professional knowledge and judgment to pick places to collect samples. Alternatively, field screening measurements can replace professional judgment when appropriate and analysis that continues to stimulate substantial methodological research. The use of ranked set sampling increases the chance that the collected samples will yield representative measurements. This results in better estimates of the mean as well as improved performance of many statistical procedures. Moreover, ranked set sampling can be more cost-efficient than simple random sampling because fewer samples need to be collected and measured. The use of professional judgment in the process of selecting sampling locations is a powerful incentive to use ranked set sampling. Optimum stratification is the method of choosing the best boundaries that make strata internally homogeneous, given some sample allocation. In order to make the strata internally homogenous, the strata should be constructed in such a way that the strata variances for the characteristic under study be as small as possible. This could be achieved effectively by having the distribution of the study variable known and create strata by cutting the range of the distribution at suitable points. If the frequency distribution of the study variable is unknown, it may be approximated from the past experience or some prior knowledge (auxiliary information) obtained at a recent study. The present investigation deals with paper the problem of optimum stratification on an auxiliary variable for proportional allocation under ranked set sampling (RSS), when the form of the regression of the estimation variable on the stratification variable given the variance function is known. A cum rule of finding approximately optimum strata boundaries has been developed. Further, empirical study has been made and presented along with relative efficiency which showed remarkable gain in efficiency as compared to unstratified RSS.

Keywords: Ranked set Sampling, approximately optimum strata boundaries, auxiliary variable optimum strata width

1. Introduction

The main aim of stratification is to produce estimators with more precision when a population characteristic is under study. The problem of obtaining optimum strata boundaries (OSB) taking study variable itself as stratification variable was first considered [1]. The study showed that for a given method of allocation, the variance is clearly a function of the strata boundaries. By minimizing the variance of the estimate of population mean sets of equations were

obtained, solutions to which gave optimum strata boundaries. Due to the implicit nature of the minimal equations, their exact solutions could not be obtained. Subsequently, various authors gave methods of obtaining approximations to the exact solutions of the minimal equations. However, the ideal situation is that the distribution of such a study variable is known and the OSB can be determined by placing boundaries on the range of this distribution at suitable cut points. For an excellent account of these investigations reference may be made [2]. When the information about the study variables are not known, the utilization of auxiliary variable as stratification variable may be considered and obtained approximate OSB under simple random sampling (SRS) studied [3]. The problem of optimum stratification for two characters under study using auxiliary information [4]. A methodology developed for obtaining AOSB using auxiliary information under compromise method of allocation [5]. The problem of obtaining OSB under proportional allocation with varying cost of each unit studied [6]. A technique proposed under Neyman allocation when the stratification is done on the two auxiliary variables under consideration [7]. Recently, the problem considered of optimum stratification for a model-based allocation under a super population model [8]. Situation considered of optimum stratification of heteroscedastic populations in stratified selection for a known allocation under SRS strategy [9]. Estimator for the population Mean under Ranked Set Sampling [10] and [11]. Situation of optimum stratification under RSS considered by [12] and [13]. Further, the selection of sample from each stratum could be taken by utilizing any sampling technique. Therefore, in the present investigation we have taken the procedure of ranked set sampling (RSS) as a method of selection of units from each stratum, which is more efficient as compared to SRS. A stratified ranked set sample (SRSS) is a sampling plan in which a population is divided into mutually exclusive strata and a RSS of n elements is quantified within each stratum. The sampling is performed independently across the strata. Therefore, one can think of an SRSS scheme as a collection of L separate ranked set samples. Originally, RSS was first suggested to estimate mean pasture and forage yields [14]. The necessary mathematical theory provided [15].

Let the population under consideration be divided into L strata and a sample n0h = (Rh x nh ) units which is selected from hth stratum is drawn using RSS, where Rh is the number of cycles and nh is sample size of each cycle. Each sample element is measured with respect to some variable Y, and estimator of the population mean is given by

nh _

' (1.1)

- = vW.

y SRSS 2

h=1 n0h

22 y^(r) .j=1 i=1

where Wh is the weight of the hth stratum and y^r^ is the sample mean based on nQh units drawn from the hth stratum.

If the finite correction is ignored, the variance of the estimate will be:

V )

_ 1

prop

n

f

.2

h" h(r) h_l

Gh(r) _

(1.2)

G2hc--— y)2 |denotes the variance of rth order statistics in hth stratum of the

2

hc

V n

random sample of size nh.

In most of these investigations related to optimum stratification, both the estimation and stratification variables are taken to be the same. Since the distribution of the estimation variable 'Y' is rarely known in practice, it is desirable to stratify on the basis of some suitably chosen concomitant variable' X'. An investigation has considered the general problem of optimum

Khalid Ul Islam Rather, S.E.H Rizvi, M. Sharma, M. Iqbal Jeelani, F. Danish RT&A, No 1 (72) APPROXIMATE OPTIMUM STRATA BOUNDARIES ..._Volume 18, March 2023

stratification based on auxiliary variables for the case of proportional allocation [16]. We consider the problem of optimum stratification on the auxiliary variable' X', assuming knowledge about the form of the regression of 'Y' on ' X' and the variance function V(y | x), minimal equations giving optimum strata boundaries have been obtained for proportional allocations under ranked set sampling. Since these equations cannot be solved easily, various methods of finding approximations to the exact solutions have been given.

In this paper, the problem of construction of strata boundaries will be dealt using classical approach when the sample is selected from the strata using RSS.

MINIMAL EQUATIONS UNDER PROPORTIONAL ALLOCATION If the regression of the estimation variable 'Y' on the stratification variable' X', in the infinite super population is given by

y = c(x) + e (2.1)

Where ' c(x)' is a function of auxiliary variable, ' e' is the error term such that E(e | x) = 0 and V(e | x) = r/(x) > 0 "x e (a, b) with (b - a) < ¥. Let f (x, y) and f (x) be the joint density function and marginal density function of (x, y) and x respectively. Then, we have

xh 1 xh

Wh = J fi(x) , Vhc = W Jc(x)fi(x) and s2hy = She , (h = 12,3...,L) ( 2.2) xh-1 h xh-1 where (xh-1, xh ) are lower and upper boundaries of the hth stratum with (x0 = a) and (xL = b),

/uhn is the expected value of j(x) and &2c is the variance of c(x) in the hth stratum. Using these relations, the variance expression (1.2), under proportional allocation

1 T L ~

= - ) +»„„ (2.3)

Let \Xh ] denote the set of optimum points of stratification on the range (a, b), for which the V(ysRss) is minimum. These points \xh ] are the solutions of the minimal equations which are

obtained by equating to zero the partial derivatives of V(ySRSS ) with respect to \xh ].

Minimization of the variance expression given in (2.3), is equivalent to the minimization of the

L L

expression 'S^WhS2}lc^r), since = /dhh is a population parameter and is a constant. On

h=1 h=\ equating to zero the partial derivative of this expression with respect to \xh ], we get

Whf( Xh )[C(Xh ) - Vhc(r) )2 - sl(r) j = W,f (Xh )[(c(Xh ) - V,c(r) )2 - sl(r) j

Wh + f (Xh )<,) = W + f (xh )sfc(r)

Therefore, using these results we get the minimal equations on simplification as

V C^SS )p

c(Xh ) =

^ №hc(r) + №ic(r) ^

i = h +1,h = 1,2,...,L -1 (2.4)

These equations are implicit functions of the strata boundaries \xh ] and their exact solutions are somewhat difficult to find. Therefore, we proceed to find the method of solving these minimal equations by conducting approximations.

2

APPROXIMATE EXPRESSIONS FOR CONDITIONAL MEAN AND VARIANCE Let the functions f (x), c(x) and hhX) are bounded away from zero and possess first two derivatives continuous "x e (a,b). Then, we have the following identities due to [17].

ii (y, x)=J (t - y )/t (t )dt=£ jk)

i+j+1

. -f(j) + O(k'+5)

Y j=0 j!(i + J +1)

where f(1 1 is the jth derivative of f (t) at t = y and k = x — y

(3.1)

I

r 3 (-k)1+1

(y, x) = f (t - y)f (t)dt = X . n f(1) + O(k'+5)

Y J!(1 + j +!)

(3.2)

O(k ) is the higher order terms with power > i

Let jUh (y, x) denote the conditional expectation of function rj(t) in the interval (y, x), so that

x

\h(t )f (t )dt

VA y x) = ~x--(3.3)

j f (t )dt

y

we have from the definition of ¡uh (y, x)

x x

M4( y, x) j f (t )dt = \h(t) f (t )dt)

therefore, we have

М„( У, x)I o( У,x) =

hi o( y, x) + I (У, x)/i!

i=i

+ O(k5)

Using the Taylor series expansions for (y, x) and I (x, y) from (3.2) and simplifying the result at point t = y, we have

M„( У,x) = h

1 + ^ k , (hf+ 2fh ) k 2 , (ff h+ffh+f h-hf2 V 3.^4, 2h "

12 fh

24 f2h

'-k3 + O(k4)

(3.4)

Proceeding in the same fashion using Taylor series expansions about the point' X, the expression for (y, x) is obtained as

M„( У, x) = h

1 k + (hf+ 2 fh ) k 2 _ {fih+ff'r+fr-hf2 2h ""

12 fh

24 f h

'-k * + O(k4)

(3.5)

Let s h(y, x) denotes the conditional variance of the function r/(t) in the interval (y, x),

we

have

_2 S h

( y,x) = Vh2( y, X)- foi y, x)f

substituting values, we get

a2h( y, x) =-

1( y)h( y)>+0(k 2)j

(3.6)

Using the above results, several other approximations can be obtained. Multiplying the series expansions for / (y, x) about the points t = y , t = x and taking the square root, we obtain

M,

.( y, x) = V ,( y),( x) [l + O(k2)]

(3.7)

From (3.3), we have

y, x)Io( y,x) = \h(t )f,(t )dt

y

Taking h(t) = t2 and using (3.7), we get

]f, (t )dt = - jt 2fi (t )dt [l + O(k2)]

y X y

Similarly expanding ^f (t) about the point t=y, we have

X 1 X

]4W)dt = k(t )dt [1+o(k2)]

(3.8)

(3.9)

APPROXIMATE SOLUTIONS OF THE MINIMAL EQUATIONS To find approximate solutions to the minimal equation (2.4), we shall obtain the series expansions

of system of equations about the point \xh ], the common boundary of hth and (h + l)ih strata. The

expansions for the two sides of the equation (2.4) are obtained by using various results proved in

the preceding section. For the expansion of the right hand side about the point xh, (y, x) is

replaced by (xh_l, xh ) while for the left hand side we replace (y, x) by (xh-l, xh ). we have

[uhc(r) - c(xh )]2 - [u,c(r) - c(xh )]2 = 0

We have from (2.13) after replacing ' y and ' X by xh and xh+l respectively.

r*2

№'c(r )

= c

c

1 +—k, + ^-J—L k + ^---J--J—L k + O(k )

2c ' 12 fc ' 24 f c '

The derivatives of c and f are evaluated at xh . Therefore, we get k

[u,c(r) - c(xh )] = k

■ + (cf'+ f ) k + O(k,2) 6f

c +

(4.1)

Similarly, we get

Vhc(r) - c(xh )]:

k. 2

c +

(c 'f' + 2f") k + O(k,2) 6f

(4.2)

Therefore from (4.1) and (4.2), we have

k 2

c +

(c f + 2fc ')

6f

k + O(ki2)

K 2

c +

(cf+ 2fC' )

6f

k + O(kr 2)

Now let us consider an expansion of the function

Bh = \c2ft (tt )dt

xh-l

■■■ Bh = c 2fkh

1 _ (c f+ 2fe" ) k

fc

2

+o(kD

multiplying it by

k_l c_ ^ 8 f

(Using Taylors expansion) and taking cube root both sides, we get

hi Bc 8 f

3 c h.

1 _ (c f + 2fc " ) 6fc

hi + O(hl)

similarly, we have

êkf Btc ] 3 ck,

_ 8 f _ 2

1 _ (cf + 2fc c ) 6fc

k, + Ok)

From the equations (4.3) and (4.4), we have 1 1

3

Or

'kh Bhc ' 3 êkh Btc 1

_ 8 f _ _ 8 f _

kh Bh = constant

(4.3)

(4.4)

(4.5)

(4.6)

(4.7)

In case it is possible to find a function Q2 {xh-l, xh ) such that

xh

k2h Bh = k2h J a2/, (t )dt

xh-l

= Q2 (xh-i, xh )[l + O(kh )2 ]

Thus the system of equations (4.6) to the same degree of accuracy can be put as

a {xh-i, xh )= Constant

the above results can be put in the form of theorem as follows. THEOREM :-

If the regression of the estimation variable 'Y' on the stratification variable ' X', in the infinite super population is given by

y = c( x) + e

i

2

Khalid Ul Islam Rather, S.E.H Rizvi, M. Sharma, M. Iqbal Jeelani, F. Danish RT&A, No 1 (72) APPROXIMATE OPTIMUM STRATA BOUNDARIES ..._Volume 18, March 2023

where ' c(x)' is a function of auxiliary variable, ' e' is the error term such that E(e | x) = 0 and V (e | x) = h( x) > 0 "x e (a, b) with (b - a) < ¥, and further if the function gx (x) fi (x) eW: then the system of equations (2.4) give strata boundaries (xh ) which correspond to the minimum of V(Y suss ) prop can be written as

kh ]c2/(t)dt.[l + O(k2h)\ 3 = k2 Jc/ (t)dt.[l + O(k2)]

vh J c /i()dt l 1 °(kh.

xh-1

xh+1 '2 rc2

xh

Neglecting the terms of order o(Sup(a b)(kh )) can be neglected; these equations can be replaced by the approximate system of equations

xh

k2h |c2f (t)dt = constant

xh-1

Or equivalently by

Q2 (xh-i, xh )= constant , kh = (xh - xh-)

02xh)[l + O(k2)] = k2h ]c2/(t)dt , i = h +1,h = 1,2,...,L

xh-1

The similar results can also be obtained by minimizing the function

L xh

± jc'fi(t)dt[l + O(k2h)\

h=l Xh-l

Thus we find that if the functionc 2 fi (x) belongs to W, the minimum value of ^ Wh02hc and

h=1

therefore V(Y stuss ) prcp, exists and the solutions of the system of equations (2.4) or equivalently of

(4.5). These equations as such are very difficult to solve and therefore it is essential to find some way out of this difficulty. It is done by replacing these systems of equations by other systems of equations which are comparatively easier to solve but are only asymptotically equivalent to the exact minimal equations. The error factor is introduced because we neglect the terms of higher powers of strata widths which is of course justifiable if the number of strata is large. We have

obtained these systems of equations after neglecting the terms of order o(Sup(a b)(kh ))4 = o(m4) where m = Sup(ab)(kh ), on both sides of the equation (4.5). If the number of strata is large and therefore terms of order 0(m4) are quite small, the error involved in the approximate systems of equations is expected to be quite small and the set of points (xh ) obtained from them shall be quite near the optimum values.

l

3

Now we proceed to develop the approximate systems of equations given in (4.6) and (4.7). Here, in finding various forms of the function Q2(Xh-\, xh), we shall keep in mind that the function Q2 (xh-1, xh ) is such that

kh )gi(t)f m = Q(xh_!,xh)[l + O(k2)]

If in (2.4) we retain only the first term on both sides of the equation and neglect the others, the two sides are equalized if

kh = constant,

b — a

and therefore xu = a +

è L - a

h , for h = 1,2,...,L

(4.8)

V

L

\h , with (x0 = a) and (xL = b)

This set of solutions cannot be expected to yield very good results as we have neglected terms of order 0(m3) on both sides of the exact minimal equations. This solution holds for all c2fi(x) provided they belong to Q and all density functions with finite range. Due to its universality of application it can be recommended in case of less information about c 2 and f (x). Apart from this, it gives the strata boundaries at once without any difficulty that may arise even in solving the approximate systems of equations. This approximate method fails if the range of ' X is infinite, but one can resort to truncation of the density function to any suitable probability level before using this approximation.

We obtain next approximate systems of equations, the optimum points of stratification are such that

h

k2h j"c2f (t)dt = c , h = 1,2,...,L

(4.9)

xh-1

The solutions obtained from this approximation are expected to be quite close to the optimum points as only terms of 0(m4 ) have been neglected. All the approximate systems that will now follow also give the points of stratification to the same degree of accuracy.

From (3.8) and equation (4.9) is obtained the following class of approximate equations. The approximations to optimum (xh ) are obtained from

J c2f (t)xdt

xh-1

For l = 1/2 and 1/3 we have

constant, h = 1,2,..., L

V

J c7 f, (t )'dt

xh-1

= c2 = constant, h = 1,2,..., L

(4.10)

For l =1/3, we have the system of equation as

x

h-1

h

3

3

xh

j [c2f, (t )dt ]

= c

h = 1,2,..., L

(4.11)

xh-1

In all these systems of equations, C s are constants to be determined and in some cases, the few equations may be meaningless such as for h = 1, i,e xh-1 = 0. Cum!^1(i) Rule

If the K (x) — c fi (t) is bounded and its first two derivative exists "x e [a, b], then for given value of L taking equal intervals on the cumulative cube root of K1 (x) will give AOSB.

EMPIRICAL STUDY:

We shall consider following distributions of auxiliary variable for evaluating the efficiency of the proposed method for obtaining optimum points of stratification. Let us assume the auxiliary variable 'x' follows certain distributions as follows:

1

I. Rectangular

II. Right-triangular

III. Exponential

f (x) = i b - a

0, otherwise

' 2(2 - x) f (x) = i (b - a)2

0, otherwise

a < x < b

f ( x) = ¿

- x+1

IV Standard normal f (x) _

1

sflp

x 2

a < x < b

1 < x <¥

—¥<X<¥

If the stratification variable follows the uniform distribution with pdf f (x) =

b - a

x î [1,2],

utilizing the cum 3 K ( x) rule, we get the stratification points as given in table I.

Table I: AOSB and Variance for uniformly distributed auxiliary variable

L AOSB Total variance nfy Lp} R.E.%

2 1.4967 0.75521 102.069

3 1.3257, 1.6624 0.75232 102.461

4 1.2483, 1.4959, 1.7466 0.75151 102.572

5 1.1983, 1.3965, 1.5898, 1.7967 0.75084 102.663

6 1.1631, 1.3289, 1.4922, 1.6543, 1.8236 0.75058 102.698

If the stratification variable follows the right triangular distribution with pdf f (x) = 2(2 — x) x î [1,2], utilizing the cum 3K1(x) rule, we get the stratification points as given in table II.

1

Table II: AOSB and Variance for Right-triangular distributed auxiliary variable

L AOSB Total Variance )pr^ } R.E.%

2 2.9662 1.01136 111.979

3 2.2935, 3.6419 0.97727 115.886

4 2.0325, 3.0001, 3.9986 0.96505 117.353

5 1.7954, 2.5879,3.4289, 4.2247 0.95773 118.250

6 1.6565, 2.3143, 2.4771, 3.6057, 4.2593 0.95589 118.477

If the stratification variable follows the Exponential distribution with pdf f (x) = e x+1 X î [1,5], utilizing the cum 3K (x) rule, we get the stratification points as given in table III.

Table III: AOSB and Variance when the auxiliary variable is exponentially distributed:

L AOSB Total Variance nfy fc«^ Lp} R.E.%

2 1.4949 0.67123 101.389

3 1.3257, 1.6558 0.66879 101.760

4 1.2589, 1.5173, 1.7814 0.66801 101.878

5 1.1981, 1.3971, 1.6452, 1.8494 0.66758 101.944

6 1.1638, 1.3277, 1.4911, 1.6562, 1.8217 0.66722 101.999

If the stratification variable follows the standard Normal distribution with pdf f ( x) =

42P

x 2

x î [0,1], utilizing the cum3K1 (x) rule, we get the stratification points as given in table IV.

Table IV: AOSB and Variance for Standard normally distributed

L AOSB Total Variance nV(VstRSS Lp } R.E.%

2 0.4939 0.08024 121.102

3 0.3282, 0.6531 0.07927 122.584

4 0.2448, 0.4947, 0.7509 0.07893 123.111

5 0.1973, 0.3943, 0.5993, 0.7986 0.07877 123.360

6 0.1687, 0.3286, 0.4944, 0.6596, 0.8269 0.07868 123.496

1

CONCLUSION

The AOSB are determined for this distribution by using cum 3K} (x) method. For each L = 2, 3, 4, 5 and 6 the variance nfy(y^ss \Iop } is calculated, which is used for the efficiency of the

stratification. The results of this investigation are given in Table I-IV. When the auxiliary variable follows uniform, right triangular, exponential and standard normal distributions the stratification points obtained has been presented in Table I-IV and percent RE as compared unstratified RSS

have been worked out. The standard normal distribution shows highest % R.E. Overall results show that the increase in the number of strata is directly proportional to the decrease in total variance. From the last column of tables it can be seen that the AOSB obtained by the proposed

method are more efficient for all L = 2,3,...,6. Thus, the proposed method of cum 3K1 (x) shows

increases gain in precision in obtaining AOSB while selecting samples using RSS.

References

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