SAMPLING DESIGNS PROPORTIONATE TO NON-NEGATIVE FUNCTIONS OF TWO QUANTILES OF AUXILIARY VARIABLE

(1)

Janusz L. Wywiał

Uniwersytet Ekonomiczny w Katowicach

SAMPLING DESIGNS PROPORTIONATE TO NON-NEGATIVE FUNCTIONS OF TWO QUANTILES OF AUXILIARY VARIABLE

1. Sampling design

Let U be a fixed population of size N. The observation of a variable under study and an auxiliary variable are denoted by y_i and x_i, i=1,...,N, respectively. Moreover, let x_i ≤x_i₊₁^,i=1,..., N−1. Our problem is estimation of the population average

= ∑

∈

U k

y

k

y N 1

.

Let us consider the sample space S of the samples s of the fixed effective size 1<n <N. The sampling design is denoted by

P (s )

. We assume that

0 ) ( ≥ s

P

for all s∈S and

∑

∈

=

s S

P ( s ) 1

.

Let

( X

_{( j}₎

)

be the sequence of the order statistics of observations of auxiliary variable in the sample s. It is well known that the sample quantile of order a

α

∈(0;1) is defined as follows:

Q

_s_,_α

= X

₍_r₎ where

r = [ n α ] + 1

, the function

[ n α ]

means the integer part of the value

[ n α ]

^,r =1,2,...,n. Let us note that

X

₍_r₎

= Q

_s_,_α for

n r n

r − ¹ ≤ α <

. In this paper it will be more conveniently to consider the order statistic than the quantile.

Let

G ( r , u , i , j ) = { s : X

₍_r₎

= x

_i

, X

₍_u₎

= x

_j

}

, r=1,...,n−1^;u=2,...,n, u

r < be the set of all samples whose r^{-th and}u-th order statistics of the auxiliary variable are equal to x_i^and

x

_j, respectively where r≤i< j≤N−n+u. Moreover,

U U

^-ⁿ ^r

N

S

+

= +

−

− +

=

r i

u n N

r u i j

j i u r

G ( , , , )

⁽¹⁾

(2)

The size of the set G(r,u,i,j) is denoted by ))

, , , ( ( )

, , ,

(r u i j Card G r u i j

g = and

⎟⎟ ⎠

⎜⎜ ⎞

⎝

⎛

−

⎟⎟ −

⎠

⎜⎜ ⎞

⎝

⎛

−

⎟⎟ −

⎠

⎜⎜ ⎞

⎝

⎛

−

= −

u n

j N r

u i j r

j i i u r

g 1

1 1

) 1 , , ,

(

⁽²⁾

Let

f ( x

_j

, x

_j

, c )

be non-negative function of values of the order statistics

X

_(r₎ and

X

_(r₎ and

⎟⎟ ⎠

⎜⎜ ⎞

⎝

⎛

−

⎟⎟ −

⎠

⎜⎜ ⎞

⎝

⎛

−

⎟⎟ −

⎠

⎜⎜ ⎞

⎝

⎛

−

= −

u n

j N r

u i j r

j i i u r

g 1

1 1

) 1 , , ,

(

⁽³⁾

The straightfoward generalization of the Wywiał’s (2009) sampling design is as follows.

Definition 1.1. The conditional sampling design proportional to the non- negative functions of the order statistics

X

₍_u₎

, X

₍_r₎ is as follows:

) , , (

) , , ) (

|

(

⁽ ⁾ ⁽ ⁾

,

z r u c

x X X c f

s

P

_r_u

=

^u ^r (4)

where i < j and r≤i≤N−n+rand r<u≤ j≤N−n+u. Particularly, let

f ( x

_j

, x

_i

, c ) = x

_j

− x

_i^and

⎩⎨

⎧

<

−

≥

−

= −

. 0

) , , ,

( for x x c

c x x for x c x

x x f

i j

i j i j i

j (5)

We say that the above sampling design is (conditional) unconditional when

)

0 ( > c

c=0. In general this concept is agree with definition of the conditional sampling design considered by Tillé (1999; 2006).

As it is well known the inclusion probability of the first order is determined

by the equations:

= ∈ = ∑

∈

} :

{ ,

,

( : ) ( | )

) , ,

(

_r_u _s_k _s _r_u

k

r u c P s k s P s c

π

, k=1,...,N

. We assume that if x≤0 then

δ ( x ) = 0

otherwise

δ ( x ) = 1

. Let us note that

)

1 ( ) 1 ( )

( x δ x − = δ x −

δ

.

Theorem 1.1. Under the sampling design

P

_r_,_u

( s )

the inclusion probabilities of the first degree are as follows. If k <r^,

) , , 1 (

1 2

2 )

, , (

) ( ) 1 ) (

, ,

( f x x c

u n

j N r

u i j r i c

u r z

k r c r

u

r _j _i

r n N

r i

u n N

r u i j

k ⎟⎟⎠

⎜⎜ ⎞

⎝

⎛

−

⎟⎟ −

⎠

⎜⎜ ⎞

⎝

⎛

−

⎟⎟ −

⎠

⎜⎜ ⎞

⎝

⎛

−

− −

= −

∑ ∑

⁻ ⁺

= +

−

− +

=

δ

π δ ⁽⁶⁾

(3)

If r≤k≤N−n+u,

− ⋅

= −

) , , (

) ( ) 1 ) (

, ,

( z r u c

k r c r

u

k r

δ π δ

⎜ +

⎜

⎝

⎛ ⎟⎟⎠

⎜⎜ ⎞

⎝

⎛

−

⎟⎟ −

⎠

⎜⎜ ⎞

⎝

⎛

−

⎟⎟ −

⎠

⎜⎜ ⎞

⎝

⎛

−

− −

−

⋅ ⁻

∑ ∑

⁺ ⁻

=

−

− +

=

) , , 1 ( 1 1

1 1

) 1 ( ) (

1 1

c x x u f

n j N r

u i j r u i

n u

k _j _i

r u k

r i

k

r u i j

δ δ

+ ⎟⎟⎠ +

⎜⎜ ⎞

⎝

⎛

−

⎟⎟ −

⎠

⎜⎜ ⎞

⎝

⎛

−

⎟⎟ −

⎠

⎜⎜ ⎞

⎝

⎛

− + −

−

∑

⁻ ⁺

=

) , , 1 ( 1 1

) 1 1

( f x x c

r u

i k r i u

n k u N

k _j _i

r u k

r i

δ

∑ ∑

⁻

= +

− +

=

⎟⎟ +

⎠

⎜⎜ ⎞

⎝

⎛

−

⎟⎟ −

⎠

⎜⎜ ⎞

⎝

⎛

−

⎟⎟ −

⎠

⎜⎜ ⎞

⎝

⎛

−

− −

−

− +

−

+ ¹

1

) , , 1 (

1 1

) 1 1 (

) (

k r i

i j u

n N

k j

c x x u f n

j N r u

i j r r i

u k u n N r

k δ δ

δ (7)

⎟⎟ +

⎠

⎜⎜ ⎞

⎝

⎛

−

⎟⎟ −

⎠

⎜⎜ ⎞

⎝

⎛

−

⎟⎟ −

⎠

⎜⎜ ⎞

⎝

⎛

− + −

− +

−

+

∑

⁻ ⁺

− +

=

) , , 1 (

1 1

) 1 1

( f x x c

u n

j N r

u k j r

k k r n

N _j _k

u n N

r u k j

δ

⎟⎟

⎠

⎟⎟ ⎞

⎠

⎜⎜ ⎞

⎝

⎛

−

⎟⎟ −

⎠

⎜⎜ ⎞

⎝

⎛

−

⎟⎟ −

⎠

⎜⎜ ⎞

⎝

⎛

−

− −

− +

−

+

∑ ∑

⁻ ⁺

+

= +

−

− +

= r n N

k i

i j u

n N

r u i j

c x x u f n

j N r

u i j r r i

k r n N

1

) , , 1 (

1 2

) 2 1 ( )

( δ

δ

If k>N−n+u,

) , , 1 (

1 1

1 )

, , (

) ) (

, ,

( f x x c

u n

j N r u

i j r i c

u r z

u n N c k

u

r _j _i

r n N

r i

u n N

r u i j

k ⎟⎟⎠

⎜⎜ ⎞

⎝

⎛

−

⎟⎟ −

⎠

⎜⎜ ⎞

⎝

⎛

−

⎟⎟ −

⎠

⎜⎜ ⎞

⎝

⎛

−

− − +

= −

∑ ∑

⁻ ⁺

= +

−

− +

=

π δ (8)

The inclusion probabilities of the second order are defined by

∑

∈ ∈

=

∈

=

} , : {

, ,

,

( , , ) ( , , ) ( : , ) ( | )

s t s k s

u r u

r k

t t

k

r u c π r u P s k s t s P s c

π

where k<t=1,...,N.

Theorem 1.2. The inclusion probabilities of the second degree of the sam- pling design

P

_r_,_u

( s | c )

are as follows.

+

∈

∈ +

∈

= +

∈

= ( , ) ( , ) ( , )

) , ,

(

₃ ₍ ₎ ₃ ₂ ₃

,_t

r u c P k t s P X

_u

x

_k

t s P k s t s

π

k

+

∈

∈ +

∈

∈ +

∈

=

+ P ( X

₍_r₎

x

_k

, t s

₃

) P ( k s

₁

, t s

₃

) P ( k s

₂

, X

₍_u₎

x

_t

)

(9)

+

∈ +

=

∈ +

=

+ P ( X

₍_r₎

x

_k

, X

₍_u₎

x

_t

) P ( k s

₁

, X

₍_u₎

x

_t

) P ( k , t s

₂

)

) , ( ) ,

( ) , ( ) ,

( X

₍ ₎

x t s

₂

P k s

₁

t s

₂

P k s

₁

X

₍ ₎

x P k t s

₁

P

_r

=

_k

∈ + ∈ ∈ + ∈

_r

∈

_t

+ ∈

+

(4)

where

( ⁻ ⁺ ⁻ ^⋅

−

= −

∈ ( k N n u )

) c , u , r ( z

) u n ) ( s t , k (

P δ 1 δ

3

. ⎟⎟⎠ + − + − + − ⋅

⎜⎜ ⎞

⎝

⎛

−

⎟⎟ −

⎠

⎜⎜ ⎞

⎝

⎛

−

⎟⎟ −

⎠

⎜⎜ ⎞

⎝

⎛

−

∑ ∑

⁻⁺ −

= +

−

− +

=

) ( _ 1 (

) , , 2 ( 2 1

1 1

1 f x x c N n u k k u

u n

j N r u

i j r i

i j r

n N

r i

u n N

r u i j

δ δ

. _⎟^⎟

⎠

⎟⎟ ⎞

⎠

⎜⎜ ⎞

⎝

⎛

−

⎟⎟ −

⎠

⎜⎜ ⎞

⎝

⎛

−

⎟⎟ −

⎠

⎜⎜ ⎞

⎝

⎛

−

∑ ∑

⁻ ⁺ −

= +

−

=

) , , 2 ( 2 1

1 1

1

c x x u f

n j N r u

i j r i

i j r

u N

r i

u n N

k j

, (10)

+ ⋅

− +

−

= −

∈

= ( , , )

) 1 (

) ( ) ) (

,

( ₍ ₎ ₃

c u r z

k u n N u n k s N

t x X

P _u _k δ δ δ

. ( , , )

1 1 1

1 1

1 f x x c

r u

i k r

i u

n k N

i j r

n k

r i

∑

⁻ ⁺

= ⎟⎟⎠

⎜⎜ ⎞

⎝

⎛

−

⎟⎟ −

⎠

⎜⎜ ⎞

⎝

⎛

−

⎟⎟ −

⎠

⎜⎜ ⎞

⎝

⎛

−

− , (11)

+ ⋅ +

−

= −

∈

∈ ( , , )

) 1 (

) ( ) 1 ) (

,

(

₂ ₃

c u r z

r u k t u n r

s u t s k

P δ δ δ

⎜ +

⎜

⎝

⎛ ⎟⎟⎠

⎜⎜ ⎞

⎝

⎛

−

⎟⎟ −

⎠

⎜⎜ ⎞

⎝

⎛

−

⎟⎟ −

⎠

⎜⎜ ⎞

⎝

⎛

− + −

− +

−

⋅

∑ ∑

⁻

=

−

− +

=

) , , 1 ( 1 2

2 1

1 1 (

1 1

c x x u f

n j N r

u i j r t i

u n

N _j _i

k r i

t r u i j

δ ⁽¹²⁾

⎟⎟⎠

⎟⎟ ⎞

⎠

⎜⎜ ⎞

⎝

⎛

−

⎟⎟ −

⎠

⎜⎜ ⎞

⎝

⎛

−

⎟⎟ −

⎠

⎜⎜ ⎞

⎝

⎛

−

− − +

−

+

∑ ∑

⁻

= +

−

− +

=

) , , 1 ( 1 2

2 1

) 1 (

1

c x x u f

n j N r

u i j r u i

n N

t _j _i

k r i

u n N

r u i j

δ ^,

+ ⋅

−

= −

∈

∈ ( , , )

) 1 (

) ( ) ) (

,

( ₍ ₎ ₃

c u r z

r k k N u s n

t x X

P _r _k

δ δ δ

⎜ +

⎜

⎝

⎛ ⎟⎟⎠

⎜⎜ ⎞

⎝

⎛

−

⎟⎟ −

⎠

⎜⎜ ⎞

⎝

⎛

−

⎟⎟ −

⎠

⎜⎜ ⎞

⎝

⎛

− + −

− +

−

⋅

∑

⁻

− +

=

) , , 1 (

1 1

) 1 1 (

) (

1

c x x u f n

j N r u

k j r

t k u n N r u k

t _j _i

t r u k j

δ

δ (13)

)

) , , 1 ( 1 1

1 1

) 1 (

) 1

( f x x c

u n

j N r u

k j r

u k n N t r k n

N _j _k

u n N

r u k

j ⎟⎟⎠

⎜⎜ ⎞

⎝

⎛

−

⎟⎟ −

⎠

⎜⎜ ⎞

⎝

⎛

−

⎟⎟ −

⎠

⎜⎜ ⎞

⎝

⎛

−

− − +

− + +

−

+

∑

⁻⁺

− +

=

δ

δ ,

(

− − + − ⋅

−

= −

∈

∈ ( ) ( )

) , , (

) ( ) 1 ) (

,

( ₁ ₃ r k t N n u

c u r z

u n s r

t s X

P δ δ δ δ

⋅

− +

−

⎟⎟ +

⎠

⎜⎜ ⎞

⎝

⎛

−

⎟⎟ −

⎠

⎜⎜ ⎞

⎝

⎛

−

⎟⎟ −

⎠

⎜⎜ ⎞

⎝

⎛

−

⋅

∑ ∑

⁻ ⁺ −

= +

−

− +

=

) ( ) 1 (

) ( ) , , 1 ( 1 1

1 2

2 f x x c r k N n u t t u

u n

j N r u

i j r i

i j r

n N

r i

u n N

r u i j

δ δ

δ

⎟⎟ +

⎠

⎜⎜ ⎞

⎝

⎛

−

⎟⎟ −

⎠

⎜⎜ ⎞

⎝

⎛

−

⎟⎟ −

⎠

⎜⎜ ⎞

⎝

⎛

−

⋅ ⁻

∑ ∑

⁺ ⁻ −

=

−

− +

=

) , , 1 (

1 1

1 2

2

1 1

c x x u f

n j N r u

i j r

i

i j r

u t

r i

t r u i j

(14)

⋅

−

− +

−

− +

−

+δ(k r 1)δ(t N n u)δ(N n r k)δ(t u)

⎟⎟

⎠

⎟⎟ ⎞

⎠

⎜⎜ ⎞

⎝

⎛

−

⎟⎟ −

⎠

⎜⎜ ⎞

⎝

⎛

−

⎟⎟ −

⎠

⎜⎜ ⎞

⎝

⎛

−

⋅ ⁻

∑ ∑

⁺ ⁻ −

+

=

−

− +

=

) , , 1 (

1 1

1 2

2

1

1 1

c x x u f

n j N r

u i j r i

i j r

u t

k i

t

r u i j

,

(5)

) , , 1 ( 1 1

1 )

, , (

) ( ) ) (

, (

1 )

(

2 f x x c

r u

i t r i u

n t N c u r z

r u u x t

X s X

P _j _i

r u t

r i t

u ⁻

∑

⁺⁻

= ⎟⎟⎠

⎜⎜ ⎞

⎝

⎛

−

⎟⎟⎠

⎜⎜ ⎞

⎝

⎛

−

⎟⎟⎠

⎜⎜ ⎞

⎝

⎛

−

− −

= −

=

∈ δ δ .

⎟⎟ +

⎠

⎜⎜ ⎞

⎝

⎛

−

⎟⎟ −

⎠

⎜⎜ ⎞

⎝

⎛

−

⎟⎟ −

⎠

⎜⎜ ⎞

⎝

⎛

−

⋅

⁻

∑ ∑

⁺ ⁻

−

=

−

− +

=

) , , 1 (

1 1

1 2

2

1 1

c x x u f

n j N r

u i j r

i

i j r

u t

r i

t r u i j

(15)

⋅

−

− +

−

− +

−

+ δ ( k r 1 ) δ ( t N n u ) δ ( N n r k ) δ ( t u )

⎟ ⎟

⎠

⎟⎟ ⎞

⎠

⎜⎜ ⎞

⎝

⎛

−

⎟⎟ −

⎠

⎜⎜ ⎞

⎝

⎛

−

⎟⎟ −

⎠

⎜⎜ ⎞

⎝

⎛

−

⋅

⁻

∑ ∑

⁺ ⁻

−

+

=

−

− +

=

) , , 1 (

1 1

1 2

2

1 1

1

c x x u f

n j N r

u i j r

i

i j r

u t

k i

t r u i j

,

=

∈

= , )

( X

₍_r₎

x

_k

X

₍_u₎

x

_t

P

(

⁽ ^, ^, ⁾

1 1 1

1 )

, , (

) 1 (

) 1

( f x x c

u n

t N r

u k t r k c

u r z

t u n N r

k

i

⎟⎟ j

⎠

⎜⎜ ⎞

⎝

⎛

−

⎟⎟ −

⎠

⎜⎜ ⎞

⎝

⎛

−

⎟⎟ −

⎠

⎜⎜ ⎞

⎝

⎛

− + −

− +

=

δ

−

δ

, (16)

− ⋅

= −

=

∈ ( , , )

) ( ) 1 ) (

,

(

₁ ₍ ₎

c u r z

r u x r

X s k

P

_u _t

δ δ

⎜⎜⎝

⎛ ⎟⎟ +

⎠

⎜⎜ ⎞

⎝

⎛

−

⎟⎟ −

⎠

⎜⎜ ⎞

⎝

⎛

−

⎟⎟ −

⎠

⎜⎜ ⎞

⎝

⎛

− + −

−

⋅

∑

⁻ ⁺

=

) , 1 ( 1 2

) 2 1 (

)

( _j _i

r u t

r i

x x r f

u i t r

i u

n t u N

t k

r

δ

(17)

, ) , 1 ( 1 2

) 2 (

) 1 (

1 ⎟⎟⎠

⎟⎟ ⎞

⎠

⎜⎜ ⎞

⎝

⎛

−

⎟⎟ −

⎠

⎜⎜ ⎞

⎝

⎛

−

⎟⎟ −

⎠

⎜⎜ ⎞

⎝

⎛

−

− − +

− +

−

⋅

∑

⁻ ⁺

+

= j i

r u t

k i

x x r f

u i t r

i u

n t k N r u t r

k

δ

− ⋅

−

− +

−

= −

∈ ( , , )

) 2 (

) 1 ) (

,

(

₂

c u r z

r u k

t r s u

t k

P δ δ

(18)

)

, , 3 (

3 1

1

1 1

c x x u f n

j N r

u i j r

i

i j k

r i

u n N

t

j

⎟⎟ ⎠

⎜⎜ ⎞

⎝

⎛

−

⎟⎟ −

⎠

⎜⎜ ⎞

⎝

⎛

−

⎟⎟ −

⎠

⎜⎜ ⎞

⎝

⎛

−

⋅ ∑ ∑

⁻

−

= +

− +

=

,

− ⋅

− +

+

−

= −

∈

∈ ( , , )

) 1 (

) 1 ) (

,

(

₍ ₎ ₂

c u r z

r u r

k n s N

t x X

P

_r _k

δ δ

) , , 2 (

2 1

1 f x x c

u n

j N r

u k j r

k

i j u

n N

r u k

j ⎟⎟

⎠

⎜⎜ ⎞

⎝

⎛

−

⎟⎟ −

⎠

⎜⎜ ⎞

⎝

⎛

−

⎟⎟ −

⎠

⎜⎜ ⎞

⎝

⎛

−

⋅ −

∑

⁻ ⁺

− +

=

,

(19)

(6)

− ⋅ +

−

= −

∈

∈ ( , , )

) )

1 (

) 1 ) (

,

(

₁ ₂

c u r z

t u n N r

u s r

t s k

P δ δ δ

⎜ ⎜

⎝

⎛ ⎟⎟ +

⎠

⎜⎜ ⎞

⎝

⎛

−

⎟⎟ −

⎠

⎜⎜ ⎞

⎝

⎛

−

⎟⎟ −

⎠

⎜⎜ ⎞

⎝

⎛

−

− −

⋅ ∑ ∑

⁻

= +

− +

=

) , , 2 (

2 2

) 2 (

1 1

c x x u f n

j N r

u i j r

k i

r

_j _i

t r i

u n N

t j

δ

(20)

, ) , , 2 (

2 2

) 2 1 (

1

1 1

⎟ ⎟

⎠

⎟⎟ ⎞

⎠

⎜⎜ ⎞

⎝

⎛

−

⎟⎟ −

⎠

⎜⎜ ⎞

⎝

⎛

−

⎟⎟ −

⎠

⎜⎜ ⎞

⎝

⎛

− + −

−

+ ∑ ∑

⁻

+

= +

− +

=

c x x u f n

j N r

u i j r

r i

k

_j _i

t k i

u n N

t j

δ

− ⋅ +

−

= −

=

∈ ( , , )

) (

) 1 ( ) 1 ) (

,

(

₁ ₍₎

c u r z

t u n N k

x r X s k

P

_r _t

δ δ δ

(21) )

, , 1 (

1 2

2

1

c x x u f n

j N r

u k j r

k

k j u

n N

t

j ⎟⎟⎠

⎜⎜ ⎞

⎝

⎛

−

⎟⎟ −

⎠

⎜⎜ ⎞

⎝

⎛

−

⎟⎟ −

⎠

⎜⎜ ⎞

⎝

⎛

−

⋅ −

∑

⁻ ⁺

+

=

,

⎜ +

⎜

⎝

⎛ ⎟⎟⎠

⎜⎜ ⎞

⎝

⎛

−

⎟⎟ −

⎠

⎜⎜ ⎞

⎝

⎛

−

⎟⎟ −

⎠

⎜⎜ ⎞

⎝

⎛

−

− −

= −

∈

∑ ∑

⁻ ⁺

= +

−

− +

=

) , 1 (

1 3

) 3 ) (

, , (

) 2 ) (

,

( ₁

r n N

r i

u n N

r u i j

i jx c x u f n

j N r u

i j r t i

c r u r z s r t k

P δ δ

).

, 1 (

1 3

) 3 1 (

∑ ∑

⁻ ⁺1 +

=

+

−

− +

=

⎟⎟

⎠

⎜⎜ ⎞

⎝

⎛

−

⎟⎟ −

⎠

⎜⎜ ⎞

⎝

⎛

−

⎟⎟ −

⎠

⎜⎜ ⎞

⎝

⎛

−

− −

−

+

^N ⁿ ^r

t i

u n N

r u i j

i j

x c x u f n

j N r

u i j r

r i δ t

(22)

2. Sampling scheme

The sampling scheme implementing the sampling design

P

_r_,_u

( s | c )

P is as follows. Firstly, population elements are ordered according to increasing values of the auxiliary variable. Let s=s₁∪{i}∪s₃ ∪{j}∪s₃, s₁ ={k:k∈U,x_k <x_i},

} ,

:

2

{ k k U x

j

x

k

x

i

s = ∈ > >

and

s

₃

= { k : k ∈ U , x

_k

> x

_j

}

. Moreover, let )

, 1 ( } { ) 1 , 1 ( } { ) 1 , 1

( i i U i j j U j N

U

U= − ∪ ∪ + − ∪ ∪ + where U(1,i−1)=

) 1 , ...

, 1

( −

= i , U(i+1, j−1)=(i+1,..., j−1), U(j+1,N)=(j+1,...,N). Let

)

);

1 , 1 (

( U i s

S −

be sample space of the sample

s

₁ of size )

);

1 , 1 ( ( ,

1SU i j s

r− + − be sample space of the sample

s

₂ of size )

);

, 1 ( ( ,

1SU j N s

r

u− − + be sample space of the sample

s

₂^{of size}n −u^. Similarly,

S = S ( U , s )

.

The sampling scheme is given by the following of probabilities:

) ( )

| ( ' ) ( )

| ( ) ( )

|

(

₁ ₁ _, ₂ ₂ _, ₃ ₃

,

s c P s p i c P s p j c P s

P

_r_u

=

_r_u _r_u (23)

(7)

where

1 3

3 1 2

2 1 1

1 , ( )

1 ) 1

( 1 ,

) 1 (

−

⎟⎟⎠

⎜⎜ ⎞

⎝

⎛

−

= −

⎟⎟⎠

⎜⎜ ⎞

⎝

⎛

−

= −

⎟⎟⎠

⎜⎜ ⎞

⎝

⎛

−

= −

u n

j s N

r P u

i s j

r P s i

P ⁽²⁴⁾

) , (

) , ,

) ( ,

| (

)

| (

) ( ,

) ( )

( , )

( )

(

, P X x c

c x X x X c P

x X x X P c i P

j u u r

j u i r u r j

u i r u

r =

=

= =

=

= (25)

∑

⁻ ⁺

=

= ^N ⁿ ^r

r i

i j j

u u r u

r f x x c g r u i j

u r c z x X P c j

p ( , ) ( , , , )

) , , ( ) 1 , (

)

| (

' _, _, ₍ ₎ ⁽²⁶⁾

) , , (

) , , , ( ) , ) (

| ( )

,

| (

) , , , (

, )

( )

(

,

z r u c

j i u r g c x x c f s P c

x X x X

P

^j ⁱ

j i u r G s

u r j

u i r u

r

∑

∈

=

(27)

In order to select the sample s, firstly the j-th element of the population should be selected, according to the probability function

p '

_r_,_u

= ( j | c )

. Next, the i-the element of the population should be drawn according to the probability function

p '

_r_,_u

= ( i | c )

. Finally, the samples

s

₁^,

s

₂^ands₃ should be selected, according to the sampling designs

P

₁

( s

₁

)

^,

P

₂

( s

₂

)

^andP₃(s₃), respectively.

3. Some sampling strategies

The well known Horvitz-Thompson (1952) estimator is as follows:

∑

∈

=

s

k k

k s

HT

y y N

π 1

, (28)

The statistic is unbiased estimator of the population mean value if

π

_k >0 for k=1,...,N. The variance and its estimator are determined by the expressions (32) and (34), respectively.

The particular case of the above estimator is the well known sampling design of the simple sample drawn without replacement is as follows:

1 0( )

−

⎟⎟⎠

⎜⎜ ⎞

⎝

=⎛ n s N

P . The variance of the mean from the simple sample

(8)

∑

∈

=

s k

s

y

k

y n 1

drawn without replacement is _s

v

_y

nN

n s N

P y

D

²

( ,

₀

( )) = −

where

∑

=

− −

=

^N

k k

y

y y

v N

1

)

2

1 (

1

_.

The results of the previous chapter lead to construction of the regression sampling strategy for the population mean ⁼

∑

ks

s yi

y n1 . We assume that

i i

i a bx e

y = + + for all i ∈U^,

∑

∈

=

U i

e

i

0

and the residuals of that linear regression function are not correlated with the auxiliary variable. The linear corre- lation coefficient between the variables y^andx will be denoted by

ρ

. Let

) ,

( X

₍_r₎

Y

_r be two dimensional random variable where

X

_(r₎ is the r-th order statistic of an auxiliary variable and Y_r is the variable under study. Wywiał (2009) considered the following estimator:

)

(

_,

, , , ,

,us HTs rus HTs

r

y b x x

y = + −

(29)

where

) ( ) ( , ,

r u

r u s

u

r X X

Y b Y

−

= − (30)

Wywiał (2009) showed that under the sampling design stated in the definition 1.1 the parameters of the following strategies

y

_r_,_u_,_s

, P

_r_,_u

( s | c ))

^{, are} approximately as follows.

( ^y ^P ^s ^c ) ^y

E

_r_,_u_,_s

,

_r_,_u

( | ) ≈

+

−

≈ ( , ( | )) 2 ( , , ( | ))

))

| ( ,

( , , , , _,

2 , ,

,

2 y P s c D y P s c bCov y x P s c

D rus ru HTs ru HTs HTs _r_u (31)

))

| ( ,

( , _,

2

2D x P s c

b HTs _r_u

+ where

⎟⎟⎠

⎜⎜ ⎞

⎝

⎛ Δ

=

∑∑

∈ ∈U

k l

l k k U

l l u k

s r s HT HT

x y c N

s P x y

Cov ^, ^, ^, ² ^,

π π

)) 1

| ( , ,

( (32)

l k l k l

k_,

= π

_,

− π , π

Δ

,

))

| ( , , ( ))

| ( ,

( , , , , _,

2 x P s c Cov x x P s c

D HTs ru = HTs HTs _r_u ,

))

| ( , , ( ))

| (

,

_, _, _, _,

,

2

y P s c Cov y y P s c

D =

_HT_s _r_u

=

_HT_s _HT_s _r_u .

SAMPLING DESIGNS PROPORTIONATE TO NON-NEGATIVE FUNCTIONS OF TWO QUANTILES OF AUXILIARY VARIABLE

Janusz L. Wywiał