1.Cochran, W.G. (1963) Sampling Techniques Survey Sampling ...

[Pages:17]STAT3014/3914 Applied Statistics-Sampling

References

Preliminary

1. Cochran, W.G. (1963) Sampling Techniques, Wiley, New York.

2. Kish, L. (1995) Survey Sampling, Wiley Inter. Science.

3. Lohr, S.L. (1999) Sampling: Design and Analysis, Duxbury Press.

4. McLennan, W. (1999) An Introduction to Sample Surveys, A.B.S. Publications, Canberra.

Section outline

1. Simple random samples and stratification. Finite population correction factor. Sample size determination. Inference over subpopulations.

2. Stratified sampling. Optimal allocation.

3. Ratio and regression estimators. Ratio estimators. Hartley-Ross estimator. Ratio estimator for stratified samples. Regression estimator.

4. Systematic sampling and cluster sampling.

5. Sampling with unequal probabilities. Probability proportional to size(PPS) sampling. The Horvitz-Thompson estimator.

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STAT3014/3914 Applied Stat.-Sampling C1-Simple random sample

1 Simple Random Samples (SRS)

1.1 The Population

We have a finite number of elements, N where N is assumed known. The population is Y1 . . . YN , where Yi is a numerical value associated with i-th element. We adopt the notation where capital letters refer to characteristics of the population; small letters are used for the corresponding characteristics of a sample.

N

Population Total: Y = Yi,

i=1

Population

Mean:

?

=

Y?

=

Y N

=

1 N

N

Yi,

i=1

Population Variance : 2 = 1 N

N

(Yi - Y? )2

and

i=1

S2 = N 2 = 1 N -1 N -1

N

(Yi - Y? )2.

i=1

These are fixed (population) quantities, to be estimated.

If we have to consider two numerical values: (Yi, Xi), i = 1, ? ? ? , N an additional population quantity of interest is

N

R

=

Yi

i=1

N

Xi

=

Y X

=

Y? X?

i=1

the ratio of totals.

SydU STAT3014 (2015) Second semester

Dr. J. Chan 2

STAT3014/3914 Applied Stat.-Sampling C1-Simple random sample

1.2 Simple Random Sampling

Focus on the numerical values Yi, i = 1, ? ? ? , N . A random sample of size n is taken without replacement : the observed values y1, ? ? ? , yn are random variables and are stochastically dependent. The sampling frame is a list of the values Yi, i = 1, ? ? ? , N.

The natural estimator for Y? is

Y? = 1 n

n

yi = y?

i=1

and hence for Y = N ? is

Y = N y?.

Distributional properties of y? are complicated by the dependence of the yi's.

Sample variance:

s2 = 1 n-1

n

(yi - y?)2

i=1

Fundamental Results E(y?) = ?.

Var( y?)

=

(1 -

n

)S2

=

(1 -

S2 f)

=

N -n

2

Nn

n N -1 n

var( y?) = (1 - n )s2

Nn

E(s2) = S2

where f is the sampling fraction and the finite population correction (f.p.c.) is 1 - f .

SydU STAT3014 (2015) Second semester

Dr. J. Chan 3

STAT3014/3914 Applied Stat.-Sampling C1-Simple random sample

1

1N

Proof: Let y = n

yi = n

yiIi where the sample membership

iS

i=1

indicator

1 if element i is in the sample, Ii = 0 if otherwise.

First, we have

n

E(Ii) = 0 ? Pr(Ii = 0) + 1 ? Pr(Ii = 1) = i = N ,

E(Ii2)

=

02

?

Pr(Ii

=

0)

+

12

?

Pr(Ii

=

1)

=

i

=

n N

,

E(IiIj) = 0 ? 0 Pr(Ii = 0&Ij = 0) + 0 ? 1 Pr(Ii = 0&Ij = 1) +

1 ? 0 Pr(Ii = 1&Ij = 0) + 1 ? 1 Pr(Ii = 1&Ij = 1)

n(n - 1) = ij = N (N - 1)

Var(Ii)

=

E(Ii2)

-

E2(Ii)

=

i(1

-

i)

=

n N

(1

-

n N

),

n(n - 1) n 2 Cov(Ii, Ij) = E(IiIj) - E(Ii)E(Ij) = ij - ij = N (N - 1) - N .

Then

1N

1N

n 1N

E(y) = n

yiE(Ii) = n

yi ? N = N

yi = Y . Unbiased

i=1

i=1

i=1

E[var(y)] d=ef. E

N - n s2y P f=.2 N - n Sy2 P=f.1 Var(y) Unbiased.

Nn

Nn

Proof 1: show that Var(y) = N - n Sy2 . Nn

SydU STAT3014 (2015) Second semester

Dr. J. Chan 4

STAT3014/3914 Applied Stat.-Sampling C1-Simple random sample

1N

Var(y) = Var n

yiIi

i=1

1 = n2

N

yi2Var(Ii) + 2

yiyjCov(Ii, Ij)

i=1

i j,i ................
................

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