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Module: Recapitulation of SRSWR and SRSWOR-II

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Subject: Statistics

Paper: Design of Experiments and Sample Survey

Module: Recapitulation of SRSWR and SRSWOR-II

Module No: SS-2

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Development Team

Principal investigator:

Dr. Bhaswati Ganguli,

Professor, Department of Statistics, University of Calcutta

Paper co-ordinator:

Dr. Bikas Kumar Sinha,

Retired Professor, Indian Statistical Institute, Kolkata

Content writer: Dr. Santu Ghosh,

Lecturer, Department of Environmental Health Engineering, Sri Ramachandra University, Chennai

Content reviewer: Dr. Sugata SenRoy,

Professor, Department of Statistics, University of Calcutta

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Disclaimer

I take this opportunity to thank Dr Bhaswati Ganguly of the Department of Statistics, Calcutta University for approaching me with a specific purpose. In this UGC initiated epathsala programme, for the subject of

’Statistics’, in her capacity as the Principal Investigator, she wanted me to act as a coordinator for the topic : Design of Experiments [DoE] and Sample Survey [SS]. I gladly accepted her proposal and volunteered to prepare all the 40 Modules as asked for. I have followed a distinctive style while preparing the modules viz., as that of a dialogue between an Instructor [Professor Bikram Kanti Sahay(BKS)] and his two students [Ms.

Sagarika Ghosh(SG) and Mr. Subhra Sankar Gupta(SSG)]. I fondly hope this instructional discourse and my efforts on two of my favorite topics in Statistics will be appreciated and found useful.

In the video recordings, I will impersonate as BKS.Mr. Samopriya Basu[MSc (Statistics), Calcutta University] andMs. Moumita Chatterjee[University of Calcutta, Kolkata] will impersonate as the students [SSG and SG] respectively.

Professor Bikas k Sinha Retired Professor of Statistics Indian Statistical Institute Kolkata

July 10, 2015

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SRSWR and SRSWOR

SSR

Sir, I think now we should address two related questions : Q1. Expression for E[1/ν];

Q2. Estimation of S2.

Also . . . now that we have explicit expressions for the variances of ¯y and

¯

y, we should have a concrete proof of the claim:

Var(¯y)>Var(¯y).

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SRSWR and SRSWOR

BKS

That’s right. We will take up one by one. Fortunately, these matters have been thoroughly studied and documented in the literature!

As to Q1, it turns out that E[1/ν] =

N

X

i=1

[i(n−1)]/Nn = 1/N+. . . .. As to Q2, we have a number of alternative solutions.

Basu was the first to hint on this result by invoking ’sufficiency principle’ in finite population sampling.

I find it a bit difficult to explain. On the other hand, we now have analytical proofs available.

More to it will follow. . .

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SRSWR and SRSWOR

BKS

Based on the entire data under SRSWR including repeats, we can have usual estimate forσ2 and hence we can obtain one forS2 via the relation between the two.

Also we can use y-values for distinct units for a sample insideCν and develop sν2 =X

i

[yi −y¯)2]/(ν−1) as an unbiased estimate forS2. But the point to be noted is the possibility of ’ν = 1’ in which case, it is ruled out!

So, what is our solution in this case ?

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SRSWR and SRSWOR

SSG

Sir, I think I can figure this out. What about

2=

0 if ν = 1;

sν2/[1−N(−n+1)] for ν >1.

In that case, direct computation yields :

E[ ˆS2] =S2×P[ν >1]/[1−N(−n+1)] =S2 since P[ν = 1] = 1/N(n−1).

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SRSWR and SRSWOR

BKS

Very Good ! You are just right.

Regarding your last point, we need to show

I σ2/n>S2[E(1/ν)1/N]

which amounts to :

I (N1)/Nn>E(1/ν)1/N

I i.e.,1/n1/Nn>E(1/ν)1/N.

There are interesting articles on this inequality. I can think ofChakrabarty (1963) which is definitely worth reading. Also I should mention about Korwar & Serfling(1970).

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SRSWR and SRSWOR

For n= 2, we can see that the two expressions are identical. Why so

?

Again, for n= 3,E[1/ν] =

N−3[N/1 + (N3−N−N(N−1)(N−2))/2 +N(N−1)(N−2)/3]

which simplifies to etc... etc...

In this context, I am tempted to mention about one interesting article by P K Sen and B K Sinha [Sankhya, 1989].

They develop nice group-theoretic argument to establish the

superiority of ¯y as against ¯y. Also they discuss many other related results and derive general expressions for negative powers of ’ν’.

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Reference

On averaging over distinct units in sampling with replacement. Bikas K Sinha and P K Sen. Sankhya Ser. B 51 (1989), 65–83.

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Thank You

References

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