Subject: Statistics
Paper: Design of Experiments and Sample Survey
Module: Recapitulation of SRSWR and SRSWOR-II
Module No: SS-2
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
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
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∗).
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. . .
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 ?
SRSWR and SRSWOR
SSG
Sir, I think I can figure this out. What about
Sˆ2=
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).
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 (N−1)/Nn>E(1/ν)−1/N
I i.e.,1/n−1/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).
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’.
Reference
On averaging over distinct units in sampling with replacement. Bikas K Sinha and P K Sen. Sankhya Ser. B 51 (1989), 65–83.