STATISTICS
IN
TRANSITION
new series
An International Journal of the Polish Statistical Association
CONTENTS
From the Editor ................................................................................................
Submission information for authors ................................................................
183
187
Sampling methods and estimation
ONYEKA A. C., A class of product-type exponential estimators of the
population mean in simple random sampling scheme ..................................
SRIVASTAVA M., GARG N., The class of estimators of finite population
mean using incomplete multi-auxiliary information .....................................
ÜNALAN T., AYHAN H. Ö., Probability sample selection method in
household surveys when current data on regional population is
unavailable ....................................................................................................
WYWIAŁ J. L., Sampling designs proportionate to sum of two order
statistics of auxiliary variable .......................................................................
189
201
217
231
Research articles
LONGFORD N. T., SALAGEAN I. C., The effect of unemployment
benefits on labour market behaviour in Luxembourg ...................................
LIBERDA B., PĘCZKOWSKI M., Households’ saving mobility in Poland ..
249
273
Other articles
MŁODAK A., Coherence and comparability as criteria of quality
assessment in business statistics ...................................................................
OKRASA W., WITEK B., Statistics as a profession – statistician as an
occupation: observations and comments from a panel of experts .................
287
319
Conferences
The regional statistics – current situation and fundamental challenges
(Borys T.) ......................................................................................................
The 22nd Didactic Conference on Teaching Quality Evaluation Methods
(Kupis-Fijałkowska A.) .................................................................................
Summer School of Baltic-Nordic-Ukrainian Network on Survey Statistics
2013 (Liberts M.) ..........................................................................................
329
337
341
Announcement
The International Year of Statistics/Statistics 2013 (Witkowski J.) ................
Volume 14, Number 2, Summer 2013
343
EDITOR IN CHIEF
Prof. W. Okrasa, University of Cardinal Stefan Wyszyński, Warsaw, and CSO of Poland
[email protected]; Phone number 00 48 22 — 608 30 66
ASSOCIATE EDITORS
Sir Anthony B. University of Oxford,
Atkinson,
UK
M. Belkindas, The World Bank,
Washington D.C., USA
Z. Bochniarz, University of Minnesota, USA
A. Ferligoj,
University of Ljubljana,
Ljubljana, Slovenia
M. Ghosh,
University of Florida, USA
Y. Ivanov,
Statistical Committee of the
Commonwealth of Independent
States, Moscow, Russia
K. Jajuga,
Wrocław University of
Economics,
Wrocław, Poland
G. Kalton,
WESTAT, Inc., USA
M. Kotzeva,
Statistical Institute of Bulgaria
M. Kozak,
University of Information
Technology and Management in
Rzeszów, Poland
D. Krapavickaite, Institute of Mathematics and
Informatics,
Vilnius, Lithuania
M. Krzyśko,
Adam Mickiewicz University,
Poznań, Poland
J. Lapins,
Statistics Department,
Bank of Latvia, Riga, Latvia
FOUNDER/FORMER EDITOR
R. Lehtonen,
A. Lemmi,
University of Helsinki, Finland
Siena University,
Siena, Italy
A. Młodak,
Statistical Office Poznań, Poland
C. A. O'Muircheartaigh,University of Chicago,
Chicago, USA
V. Pacakova,
University of Economics,
Bratislava, Slovak Republic
R. Platek,
(Formerly) Statistics Canada,
Ottawa, Canada
P. Pukli,
Central Statistical Office,
Budapest, Hungary
S.J.M. de Ree,
Central Bureau of Statistics,
Voorburg, Netherlands
I. Traat,
University of Tartu, Estonia
V. Verma,
Siena University,
Siena, Italy
V. Voineagu,
National Commission for Statistics,
Bucharest, Romania
J. Wesołowski,
Central Statistical Office of Poland,
and Warsaw University of
Technology,
Warsaw, Poland
G. Wunsch,
Université Catholiąue de Louvain,
Louvain-la-Neuve, Belgium
J. L. Wywiał,
University of Economics in
Katowice, Poland
Prof. J. Kordos, Formerly Central Statistical Office, Poland
EDITORIAL BOARD
Prof. Janusz Witkowski (Chairman), Central Statistical Office, Poland
Prof. Jan Paradysz (Vice-Chairman), Poznań University of Economics
Prof. Czesław Domański, University of Łódź
Prof. Walenty Ostasiewicz, Wrocław University of Economics
Prof. Tomasz Panek, Warsaw School of Economics
Prof. Mirosław Szreder, University of Gdańsk
Władysław Wiesław Łagodziński, Polish Statistical Association
Editorial Office
Marek Cierpiał-Wolan, Ph.D.: Scientific Secretary
[email protected]
Beata Witek: Secretary
[email protected]. Phone number 00 48 22 — 608 33 66
Rajmund Litkowiec: Technical Assistant
Address for correspondence
GUS, al. Niepodległości 208, 00-925 Warsaw, POLAND, Tel./fax:00 48 22 — 825 03 95
ISSN 1234-7655
STATISTICS IN TRANSITION-new series, Summer 2013
183
STATISTICS IN TRANSITION-new series, Summer 2013
Vol. 14, No. 2, pp. 183–185
FROM THE EDITOR
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A CLASS OF PRODUCT-TYPE EXPONENTIAL
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STATISTICS IN TRANSITION-new series, Summer 2013
Vol. 14, No. 2, pp. 201–216
THE CLASS OF ESTIMATORS OF FINITE
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REFERENCES
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STATISTICS IN TRANSITION-new series, Summer 2013
APPENDIX
Data Set I
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217
STATISTICS IN TRANSITION-new series, Summer 2013
Vol. 14, No. 2, pp. 217–230
PROBABILITY SAMPLE SELECTION METHOD IN
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STATISTICS IN TRANSITION new series, Summer 2013
231
STATISTICS IN TRANSITION new series, Summer 2013
Vol.14, No. 2, pp. 231-248
SAMPLING DESIGNS PROPORTIONATE
TO SUM OF TWO ORDER STATISTICS
OF AUXILIARY VARIABLE
Janusz L. Wywiał1
ABSTRACT
In this paper the case of a conditional sampling design proportional to the
sum of two order statistics is considered. Several strategies including the
Horvitz-Thompson estimator and ratio-type estimators are discussed. The
accuracy of these estimators is analyzed on the basis of computer simulation
experiments.
Key words: sampling design, order statistic, sample quantile, auxiliary variable, Horvitz-Thompson statistic, inclusion probabilities, sampling scheme,
ratio estimator.
1. Introduction
Sampling designs are usually constructed on the basis of an auxiliary variable observations in order to improve the accuracy of estimation of population parameters. For instance, the sampling design of Lahiri (1951), Midzuno
(1952) and Sen (1953) proportional to the sample mean of the positive valued auxiliary variable leads to unbiasednees of the well known ordinary ratio
estimator. Wywial (2008, 2009) proposed sampling designs dependent on
order statistics of the auxiliary variable. Here that approach is continued because the sampling design proportional to the sum of two auxiliary variable
order statistics is proposed. The conditional version of this sampling design
is considered in order to improve the estimation effects. The review of the
sampling designs or schemes dependent on auxiliary variables and their conditional versions is considered by Till´e (2006).
1
Katowice University of Economics, Poland, [email protected].
232
Janusz L. Wywiał: Sampling designs proportionate to sum ...
2. Sampling design
Let U be a fixed population of the size N . The observation of a variable under study and of an auxiliary variable are denoted by yi and xi , i = 1, . . . , N ,
respectively. Moreover, let 0 < xi ≤ xi+1 , i = 1, . . . , N − 1. Our problem
is the estimation of the population average y¯ = N1 k∈U yk . 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), so P (s) ≥ 0 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 the
order α ∈ (0; 1) is defined as follows: Qs,α = X(r) where r = [nα] + 1, the
function [nα] means the integer part of the value nα, r = 1, 2, ..., n. Let us
note that X(r) = Qs,α for r−1
≤ α < nr . In this paper it will be more conven
nient to consider the order statistic rather than the quantile. Let G(r, u, i, j) =
{s : X(r) = xi , X(u) = xj }, r = 1, .., n − 1; u = 2, ..., n, r < u be the set of
all samples whose r -th and u -th order statistics of the auxiliary variable are
equal to xi and xj , respectively, where r ≤ i < j ≤ N − n + u.
N −n+r N −n+u
G(r, u, i, j) = S.
i=r
(1)
j=i+u−r
The size of the set G(r, u, i, j) is denoted by g(r, u, i, j) = Card(G(r, u, i, j))
and
i−1
j−i−1
N −j
g(r, u, i, j) =
,
(2)
r−1 u−r−1
n−u
N
n
N −n+r N −n+u
= Card(S) = Card
j=i+u−r
N −n+r N −n+u
=
=
G(r, u, i, j)
i=r
N −n+r N −n+u
Card(G(r, u, i, j)) =
i=r
j=i+u−r
g(r, u, i, j),
i=r
P X(r) = xi , X(u) = xj =
j=i+u−r
g(r, u, i, j)
N
n
.
(3)
Let h(xj , xi ) be a non-negative function of values xj , xi of the order statistics X(u) and X(r) , respectively. Moreover let
f (xj , xi , c) =
h(xj , xi ) if h(xj , xi ) ≥ c,
0 f or h(xj , xi ) < c.
(4)
STATISTICS IN TRANSITION new series, Summer 2013
233
and
N −n+r N −n+u
z(r, u, c) =
f (xj , xi , c)g(r, u, i, j).
i=r
(5)
j=i+u−r
The straightfoward generalization of the Wywiał’s (2009) sampling design
is as follows.
Definition 1.1. The conditional sampling design proportional to the nonnegative functions of the order statistics X(u) , X(r) is as follows:
Pr,u (s|c) =
f (xj , xi , c)
z(r, u, c)
for
s ∈ G(r, u, i, j)
(6)
where i < j, r ≤ i ≤ N − n + r and r < u ≤ j ≤ N − n + u,
0 ≤ c ≤ c0 .
As it is well known, the inclusion probability of the first order is determined
by the equations: πk (r, u, c) = Pr,u (s : k ∈ s|c) = {s:k∈s} Pr,u (s|c), k =
1, ..., N.
Now, the upper possible value c0 of the constant c should be stated in such
a way that πk (r, u, c) > 0 for all k = 1, ..., N . It is important because under the just defined condition the well known Horvitz-Thompson estimator
(considered in the sections 3) is unbiased for the a population mean.
The above defined sampling design is treated as conditional (unconditional) when c > 0 (c = 0), see the definition of the conditional sampling
design considered by Till´e (1999, 2006).
Particularly, let h(xj , xi ) = xj + xi . Thus
f (xj , xi , c) =
xj + xi f or xj + xi ≥ c,
0 f or xj + xi < c.
(7)
We have to assume that 0 ≤ c ≤ c0 = x1 + xN . Thus, under this assumption the inclusion probabilities πN (r, u, c) > 0 for all i = 1, ..., N . When
c > c0 , x1 + xi < c for all i = 2, ..., N and π1 (r, u, c) = 0 and πk (r, u, c) ≥ 0
for k = 2, ..., N . In this case, as it is well known, the Horvitz-Thompson’s
statistic is a biased estimator of the population mean.
Let δ(x) be such a function that if x ≤ 0 then δ(x) = 0 otherwise δ(x) = 1.
Moreover, δ(x)δ(x − 1) = δ(x − 1). The following two theorems are the
straightforward generalizations of those ones proved by Wywiał(2009).
234
Janusz L. Wywiał: Sampling designs proportionate to sum ...
Theorem 1. The inclusion probabilities of the first order for the conditional
sampling design Pr,u (s|c) are as follows:
πk (r, u, c) =
1
=
δ(r − 1)
z(r, u, c)
j−i−1
u−r−1
N −n+r
N −n+u
i=rδ(r−k)+(k−1)δ(k+1−r)δ(N −n+r−k) j=i+u−r
i−2
r−2
N −j
f (xj , xi , c) + δ(k − r)δ(N − n + u − k)δ(u − r − 1)
n−u
min(k−1,N −n+r)
N −n+u
i=r
j=max(i+u−r,k+1)
i−1
r−1
j−i−2
u−r−2
N −j
f (xj , xi , c)+
n−u
+ δ(k − u)δ(n − u)δ(N − n + u − k + 1)
k−u+r−1
k−1
i=r
j=i+u−r
i−1
r−1
j−i−1
u−r−1
N −n+r N −n+u
δ(k−N +n−u)
i=r
j=i+u−r
i−1
r−1
N −j−1
f (xj , xi , c) + δ(n − u)
n−u−1
j−i−1
u−r−1
+ δ(k + 1 − r)δ(N − n + r − k + 1)
N −n+u
j=k+u−r
j−k−1
u−r−1
N −j−1
f (xj , xi , c)+
n−u−1
k−1
r−1
N −j
f (xj , xk , c)+δ(k−u+1)δ(N −n+u−k+1)
n−u
N −k
n−u
k−u+r
i=r
i−1
r−1
k−i−1
f (xk , xi , c)
u−r−1
(8)
Theorem 2. The inclusion probabilities of the second order for the conditional sampling design Pr,u (s|c) are as follows:
πk,t (r, u, c) = P (k, t ∈ s1 )+P k ∈ s1 , X(r) = xt +P (k ∈ s1 , t ∈ s2 ) +
+ P k ∈ s1 , X(u) = xt + P (k ∈ s1 , t ∈ s3 ) + P X(r) = xk , t ∈ s2 +
+ P X(r) = xk , X(u) = xt + P X(r) = xk , t ∈ s3 + P (k, t ∈ s2 ) +
+ P k ∈ s2 , X(u) = xt + P (k ∈ s2 , t ∈ s3 ) + P X(u) = xk , t ∈ s3 +
+ P (k, t ∈ s3 ) (9)
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STATISTICS IN TRANSITION new series, Summer 2013
where
P (k, t ∈ s1 ) =
δ(r − 2)δ(N − n + r − t)
·
z(r, u)
N −n+r
N −n+u
i−3
r−3
·
i=max(r,t+1) j=i+u−r
j−i−1
u−r−1
N −j
f (xj , xi , c).
n−u
P k ∈ s1 , X(r) = xt =
=
δ(r − 1)δ(N − n + r − k)δ(N − n + r + 1 − t)δ(t + 1 − r)
·
z(r, u)
·
t−2
r−2
N −n+u
j−t−1
u−r−1
j=t+u−r
N −j
f (xj , xt , c),
n−u
P (k ∈ s1 , t ∈ s2 ) = δ(N − n + r − k)·
·
δ(t − r)δ(r − 1)δ(u − r − 1)δ(N − n + u − t)δ(t − k − 1)
·
z(r, u)
min(t−1,N −n+r)
N −n+u
i=max(r,k+1)
j=max(t+1,i+u−r)
·
i−2
r−2
j−i−2
u−r−2
N −j
f (xj , xi , c),
n−u
P k ∈ s1 , X(u) = xt = δ(N − n + r − k)·
·
δ(t + 1 − u)δ(N − n + u + 1 − t)δ(t − k − u + r)δ(r − 1)
·
z(r, u)
N −t
·
n−u
t−u+r
i=max(r,k+1)
i−2
r−2
t−i−1
f (xt , xi , c),
u−r−1
P (k ∈ s1 , t ∈ s3 ) =
=
δ(N − n + r − k)δ(t − u)δ(r − 1)δ(n − u)δ(t − k − u + r − 1)
·
z(r, u)
min(t−u+r−1,N −n+r) min(t−1,N −n+u)
·
i=max(r,k+1)
j=i+u−r
i−2
r−2
j−i−1
u−r−1
N −j−1
·
n−u−1
· f (xj , xi , c),
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Janusz L. Wywiał: Sampling designs proportionate to sum ...
P X(r) = xk , t ∈ s2 =
=
δ(k + 1 − r)δ(N − n + r + 1 − k)δ(t − r)δ(N − n + u − t)δ(u − r − 1)
·
z(r, u)
·
k−1
r−1
N −n+u
j=max(t+1,k+u−r)
j−k−2
u−r−2
N −j
f (xj , xk , c),
n−u
P X(r) = xk , X(u) = xt =
=
δ(k + 1 − r)δ(N − n + r + 1 − k)δ(t + 1 − u)δ(N − n + u − t + 1)
·
z(r, u)
k−1
t−k−1
N −t
· δ(t + 1 − k − u + r)
f (xt , xk , c),
r−1
u−r−1
n−u
P X(r) = xk , t ∈ s3 =
=
δ(k + 1 − n)δ(N − n + r + 1 − k)δ(t − u)δ(n − u)δ(t − k − u + r)
·
z(r, u)
·
k−1
r−1
P (k, t ∈ s2 ) =
min(t−1,N −n+u)
j−k−1
u−r−1
j=k+u−r
N −j−1
f (xj , xk , c),
n−u−1
δ(k − r)δ(N − n + u − t)δ(u − r − 2)
·
z(r, u)
min(k−1,N −n+r)
N −n+u
i=r
j=max(t+1,i+u−r)
·
i−1
r−1
j−i−3
u−r−3
N −j
f (xj , xi , c),
n−u
P k ∈ s2 , X(u) = xt =
=
δ(k − r)δ(N − n + u − k)δ(t + 1 − u)δ(N − n + u + 1 − t)δ(u − r − 1)
·
z(r, u)
N −t
·
n−u
min(k−1,t−u+r)
i=r
i−1
r−1
t−i−2
f (xt , xi , c),
u−r−2
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STATISTICS IN TRANSITION new series, Summer 2013
P (k ∈ s2 , t ∈ s3 ) =
=
δ(k − r)δ(N − n + u − k)δ(t − u)δ(t − k − 1)δ(u − r − 1)δ(n − u)
·
z(r, u)
min(k−1,t−u+r−1,N −n+r) min(t−1,N −n+u)
i−1
r−1
·
i=r
j=max(i+u−r,k+1)
j−i−2
u−r−2
N −j−1
·
n−u−1
· f (xj , xi , c),
P X(u) = xk , t ∈ s3 =
=
δ(k + 1 − u)δ(N − n + u + 1 − k)δ(t − u)δ(n − u)
·
z(r, u)
N −k−1
·
n−u−1
P (k, t ∈ s3 ) =
k−n+r
i=r
i−1
r−1
k−i−1
f (xk , xi , c),
u−r−1
δ(k − u)δ(n − u − 1)
·
z(r, u)
min(N −n+r,k−1−u+r) min(k−1,N −n+u)
·
i=r
j=i+u−r
i−1
r−1
j−i−1
u−r−1
N −j−2
·
n−u−2
· f (xj , xi , c).
3. Sampling scheme
The sampling scheme implementing the sampling design Pr,u (s|c) is as
follows. Firstly, population elements are ordered according to increasing values of the auxiliary variable. Let s = s1 ∪ {i} ∪ s3 ∪ {j} ∪ s3 where s1 =
{k : k ∈ U, xk < xi } is the simple random sample of the size r − 1 drawn
without replacement from the subpopulation U (1, i − 1) = (1, ..., i − 1),
s2 = {k : k ∈ U, xj > xk > xi } is the simple random sample of the size
u − r − 1 drawn without replacement from U (i + 1, j − 1) = (i + 1, ..., j − 1)
and s3 = {k : k ∈ U, xk > xj } s the simple random sample of the size
n − u drawn without replacement from U (j + 1, N ) = (j + 1, ..., N ). Let us
note that U = U (1, i − 1) ∪ {i} ∪ U (i + 1, j − 1) ∪ {j} ∪ U (j + 1, N ). Let
S (U (1, i − 1); s) be sample space of the sample s1 , let S (U (i + 1, j − 1); s)
be sample space of the sample s2 and let S (U (j + 1, N ); s) be sample space
of the sample s3 . Moreover, S = S (U, s)).
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Janusz L. Wywiał: Sampling designs proportionate to sum ...
The sampling scheme is given by the following probabilities:
Pr,u (s|c) = P1 (s1 |i)pr,u (i|c)P2 (s2 |i, j)pr,u (j|i, c)P3 (s3 |j)
(10)
where
P1 (s1 |i) =
i−1
r−1
−1
,
P2 (s2 |i, j) =
j−i−1
u−r−1
−1
,
P3 (s3 |j) =
N −j
n−u
pr,u (i, j|c)
,
pr,u (i|c)
f (xj , xi , c)g(r, u, i, j)
,
Pr,u (s) =
z(r, u, c)
pr,u (j|i, c) =
pr,u (i, j|c) =
s∈G(r,u,i,j)
−1
,
(11)
(12)
N −n+u
1
pr,u (i|c) =
f (xj , xi , c)g(r, u, i, j).
z(r, u, c) j=i+u−r
(13)
In order to select the sample s, firstly the i-th element of the population should
be selected according to the probability function pr,u (i|c). Next, the j-the element of the population should be drawn according to the probability function
pr,u (j|i, c). Finally, the samples s1 , s2 and s3 should be selected according to
the sampling designs P1 (s1 ), P2 (s2 ) and P3 (s3 ), respectively.
4. Some sampling strategies
The well known Horvitz-Thompson estimator (1952) is as follows:
y¯HT,s =
1
N
k∈s
yk
.
πk
(14)
The statistic is unbiased estimator of the population mean value if πk > 0 for
k = 1, ..., N . The variance and its estimator are determined by the expressions (20) and (22), respectively.
The particular case of the above estimator is the well known sampling design of the simple random sample drawn without replacement whose sam−1
pling design is: P0 (s) = Nn
. The variance of the mean from the simple
1
random sample y¯s = n k∈s yk drawn without replacement is D2 (¯
ys , P0 (s)) =
N
N −n
1
2
v(y) where v(y) = N −1 k=1 (yk − y¯) .
nN
Let us construct the following ratio sampling strategy for the population
mean y¯ = N1 i∈U yi . We assume that yi = bxi + ei for all i ∈ U, i∈U ei =
239
STATISTICS IN TRANSITION new series, Summer 2013
0 and the residuals of that linear regression function are not correlated with
the auxiliary variable. Thus, it has been assumed that the intercept of the
linear regression function is equal to zero. The linear correlation coefficient
between the variables y and x will be denoted by ρ. Let X(r) , Yr be two
dimensional random variables where X(r) is the r-th order statistic of an auxiliary variable and Yr is the variable under study. Let us define the following
ratio type estimator:
y¯r,u,s = y¯s
E X(r) + X(u) |c
X(r) + X(u)
(15)
where on the basis of the expressions (3) and (5) we have
E X(r) + X(u) |c =
N −n+r N −n+u
=
f (xj , xi , c)P X(r) = xi , X(u) = xj |X(r) + X(u) ≥ c =
i=r
j=i+u−r
N −n+r N −n+u
f (xj , xi , c)
i=r
=
j=i+u−r
N −n+r
i=r
N −n+r
i=r
N −n+u
j=i+u−r
N −n+u
j=i+u−r
P X(r) = xi , X(u) = xj
=
P X(r) + X(u) ≥ c
f (xj , xi , c)g(r, u, i, j)
γ(xj , xi , c)g(r, u, i, j)
N −n+r N −n+u
α(r, u, c) =
γ(xj , xi , c)g(r, u, i, j) =
i=r
j=i+u−r
γ(xi , xj , c) =
1
0
for
for
=
z(r, u, c)
, (16)
α(r, u, c)
N
P X(r) + X(u) ≥ c ,
n
xi + xj ≥ c
xi + xj < c.
Let Sc = {s : x(r) + x(u) ≥ c} and Sc¯ = {s : x(r) + x(u) < c} =
S − Sc where (x(r) , x(u) ) are values of the order statistics (X(r) , X(u) ). More1
1
over, let y¯c = Card(S
¯s , y¯c¯ = Card(S
¯s where Card(Sc ) =
s∈Sc y
s∈Sc¯ y
c)
c
¯)
α(r, u, c). Under the stated assumptions we have:
E (¯
yr,u,s , Pr,u (s|c)) =
y¯s
s∈Sc
=
y¯s
s∈Sc
E X(r) + X(u) |c
Pr,u (s|c) =
x(r) + x(u)
x(r) + x(u)
z(r, u, c)
1
y¯s = y¯c .
=
Card(Sc ) s∈S
α(r, u, c) x(r) + x(u) z(r, u, c)
c
240
Janusz L. Wywiał: Sampling designs proportionate to sum ...
Particularly, if c = 0 then Pr,u (s|0) = Pr,u (s) and
E (¯
yr,u,s , Pr,u (s)) =
y¯s
s∈S
E X(r) + X(u)
1
Pr,u (s) = N
x(r) + x(u)
n
y¯s = y¯.
s∈S
These results and the decomposition:
y¯ =
1
N
n
1
y¯s =
N
n
s∈S
=
1
N
n
y¯s +
s∈Sc
y¯s
=
s∈Sc¯
(α(r, u, c)¯
yc + (1 − α(r, u, c))¯
yc¯) =
= y¯c P X(r) + X(u) ≥ c + y¯c¯P X(r) + X(u) < c
lead to the following expression:
E (¯
yr,u,s , Pr,u (s|c)) =
y¯ for c = 0
y¯ + (¯
yc − y¯c¯) P X(r) + X(u) < c
c > 0.
(17)
the strategy
for
Hence, under the unconditional sampling design Pr,u (s)
(¯
yr,u,s , Pr,u (s|c)) is unbiased.
The next ratio type estimator, see e.g. S¨arndal at. all (1992), is as follows:
y˜s = y¯HT,s
x¯
(18)
x¯HT,s
The parameters of the strategy (˜
ys , Pr,u (s|c)) are approximately as follows:
E (˜
ys , Pr,u (s|c)) ≈ y¯,
ys , Pr,u (s|c)) ≈ D2 (¯
yHT,s , Pr,u (s|c))−2hCov (¯
yHT,s , x¯HT,s , Pr,u (s|c)) +
D2 (˜
xHT,s , Pr,u (s|c)) (19)
+ h2 D2 (¯
where h =
y¯
x
¯
and
Cov (¯
yHT,s , x¯HT,s , Pr,u (s|c)) =
1
N2
Δk,l
k∈U l∈U
yk xl
πk πl
,
(20)
D2 (¯
xHT,s , Pr,u (s|c)) = Cov (¯
xHT,s , x¯HT,s , Pr,u (s|c)) ,
Δk,l = πk,l −πk πl ,
2
D (¯
yHT,s , Pr,u (s|c)) = Cov (¯
yHT,s , y¯HT,s , Pr,u (s|c)) .
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STATISTICS IN TRANSITION new series, Summer 2013
The variance: D2 (¯
yr,u,s , Pr,u (s|c)) can be estimated by the following approximately unbiased estimator:
ˆ 2 (˜
ˆ 2 (¯
D
ys , Pr,u (s|c)) = D
yHT,s , Pr,u (s|c)) +
ˆ 2 (¯
xHT,s , Pr,u (s|c)) (21)
yHT,s , x¯HT,s , Pr,u (s|c)) + h2r,u,s D
− 2hr,u,s Cov (¯
where
hr,u,s =
Cov (¯
yHT,s , x¯HT,s , Pr,u (s|c)) =
y¯HT,s
,
x¯HT,s
1
N2
Δ∗,k,l
k∈s l∈s
yk xl
πk πl
,
(22)
Δk,l
ˆ 2 (¯
,
D
xHT,s , Pr,u (s|c)) = Cov (¯
xHT,s , x¯HT,s , Pr,u (s|c)) ,
Δ∗k,l = πk,l
2
ˆ (¯
yHT,s , Pr,u (s|c)) = Cov (¯
yHT,s , y¯HT,s , Pr,u (s|c)) .
D
Let us remind the following ordinary ratio estimator.
x¯
yˆs = y¯s .
(23)
x¯s
The approximate value of the variance is as follows:
N (N − n)
D2 (ˆ
ys , P0 (s)) ≈
v(y) + h2 v(x) − 2hv(x, y)
n
¯)(yk − y¯) and particularly v(y) = v(y, y),
where v(x, y) = N 1−1 N
k=1 (xk − x
v(x) = v(x, x).
The approximately unbiased estimator of the variance is as follows:
N (N − n)
ˆ 2 (ˆ
D
vs (y) + h2s vs (x) − 2hs vs (x, y) ,
ys , P0 (s)) =
n
1
where hs = xy¯¯ss , vs (x, y) = n−1
¯s )(yk − y¯s ) and particularly
k∈s (xk − x
ys , P0 (s)) is approximately
vs (y) = vs (y, y), vs (x) = vs (x, x). The strategy (ˆ
unbiased for the population mean y¯.
It is well known that the strategy (ˆ
ys , P1 (s)) is unbiased for y¯ where
x¯s
(24)
P1 (s) = N
n
is the sampling design of Lahiri (1951), Midzumo (1952) and Sen (1953).
5. Simulation analysis of strategies’ accuracy
The population taken into account consists of the municipalities in Sweden
whose number is N = 284. The value xk , k = 1, ..., N , of the auxiliary
variable x is equal to the size (in thousands) of people population in the k-th
242
Janusz L. Wywiał: Sampling designs proportionate to sum ...
municipality in 1975. The value yk , k = 1, .., N , of the variable under study
y is the taxation revenues (in millions of kronor) from the k-th municipality in
1985. Their observations were published by S¨arndal, Swenson and Wretman
(1992), pp. 652-659. number is N = 284.
Figure 1. Scatterplot of y versus x.
There are three outlier observations of the variable as it is shown in Figure
1. Let σ and β3 be the standard deviation and the skewness coefficient, respectively, of the auxiliary variable in the population. In case of data without
outliers (the number of municipalities is 281) x¯ = 24, 263, σ = 24, 153 and
β3 = 0, 043. In case of data with outliers (the number of municipalities is
N=284) x¯ = 28.810, σ = 52, 873 and β3 = 8, 427. Thus, the distribution
of the variable without the outliers is almost symmetric. But the distribution
of the variables with outliers is highly right skewed. The samples according to the preassigned sampling design were drawn from the just presented
population. The samples were replicated 1000 times.
The Figure 1 shows that the dependence between the variable under study
and the auxiliary variable can be approximated by means of a linear regression with its constance equal to zero. In this case, as it is well-known, the
accuracies of regression type estimators of a population mean are similar to
the accuracy of the ratio type estimators. Moreover, the ratio estimators are
simpler than the regression ones. Thus, that is why we consider only the ratio
estimator in the analysis.
243
STATISTICS IN TRANSITION new series, Summer 2013
Let M SE(t, P (s)) be the mean square error of the strategy (t, P (s)) used
to estimate the population mean y¯. The coefficient of the relative efficiency
is defined as follows:
M SE(t, P (s))
100%
D2 (¯
ys , P0 (s))
The results of the simulation analysis are presented by Tables and Figures
2, 3. The tables show the relative efficiency coefficients. The distributions of
the estimators, generated on the basis of samples of the size n = 14 drawn
by means of appropriate sampling schemes are presented by means of the
well-known box-plots on Figures 2 and 3. Table 1 shows the notation of the
strategies.
e(t, P (s)) =
Table 1. The symbols of the strategies.
strategy
y¯s , P0 (s)
yˆs , P0 (s)
yˆs , P1 (s)
y¯HT,s , P1 (s)
y¯HT,s , Pn−1,n (s)
y¯HT,s , Pn−1,n (s|3¯
x)
y˜s , Pn−1,n (s)
y˜s , Pn−1,n (s|3¯
x)
y¯n−1,n,s , Pn−1,n (s)
y¯n−1,n,s , Pn−1,n (s|3¯
x)
symbol
ty0
ty1
ty2
ty3
ty4
ty4d3
ty5
ty5d3
ty6
ty6d3
efficiency
−
e1
e2
e3
e4
e4
e5
e5
e6
e6
Table 2. The relative efficiency coefficients (%) of the strategies.
N:
n
2 (0,7%)
3 (1%)
6 (2%)
9 (3%)
14 (5%)
29 (10%)
e1
2.50
2.88
3.09
3.28
3.42
3.22
281
e2
2.24
2.58
2.86
3.09
3.30
3.14
e3
17.62
28.10
46.62
57.95
73.59
84.51
e1
1.56
1.88
2.73
3.78
5.04
7.22
284
e2
1.15
1.35
1.96
2.66
3.62
5.92
e3
5.60
8.64
14.87
20.49
29.33
47.44
Firstly, let us consider the strategies under the unconditional sampling designs. Table 2 shows the relative efficiency coefficients of the strategies which
244
Janusz L. Wywiał: Sampling designs proportionate to sum ...
do not depend on the sampling design Pr,u (s|c). The ratio estimator under
the sampling design P1 (s) is slightly better than the ratio estimator under the
simple random sample and they are both significantly more accurate than the
Horvtiz-Thompson estimator under the sampling design P1 (s). In general,
Tables 2, 3 and 4 let us infer that the ratio type strategies are significantly better than the Horvitz-Thompson ones. This conclusion is strongly confirmed
by the box-plots shown by Figures 2 and 3. The strategy (ˆ
ys , P1 (s)) is the
best among the six considered strategies except for the case of the population
with outliers when the strategy (˜
ys , Pn−1,n (s)) is the best for n = 14 n = 29.
Wywiał(2007) considered the accuracy of estimation on the basis of the
sampling design proportional to one order statistic denoted by Pr (s). For
some of possible values r of the sampling design Pr (s) the mean squares of
the estimators were determined on the basis of the simulation analysis. The
results of the analysis lead to the conclusion that the considered strategies
dependent on sampling design Pr (s) are the most accurate when r = n − 1
or r = n. That is why Tables 3 and 4 deal only with the case when r = n−1 or
r = n. In general, all the inclusion probabilities of the first order are expected
to be proportional to the appropriate values of a positively valued auxiliary
variable. Thus, in the case considered we can suppose that if r < n − 1 and
r < u ≤ n, the inclusion probabilities πk (r, u, c) are not so proportional to
xk as in the case when r = n − 1 and u = n for all k = 1, ..., N .
Table 3. The relative efficiency coefficients (%) of the conditional strategies for Pn−1,n (s|k¯
x), k = 0, 1, 2, 3. The population with outliers, N = 284.
k¯
x
n
2
3
6
9
14
29
e4
6.5
9.2
18.2
21.4
25.6
34.5
0
e5
1.3
1.5
2.3
2.9
3.4
4.5
e6
2.0
1.9
3.0
5.5
13.9
56.8
e4
1.9
3.5
11.8
17.8
23.2
34.9
x¯
e5
8.6
1.1
2.0
2.8
3.2
4.6
2¯
x
e6
e4
e5
2.2 4.3 0.7
2.0 3.9 0.9
2.9 5.6 1.5
5.4 9.0 2.0
13.5 15.3 2.8
58.2 29.1 4.4
3¯
x
e6
e4
e5
3.2 9.8 5.4
3.0 6.3 0.5
3.0 5.7 0.9
4.4 6.2 1.3
10.0 8.4 1.9
48.7 19.8 3.6
e6
3.2
3.5
3.8
4.8
7.7
35.1
The analysis of Tables 3, 4 and Figures 2, 3 leads to the following conclusion. The relative accuracies of the sampling strategies for sampling designs
Pr,u (s|c) are usually better in case of the population with outliers or extreme
values.
245
STATISTICS IN TRANSITION new series, Summer 2013
The Figures 2 and 3 let us infer that in the case of the population without
outliers values the distributions of the estimators are almost symmetric. In
the case of the population with outliers values the estimators are distributed
with a large number of outliers. In this situation we cannot expect an accurate estimation. But the analysis of the Figures 2 and 3 lets us say that
in the case of the population with outliers the conditional sampling design
(for c > 0) leads to reduction of the number of outliers observations in the
distribution of appropriate estimators. The conditional strategies which depend on the sampling design Pr,u (s|c) are usually more accurate than their
appropriate unconditional versions (for c = 0). The estimators of the conditional strategies are unbiased or negligible biased estimators of the population mean. When the conditioning value c increases (the considered levels: c = 0, c = x¯, c = 2¯
x, and c = 3¯
x) the accuracies of the strategies
(˜
ys , Pr,u (s|c)) and (¯
yHT,s , Pr,u (s|c)) usually increase, too. The accuracy of
the strategy (˜
ys , Pr,u (s|c)) is the best among the conditional ones. The strategy (¯
yr,u,s , Pr,u (s|c)) is not worse than (¯
yHT,s , Pr,u (s|c)) except for the case
of the population with outliers and of the sample size n = 29.
Table 4. The relative efficiency coefficients (%) of the conditional strategies for Pn−1,n (s|k¯
x), k = 0, 1, 2, 3. The population without outliers,N = 281.
k¯
x
n
2
3
6
9
14
29
e4
17.7
24.5
41.8
51.8
63.8
81.7
0
e5
2.3
2.3
2.5
2.8
3.1
3.1
e6
2.7
3.1
12.7
25.3
44.6
84.3
e4
3.8
8.0
30.3
50.6
59.9
75.0
x¯
e5
1.6
1.7
2.2
2.8
3.0
3.1
e6
2.1
3.0
11.8
22.8
39.6
82.9
e4
17.6
15.9
17.1
26.3
39.9
71.8
2¯
x
e5
2.5
2.0
2.2
2.3
2.8
2.9
e6
1.2
2.2
2.3
16.8
28.7
69.1
e4
32.3
14.3
14.8
26.8
36.4
56.9
3¯
x
e5
3.0
2.4
2.3
2.3
2.2
2.9
e6
0.8
1.9
1.8
11.3
22.1
52.4
246
Janusz L. Wywiał: Sampling designs proportionate to sum ...
Figure 2. Boxplot of the estimator distributions in the population without outliers for n=14
Figure 3. Boxplot of the estimator distributions in the population without outliers for n=14
STATISTICS IN TRANSITION new series, Summer 2013
247
6. Conclusions
The inclusion probabilities of the conditional sampling design proportionate to the sum of two order statistics are presented. They let us determine the
variance of the Horvitz-Thompson estimator as well as its estimate.
The simulation analysis lets us expect the estimation strategies with the
sampling design Pn−1,n (s|c) not to be less accurate than the strategies with
sampling design Pr,u (s|c).
In general, the accuracies of the considered ratio type strategies:
(˜
ys , Pn−1,n (s|c)), (ˆ
ys , P1 (s)) or (ˆ
ys , P0 (s)) are the best among the all strategies considered in the analysis. The accuracies of these three strategies are
comparable. Moreover, the conditional strategies for c = k¯
x, k > 0 are
slightly better than the appropriate unconditional ones.
Let us underline that the above conclusions cannot be treated as sufficiently
general because they have been derived on the basis of a partial computer simulation analysis based on special data set taken into account. But it seems that
those results can be an inspiration for larger simulation analyses or studies of
the formal properties of the sampling design.
Acknowledgement
The research was supported by the grant number N N111 434137 from the
Ministry of Science and Higher Education.
248
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/$+,5, * : $ PHWKRG IRU VDPSOH VHOHFWLRQ SURYLGLQJ XQELDVHG UDWLR
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7,//( < Sampling algorithms 6SULQJHU
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WKH RUGHU VWDWLVWLF RI DQ DX[LOLDU\ YDULDEOH Statistics in Transition-new series,
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DX[LOLDU\ YDULDEOH Statistical Papers, 9RO 1R SS ±
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STATISTICS IN TRANSITION-new series, Summer 2013
249
STATISTICS IN TRANSITION-new series, Summer 2013
Vol. 14, No. 2, pp. 249–272
THE EFFECT OF UNEMPLOYMENT BENEFITS ON
LABOUR MARKET BEHAVIOUR IN LUXEMBOURG
Nicholas T. Longford , Ioana C. Salagean ABSTRACT
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1. Introduction
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REFERENCES
$%$',( $ ,0%(16 * /DUJH VDPSOH SURSHUWLHV RI PDWFKLQJ
HVWLPDWRUV IRU DYHUDJH WUHDWPHQW HIIHFWV Econometrica ±
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VSHOOV HYLGHQFH IURP WKH 1HZ -HUVH\ H[WHQGHG EHQHILW SURJUDP Journal of
Public Economics ±
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ZLWK OLPLWHG RYHUODS LQ HVWLPDWLRQ RI DYHUDJH WUHDWPHQW HIIHFWV Biometrika ±
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273
STATISTICS IN TRANSITION-new series, Summer 2013
Vol. 14, No. 2, pp. 273–286
HOUSEHOLDS’ SAVING MOBILITY IN POLAND
Barbara Liberda , 0DUHN 3ĊF]NRZVNL ABSTRACT
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REFERENCES
%$;7(5 0 -(50$11 8 - +RXVHKROG 3URGXFWLRQ DQG WKH ([FHVV
6HQVLWLYLW\ RI &RQVXPSWLRQ WR &XUUHQW ,QFRPH American Economic Review
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&$03%(// - < 'RHV 6DYLQJ $QWLFLSDWH 'HFOLQLQJ /DERXU ,QFRPH"
$Q $OWHUQDWLYH 7HVW RI WKH 3HUPDQHQW ,QFRPH +\SRWKHVLV Econometrica, SS ±
&$552// & ' +RZ 'RHV )XWXUH ,QFRPH $IIHFW &XUUHQW
&RQVXPSWLRQ" Quarterly Journal of Economics, )HEUXDU\ SS ±
&$552// & ' 3UHFDXWLRQDU\ 6DYLQJ DQG WKH 0DUJLQDO 3URSHQVLW\ WR
&RQVXPH 2XW RI 3HUPDQHQW ,QFRPH Journal of Monetary Economics,
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)/$9,1 0 $ 7KH $GMXVWPHQW RI &RQVXPSWLRQ WR &KDQJLQJ
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+$// 5 ( 6WRFKDVWLF ,PSOLFDWLRQV RI WKH /LIH &\FOH3HUPDQHQW
,QFRPH +\SRWKHVLV 7KHRU\ DQG (YLGHQFH Journal of Political Economy 9RO
1R SS ±
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-(1.,16 6 3 9$1 .(50 3 7UHQGV LQ LQFRPH LQHTXDOLW\ SURSRRU
LQFRPH JURZWK DQG LQFRPH PRELOLW\ Oxford Economic Papers SS ±
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STATISTICS IN TRANSITION-new series, Summer 2013
287
STATISTICS IN TRANSITION-new series, Summer 2013
Vol. 14, No. 2, pp. 287–318
COHERENCE AND COMPARABILITY AS CRITERIA
OF QUALITY ASSESSMENT IN BUSINESS
STATISTICS
$QGU]HM 0áRGDN ABSTRACT
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1. Introduction
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DV IROORZV (XURVWDW European statistics should be consistent internally, over time and comparable
between regions and countries; it should be possible to combine and make joint
use of related data from different sources.
Indicators
x Statistics are internally coherent and consistent (e.g. arithmetic and
accounting identities observed).
x Statistics are coherent or reconcilable over a reasonable period of time.
x Statistics are compiled on the basis of common standards with respect to
scope, definitions, units and classifications in the different surveys and
sources.
x Statistics from the different surveys and sources are compared and reconciled.
x Cross–national comparability of the data is ensured through periodical
exchanges between the European statistical system and other statistical
systems; methodological studies are carried out in close cooperation between
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STATISTICS IN TRANSITION-new series, Summer 2013
319
STATISTICS IN TRANSITION-new series, Summer 2013
Vol. 14, No. 2, pp. 319–328
STATISTICS AS A PROFESSION –
STATISTICIAN AS AN OCCUPATION:
observations and comments from
a panel of experts
Wlodzimierz Okrasa 1, Beata Witek 2
1. Introduction and background
Statisticians rarely devote as much attention to their profession as it does
deserve, including such fundamental questions as what actually constitutes
statistics today - as a discipline in relation to others, primarily to mathematics and
observation-based sciences (theoretically or applied oriented), on one side, and in
researching and teaching, on the other. Especially, given its inherent dynamics
and externally caused transition to new stages of its permanent development, and
what is their – statisticians' – own view of their occupational status, including
who actually should unambiguously be considered statistician. And how s/he
ought to be prepared through education and training system to play this important
role in various domains – in academia and policy making, as well as in private life
and business management. Therefore, any occasion to exchange views on such
dual aspects (disciplinary and occupational status) among experts during scientific
meetings seems to be worth of reporting. One of such meeting took recently place
at the conference on Methods of Assessment of Quality of Teaching held in the
University of Lodz last June (see note on it in this issue below) during which a
discussion panel was organized to address some of the above issues. As the
panel’s organizers, we feel deeply indebted to all its participants for sharing their
thoughts and opinions: Prof. Prof. Czeslaw Domanski, Jozef Dziechciarz,
Miroslaw Krzysko, Marek Rocki, and Janusz Wywial. As a part of our
appreciation of their generosity and of the quality of the panel’s output, their
voices are summarized here, extended a bit by introductory and concluding
remarks, while taking into account the voices of the highly competent audience3,
composed of academic teachers and researchers.
1
Central Statistical Office of Poland and the University of Cardinal Stefan Wyszynski in Warsaw.
Central Statistical Office of Poland and the University of Cardinal Stefan Wyszynski in Warsaw.
3
Discussants: Prof. Prof. K. Jajuga, S. M. Kot, A. Sokolowski, L. Tomaszewicz, Tadeusz
Gerstenkorn. In addition, A. Kupis-Fijalkowska, PhD, was an invited discussant for presenting the
Eurostat's European Master in Official Statistics (EMOS) initiative.
2
320
W. Okrasa, B. Witek: Statistics as a profession …
While statistics as a profession – meant as a domain of scientific activity,
including education – is primarily an object of methodological reflection,
statistician as an occupation is basically an example of a labour market category
characterized also in sociological terms (status, prestige, ethos, etc.). Although the
former was the main focus of the panel discussion, some occupation-related issues
are worthwhile to be mentioned here too. In the vein of Neyman's saying "[S]tatistics is the servant to all sciences" (cf. Chiang 1), what implies its presence
across all subject matter disciplines, through inter alia necessary involvement of
representatives of those disciplines in applying statistical methods – statisticians
constitute one of the most heterogeneous categories in occupational statistics. For
instance, US Bureau of Labor Statistics counts about 200 specific occupations
under this title, defined as follows: "Statisticians. Develop or apply mathematical
or statistical theory and methods to collect, organize, interpret, and summarize
numerical data to provide usable information. ... Includes mathematical and
survey statisticians. Excludes “Survey Researchers” (US BLS 2010, p. 23) 2. This
is supplemented by definition: "Survey Researchers. Plan, develop, or conduct
surveys. May analyze and interpret the meaning of survey data, determine survey
objectives, or suggest or test question wording. Includes social scientists who
primarily design questionnaires or supervise survey teams. Excludes "Market
Research Analysts and Marketing Specialists" (ibidem, 35). Such a broad
interpretation of the occupation accords with perhaps the most widely accepted
among statisticians answer to the question 'Who is the statistician?' given by
Platek and Särndal about decade ago (2001 3). Starting with considerations of what
can realistically be expected from statisticians in terms of quality products
generated by national statistical agencies, the authors arrived with the following
definition: "[T]he statistician ... is anyone who contributes to the ultimate delivery
of statistics and data to users.", and specify the main categories of this occupation:
"theoretical statistician – survey methodologist – subject matter specialist –
information technologist – and survey manager" (ibidem, p. 3). Keeping in mind
such a broad interpretation of both 'statistics' and 'statistician' we began the panel
discussion with a concern about the quality of the process of creating new
generations of performers in the scene of this profession.
1
“Statisticians in History”, http://www.amstat.org/about/statisticiansinhistory/index.cfm?fuseaction
=biosinfo&BioID=11.
2
US BLS 2010 SOC Definitions U.S. Bureau of Labor Statistics On behalf of the Standard Occupational
Classification Policy Committee (SOCPC), http://www.bls.gov/soc/soc_2010_definitions.pdf.
3
R Platek and C-E Särndal, 2001. Can a Statistician Deliver? Journal of Official Statistics, Vol. 17,
No. 1. pp. 1–20.
STATISTICS IN TRANSITION-new series, Summer 2013
321
2. Scoping panel’s perspective
All the panelists and discussants agreed that there is a tremendous demand for
statisticians and a big need to prepare new cohorts of specialists in the art of using
data for sectors of education, government, and industry in a way readying them to
meet the challenges from the technologically advanced society. It makes this
occupation both an attractive path of carrier for new alumni – along with Tukey’s
view: "[T]he best thing about being a statistician is that you get to play in
everyone else's backyard." 1 – and a highly respected as a job in view of the
general public. For instance, according to a US survey of occupational status,
statistician is ranked fifth out of about two hundred (together with mathematician
and engineer – Kennett, 2011 2).
Much of the panelists' attention revolved around the issue of what? to do and
how?, in order to equip the new generations of statisticians in tools and abilities
assuring the highest standards of professional quality, given the growing
expectation concerning the statisticians' deliverables (in Platek and Särndal's
meaning) on the one hand, and the existing drawbacks on the other. Especially,
the lack of mathematical background among the majority of students as
a consequence of the earlier reform of the high school curriculum, and subsequent
lowering requirements from candidates for studying statistics, being often tough
in the standard-liberal environment. This concerns the whole process of
education, including textbooks and other means and conditions of teaching, which
are summarized here as they emerged in the panelists' presentations:
(i) the means and conditions of teaching statistics;
(ii) the quality of teaching;
(iii) the problem of curriculum;
(iv) professional and occupational aspects of statistics.
3. The means and conditions of teaching statistics
The problem of quality of textbooks for teaching statistics
Because of the great importance and influence of the quality of textbooks on
teaching and students' knowledge, this issue was one of the key ones discussed
during the Panel. The topic was initiated by Prof. M. Krzysko, who referred to the
first Polish textbook on statistics entitled “Outline of statistical methods as
applied to anthropology” by Jan Czekanowski (1913), as an exemplary model.
The speaker also drew attention to the limitations and weaknesses of modern
1
2
American Statistical Association, http://www.amstat.org/careers/whatisstatistics.cfm.
R S Kenneth, Statistics As a Profession.
322
W. Okrasa, B. Witek: Statistics as a profession …
textbooks on statistics: outdated content, excessive focus on descriptive statistics,
presence of elementary errors, unfair reviews allowing authorizing the publication
of low-quality books, collections of tasks that relate to outdated data and obsolete
problems. Concern can be raised by textbooks on statistics in general secondary
schools – it turns out that the textbooks approved by the Ministry of Science and
Higher Education are not necessarily a good basis to start education in this area.
Agreeing with the above, Prof. K. Jajuga stressed that textbooks should be
tailored to different areas of expertise, including consultations with the
appropriate persons about the subject matter related to statistics, for example, with
economists in the econometric issues.
Prof. Cz. Domanski, as the President of the Polish Statistical Association
(PTS), considered creating by PTS (in cooperation with other statisticians) a
suitable consultative team for evaluating textbooks, or finding such a good author
as, for example, Marek Fisz. He recalled that “a statistician seeks the truth” and
one should not accept inappropriate behavior at meetings of a council (those
related to, for example, lowering the number of hours or unacceptable
combination of mathematics with statistics).
Supporting these suggestions, Prof. Okrasa stressed that it would be desirable
in terms of assurance of quality of teaching statistics to set up a PTS’s council or
a team for new textbooks matters. Another valuable idea would be the one of
creating (along other countries, for instance, the United States) up-to-date
textbooks for practitioners, under the name of Best Statistical Practices, that
would keep track of new methods and techniques in the field of applied statistics
and lay particular emphasis on the needs of official statistics. This type of
instructions for daily work of a statistician would help to raise the prestige of both
statisticians and institutions employing them (both public and private ones, such
as think-tanks).
Initiation into the profession - motivation and preparation
Learning statistics should start at earlier stage of education than higher
education, according to the panelists. Elements of statistics should be taught
already in general secondary schools (especially in economic schools), as argued
by Prof. J. Dziechciarz. Referring to the need to elicit emotional motivation to
follow this difficult field of study (mentioned by Prof. Domanski) Prof. M. Rocki
stressed the need to create such attitudes (emotions) even at the level of primary
school (which was also addressed by Prof. J.L. Wywial), pointing at the same
time to economic universities, such as Warsaw School of Economics which
pioneered with some initiatives to this aim, launching programs such as Children's
Economic University and the Academy of Young Economist for students of
STATISTICS IN TRANSITION-new series, Summer 2013
323
primary and general secondary schools. And at the subject contests organized to
enable the dissemination of issues among young people and to test the knowledge
of statistics (e.g. the statistics contest or complement of mathematics or
entrepreneurship contests). A good place to stimulate motivation to study
statistics could be, according to Prof. Domanski, the Polish Statistical
Association, which takes initiatives to “stimulate emotions” and organizes
statistical competitions in several Polish cities.
Learning-teaching conditions
The use of larger amount of current, real-world data stored on CDs and
conducting exercises in computer rooms which require not only passive use of
statistical software, but making conscious choices based on theoretical knowledge
was postulated by Prof. M. Krzysko (seconded by Prof. S.M. Kot). Recalling the
limitations associated with the need to join groups because of financial
constraints, Prof. L. Tomaszewicz stressed the importance of the learning
conditions – which consists also of insufficient length of studies – the worst
candidate can be made a gem would s/he be met with adequate teaching
environment. Noting that the 6-semester bachelor studies do not provide a
complete study program that could educate analysts with good knowledge of
statistics, Prof. Rocki stressed that the sequence of the subjects itself – analysis,
algebra, probability theory, mathematical statistics and econometrics – requires
five semesters, which leaves no room for other subjects enriching the knowledge
of the graduate. Possible solutions are (i) the struggle for a uniform courses
leading to a master’s degree, and (ii) the formulation of the university
qualification framework so as to ensure proper education. In addition, efforts
should be taken for participation in determining the title of the professional – the
proper nomenclature should reflect the knowledge and skills of graduates (e.g.,
along University of Minnesota offering a Master of Statistics degree). Bearing in
mind the regulation defining the duration of studies of “at least” six semesters,
Prof. Domanski shared this view and pointed to the possibility of extending the
studies and encouraged taking the initiative and striving for uniform courses
leading to a master’s degree.
The bottom line: Poor quality of modern textbooks of statistics in Poland
results from the use of out-of-date information, referring to obsolete problems, too
much focus on descriptive statistics and the lack of a fair selection of textbooks,
the consequence of which is the presence of elementary errors in them. In view of
this situation it would be reasonable to appoint a consultative team of experts
charged with responsibility assess and recommend for distribution only the
highest quality textbooks. Providing background and interest in knowledge
324
W. Okrasa, B. Witek: Statistics as a profession …
oriented towards statistics should be preferably taken as early as in primary
school. One should also take steps to extend the duration of the studies, ideally
bringing them to a uniform course (that leads to a master’s degree).
4. The quality of teaching
The quality of university teachers
As a precondition for producing statisticians as good professionals the
availability of quality teachers must be seen, and the Polish educational system
has not worked out the mechanism for preparing people to teach statistics
properly at each level. It was the main concern of Prof. J. Dziechciarz, who
addressed a gap between real-world problems and formal approaches due to the
fact that persons teaching statistics are typically graduated in mathematics and
educated in the area of advanced statistics. This situation causes two kinds of
obstacles:
(i) removing from the curriculum the basics of statistics, mainly tools of
descriptive statistics, (ii) the lack of teaching suited to the subject matter
disciplines, to the specific needs of economics, medicine, social sciences, etc.
In teaching statistics there is a necessity to become familiar with the specific
objectives of a given field (which was emphasized also by Prof. Kot).
A step in overcoming problems related to the lack of subject matter specialists
in the profession of a statistician has been made by universities which begun
introducing new field of studies such as “Commercial Analyst” or “Business
Analyst,” taking into account all the elements essential to practicing statistician. It
is worth noting that members of the Polish Accreditation Committee should
conduct the assessment of the quality of education in a way comprising
verification of the competence of all teachers given courses in statistics (not only
of the staff members, being included into so-called the academic minimum).
The quality of students – the recruitment policy
One of the causes of deterioration of quality of students was seen by the
panelists in the student recruitment policy due to the lack of an entry exam that
would make it possible to select those who are predestined to study specific
subjects from those who simply want to study. According to Prof. Rocki, who
addressed this concern, confining the recruitment criteria to the obligatory
‘matura’ examination results in admitting students who are good at graduation
subjects but not necessarily prepared to study the specific subject.
Apart from the selection of candidates, flexibility and multidisciplinarity is
advisable in teaching, whereas a statutory need is to enroll the student on a
STATISTICS IN TRANSITION-new series, Summer 2013
325
particular course of study, which may result in wrong choices and waste of public
money in the course of studying. Prof. Rocki indicated to results of a survey
carried out by the Warsaw School of Economics which showed that in significant
proportion the students entering university do not know what choice to make,
while among those who have made a choice, one third switches to other fields
than the one previously chosen. Therefore, flexibility in studying is necessary, as
well as the waver from the requirement to declare the field of study at an early
stage – this would give the opportunity for a more informed and consciously
made choice of statistics as a main subject of study.
In conclusion: The practical problems are the lack of mechanisms to prepare
for training statisticians, the teaching process not suited to the different subject
matter areas, as well as the lack of mechanisms for selecting candidates for
studies and insufficient flexibility to selecting and changing a field of study.
Therefore, the necessity to take into account the specificity of particular field of
study is stressed, along with including in the curriculum all the key elements and
special tools enabling students to practice statistics. The quality of a student
should be improved by introducing university entrance examination and removing
the need for the selection of the field of study at an early stage of education.
5. The issue of curriculum
Apart from the question of who is to teach statistics, it is important to ask
what? should be included in the curriculum. Prof. Wywial talked about dubious
legitimacy of the profession that requires specialized education in higher
education, in case of the absence of a subject (both I and II degree) that teaches
basics of statistical inference in a reliable way. Computer science and
econometrics were to be such a subject, which was, however, targeted towards
experts in the application of quantitative methods in economics and towards
computer science (which was also addressed by Prof. Rocki).
Another problem arises from sometimes observed attempts to remove from
the curriculum quantitative methods, and statistics in particular, due to
commercial focus on rapid training of graduates (which was also pointed by Prof.
L. Tomaszewicz supported by other panelists). What is important, according to
Prof. K. Jajuga, is also teaching statistics in subjects other than computer science
and econometrics, and especially in these with the reduced number of hours.
Prof. Tomaszewicz stressed that the statistical and econometric core of the fields
of study such as computer science and econometrics should be maintained by
inter-subject actions, even in defining the occupation of a statistician by means of
effects for statistical training, introduced pursuant to the National Qualifications
326
W. Okrasa, B. Witek: Statistics as a profession …
Framework. This would refine the graduate profile, defined differently depending
on a variety of specialties, which cover the most difficult challenges of modern
data analysis. The requirement set out in the description of a graduate profile,
such as “knows basic statistical methods...” is not enough to create a statisticsbased program on this basis, as pointed out by Prof. Rocki.
6. Professional and occupational aspects of statistics
If we assume that there is no single profession of a statistician, we recognize
that there is no single model of education, and that there is a need to adjust
teaching to the type of working areas. An example may be the category of official
statistician invoked by the panel organizer. The occupation of a statistician
working in public statistics institutions has been recognized by Eurostat as
important and specific, and an initiative for the program called EMOS “The
European Masters in Official Statistics” was undertaken. It was initiated by
Eurostat currently conducting a series of meetings with the National Statistical
Institutes. As explained by Ms. A. Kupis-Fijalkowska (assistant of Prof.
Domanski, an expert for Poland and neighboring countries), EMOS project is to
bring to universities, starting from 2014, an additional educational module for the
final year of Master's Degree, which would allow for educating a statistician
prepared to work in the statistical office, or at Eurostat.
A model statistician as seen by mathematical statistics may not be the same as
the official statistician. They both are needed but, according to Prof. Okrasa, one
cannot expect the same qualifications from both of them. It is worth to quote the
conclusions of one of the surveys on what statisticians think about their
profession, presenting desirable skills of a statistician: (1) mathematical basics,
(2) the ability of critical thinking, (3) the ability of active learning, interacting
with representatives of other areas, (4) the ability of active listening
(communication and contact with a user). Moreover, statisticians asked in surveys
why they want to be statisticians generally appreciate independence (higher than
salary), autonomy of their work and high esteem among various types of
employers (generally higher than of other staff), as well as prestige in the society.
When considering questions about the nature of statistics, Prof. Wywial came
to the conclusion that statistics should be considered primarily in terms of
profession (referring to the Polish language dictionaries which define profession
requiring acquired qualifications in higher education institutions as a concept
somewhat different from the term “occupation”). Its subject is the empirical
verification of theories produced in other sciences and support in recognition of
the characteristics of population, which are the target of studies in other
STATISTICS IN TRANSITION-new series, Summer 2013
327
disciplines. A real solution could be the introduction of elite ordered studies and
improvement or maintaining the level of studies on existing subjects, such as
computer science and econometrics as well as economic analysis.
Given the high prestige of the profession of a statistician and the growing
concern about bringing the state-of-the-art statistical knowledge and skills to
official statistics it was suggested (W. Okrasa) to consider launching in Poland a
kind of competition-based scholarship for researchers with proven achievements
(modeled on the National Science Foundation’s program of Senior Research
Fellowship at the US Bureau of Labour Statistics and at the US Bureau of
Census), who would be working on problems being currently of the focus of
official statistics. Such problems are, for example, new modes of conducting
census or administrative registers vs. statistical data collection system, or whether
and how electronic future can provide a threat to statisticians as professionals and
to the institutions of official statistics, given that big data, generated by other
systems of information, are channeled outside of the area remaining under the
control of institutions responsible for public statistics.
In spite of being discussed as essentially country-specific, the above issues
have been actually internationally recognized for decades, just to mention the
presentation by Hartley as the American Statistical Association (ASA) President
(1979 – entitled Statistics As a Science and As a Profession), who tried to solve a
traditional trade-off between professional equipment of mathematical and applied
statisticians. While rejecting claims for 'more mathematics' to assure the quality
solution of a real-world problem – or to blame its insufficiency for criticism
("standard of our papers is low", he quotes) – he pointed to cooperation between
statistician and subject matter specialist as a way to balance between deductive
(formal) and inductive (empirical) components of the statistics as a profession
(science) and as an occupation (in terms used here). He was seconded by one of
his distant successor, J. Stuart Hunter (the ASA President in 1993) who
emphasized in his presidential address that while "a professional in statistics is a
person whose everyday work consisted of making sense of data", there are also
others – "the builders of statistical theory and makers of statistical tools" – who
are "vital to the health of the statistical profession" [italic added] 1.
Avoiding temptation to go beyond the scope of the referred panel's discussion,
one may indicate the numerous and thematically reach sessions devoted to that
issues being vigorously debated at such prominent meetings as the World
Statistics Congress held last August in Hong Kong. It does prove that new
approaches are continuously sought in most of the nations as some titles inform
1
J. S. Hunter, 1994. Statistics As a Profession. Journal of the American Statistical Association
Vol. 89, Issue 425, pages 1–6.
328
W. Okrasa, B. Witek: Statistics as a profession …
about that – let us mention only a few out of several dozen presented over there1,
for instance: Research on the Modes of Statistical Education and Training in
China (Xia Rongpo) – Changing Educational Framework in the Transition to New
Educational Standards at Russian Universities of Life Science and their Impact on
the Teaching of Statistics (Galina Kamyshova and Lyman McDonald, Russia) –
Engaging Students in Statistics Education: situated learning in statistics projects
(Pieternel S. Verhoeven, the Netherlands) – Good Practice in Using Statistics in
Statistics Education Research (Neville Davies and Gemma Parkinson UK) – New
Perspectives: A Statistician and a Statistics Educator Discuss the Lessons Learned
from Cross Disciplinary Sojourns (Jennifer J. Kaplan et al., USA) – Radical
Statistics: Teachers and Students on the Highwire (Bruno de Sousa et al., Portugal
and Spain).
A systematic overview of the problems and approaches discussed on the
global scale – i.e. at meetings like the 59th WSC – would provide the needed
contextualization for those identified as of the key importance during the panel.
1
59th ISI World Statistics Congress in Hong Kong, http://www.isi2013.hk/en/index.php.
329
STATISTICS IN TRANSITION-new series, Summer 2013
STATISTICS IN TRANSITION-new series, Summer 2013
Vol. 14, No. 2, pp. 329±336
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