Confidence intervals and significance tests for mean of symmetrical population with smallest or largest outlier observatio

Walsh, J.

English

NASA Technical Reports Server

General Disclaimer 
One or more of the Following Statements may affect this Document 
 
 This document has been reproduced from the best copy furnished by the 
organizational source. It is being released in the interest of making available as 
much information as possible. 
 
 This document may contain data, which exceeds the sheet parameters. It was 
furnished in this condition by the organizational source and is the best copy 
available. 
 
 This document may contain tone-on-tone or color graphs, charts and/or pictures, 
which have been reproduced in black and white. 
 
 This document is paginated as submitted by the original source. 
 
 Portions of this document are not fully legible due to the historical nature of some 
of the material. However, it is the best reproduction available from the original 
submission. 
 
 
 
 
 
 
 
Produced by the NASA Center for Aerospace Information (CASI) 
https://ntrs.nasa.gov/search.jsp?R=19710016615 2020-03-11T20:10:18+00:00Z
EXACT INTERVALS AND TESTS FOR MEA 14 OF SYMMETRI4AL POPULATION
WIDEN ONE "SAMP: E" VALUE POSSIBLY AN OUTLIER
by
John E. Walsh
Technical Report No. 88
Department of Statistics 0 14R Contract
Reproduction in whole or in part is permitted
for any purpose of the United States Government
This document ha ,. been approved for public release
and sale: its distribution is unlimited.
DEPARTMENT OF STATISTICS
Southern Methodist University
Dallas, Texas 75222
N7	 ^^fJ	
^
fl	 _
 '-
.s. 9 1
A S J. C R R 1 M X G R A D N!11
 ^'	 —----------- --
^^ ^ 91011^^,^'
C^j	JUN 1971
fit! ^^^^^1^
EXACT INTERVALS AND TESTS FOR MEAN OF SYMMETRICAL POPULATION
WHEN 'ENE "SAMPLE" VALUE POSSIBLY AN OUTLIER
r
by
John Z. Walsh
d
Technical Report No. 88
Department of Statistics ONR Contract
December 1, 1970
Research sponsored by t:;e Office of Naval Research
Corit^ avt N00014-68-A-0515
Project NR 042-260
Reproduction in whole or in part is permitted
for any purpose of the United States Government.
,D
This document has been approved For public release
and sale; its distribution is unlimited.
DEi ^aRTMENT OF STATISTICS
Southern Methodist University
9
d	 !
EXACT 1NTERVALS AND TESTS F01 MEAK OF SYMMETRICAL POPULATION
WHEN ONE "SAMPLE" VALUE POSSIBLY AN OUTLIER
John E. Walsh
Southern Methodist University*
4
r
l
t
ABSTRACT
The (continuous) data are n observations that are believed to be
a random sample from a symmetrical population. Confidence intervals and
significance tests for the population mean are desired. There is,
however, the possibility that either the smallest observation or the
largest observation is an outlier. That is, the population providing
this observation differs from the symmetrical popu]a".ion providing the
other n - 1 observations. If thin occurs ; intervals and tests are
desired for the mean ci the population providing the other n - 1 obser-
vations. Some investigation difficulties can be overcome if intervals
and tests can be developed that are simultaneously usable for all. of
these three situations (a confidence coefficient, or significance level,
has the same value for all three situations). Two kinds of intervals
and tests with this property are developed. These results always in-
volvc: both the next to largest and next to smallest observations and
should have at least moderately high efficiencies. Also, some extensions
are considered, such as allowing each observation to be from a different
population.
*	 Research partially supported by Mobil Research and Development
Corporation and by NASA Grant NGR 44-007-028. Also associated with ONR
Contract NOOO14-68-A-0515.
1
n
R	 1
(2 )
INTRODUCTION AND DISCUSSION
The data are n independent observations that are continuous data and
are believed to be a ranftm sample from a symmetrical population. The
order statistics of these observations are
X 1 < ... <Xn.
Desired are confidence intervals and significance tests for the mean µ of
the population sampled. However, the experimental situation is such
that X 1 or Xn.9 but not both, could possibly be an outlier. More
specifically, the population providing the observation that is X1 , or
is Xn, differs from the symmetrical population that provides the other
n - 1 observations. inequality of means is the difference of predominant
interest but lack of symmetry for the population providing the outlier can
also be of interest. When this outlier situation occurs, intervals and
tests are desired for the mean p of the symmetrical population providing
the other n - 1 observations. The motivation for considering this outlier
possibility could be due to almost anything (previous experience with
data of this type, an examination of the observed values, a desire to
protect against this possibility, etc.).
If X 1 is an outlier, X2 .9 000 .9 Xn, are the order statistics of a
random sample of size n - 1 from a symmetrical population. Here, X2
represents the smallest sample value,, etc. If X  is an outlier, X l.9 •••,
Xn 1 are the order statistics of a random sample of size n - 1 from a
symmetrical population.
Given a satisfactory procedure for deciding on the rejection of an
outlier, one approach would be to first use this procedure to decift
highly efficient'for almost any kind of underlying distribution.
i
( 3 )
which of the three situations (random sample, X 1 an outlier, X  an out-
lier) exists. A procedure for rejection of outlying observations for a
general continuous symmetrical population is given in ref. 1. However,
this procedure is not usable for a single outlier. Moreover, the
decision reached might be erroneous even if a satisfactory procedure for
rejection of an outlier were available.
A more desirable approach is to obtain intervals and tests that
simultaneously are applicable to all three situations. More specifically,
an interval has the same confidence coefficient, and a test has the same
significance level, for the three situations. Two-sided intervals and
tests with this property can be developed and two types are given. For
brevity, only confidence intervals are considered. However, the corres-
ponding tests can be obtained from these intervals and the usual manner.
The normal distributions are examples of symmetrical distributions.
Thus, the results presented can be used to investigate p for the normal-
ity case. These results should have reasonably high efficiency for each
of the three situations when the population providing n, or n - 1, sample
values is normal. This efficiency property is indicated by some power
computations given in ref. 2 for the case where X 2
 and Xn l are used
(appropriately) for a sample o size n. Of course, comparison with
the corresponding best results separately for each of the three situations
is somewhat unfair to the results given here. That is, the best results
for a situation are only usable for that situation. A more meaningful
comparison would be with the best results that are simultaneously usable
for all three situations. Then, the results given here would seem to be
(4)
The intervals and tests are of a symmetrical nature (same probabil-
ity for each "tail") when the random sample situation occurs. The
results for the outlier situations are two-sided but the probability for
one tail does not even roughly equal the probability for the other tail.
Thus, an unequal emphasis occurs for each outlier situation and should
be taken into consideration when the results given here are candidates
for use.
The next section contains a statement of the two types of confidence
intervals and verification that a confidence coefficient has the same
value for the three situations. An outline of some extensions, including
use when each observation can be from a different population, is given in
the final section.
RESULTS AND VERIFICATION
Direct use of material given in refs. 2 and 3 shows that, for a
random sample of size n from a continuous symmetrical population,
r	 P [ (X2 + Xi+l)/2 s w s (X n-1 + Xn-i)/2) = 1 - (n + 2 - 1)2-(n
P {min [X3, (X2 + Xi+1) /2 ] 15 e'
	[Xn-2 )' (Xn-1 + Xn-i) /21)
= 1- [ni- (d + 1) (d - 2)/212- (n - 1) 0
These relationships define the two types of confidence intervals that
are considered, and furnish their confidence coefficients
(l .1
	
Sn/2 -1).
Now, verification is given for the assertion that a confidence
coefficient has the same value for the three situations. Proof for the
r
(5)
situation where 
x  
is an outlier is sufficient, since a completely
analogous proof holds for the situation of X1 an outlier.
When X  is an outlier )
 the sample size is n - 1 and XV 000.9 Xn-1
are the sample order statistics . Use of material on page 160 of ref. 4.,
and some of the above results, for sample size n - 1 shows that
P[(X2 + Xi+l )/2 > µ] _ (n + 1 - i)2-(n
	 i)
P[Xn-1 + Xn-i)/2 < p) = 2-(n
-i)0
P (min [XV (X2 + Xi+1 )/2] < µ} _ [(n - 1)i - (d + 1) (d - 2)/2]2-(n - 1)P
P (max [Xn-2 (Xn-1 + Xn-i)/2] > µ} = 1 2-
(n
^
	
- 1)
Summation of the two tail probabilities for each type and subtraction
from unity verifies that a confidence coefficient for this outlier
situation equals the corresponding confidence coefficient for the random
sample situation.
COMMENTS ON EXTENSIONS
The results given are expressed in a way that is ordinarily used
when the possibility of an outlier is cinsidered. The requirements of
random samples are, however, not needed. The two types of intervals and
tests are usable under more general circumstances. That is, they are
applicable when the conditions are such that the observations are inde-
pendent and from continuous symmetrical populations that are believed to
have a common mean p. However, either X 1 or Xn could be an outlier.
Here, "outlier" implies that the population providing this observation
(6)
has a mean differant from the common mean µ for the synmetrical
populations providing the other r - 1 observations, and/or, that the
outlier population is not synanetrical.
The mean might not exist for so..* populations. Then, the median
can be investigated and p represents the central median (center median
in interval of median values, when the inedisn 4s; not unique) .
Continuous populations are not required if ties are resolved by
use of an independent probability process (for example, see pages 139-
140 of ref. 4). In any case, the true confidence coefficient is at
least as large as the value for continuous populations.
Finally, other types of results likely could be developed that have
the same confidence coefficient value for the three situations. The
two types presented have the advantage of simplic4ty and of using both
X2 and Xn-1
t j, i
(7)
•	 1. John E. Walsh, "Some nonparametric tests of whether the largest
observations of a set area too large or too mall," Annals of Mathe-
matical Statistics, Vol. 21 (19501 0 pp. 583-592.
2. John Edward Walsh, Some Significance Tests for the Median Which Are
Valid Under Very General Conditions., Unpublished Doctoral Thesis
Princeton University, 1947, 87pp.
3. John E. Walsh, "Some significance tests for the median which Pre
valid under very general conditicne," Annals of Mathematical
Statistics, Vol. 20 (1949) 0 pp. 64-81.
4. John E. Walsh, Handbook of Nonparametric . Statistics, D. Van Nostrand
Co., Inc., Princeton, N. J., 1962, 575 pp.
1
1
Ii
IlMrT,a_gCT£T n—
DOCUMENT CONTROL DATA • R b D
Secwst y elasrllientlon at title, hot tl	 ul .rh,.r ►.,rI lout tred.-irmg annotnlitrl muAt he or. turd when III$* uvorett ► i-port I . I V1 . Alllo.11
I
	 Oft f QINA TING ACTIVITY (Carporefe #8601010r) do. NEPORT 'SECURIT- CI.AWOICAII O N
UNCLASSIFIED
SOUTHERN METHODIST UNIVERSITY 2b. GROUP
UNCLASSIFIED
REPORT TITLE
"Exact intervals and tests for mean of symmetrical population when one "sample"
value possibly an outlier."
4 DESC RIP TIVE NOTES (?ype of report ated.inelu•ivo dat*s)
Technical Report
s, Au THORIS) (Fi rst name, mi dlo iniflol, last name)
John E. Walsh
• REPORT DATE 71. TOTAL NO. OF PAGES 7b. NO. OF REFS
December 1, 1970 7 4
411. CONTRACT OR GRANT NO. Sa. ORIGINATOR'$ REPORT NUMBERIS)
N00014-68-A-0515
b. PROJXC T NO.
NR 042-260
88
^.
d.
98. OTHER REPORT N0131 (Any other numbers that may be assigned
this report)
10. CISTRIBUTION STATEMENT
This document has been approved for public release and sale; its distribution is
unlimited.
	 Reproduction in whole or in
United States Government.
part	 -a permitted for any purpose of the
It. SUPPLEMENTARY NOTES 12. SPONSORING MILITAnV ACTIVITY
Office of Naval Research
13. APSTRACT
The (continuous) data are n observations that are believed to be a random
sample from a symmetrical population. Confidence intervals and significance tests
for the population mean are desired. There is, however, the possibility that either
the smallest observation or the largest observation is an outlier.	 That i3, the
population providing this observation differs from the symmetrical population pro-
viding the other n - 1 observations. If this occurs, intervals and tests are
desired for the mean of the population providing the other n - 1 observations. 	 Some
investigation difficulties can be overcome if intervals and tests can be developed
that are simultaneously usable for all of these three situations (a confidence
coefficient, or significance level, has the same value for all three situations).
Two kinds of intervals and tests with this property are developed.
	 These results
always involve both the next to largest and next to smallest observations and
should have at least moderately high efficiencies. Also, some extensions are
considered, such as allowing each observation to be from a different population.
IASSIFIED
Security ClasSiticatinn A
	 halt
I


Exact intervals and test for mean of symmetrical population when one sample value possibly an outlier

Abstract

Similar works

Full text

Available Versions

NASA Technical Reports Server