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Tests of significance using selected sample quantiles
Statistics & Probability Letters 2 (1984) 295-297 North-Holland
October 1984
TESTS OF SIGNIFICANCE USING SELECTED SAMPLE QUANTILES
A.K.MD. Ehsanes SALEH Carleton University, Ottawa, Canada
M. Masoom ALI and Dale U M B A C H Bali State University, Muncie, IN, USA Received December 1983 Revised May 1984
Abstract: Large sample tests of significance for the location parameter, the scale parameter, and quantiles for a location-scale family of distributions based on a few optimally chosen sample quantiles are considered.
Keywords: quantiles, tests of hypotheses.
1. Introduction and preliminaries
respectively, by minimizing
Let X(1), X(2) . . . . ,X(n) be the order statistics of a random sample of size n from a location-scale family of distributions with distribution function F~,o(x) = F ( ( x - / x ) / o ) , and density function f~,,o(X) = ( 1 / o ) f ( ( x - p)/o), where - oo < # < oo and o > O. Consider k ( ~ n ) sample quantiles, X(nO, X(n z),- • -, X(n k), whose ranks satisfy the relation l~
~= ( X(.)-~l-ou)'W-l(
(ui, u2,... ,uk)', and W = (wij), with u i = F - l ( p i ) and w ~ j = p ~ ( 1 - P j ) / ( f i f j ) and f i = f ( u i ) , for i = 1, 2 , . . . , k , j = 1, 2 , . . . , k , and i < j . Thus, for a fixed spacing, we find (see Ogawa (1962a) for details) the ABLUE's of p and a, say/2 and 6,
X(.)-~l-ou). (1.1)
With fk+ 1 =f0 = Uk + l f k + l -~ Uofo = O, and p (i) = Pi - P ~ - I , u ( ° = u , f , - u,_if~_~, f ( ' ) = L - L - ~ , and X (i) -~ X ( n i ) f i
and
- X(ni_l)fi_l,
A=K, K2-K~,
(1.2)
k+l
k+l
K1 = E ( f ( O ) z / P (0,
r : = E (.('))2/p('), i=l
i=1 k+l
K 3 = 2~, f ( i ) u ( i ) / / P (i), i=1
(1.3) k+l
x=
k+l
f(')x(')/p ('),
Y=
i+1
u(i)x(')/p (i), i~l
(1.4) the ABLUE's of p and a are given by /2= A-I(K2X-
ga Y)
and
~ = A-1(K1Y-
K3X).
Also, the A B L U E of the ~-th quantile, x~ = p + o F - I ( ~ ) , for 0 < ~ < 1, is given by ~ =/2 +
016%7152/84/$3.00 © 1984, Elsevier Science Publishers B.V. (North-Holland)
295
Volume 2, Number 5
STATISTICS & PROBABILITY LETTERS
aF-l(~). The asymptotic variances of/2 and a are given by V a r ( g ) f ( n A ) - l o 2 K 2 and V a r ( a ) = ( n A ) - l o 2 K 1 . With Cov(/2, #)ffi - ( n A ) - l o 2 K 3 , we have V a r ( ~ ) = ( n A ) - l o 2 L ~ . , where L¢ = KI(F-I(~))
2 - 2K3F-l(~)
October 1984
Following Anderson (1958, p. 189-190), we find the minimum of $2 subject to the restriction #=#0, say fl(#0), as f l ( # o ) = 1 2 0 + ( # # o ) 2 A / K 2 . Thus, the ratio defined by
+ K 2.
W e note that W -1 =(v~j), where v,. = fi2(1/p(i+l)+ l / p (0) for i = l , 2 , . . . , k , vi,i+l = Vi+l. i = f i f i + l / p (i+1), a n d vi i = 0 for li - J l > 2 with i, j = 1, 2, .... k. By the asymptotic normality of X(-), it follows that n12/o 2 has a chi-square distribution with k degrees of freedom. Also, we note that 12 can be orthogonally partitioned as 12 = I20 + I~1, where
I'21= KI(/~- #)2 + 2K3(/2- #)(~ - o )
(1.5) and f~o -- (X(-) -/il - au)'W-1 (X(.) - #I - ou).
(1.6) Thus, n ~ o / a 2 has a chi-square distribution with k - 2 degrees of freedom and n~i/o 2 has a chisquare distribution with 2 degrees of freedom.
2. T e s t s o f s i g n i f i c a n c e
In this section we consider tests of significance for hypotheses concerning the parameters and quantiles based on k ( < n) optimally chosen sample quantiles for a location-scale family of distributions, namely //1: # = / t o , o unspecified, /-/2: o - o0, # unspecified, and /'/3: x~ = x~, # and o otherwise unspecified. We first consider hi: # = #0, o unspecified.
ll= ~/f~2(~-#o)/~o/(k-2 )
has a central Student's t-distribution with k - 2 degrees of freedom under H 1. Under the alternative hypothesis t 1 has a noncentral t-distribution with noncentrality parameter 8~ = ( n A / K 2)1/2(# - #o)/O. Hence, the spacing ( p l , P2 . . . . . P k ) which maximizes the power of the test is obtained by maximizing A / K 2 with respect to u = (u 1, u 2 . . . . , Uk)'. This results in the same spacing as the optimal spacing for the ABLUE of # since Var(ft) = ( h A ) - lo 2K2. If we restrict ourselves to symmetric distributions and a symmetric spacing, then we choose u to maximize K1, since for this c a s e K 3 --- 0 and hence A / K 2 = K 1. In a similar manner we derive tests for H 2 and H 3. These are each t-tests and are summarized in Table 1, along with the respective noncentrality parameters and variances. From Table 1, it is clear that the spacings which maximize the power of the various tests are the same as the spacings which maximize the variances of the corresponding estimates. This is the case since the power is maximized when the absolute value of the noncentrality parameter is maximized. The optimal spacings for various distributions are available in Sarhan and Greenberg (1962) and in many other references, including Ogawa (1951), Kubat and Epstein (1980), Saleh (1981), and Hassanein (1971, 1972, 1974).
Table 1 Test statistics, noneentrality parameters, and the variance of the corresponding estimators for the three hypotheses Hypothesis
H1: pffipo
-,,,,o)
,,Id'o/(k-2)
Noncentrality
~
t, ffi
296
02
Var(~) - - ~ K2
oo)
,/ao/(k- 2) nA
P - P0.
) variance
H3:x~ffi x£o
H2:offi %
T t,t ti,ti
8o-
(2.1)
o-
%
(-g-)
G2
Vat(~) ffi~-~K~
I,,A 0 2
var(~)-- ~-~z,~
Volume 2, Number 5
STATISTICS & PROBABILITY LETTERS
References Anderson, T.W. (1958), Introduction to Multivariate Statistical Analysis, John Wiley, New York. Hassanein, K.M. (1971), Percentile estimators for the parameters of the Weibull distribution, Biometrika 58, 673-676. Hassanein, K.M. (1972), Simultaneous estimation of the parameters of the extreme value distribution by sample quantiles, Technometrics 14, 63-70. Hassanein, K.M. (1974), Linear estimation of the parameters of the logistic distribution by selected order statistics for very large samples, Statische Hefte 15, 65-70. Kubat, P. and B. Epstein (1980), Estimation of quantiles of location-scale distributions based on two or three order statistics, Technometrics 22, 575-581.
October 1984
Ogawa, J. (1951), Contributions to the theory of systematic statistics, Osaka Math. J. 3, 172-213. Ogawa, J. (1962a), Estimation of the location and scale parameters by sample quantiles (for large samples), In: A.E. Sarhan and B.G. Greenberg, eds., Contributions to Order Statistics, John Wiley, New York, pp. 47-55. Ogawa, J. (1962b), Optimum spacing and grouping for the exponential distribution, In: A.E. Sarhan and B.G. Greenberg, eds., Contributions to Order Statistics, John Wiley, New York, pp. 371-380. Saleh, A.K.Md.E. (1981), Estimating quantiles of exponential distribution. In: M. Cstrgt, D. Dawson, J.N.K. Rao and A.K.Md.E. Saleh, eds., Statistics and Related Topics, Amsterdam, North-Holland, pp. 145-151. Sarhan, A.E. and B.G. Greenberg, eds. (1962), Contributions to Order Statistics, John Wiley, New York.
297
October 1984
TESTS OF SIGNIFICANCE USING SELECTED SAMPLE QUANTILES
A.K.MD. Ehsanes SALEH Carleton University, Ottawa, Canada
M. Masoom ALI and Dale U M B A C H Bali State University, Muncie, IN, USA Received December 1983 Revised May 1984
Abstract: Large sample tests of significance for the location parameter, the scale parameter, and quantiles for a location-scale family of distributions based on a few optimally chosen sample quantiles are considered.
Keywords: quantiles, tests of hypotheses.
1. Introduction and preliminaries
respectively, by minimizing
Let X(1), X(2) . . . . ,X(n) be the order statistics of a random sample of size n from a location-scale family of distributions with distribution function F~,o(x) = F ( ( x - / x ) / o ) , and density function f~,,o(X) = ( 1 / o ) f ( ( x - p)/o), where - oo < # < oo and o > O. Consider k ( ~ n ) sample quantiles, X(nO, X(n z),- • -, X(n k), whose ranks satisfy the relation l~
~= ( X(.)-~l-ou)'W-l(
(ui, u2,... ,uk)', and W = (wij), with u i = F - l ( p i ) and w ~ j = p ~ ( 1 - P j ) / ( f i f j ) and f i = f ( u i ) , for i = 1, 2 , . . . , k , j = 1, 2 , . . . , k , and i < j . Thus, for a fixed spacing, we find (see Ogawa (1962a) for details) the ABLUE's of p and a, say/2 and 6,
X(.)-~l-ou). (1.1)
With fk+ 1 =f0 = Uk + l f k + l -~ Uofo = O, and p (i) = Pi - P ~ - I , u ( ° = u , f , - u,_if~_~, f ( ' ) = L - L - ~ , and X (i) -~ X ( n i ) f i
and
- X(ni_l)fi_l,
A=K, K2-K~,
(1.2)
k+l
k+l
K1 = E ( f ( O ) z / P (0,
r : = E (.('))2/p('), i=l
i=1 k+l
K 3 = 2~, f ( i ) u ( i ) / / P (i), i=1
(1.3) k+l
x=
k+l
f(')x(')/p ('),
Y=
i+1
u(i)x(')/p (i), i~l
(1.4) the ABLUE's of p and a are given by /2= A-I(K2X-
ga Y)
and
~ = A-1(K1Y-
K3X).
Also, the A B L U E of the ~-th quantile, x~ = p + o F - I ( ~ ) , for 0 < ~ < 1, is given by ~ =/2 +
016%7152/84/$3.00 © 1984, Elsevier Science Publishers B.V. (North-Holland)
295
Volume 2, Number 5
STATISTICS & PROBABILITY LETTERS
aF-l(~). The asymptotic variances of/2 and a are given by V a r ( g ) f ( n A ) - l o 2 K 2 and V a r ( a ) = ( n A ) - l o 2 K 1 . With Cov(/2, #)ffi - ( n A ) - l o 2 K 3 , we have V a r ( ~ ) = ( n A ) - l o 2 L ~ . , where L¢ = KI(F-I(~))
2 - 2K3F-l(~)
October 1984
Following Anderson (1958, p. 189-190), we find the minimum of $2 subject to the restriction #=#0, say fl(#0), as f l ( # o ) = 1 2 0 + ( # # o ) 2 A / K 2 . Thus, the ratio defined by
+ K 2.
W e note that W -1 =(v~j), where v,. = fi2(1/p(i+l)+ l / p (0) for i = l , 2 , . . . , k , vi,i+l = Vi+l. i = f i f i + l / p (i+1), a n d vi i = 0 for li - J l > 2 with i, j = 1, 2, .... k. By the asymptotic normality of X(-), it follows that n12/o 2 has a chi-square distribution with k degrees of freedom. Also, we note that 12 can be orthogonally partitioned as 12 = I20 + I~1, where
I'21= KI(/~- #)2 + 2K3(/2- #)(~ - o )
(1.5) and f~o -- (X(-) -/il - au)'W-1 (X(.) - #I - ou).
(1.6) Thus, n ~ o / a 2 has a chi-square distribution with k - 2 degrees of freedom and n~i/o 2 has a chisquare distribution with 2 degrees of freedom.
2. T e s t s o f s i g n i f i c a n c e
In this section we consider tests of significance for hypotheses concerning the parameters and quantiles based on k ( < n) optimally chosen sample quantiles for a location-scale family of distributions, namely //1: # = / t o , o unspecified, /-/2: o - o0, # unspecified, and /'/3: x~ = x~, # and o otherwise unspecified. We first consider hi: # = #0, o unspecified.
ll= ~/f~2(~-#o)/~o/(k-2 )
has a central Student's t-distribution with k - 2 degrees of freedom under H 1. Under the alternative hypothesis t 1 has a noncentral t-distribution with noncentrality parameter 8~ = ( n A / K 2)1/2(# - #o)/O. Hence, the spacing ( p l , P2 . . . . . P k ) which maximizes the power of the test is obtained by maximizing A / K 2 with respect to u = (u 1, u 2 . . . . , Uk)'. This results in the same spacing as the optimal spacing for the ABLUE of # since Var(ft) = ( h A ) - lo 2K2. If we restrict ourselves to symmetric distributions and a symmetric spacing, then we choose u to maximize K1, since for this c a s e K 3 --- 0 and hence A / K 2 = K 1. In a similar manner we derive tests for H 2 and H 3. These are each t-tests and are summarized in Table 1, along with the respective noncentrality parameters and variances. From Table 1, it is clear that the spacings which maximize the power of the various tests are the same as the spacings which maximize the variances of the corresponding estimates. This is the case since the power is maximized when the absolute value of the noncentrality parameter is maximized. The optimal spacings for various distributions are available in Sarhan and Greenberg (1962) and in many other references, including Ogawa (1951), Kubat and Epstein (1980), Saleh (1981), and Hassanein (1971, 1972, 1974).
Table 1 Test statistics, noneentrality parameters, and the variance of the corresponding estimators for the three hypotheses Hypothesis
H1: pffipo
-,,,,o)
,,Id'o/(k-2)
Noncentrality
~
t, ffi
296
02
Var(~) - - ~ K2
oo)
,/ao/(k- 2) nA
P - P0.
) variance
H3:x~ffi x£o
H2:offi %
T t,t ti,ti
8o-
(2.1)
o-
%
(-g-)
G2
Vat(~) ffi~-~K~
I,,A 0 2
var(~)-- ~-~z,~
Volume 2, Number 5
STATISTICS & PROBABILITY LETTERS
References Anderson, T.W. (1958), Introduction to Multivariate Statistical Analysis, John Wiley, New York. Hassanein, K.M. (1971), Percentile estimators for the parameters of the Weibull distribution, Biometrika 58, 673-676. Hassanein, K.M. (1972), Simultaneous estimation of the parameters of the extreme value distribution by sample quantiles, Technometrics 14, 63-70. Hassanein, K.M. (1974), Linear estimation of the parameters of the logistic distribution by selected order statistics for very large samples, Statische Hefte 15, 65-70. Kubat, P. and B. Epstein (1980), Estimation of quantiles of location-scale distributions based on two or three order statistics, Technometrics 22, 575-581.
October 1984
Ogawa, J. (1951), Contributions to the theory of systematic statistics, Osaka Math. J. 3, 172-213. Ogawa, J. (1962a), Estimation of the location and scale parameters by sample quantiles (for large samples), In: A.E. Sarhan and B.G. Greenberg, eds., Contributions to Order Statistics, John Wiley, New York, pp. 47-55. Ogawa, J. (1962b), Optimum spacing and grouping for the exponential distribution, In: A.E. Sarhan and B.G. Greenberg, eds., Contributions to Order Statistics, John Wiley, New York, pp. 371-380. Saleh, A.K.Md.E. (1981), Estimating quantiles of exponential distribution. In: M. Cstrgt, D. Dawson, J.N.K. Rao and A.K.Md.E. Saleh, eds., Statistics and Related Topics, Amsterdam, North-Holland, pp. 145-151. Sarhan, A.E. and B.G. Greenberg, eds. (1962), Contributions to Order Statistics, John Wiley, New York.
297