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Quantile-Quantile plots under random censorship
Journal of Statistical Planning and Inference 15 (1986) 123-128 North-HoUand
123
QUANTILE-QUANTILE PLOTS UNDER RANDOM CENSORSHIP Emad-Eldin A.A. ALY* Department of Statistics and Applied Probability, The University o f Alberta, Edmonton, Alberta, Canada T6G 2G1 Received 17 August 1984; revised manuscript received 13 November 1985 Recommended by P.K. Sen
Abstract: We obtain strong approximation re.suits for the product-limit Quantile-Quantile (PL-Q-Q) process. In addition, product-limit confidence bands for the theoretical Q-Q plot are constructed.
AMS Subject Classification: Primary 62G15; Secondary 60F15. Key words and phrases: PL-Q-Q processes; Kiefer process; confidence bands for Q-Q plots.
1. Introduction and preliminaries
Consider the problem of comparing two populations on the basis of two independent random samples (rs) each randomly censored from the right. Quantile-Quantile (Q-Q) plots are very useful graphical tools in the noncensored case. In the present paper we extend Q-Q plots to the censored case as well. Let {Xo}~ 1 and {Zu}~I, j = l , 2 , be four independent rs. The distribution function (DF) of X U is Fj and that of Z U is/-/j, j-- 1, 2. In the random censorship model from the right, the X Umay be censored by the ZU, so that one observes only the pairs { (X~i~ (~ij) } in~l , where X~ij= Xo A Zij ( = min(Xo, Zq )) and 6 U is the indicator function of the event {Xij<_Zu}, i= 1,2, ...,nj and j= 1,2. Thus {X~U}in~=l,j= 1,2 are two independent rs with DF's Fj°, j = 1,2, respectively, where 1 - F j°= ( 1 - F j ) ( 1 - H i ) , j= 1,2. The subdistribution function, ~ , of the uncensored X~u observations is given by
fJ(t)=P{X~#
(1.1)
* This research is partially supported by an NSERC Canada grant at the University of Alberta. Part of this work was done while the author was an Assistant Professor at King Saud University, Riyadh, Saudi Arabia. 0378-3758/86/$3.50 © 1986, Elsevier Science Publishers B.V. (North-Holland)
E.-E.A.A. AIy / Quantile-quantile plots
i24
For any D F L , let L-~(.):=inf{t: L(t)>_. } be the quantile function (QF) of L and S(L):= {t: 0 < L ( t ) < 1} be the support of L. For all the DF's considered here we will always assume that S(L)= (t L, TL). Let Fj be the (Kaplan-Meier (1958)) product-limit (PL) estimator of Fj based on {(X°t~ij)}n~l and l~j-l=gj be the corresponding QF, j = 1,2. The PL-Q-Q plot of F2 against FI is zl(. ) = Q2(Fl(. )) and the corresponding PL-Q-Q process, f(-), is defined by
f'(x) = Nl/2f2 Q2(Fl (X))(O.2(f~(x)) - Q2(Fl(x)) ),
x ~ S(FI),
(1.2)
where N=nl+n2, Q2=Ff 1, f2=F~ is assumed to exist, f2QE(.)=f2(Q2(.)) and 02(0) := min{X~/2:6i2 = 1}. For detailed discussions on the PL technique we refer to Burke et al. (1981), Aly et al. (1985) and the references contained therein. For strong approximation results as well as further references on Q-Q processes we refer to Aly (1986) and Aly and Bleuer (1986). The techniques of the latter four papers are used heavily in the sequel. It will be a convenient device to define a new set of rv by setting U~/=Fj(Xij), W/j=Fj(Zu) and U~ij=Fj(X~)=UiyAWij, i = l , 2 , . . . , n j and j = l , 2 . Let ]~j be the PL-estimator based on {(U~ (~iJ)}~l and Uj be the corresponding PL-QF, j = 1, 2. Next define
f*(y)= N1/2(O2(El(y))- y),
O_y_
(1.3)
and
fl(y)=f(F~1(y))=N~/Zf2Q2(y)(Qa(El(y))-Q2(y)),
0 < y < 1.
(1.4)
We will assume that nl/N--)Ae(O,l) as niAna--)oo. We will write logEN to denote log log N, and supi~b) to denote SUPa__.
r(N)=N-I/4(logaN)I/4(logN)I/a
and
e(N)=N-llog2 N,
(1.5)
The following result is a generalization of Theorem 2.1 of Aly (1986) and is proved similarly. Theorem 1.1. There exists a probability space on which two independent Kiefer processes, Ky(., . ), j = 1, 2, are defined such that
sup lf*(Y) - ;t-l/2n~l/2Kl(y, nl) - (1 - A)-l/2n21/2K2(Y, n2)l = O(r(N)) a . s . , 10,p0] (1.6)
where r(N) is as in (1.5), po
= (1 - . ) ( 1 - HAd-I(.))),
EKj(t,s)=0
and EKj(t, s)Kj (x, y) = (s Ay)(1 - t)(1 -- x)d7 (t Ax)
E.-E.A.A. Aly / Quantile-quantile plots
125
with ?t
d;(t)= 1o (1 -
u)-2(1 - Hj(Fj-I(u)))-I du.
(I .7)
The following theorem is an analogue of Theorem 3.2 of Cs~ki (1977).
Theorem 1.2. Let Po be as in (1.6) and e(N) as in (1.5). Then, f o r some positlve constants C~ and C 2
lim
sup
{ylog2N}-W2]f*(y)l<_C 2 a.s.
(1.8)
n l A rt2"-+Go [CI e(N), P0]
In the following we list a set of conditions of which subsets will be used at different stages in the sequel: b) is twice differentiable and f j = F j > 0 on (t~, T6), for some yy>0 and pTe (0, II,
sup IfjF;l(Y)t to,p;] ffF;I(Y) -
0 < limfj(x) < oo,
(1.9) (1.10) (1.11)
x~t~
if limfj(x) =0, then J)is nondecreasing on (%,t0) for some to>tFj. (1.12) The following result is a generalization of Theorem 4.3 of Aly et al. (1985) and Theorem 3.1 of Aly (1986). The latter two results are generalizations of Theorem 3 of Cs6rg6 and R6v6sz (1978) and Theorem 3.1 of Cs6rg~ et al. (1982).
Theorem 1.3. Assume that F 2 satisfies (1.9) and (1.10) and let Po be as in (1.6), p~' and 72 as in (1.10), e(N) as in (1.5), and p 0 such that sup [ f * ( y ) - f l o , ) l = O ( N - l / 2 1 o g 2 N ) a.s. (1.13) ICe(N),Pl
If, in addition to (1.9) and (1.10), we also assume that F2 satisfies (1.11) or (1.12), then we respectively have
sup If*(y)-fl(y)l (0,N
=O(N-l/21og2
N)
a.s.
(1.14)
and f o r arbitrary e > 0,
(-O(N- 1/2 log2 N )
if )'2< 1, sup I f * 0 ' ) - fl0')l = ~ O(N-l/2(log2 N) 2) if ~'2= 1, (0,pl [ O ( N - l/2(log2 N)r2(log N)(l +e)(r2-1)) if ))2> 1.
a.s.
(1.15)
E.-E.A.A. Aly / Quantile-quantile plots
126
The following result is an analogue of Theorem 6 of Cs6rg6 and R6v6sz (1978) and it follows by combining Theorem 1.1 and Theorem 1.3. Theorem 1.4. Using the same notation o f Theorem 1.3 and on the same probability space o f Theorem 1.1 we have: (i) I f F2 satisfies (1.9) and (1.10), p*=F~1(p), and J*(N)=F~I(Ce(N)), then
sup
[ f ( x ) - 2-1/2nll/2Kl(Fl(x),nl)-(1 - 2)-l/2n21/2K2(Fl(X),n2)[
[J*(N),p*]
=O(r(N))
a.s.,
(1.16)
where r(N) is as in (1.5) and KI,K2 are those o f (1.6). (ii) l f F2 satisfies (1.9), (1.10) and one of(1.11) and (1.12), then the sup in (1.16) can be taken over (-o0, p*].
2. PL-confidence bands for Q-Q plots
The main result of this section is a PL-version of Theorem 3.1 of Aly and Bleuer (1986). First, we will introduce some notations and estimators. Let d j ( . ) = d T ( F j ( . ) ), j = 1,2, where dj* is as in (1.7), and introduce the estimators f)°y(x) := nj- l#{ill<_i<_n j , X~
dJ(t)=I'-o, (1-~jj(s))-2dffJ'n'(s)'
j=l,2.
Theorem 2.1. On the probability space o f Theorem 1.1 and on assuming any one o f the conditions (i) and (ii) o f Theorem 1.4, we have lim n I A n2--* oo
P{ Q_.2(Fl(X)-CTN(fl(X)))~Q2(Fl(X))~_9_~2(fl(X)-[-~lPN(Fl(X))); k[0,II
provided that F 1 satisfies (1.9) and (1.10), where p is as in (1.13), 0 < J < ½, c>0, W(. ) is a standard Wiener process, and
N6v) = (i -y){n?la
(p)) +
i,,2.
E.-E.A.A. Aly / Quantile-quantile plots
127
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128
E.-E.A.A. Aly / Quantile-quantile plots
The proof of Theorem 2.1 is a combination of the proofs of Theorem 3.1 of Aly and Bleuer (1986), and Theorem 4.3 of Aly et al. (1985). The following representation is also used: {Kj(t,S); 0___t_< 1, s > 0 } D {(1 -t)Wj(dj(Qj(t)),s); O<_t<_1, s > 0 } , where Wj(.,. ), j = 1,2, are independent two-parameter Wiener processes and =D denotes equality in distribution. We mention here that Cs6rg~ and Horvfith (1982) have tabulated P{suP[0, q [W(t)]<_c} for 0.301_
3. An example We have simulated four independent rs for which Fj(x) = q~(x) , /-/2(x) = 1 - e -x, j = 1,2, where q~(-) is the standard normal DF, and nl = n2 = 100. The corresponding PL-Q-Q plot together with a 90°70 confidence band are given in Figure 1. We observe that the straight line through the origin with unit slope fits very well into the band. Finally, we mention that, in this example, 28 observations from the {Xil}] °° sample and 25 from the {Xi2}~O0 sample are censored.
Acknowledgement I wish to thank an associate editor for his suggestions which improved the presentation of this paper.
References Aly, E.-E. (1986). Strong approximations of the Q-Q-process. J. Multivariate AnaL, to appear. Aly, E.-E. and S. Bleuer (1986). Confidence bands for Q-Q plots. Statistics and Decisions, 4, 205-225. Aly, E.-E., M. Cs6rg¢] and L. Horwlth (1985). Strong approximations of the quantile process of the product-limit estimator. J. Multivariate Anal. 16, 185-210. Burke, M.D., S. Cs6rg¢3 and L. Horv~ith (1981). Strong approximations of some biometric estimates under random censorship. Z. Warsch. Verw. Gebiete 56, 187-212. Cs~iki, E. (1977). The law of iterated logarithm for normalized empirical distribution function. Z. Wahrsch. Verw. Gebiete 38, 147-167. Csfrg~, M., S. Cs6rg~, L. Horv~ith and P. R~v6sz (1982). On weak and strong approximations of the quantile process. In: Proceedings o f the Seventh Conference on Probability Theory, Brasov, Aug. 29Sept. 4, 1982, to appear. Cs6rg~, M. and P. R6v6sz (1978). Strong approximations of the quantile process. Ann. Statist. 6, 882-894. Cs6rg¢], S. and L. Horv~ith (1982). Empirical Efron transforms of the product-limit process. Carleton Math. Lecture Note No. 38. Kaplan, E.L. and P. Meier (1958). Nonparametric estimation from incomplete observations. J. Amer. Statist. Assoc. 53, 457--481.
123
QUANTILE-QUANTILE PLOTS UNDER RANDOM CENSORSHIP Emad-Eldin A.A. ALY* Department of Statistics and Applied Probability, The University o f Alberta, Edmonton, Alberta, Canada T6G 2G1 Received 17 August 1984; revised manuscript received 13 November 1985 Recommended by P.K. Sen
Abstract: We obtain strong approximation re.suits for the product-limit Quantile-Quantile (PL-Q-Q) process. In addition, product-limit confidence bands for the theoretical Q-Q plot are constructed.
AMS Subject Classification: Primary 62G15; Secondary 60F15. Key words and phrases: PL-Q-Q processes; Kiefer process; confidence bands for Q-Q plots.
1. Introduction and preliminaries
Consider the problem of comparing two populations on the basis of two independent random samples (rs) each randomly censored from the right. Quantile-Quantile (Q-Q) plots are very useful graphical tools in the noncensored case. In the present paper we extend Q-Q plots to the censored case as well. Let {Xo}~ 1 and {Zu}~I, j = l , 2 , be four independent rs. The distribution function (DF) of X U is Fj and that of Z U is/-/j, j-- 1, 2. In the random censorship model from the right, the X Umay be censored by the ZU, so that one observes only the pairs { (X~i~ (~ij) } in~l , where X~ij= Xo A Zij ( = min(Xo, Zq )) and 6 U is the indicator function of the event {Xij<_Zu}, i= 1,2, ...,nj and j= 1,2. Thus {X~U}in~=l,j= 1,2 are two independent rs with DF's Fj°, j = 1,2, respectively, where 1 - F j°= ( 1 - F j ) ( 1 - H i ) , j= 1,2. The subdistribution function, ~ , of the uncensored X~u observations is given by
fJ(t)=P{X~#
(1.1)
* This research is partially supported by an NSERC Canada grant at the University of Alberta. Part of this work was done while the author was an Assistant Professor at King Saud University, Riyadh, Saudi Arabia. 0378-3758/86/$3.50 © 1986, Elsevier Science Publishers B.V. (North-Holland)
E.-E.A.A. AIy / Quantile-quantile plots
i24
For any D F L , let L-~(.):=inf{t: L(t)>_. } be the quantile function (QF) of L and S(L):= {t: 0 < L ( t ) < 1} be the support of L. For all the DF's considered here we will always assume that S(L)= (t L, TL). Let Fj be the (Kaplan-Meier (1958)) product-limit (PL) estimator of Fj based on {(X°t~ij)}n~l and l~j-l=gj be the corresponding QF, j = 1,2. The PL-Q-Q plot of F2 against FI is zl(. ) = Q2(Fl(. )) and the corresponding PL-Q-Q process, f(-), is defined by
f'(x) = Nl/2f2 Q2(Fl (X))(O.2(f~(x)) - Q2(Fl(x)) ),
x ~ S(FI),
(1.2)
where N=nl+n2, Q2=Ff 1, f2=F~ is assumed to exist, f2QE(.)=f2(Q2(.)) and 02(0) := min{X~/2:6i2 = 1}. For detailed discussions on the PL technique we refer to Burke et al. (1981), Aly et al. (1985) and the references contained therein. For strong approximation results as well as further references on Q-Q processes we refer to Aly (1986) and Aly and Bleuer (1986). The techniques of the latter four papers are used heavily in the sequel. It will be a convenient device to define a new set of rv by setting U~/=Fj(Xij), W/j=Fj(Zu) and U~ij=Fj(X~)=UiyAWij, i = l , 2 , . . . , n j and j = l , 2 . Let ]~j be the PL-estimator based on {(U~ (~iJ)}~l and Uj be the corresponding PL-QF, j = 1, 2. Next define
f*(y)= N1/2(O2(El(y))- y),
O_y_
(1.3)
and
fl(y)=f(F~1(y))=N~/Zf2Q2(y)(Qa(El(y))-Q2(y)),
0 < y < 1.
(1.4)
We will assume that nl/N--)Ae(O,l) as niAna--)oo. We will write logEN to denote log log N, and supi~b) to denote SUPa__.
r(N)=N-I/4(logaN)I/4(logN)I/a
and
e(N)=N-llog2 N,
(1.5)
The following result is a generalization of Theorem 2.1 of Aly (1986) and is proved similarly. Theorem 1.1. There exists a probability space on which two independent Kiefer processes, Ky(., . ), j = 1, 2, are defined such that
sup lf*(Y) - ;t-l/2n~l/2Kl(y, nl) - (1 - A)-l/2n21/2K2(Y, n2)l = O(r(N)) a . s . , 10,p0] (1.6)
where r(N) is as in (1.5), po
= (1 - . ) ( 1 - HAd-I(.))),
EKj(t,s)=0
and EKj(t, s)Kj (x, y) = (s Ay)(1 - t)(1 -- x)d7 (t Ax)
E.-E.A.A. Aly / Quantile-quantile plots
125
with ?t
d;(t)= 1o (1 -
u)-2(1 - Hj(Fj-I(u)))-I du.
(I .7)
The following theorem is an analogue of Theorem 3.2 of Cs~ki (1977).
Theorem 1.2. Let Po be as in (1.6) and e(N) as in (1.5). Then, f o r some positlve constants C~ and C 2
lim
sup
{ylog2N}-W2]f*(y)l<_C 2 a.s.
(1.8)
n l A rt2"-+Go [CI e(N), P0]
In the following we list a set of conditions of which subsets will be used at different stages in the sequel: b) is twice differentiable and f j = F j > 0 on (t~, T6), for some yy>0 and pTe (0, II,
sup IfjF;l(Y)t to,p;] ffF;I(Y) -
0 < limfj(x) < oo,
(1.9) (1.10) (1.11)
x~t~
if limfj(x) =0, then J)is nondecreasing on (%,t0) for some to>tFj. (1.12) The following result is a generalization of Theorem 4.3 of Aly et al. (1985) and Theorem 3.1 of Aly (1986). The latter two results are generalizations of Theorem 3 of Cs6rg6 and R6v6sz (1978) and Theorem 3.1 of Cs6rg~ et al. (1982).
Theorem 1.3. Assume that F 2 satisfies (1.9) and (1.10) and let Po be as in (1.6), p~' and 72 as in (1.10), e(N) as in (1.5), and p
If, in addition to (1.9) and (1.10), we also assume that F2 satisfies (1.11) or (1.12), then we respectively have
sup If*(y)-fl(y)l (0,N
=O(N-l/21og2
N)
a.s.
(1.14)
and f o r arbitrary e > 0,
(-O(N- 1/2 log2 N )
if )'2< 1, sup I f * 0 ' ) - fl0')l = ~ O(N-l/2(log2 N) 2) if ~'2= 1, (0,pl [ O ( N - l/2(log2 N)r2(log N)(l +e)(r2-1)) if ))2> 1.
a.s.
(1.15)
E.-E.A.A. Aly / Quantile-quantile plots
126
The following result is an analogue of Theorem 6 of Cs6rg6 and R6v6sz (1978) and it follows by combining Theorem 1.1 and Theorem 1.3. Theorem 1.4. Using the same notation o f Theorem 1.3 and on the same probability space o f Theorem 1.1 we have: (i) I f F2 satisfies (1.9) and (1.10), p*=F~1(p), and J*(N)=F~I(Ce(N)), then
sup
[ f ( x ) - 2-1/2nll/2Kl(Fl(x),nl)-(1 - 2)-l/2n21/2K2(Fl(X),n2)[
[J*(N),p*]
=O(r(N))
a.s.,
(1.16)
where r(N) is as in (1.5) and KI,K2 are those o f (1.6). (ii) l f F2 satisfies (1.9), (1.10) and one of(1.11) and (1.12), then the sup in (1.16) can be taken over (-o0, p*].
2. PL-confidence bands for Q-Q plots
The main result of this section is a PL-version of Theorem 3.1 of Aly and Bleuer (1986). First, we will introduce some notations and estimators. Let d j ( . ) = d T ( F j ( . ) ), j = 1,2, where dj* is as in (1.7), and introduce the estimators f)°y(x) := nj- l#{ill<_i<_n j , X~
dJ(t)=I'-o, (1-~jj(s))-2dffJ'n'(s)'
j=l,2.
Theorem 2.1. On the probability space o f Theorem 1.1 and on assuming any one o f the conditions (i) and (ii) o f Theorem 1.4, we have lim n I A n2--* oo
P{ Q_.2(Fl(X)-CTN(fl(X)))~Q2(Fl(X))~_9_~2(fl(X)-[-~lPN(Fl(X))); k[0,II
provided that F 1 satisfies (1.9) and (1.10), where p is as in (1.13), 0 < J < ½, c>0, W(. ) is a standard Wiener process, and
N6v) = (i -y){n?la
(p)) +
i,,2.
E.-E.A.A. Aly / Quantile-quantile plots
127
X
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i°
°~ 0
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•
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I
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I i°
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i
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m
m to i
128
E.-E.A.A. Aly / Quantile-quantile plots
The proof of Theorem 2.1 is a combination of the proofs of Theorem 3.1 of Aly and Bleuer (1986), and Theorem 4.3 of Aly et al. (1985). The following representation is also used: {Kj(t,S); 0___t_< 1, s > 0 } D {(1 -t)Wj(dj(Qj(t)),s); O<_t<_1, s > 0 } , where Wj(.,. ), j = 1,2, are independent two-parameter Wiener processes and =D denotes equality in distribution. We mention here that Cs6rg~ and Horvfith (1982) have tabulated P{suP[0, q [W(t)]<_c} for 0.301_
3. An example We have simulated four independent rs for which Fj(x) = q~(x) , /-/2(x) = 1 - e -x, j = 1,2, where q~(-) is the standard normal DF, and nl = n2 = 100. The corresponding PL-Q-Q plot together with a 90°70 confidence band are given in Figure 1. We observe that the straight line through the origin with unit slope fits very well into the band. Finally, we mention that, in this example, 28 observations from the {Xil}] °° sample and 25 from the {Xi2}~O0 sample are censored.
Acknowledgement I wish to thank an associate editor for his suggestions which improved the presentation of this paper.
References Aly, E.-E. (1986). Strong approximations of the Q-Q-process. J. Multivariate AnaL, to appear. Aly, E.-E. and S. Bleuer (1986). Confidence bands for Q-Q plots. Statistics and Decisions, 4, 205-225. Aly, E.-E., M. Cs6rg¢] and L. Horwlth (1985). Strong approximations of the quantile process of the product-limit estimator. J. Multivariate Anal. 16, 185-210. Burke, M.D., S. Cs6rg¢3 and L. Horv~ith (1981). Strong approximations of some biometric estimates under random censorship. Z. Warsch. Verw. Gebiete 56, 187-212. Cs~iki, E. (1977). The law of iterated logarithm for normalized empirical distribution function. Z. Wahrsch. Verw. Gebiete 38, 147-167. Csfrg~, M., S. Cs6rg~, L. Horv~ith and P. R~v6sz (1982). On weak and strong approximations of the quantile process. In: Proceedings o f the Seventh Conference on Probability Theory, Brasov, Aug. 29Sept. 4, 1982, to appear. Cs6rg~, M. and P. R6v6sz (1978). Strong approximations of the quantile process. Ann. Statist. 6, 882-894. Cs6rg¢], S. and L. Horv~ith (1982). Empirical Efron transforms of the product-limit process. Carleton Math. Lecture Note No. 38. Kaplan, E.L. and P. Meier (1958). Nonparametric estimation from incomplete observations. J. Amer. Statist. Assoc. 53, 457--481.