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Estimators for the inverse powers of a normal mean
Journal of Statistical Planning and Inference 143 (2013) 441–455
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Estimators for the inverse powers of a normal mean Christopher S. Withers a, Saralees Nadarajah b,n a b
Applied Mathematics Group, Industrial Research Limited, Lower Hutt, New Zealand School of Mathematics, University of Manchester, Manchester M13 9PL, UK
a r t i c l e i n f o
abstract
Article history: Received 7 October 2011 Received in revised form 17 June 2012 Accepted 19 June 2012 Available online 28 June 2012
Given a sample from a normal population unbiased estimators are obtained for positive powers of the mean and estimators of almost exponentially small bias are obtained for negative powers of the mean. Simulation studies show superior performance of these estimators versus known ones. & 2012 Elsevier B.V. All rights reserved.
Keywords: Bias reduction Normal Powers of the mean
1. Introduction For a random sample of size n from a normal distribution with mean m and variance v, let x and s denote the sample mean and the sample standard deviation, respectively. Consider the problem of estimating 1=m. This problem arises in many areas, including 1. In experimental nuclear Physics, a charged particle momentum p ¼ 1=m when m is the track curvature of a particle (Lamanna et al., 1981; Treadwell, 1982); 2. In the one-dimensional special case of the single period control problem, as discussed by Zellner (1971) and Zaman (1981b); 3. In estimation of structural parameters of a simultaneous equation as recognized in Zellner (1978) and described in Zaman (1981a); 4. In estimating various parameters of economic importance: for example, the investment multiplier, given an estimate of the marginal propensity to consume in a simple Keynesian model; or the long-run supply elasticity from Nerlove’s supply response model (see, for example, Braulke, 1982, Eq. (5)).
A related problem is the estimation of the inverse of the coefficient of variation, that is requiring this estimate:
pffiffiffi v=m. There are many situations
1. When dealing with the coefficient of variation of the normal distribution directly, the expected value of s9x, is infinite (see Johnson and Kotz, 1970, p. 75). This difficulty can be removed by estimating the inverse of the coefficient of variation (Chaturvedi and Rani, 1996); n
Corresponding author. E-mail address: [email protected] (S. Nadarajah).
0378-3758/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jspi.2012.06.018
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2. The exact sampling distribution of the coefficient of variation is quite difficult to obtain for non-normal distributions. In some cases, it is easier to work with the inverse of the coefficient of variation (Sharma and Krishna, 1994; Ng, 2005); 3. In medical imaging, the inverse of the coefficient of variation is used to discriminate rolling leukocytes from cluttered environments (Dong et al., 2005; Sahoo et al., 2006); 4. In finance, the inverse of the coefficient of variation measures the performance index of banks or firms (Baker and Edelman, 1991; Turen, 1996), the risk/return ratio (Powers and Powers, 2009) and the risk-adjusted return (McAuliffe, 2010). The inverse of the coefficient of variation is also known as the Sharp ratio in finance (Kaluszka, 2003); 5. In electrical and electronic engineering, the inverse of the coefficient of variation is used to estimate signal reliability or signal-to-noise ratios (Brown et al., 2001; Bergemann and Zhao, in press); 6. In Physics, the inverse of the coefficient of variation is used to characterize the mixing efficiency of a flat cylinder that is driven by two surface acoustic waves (Frommelt et al., 2008); 7. Chauvet and Potter (2001) use the inverse of the coefficient of variation to examine whether the US business cycle expansion that started in March 1991 is a onetime unique event or whether its length is a result of a change in the stability of the US economy; 8. In structural design and construction, the inverse of the coefficient of variation is often used a reliability index (Zubeck and Kvinson, 1996; Duerr, 2008); 9. Gauthier and Horne (2004) find that the inverse of the coefficient of variation emphasizes the potential for species discrimination; 10. In remote sensing, the inverse of the coefficient of variation provides an indication of the separation between two populations (Ahern, 1988). The inverse of the coefficient of variation is also used as a descriptive parameter in faculty evaluation studies (Rousseau, 1998), to enhance permanent scatterers identification (Refice et al., 2003), to detect robust patterns in the spread of epidemics (Crepey and Barthelemy, 2007), and to model genetic variation in humans (Pavan et al., 2009). Another related problem is the estimation of the ratio of two independent normal means. This problem arises frequently in sample surveys and in biological sciences, for example, with respect to safety assessment of a substance such as a new pharmaceutical compound or a pesticide relative to a vehicle or negative control. Kotz and Johnson (1986, pp. 639–646) provide an excellent discussion of the application areas. There have been only a small number of approaches to estimate 1=m when v is assumed known. The maximum likelihood estimator for 1=m is 1 , x
ð1:1Þ
an estimator with infinite variance. Zellner (1978) provided the following improved estimator of 1=m: x x 2 þ v=n
,
ð1:2Þ
again a biased estimator, in the context of minimizing posterior expected loss. Srivastava and Bhatnagar (1981) considered the class of estimators taking the form of (1.2) x x 2 þ gv=n
,
ð1:3Þ
where g is a constant, and showed that these estimators have bias of the order of Oðn2 Þ. Voinov (1985) provided unbiased estimators for powers of 1=m under the assumption that the sign of m is known. In particular, Voinov (1985) provided the following unbiased estimator of 1=m: ( ) pffiffiffi pffiffiffiffiffiffiffiffiffi nx 2np nðxÞ2 pffiffiffi exp F pffiffiffi , ð1:4Þ 2v v v where FðÞ denotes the standard normal distribution function. pffiffiffi Assuming both m and v are unknown, the estimators for v=m corresponding to (1.1), (1.3) and (1.4) are s , x sx x 2 þ gs2 =n and
,
( ) pffiffiffi pffiffiffiffiffiffiffiffiffi nx nðxÞ2 , F 2np exp s 2s2
respectively.
ð1:5Þ
ð1:6Þ
ð1:7Þ
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443
Confidence intervals for ratios of normal means have been developed by many authors over the past 50 years. We mention Fieller (1954), Bennett (1963), Chakravarti (1971), Steffens (1971), James et al. (1974), Sevin (1978), Mendoza and Gutierrez-Pena (1999), Tsao and Hwang (1999), Li and Zhang (2003), Diaz-Frances and Sprott (2004), Lee and Lin (2004), Kim (2005), Mendoza (2005), Dilba et al. (2006) and Shi et al. (2008). Confidence intervals for ratios of means from correlated normals have also been studied, see Bose (1942), Roy and Potthoff (1958), Malley (1982), Hannig et al. (2003) and Hannig (2006). The focus of this paper will be with point estimators although the theory developed here can be used to obtain confidence intervals. In this paper, we consider the general problem of estimating vI mJ given a sample from Nðm,vÞ for I real and J an integer. We show how to construct estimators with bias of the order Oðnk Þ for any k 4 1. In particular, we provide unbiased estimators for mJ when J Z0 and estimators of exponentially small bias for mJ when J o0. We believe that this is the first time such general results have been obtained for functions of the normal mean and normal variance. The results are organized as follows. In Section 2, we suppose for given ðn,f Þ we observe independent variables 2
fs vw2f ,
x Nðm,v=nÞ,
ð1:8Þ
where I 4f =2 and means ‘‘distributed as’’. For J Z 0, unbiased estimators are given. For J o0, we give estimators with Q J almost exponentially small bias as n-1. In particular, we estimate 1=m and m2 v. Section 3 gives estimators of pi¼ 1 vIi i mi i based on independent samples of size ni from Nðmi ,vi Þ, i ¼ 1; 2, . . . ,p. For fJ i Z 0g they are unbiased; otherwise their bias is almost exponentially small as min ni -1. Finally, Section 4 performs simulations to show that our estimators for 1=m and pffiffiffi v=m are better than (1.1), (1.3), (1.4), (1.5), (1.6) and (1.7). Throughout this paper, we use the notation a‘ b‘ to mean that a‘ =b‘ -1 as ‘ approaches its limiting value. As noted, we had also used to mean ‘‘distributed as’’. The distinction between these two should be clear from the context. 2. One sample case Suppose (1.8) holds. Then E1=x does not exist though its principal values does. The problem of estimating the latter seems a difficult one—see Withers and Nadarajah (2010). We shall give estimators for t ¼ vI mJ ,
ð2:1Þ
for v m when I 4 f =2 and J is an integer. These are unbiased if J Z 0 and have almost exponentially small bias if J o 0. We assume that if I
J
J o 0,
0 o m0 o 9m9,
ð2:2Þ
where m0 is known. If J o0 we base our estimator on
mn ¼ xð1In Þ þcIn ,
ð2:3Þ
where In ¼ Ið9x9 r m0 Þ, and 0 o 9c9 r 1
is arbitrary,
for example, c ¼ m0 sign x. Set ( x if J Z 0, b¼ h ¼ ðm,vÞ, m mn if J o 0,
ð2:4Þ
b ,s2 Þ, hb ¼ ðm
ð2:5Þ
lI ¼ Eðw2f =f ÞI ¼ ðf =2ÞI Gðf =2þ IÞ=Gðf =2Þ,
ð2:6Þ
J EN2i =lI þ i : aIJi ¼ 2i
ð2:7Þ
Let N Nð0; 1Þ, so EN2i ¼ 1 3 5 ð2i1Þ ¼ ð2iÞ!2i =i!. Theorem 2.1. Suppose (2.1)–(2.7) hold. For k Z1, set X IJ t IJnk ðhÞ ¼ vI mJ ai ðn1 vm2 Þi :
ð2:8Þ
0riok
We have the following:
bÞ estimates t with bias nk as n-1; (a) t IJnk ðh b Þ is unbiased; (b) If J Z0 and k ¼ J=2 þ 1 then t IJnk ðh b Þ estimates vI mJ with bias (c) Suppose J o 0. Fix 0 o A o1 and set kn ¼ 2½nA =2. Then t IJnkn ðh expfð1AÞnA ln ng, as n-1.
ð2:9Þ
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If J o 0 then t IJn1 ðyÞ converges or diverges as vm2 o or 4n=f . The proof of this theorem is given in Appendix A. We now give some examples. For r Z0, set ðJÞr ¼ JðJ1Þ ðJr þ 1Þ, so ( J!=ðJrÞ! if J Z 0, ðJÞr ¼ ð1Þr ðJ þ r1Þr if J o 0:
b Þ, where Example 2.1. If J 4 0 an unbiased estimator of mJ is AJ ðh X i ðJÞ2i ðf =2þ i1Þ1 AJ ¼ AJ ðhÞ ¼ mJ i ðD=4Þ =i!
ð2:10Þ
0 r 2i r J
and D ¼ fn
1
vm2 . The first few are A2 ¼ m2 ð1 þ Df
A1 ¼ m,
A4 ¼ m4 ð1 þ 6Df
1
1
þ 3D2 f
A3 ¼ m3 ð1 þ 3Df
Þ,
1
1
Þ,
ðf þ 2Þ1 Þ,
A5 ¼ m5 ð1 þ 10Df
1
þ 15D2 f
1
ðf þ2Þ1 Þ,
A6 ¼ m6 ð1 þ 15Df
1
þ 45D2 f
1
ðf þ 2Þ1 þ15D3 f
A7 ¼ m7 ð1 þ 21Df
1
þ 105D2 f
1
1
ðf þ 2Þ1 þ 105D3 f
ðf þ 2Þ1 ðf þ4Þ1 Þ, 1
ðf þ2Þ1 ðf þ 4Þ1 Þ:
b Þ, where Example 2.2. If 0 oJ o f an unbiased estimator of ðv1=2 mÞJ is BJ ðh X i ðJÞ2i g 0 g 1 BJ ¼ BJ ðhÞ ¼ DJ1 J2i ðD=4Þ =i!,
ð2:11Þ
0 r 2i r J
where g j ¼ Gððf jÞ=2Þ, D1 ¼ ðvf =2Þ1=2 m, D ¼ fn ( if j Z 0, ðf =21Þj g 0 g 1 2j ¼ ðf =21jÞ1 if j o 0, j
1
vm2 ¼ 2=ðnD21 Þ. Since
we have B1 ¼ D1 g 0 g 1 1 ,
B2 ¼ D21 ðf =21 þ D=2Þ,
1 B3 ¼ D31 g 0 ðg 1 3 þ 3g 1 D=2Þ,
B4 ¼ D41 fðf =21Þ2 þ 3ðf =21ÞD þ3D2 =4g, 1 1 2 B5 ¼ D51 g 0 ðg 1 5 þ 5g 3 D=4 þ15g 1 D =4Þ,
B6 ¼ D6 fðf =21Þ3 þ 15ðf =21Þ2 D=2þ 45ðf =21ÞD2 =4 þ15D3 =8g, and so on. b Þ of (2.10) with 0 r2ir J replaced by 0 r io kn ¼ ½nA , Example 2.3. If J o 0 an estimator of mJ with bias as in (2.9) is AJ ðh where 0 o A o1. In particular, A1 ¼ m1
kX n 1
i ð2iÞ!ðf =2 þ i1Þ1 i ðD=4Þ =i!
ð2:12Þ
i¼0
and A2 ¼ m2
kX n 1
i ð2i þ1Þ!ðf =2þ i1Þ1 i ðD=4Þ =i!,
i¼0 1
where D ¼ fn
vm2 .
b Þ of (2.11) with 0 r2ir J replaced by Example 2.4. If J o 0 an estimator of ðv1=2 mÞJ with bias as in (2.9) is BJ ðh A m 2m 0 ri okn ¼ ½n , where 0 o Ao 1. In particular, for J¼2m, even, v =m is estimated using B2m ¼ D2m 1
kX n 1
i ð2m þ 2i1Þ2i ðf =2 þi þ m1Þ1 i þ m ðD=4Þ =i!,
i¼0
where D1 ¼ ðvf =2Þ1=2 m, D ¼ fn
1
vm2 ¼ 2=ðnD21 Þ.
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445
b Þ is determined by I, J, f and l ¼ ðv=nÞ1=2 9m9, the coefficient of From Theorem 2.1 it follows that the relative bias of t IJnk ðh variation. 3. Multi-sample case Suppose x i Nðmi ,vi =ni Þ and f i s2i =vi w2f for i ¼ 1; 2, . . . ,p with all variables independent. We wish to estimate i Q J tðhÞ ¼ pi¼ 1 vIi i mi i , where Ii 4 f i =2 and Ji is an integer. If Ji o 0 assume 0 o m0i o 9mi 9, where m0i is known. Set ( x i if J i Z 0, bi ¼ h ¼ ðl,vÞ, m mni if Ji o 0,
mni ¼ x i ð1Ini Þ þ ci Ini , Ini ¼ Ið9x i 9 r m0i Þ, vb i ¼ s2i , where 0 r9ci 9r 1 is arbitrary. For k 2 Np , where N ¼ f0; 1,2, . . .g, set t IJnk ðhÞ ¼
p Y i¼1
IJ
t ni ki ðhi Þ, i i
of (2.8) with hi ¼ ðmi ,vi Þ. Then Theorem 2.1 implies. Theorem 3.1. With the notation set as above, we have the following: b Þ estimates tðhÞ with bias Pp nki ; (a) t IJnk ðh i¼1 i b Þ is unbiased; (b) If J Z 0 and k ¼ J=2þ 1 then t IJnk ðh (c) Set ( 2½nAi i =2, where 0 o Ai o 1 if Ji o 0, ki ¼ if Ji Z 0: Ji =2 þ1 b Þ has bias Then t IJnk ðh 8 9 < X = Ai ð1Ai Þni ln ni exp : J o0 ; i
as min ni -1. Example 3.1. An estimator of m1 =m2 with bias expfð1AÞnA2 ln n2 g as n2 -1 is x 1 Af 2 n2 ðmn2 ,s22 Þ, where Afn ðhÞ ¼ A1 of (2.12). 4. A simulation study Here, we perform simulations to compare the estimators proposed in Section 2 with known ones. We conduct two simulation studies. The R (R Development Core Team, 2011) codes used to perform these simulation studies can be obtained from the corresponding author, email: [email protected] Firstly, we compare the estimator for 1=m given by (2.12) assuming that v is unknown versus those given by (1.1), (1.3) and (1.4) assuming that v is known. We use two criteria for comparing the four estimators: mean squared error and bias. We computed (2.12), (1.1), (1.3) and (1.4) by simulating ten thousand replications of samples of size n from N (1, v) for v ¼0.2, 0.5, 2, 10 and n¼ 10, 20, y, 1000. We take f¼n 1, m0 ¼ 0:5 and c ¼ m0 sign x in (2.12). Note that m0 is half the true value of m. This is not an unreasonable lower bound on m. Even smaller values can be taken for m0 , but they would decrease the efficiency of the proposed estimators. In practice, m0 should be chosen as close as possible to m but not too large that m0 is not a lower bound. The constants g in (1.3) and A in (2.12) for a given v are chosen to minimize the respective variances. For g, we follow the procedure:
select a range of values for g 4 0; for each g, compute 0 12 000 10 000 X x 1 10X x 1 j k @ A En ¼ 10 000 k ¼ 1 x 2k þ gv=n 10 000 j ¼ 1 x 2j þ gv=n for all n¼10, 20, y, 1000, where x k denotes the sample mean for the kth replication;
choose the optimal g as the one that minimizes E10 þ E20 þ þ E1000 .
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C.S. Withers, S. Nadarajah / Journal of Statistical Planning and Inference 143 (2013) 441–455
For A, we follow the procedure:
select a range of values for A 2 ð0; 1Þ; for each A, compute 0 12 000 000 1 10X 1 10X ðkÞ ðjÞ A @ A1 A En ¼ 10 000 k ¼ 1 10 000 j ¼ 1 1
for all n ¼10, 20, y, 1000, where AðkÞ 1 is the A1 given by (2.12) for the kth replication; choose the optimal A as the one that minimizes E10 þ E20 þ þ E1000 .
The same ten thousand replications are used for every g and for every A. The optimal choices given by these procedures allow the estimators, (1.3) and (2.12), to realize their greatest potential. The optimal choices for different v are different. However, the optimal choices of g and A appeared to take values around 1 and 1/4, respectively. Hence, for a fair comparison of the simulation results, we shall choose g ¼1 and A ¼1/4 for all v. The plots of the mean squared error and the bias versus n Z 10 for the selected v and for the four estimators are shown in Figs. 1 and 2. The x-axes are plotted on log scale. The following observations can be drawn from the figures:
1. the mean squared errors generally decrease to zero with increasing n; 2. the smallest mean squared errors for either v¼0.2, 0.5 and all n or v¼2, 10 and large n are by the new estimator given by (2.12); 3. the second smallest mean squared errors for either v¼ 0.2, 0.5 and all n or v ¼2, 10 and large n are by (1.3); 4. the third smallest mean squared errors for either v ¼0.2, 0.5 and all n or v¼ 2, 10 and large n are by (1.4); 5. the largest mean squared errors for either v¼0.2, 0.5 and all n or v ¼2, 10 and large n are by (1.1); 6. the smallest mean squared errors for v ¼2, 10 and small n are by (1.3); 7. the second smallest mean squared errors for v¼ 2, 10 and small n are by (1.4) and the new estimator given by (2.12); 8. the largest mean squared errors for v ¼2, 10 and small n are by (1.1); 9. the mean squared errors generally increase with increasing v; 10. the biases generally approach zero with increasing n; 11. the biases are generally positive for (1.1); 12. the biases are generally negative for (1.3) and the new estimator given by (2.12); 13. the smallest biases for all v and large n are by (1.3), (1.4) and the new estimator given by (2.12); 14. the smallest biases are for all v and small n are by (1.4); 15. the second smallest biases for all v and small n are by (1.3); 16. the third smallest biases for all v and small n are by the new estimator given by (2.12); 17. the largest biases for all v and for all n are by (1.1); 18. the biases generally increase with increasing v. The new estimator gives the best performance with respect to mean squared error for all large n. It also gives the best performance with respect to mean squared error for small v and for all n. The best performance with respect to bias is by (1.4), an unbiased estimator. We noted earlier in Section 1 that the estimator (1.1) has infinite variance. So, the mean squared error of (1.1) is also infinite. The estimates reported in Fig. 1 are finite because they are based on a finite number of replications. The estimates appear fairly stable at least for large n. To explain this, we have plotted the mean squared error of (1.1) versus the number of replications, see Fig. 3. We have taken n ¼20, m ¼ 1 and v ¼10. We can see that the divergence of the mean squared error to infinity is very slow even for one hundred thousand replications. Hence, the mean squared errors for (1.1) reported in Fig. 1 should be treated as underestimates. A refined estimator for 1=m based on (2.12) is 8 > < 2 A1 > :2
if A1 o2, if 2r A1 r 2, if A1 42:
This estimator suggested by a referee – for whom we are most grateful – will have smaller mean squared errors than (2.12). We hope to follow up this estimator in a future study. pffiffiffi The second simulation study is to compare the estimators for v=m given by Example 2.4, (1.5), (1.6) and (1.7), assuming that both m and v are unknown. We conducted the simulation as in the first study with f ¼ n1, m0 ¼ 0:5, c ¼ m0
C.S. Withers, S. Nadarajah / Journal of Statistical Planning and Inference 143 (2013) 441–455
v = 0.2
0.025
v = 0.5
0.08
0.015
Mean Squared Error
Mean Squared Error
0.020
0.010
0.06
0.04
0.005
0.02
0.00
0.00 10
20
50
100 n
200
500
1000
10
20
50
v= 2 1.0
0.4
0.8
0.3
0.2
0.0
0.0 50
100 n
500
1000
200
500
1000
0.4
0.2
20
200
0.6
0.1
10
100 n v = 10
0.5
Mean Squared Error
Mean Squared Error
447
200
500
1000
10
20
50
100 n
Fig. 1. Mean squared error of the estimators of 1=m given by (1.1) (in black), (1.3) (in brown), (1.4) (in blue) and (2.12) (in red) for m ¼ 1 and v¼ 0.2, 0.5, 2, 10. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this article.)
sign x, v ¼0.2, 0.5, 2, 10 and n ¼10, 20,y,1000. The constants g in (1.6) and A in Example 2.4 for a given v are chosen similarly to the first simulation study. For g, we follow the procedure:
select a range of values for g 4 0; for each g, compute 0 12 000 10 000 X s x 1 10X s x 1 j j k k @ A En ¼ 10 000 k ¼ 1 x 2k þ gs2 =n 10 000 j ¼ 1 x 2j þ gs2 =n k
j
for all n ¼10, 20,y,1000, where x k denotes the sample mean for the kth replication and sk denotes the sample standard deviation for the kth replication; choose the optimal g as the one that minimizes E10 þ E20 þ þ E1000 .
448
C.S. Withers, S. Nadarajah / Journal of Statistical Planning and Inference 143 (2013) 441–455
v = 0.2
v = 0.5
0.06
0.020
0.04 Bias
Bias
0.015
0.010 0.02
0.005 0.00 0.000
10
20
50
100 n
200
500
1000
10
20
50
v= 2
100 n
200
500
1000
v = 10
0
0.4
−2
Bias
Bias
0.2
0.0
−4
−0.2
−6
10
20
50
100 n
200
500
1000
10
20
50
100 n
200
500
1000
Fig. 2. Bias of the estimators of 1=m given by (1.1) (in black), (1.3) (in brown), (1.4) (in blue) and (2.12) (in red) for m ¼ 1 and v¼0.2, 0.5, 2, 10. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this article.)
For A, we follow the procedure:
select a range of values for A 2 ð0; 1Þ; for each A, compute 0 12 000 10 000 X 1 10X 1 ðkÞ ðjÞ @B B A En ¼ 1 10 000 k ¼ 1 10 000 j ¼ 1 1 for all n ¼10, 20, y, 1000, where BðkÞ 1 is the B1 given by Example 2.4 for the kth replication;
choose the optimal A as the one that minimizes E10 þ E20 þ þ E1000 . The optimal choices given by these procedures are also different for different v. Again the optimal choices of g and A appeared to take values around 1 and 1/4, respectively, so we shall take g ¼1 and A¼ 1/4 for all v.
C.S. Withers, S. Nadarajah / Journal of Statistical Planning and Inference 143 (2013) 441–455
449
Mean Squared Error
10000 8000 6000 4000 2000 0 0e+00
2e+04
4e+04 6e+04 Number of Replications
8e+04
1e+05
Fig. 3. Mean squared error of (1.1) versus the number of replications for n ¼20, m ¼ 1 and v¼10.
v = 0.2
v = 0.5
0.015
Mean Squared Error
Mean Squared Error
0.06
0.010
0.005
0.04
0.02
0.00
0.000 10
20
50
100
200
500
1000
10
20
50
0.8
4
0.6
0.4
0.0
0 100 n
1000
200
500
1000
2
1
50
500
3
0.2
20
200
v = 10 5
Mean Squared Error
Mean Squared Error
v= 2 1.0
10
100 n
n
200
500
1000
10
20
50
100 n
pffiffiffi Fig. 4. Mean squared error of the estimators of v=m given by (1.5) (in black), (1.6) (in brown), (1.7) (in blue) and Example 2.4 (in red) for m ¼ 1 and v¼ 0.2, 0.5, 2, 10. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this article.)
450
C.S. Withers, S. Nadarajah / Journal of Statistical Planning and Inference 143 (2013) 441–455
The plots of the mean squared error and the bias versus n Z 10 for the selected v and for the four estimators are shown in Figs. 4 and 5. Again the x-axes are plotted on log scale. The observations from these figures are the same as those from Figs. 1 and 2 except that 1. 2. 3. 4. 5. 6. 7.
the smallest mean squared errors for v ¼0.2 and small n are by (1.6) and (1.7); the second smallest mean squared errors for v ¼0.2 and small n are by the new estimator given by Example 2.4; the smallest mean squared errors for v ¼0.4 and small n are by (1.6) and the new estimator given by Example 2.4; the second smallest mean squared errors for v ¼0.4 and small n are by (1.7); the largest mean squared errors for v ¼0.2, 0.4 and small n are by (1.5); the smallest mean squared errors for v ¼2, 10 and small n are by (1.6); the second smallest mean squared errors for v ¼2, 10 and small n are by (1.7) and the new estimator given by Example 2.4; 8. the largest mean squared errors for v ¼2, 10 and small n are by (1.5); 9. the smallest mean squared errors for all large n are by the new estimator given by Example 2.4; 10. the second smallest mean squared errors for all large n are by (1.6); v = 0.2
v = 0.5 0.02
0.000
0.01
Bias
Bias
−0.004 0.00
−0.01
−0.008
−0.02 −0.012 −0.03 10
20
50
100 n
200
500
1000
10
20
50
v= 2
100 n
200
500
1000
200
500
1000
v = 10 4
0.2 0.1
2
−0.1
Bias
Bias
0.0 0
−0.2 −2 −0.3 −0.4
−4
10
20
50
100 n
200
500
1000
10
20
50
100 n
pffiffiffi Fig. 5. Bias of the estimators of v=m given by (1.5) (in black), (1.6) (in brown), (1.7) (in blue) and Example 2.4 (in red) for m ¼ 1 and v¼ 0.2, 0.5, 2, 10. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this article.)
C.S. Withers, S. Nadarajah / Journal of Statistical Planning and Inference 143 (2013) 441–455
11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23.
the the the the the the the the the the the the the
451
third smallest mean squared errors for all large n are by (1.7); largest mean squared errors for all large n are by (1.5); biases are generally negative for (1.6) and (1.7); smallest biases for v ¼0.2, 0.5 and for all n are by the new estimator given by Example 2.4; second smallest biases for v ¼0.2, 0.5 and for all n are by (1.5); third smallest biases for v ¼0.2, 0.5 and for all n are by (1.7); largest biases for v ¼0.2, 0.5 and for all n are by (1.6); smallest biases for v ¼2, 10 and small n are by (1.7); second smallest biases for v ¼2, 10 and small n are by (1.5); third smallest biases for v ¼2, 10 and small n are by (1.6); largest biases for v ¼2, 10 and small n are by the new estimator given by Example 2.4; smallest biases for v ¼2, 10 and large n are by (1.6) and (1.7), and the new estimator given by Example 2.4; largest biases for v ¼2, 10 and large n are by (1.5).
The new estimator again gives the best performance with respect to mean squared error for large n. The best performance with respect to bias for small v and all n is by the new estimator. The best performance with respect to bias for large v and large n is by (1.6) and (1.7), and the new estimator. In summary, the estimators given by Theorem 2.1 show the best performance with respect to mean squared error for all large n. With respect to bias, the estimators in Theorem 2.1 give the best performance at least for all large n. They also give the smallest biases for some small n. Finally, we like to make a point about the procedures for choosing optimal g and A. Normally, the practitioner will not have the luxury of doing the required simulation for these procedures, even if there is some model that describes the observed data. It would be ideal to have some simple expressions for g or A based only on n, x, and s. Such expressions could be obtained, for example, by minimizing the first few terms of an asymptotic expansion for the mean squared error. In this paper, we have not studied asymptotic properties of the mean squared error of the estimators. Such a study will require substantially more work, so a topic for future work.
Acknowledgments The authors would like to thank the Editor and the two referees for careful reading and for their comments which greatly improved the paper. Appendix A. Proof of Theorem 2.1 p
We precede the proof with some lemmas. Set d ¼ m0 =9m9 2 ð0; 1Þ, d ¼ m1 ðv=nÞ1=2 and 9X9p ¼ ðE9X9 Þ1=p . Lemma A.1. We have PðIn ¼ 1Þ n1=2 expðlnÞ as n-1, where l ¼ ð1dÞ2 m2 =2v. b =m1Þ=d Nð0; 1Þ and a 7 ¼ ð1 7 dÞ=9d9. Then PðIn ¼ 1Þ ¼ Pð91 þ dN9 o dÞ ¼ Fða ÞFða þ Þ fða Þ= Proof. Set N ¼ ðm a n1=2 expðlnÞ, where fðÞ is the derivative of FðÞ. & Lemma A.2. For J and k non-negative integers ð1xÞJ ¼ AJk ðxÞ þBJk ðxÞ, where AJk ðxÞ ¼
k1 X
J j
i¼0
! ðxÞj ,
and BJk ðxÞ ¼
J1 X
J1 þ k
j¼0
jþk
! xj þ k ð1xÞ1j :
Proof. Differentiate ð1xÞ1 ¼
p1 X j¼0
J times.
&
xj þxp ð1xÞ1
452
C.S. Withers, S. Nadarajah / Journal of Statistical Planning and Inference 143 (2013) 441–455 1
Lemma A.3. If x ¼ dN, Q ¼ J1 þk and z ¼ 21=2 d 9ð1In ÞBJk ðxÞ91
¼ 2d
1
9d9 r 1=2 then
1=2 k
9 =ðk=2Þ!:
Q !9d2
ðA:1Þ
Proof. The left hand side of (A.1) is less than or equal to ! J1 X Q jþk 1j E9x9 : d j þk j¼0
Note that Q i i i E9x9 ¼ Q !9d21=2 9 =fðQ iÞ!ði=2Þ!g r Q !9d21=2 9 =ðk=2Þ!, i for iZ k. So, the left hand side of (A.1) is less than or equal to 1
Q !d
ðk=2Þ!1 9d21=2 9
k
J1 X
zj ,
j¼0
which is less than or equal to the right hand side of (A.1).
&
Lemma A.4. Assume (2.5). For j Z 0 set J EN2j : K Jj ¼ vj mJ2j nJj , nJj ¼ 2j Then for J Z 0, k1 X
Ex J ¼
( nj K Jj þ
j¼0
Oðnk Þ 0
if k1 oJ=2, if k1 ZJ=2,
ðA:2Þ
and b Em
J
¼
k1 X
nj K J þ Enk , j
ðA:3Þ
j¼0
where
Enk ¼ mJ Efð1In ÞBJ2k ðxÞ þ In C J2k ðxÞg ¼ Oðnk Þ and C J2k ðxÞ ¼ cJ AJ2k ðxÞ. Proof. Note that x ¼ mð1 þ dNÞ, so (A.2) follows by the binomial expansion. Note that b J ¼ x J ð1In Þ þcJ In ¼ mJ fAJ2k ðxÞ þ ð1In ÞBJ2k ðxÞ þIn C J2k ðxÞg ¼ LA þ LB þ Lc m say, where x ¼ dN. Note that mJ EAJ2k ðxÞ is the first term on the right hand side of (A.3). Also ELB nk by Lemma A.3. Fix 1 o p o1, 1=p þ1=q ¼ 1. Then 9ELC 9 r 9In 9p 9C J2k ðxÞ9q n1=2 expðln=pÞ by Lemma A.1. & Proof of Theorem 2.1. For J Z 0, ( k1 X Oðnk Þ 2I J j I J Es x ¼ n lI v K j þ 0 j¼0
if k1o J=2, if k1Z J=2,
and Es2I x J ¼
k1 X
nj lI vI K J þ Oðnk Þ, j
j¼0
so (a) and (b) follow upon substitution. Put J 0 ¼ J. Then k1 3 X X J bÞ ¼ EtIJnk ðh Akli , EN 2i ðnÞi vI þ i Eðmn ÞJ2i ¼ t 2i i¼0 i¼1 corresponding to 0
ðmn ÞJ ¼ mJ
0
3 X
AJ0li ,
i¼1
equalling the right hand side of (A.3) with ðJ,kÞ ¼ ðJ 0 ,2lÞ.
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453
2
Choose l ¼ 2½nA =2. Set gm ¼ ðJ0 þ 2m1Þ!ðJ 0 1Þ!1 ðd =2Þm =m!. Then kX þ l2 l1 kX þ l2 X X m Akl1 ¼ gm ð1Þi gm C mk þ gm bmlk , ¼ 1þ i i¼0 k l where the inner sum is over 0 ri r m, i o k and mi o l, k1 X m C mk ¼ ð1Þi i i¼0 is bounded by
m k1
,
k1 X
bmlk ¼
ð1Þi
i ¼ ml þ 1
m i
¼ bm,ml þ 1 þb0m,mk þ 1 ,
and bmN ¼
X0
ð1Þi
N
00 mN X m ð1Þi , b0mN ¼ , i i
m
P P sums to [m/2] halving the last term if m is even, 00 sums from m/2 if m is even, halving the last term, and 00 where sums from (mþ1)/2 if m is odd. m So, b0mN ¼ ð1Þm bmN . For m ¼2M, 0 r bmN r m , so 0 rb r m þ k1 . For m¼2M þ1, 0 rð1ÞM bmN r mNMþ M , so l1 þ MN k1 þ M mlkl1 þl1 bmlk ¼ bm,ml þ 1 bm,mk þ 1 is bounded by þ 2 M M . So, M M kX þ l2 KX þ L2 m 2M l1 þM 9Akl1 19 r gm g2M ¼ S1 þS2 þS3 þ þ2g2M þ 1 k1 l1 M L k P0
say, where K ¼ k=2 and L ¼ l=2. 2 Set r n ¼ d =2. Then gM fr n 4 expð1Þmgm mJ1 gm ðk þ l2ÞJ1 g ¼ rn 4 expð1Þðkþ l2Þ. So, þ l2 k þl2 k X S1 gm ðk þlÞk þ l þ J1:5 k0:5k l0:5l gk T, k1 k
for
m r k þl2
by
Stirling’s
formula,
where
where l2 X
T¼
gi 1,
i¼0
as g-0. So, ln S1 r constant kln nð1AÞð1 þoð1ÞÞ, so S1 behaves as nA1 . Also þ L2 k þl4 K X g2M ðkþ l4ÞJ1 S1 S2 l1 L and S3
KX þ L2
r n g2M ðk þ l4ÞJ lrn S2 S2 ,
L
so Akl1 1 behaves as nA1 . By Lemma A.3, ! k1 X J 0 2 9Akl2 9 ¼ EN2i ðd Þi Eð1In ÞBJ02lþ 2i ðxÞ expfl ln nð1AÞð1 þ oð1ÞÞg, 2i i¼0 by Stirling’s formula. Note that the right hand side of (A.4) behaves (2.9). Also for 1 o p o1 and 1=p þ 1=q ¼ 1, ! ( ) J0 þ 2k1 J 0 J0 J0 2k2 9EIn C 2k ðxÞ9 r 9In 9p 9C 2k ðxÞ9q r9In 9p 9C9 þ 2 9N 9q J0 1 for d r 1=2 since Akl3 ¼
k1 X i¼0
L þ j L
J0 2i
and 9Nj 9q increase with j. So, ! 2
EN 2i ðd Þi EIn C J02lþ 2i ðxÞ
ðA:4Þ
454
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is bounded by 9In 9p
J0 þ 2k2 J 0 1
! EN
( 2k2
29C9
J 0
þ 49N
2l2
9q
J 0 þ 2k þ2l3 J0 1
!) :
2
For d r 1=2 and d rc2 =2, this bound behaves as n1=2 expfnlg by Lemma A.1 using 9Nj 9q fqj expð1Þg5=2 and
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Contents lists available at SciVerse ScienceDirect
Journal of Statistical Planning and Inference journal homepage: www.elsevier.com/locate/jspi
Estimators for the inverse powers of a normal mean Christopher S. Withers a, Saralees Nadarajah b,n a b
Applied Mathematics Group, Industrial Research Limited, Lower Hutt, New Zealand School of Mathematics, University of Manchester, Manchester M13 9PL, UK
a r t i c l e i n f o
abstract
Article history: Received 7 October 2011 Received in revised form 17 June 2012 Accepted 19 June 2012 Available online 28 June 2012
Given a sample from a normal population unbiased estimators are obtained for positive powers of the mean and estimators of almost exponentially small bias are obtained for negative powers of the mean. Simulation studies show superior performance of these estimators versus known ones. & 2012 Elsevier B.V. All rights reserved.
Keywords: Bias reduction Normal Powers of the mean
1. Introduction For a random sample of size n from a normal distribution with mean m and variance v, let x and s denote the sample mean and the sample standard deviation, respectively. Consider the problem of estimating 1=m. This problem arises in many areas, including 1. In experimental nuclear Physics, a charged particle momentum p ¼ 1=m when m is the track curvature of a particle (Lamanna et al., 1981; Treadwell, 1982); 2. In the one-dimensional special case of the single period control problem, as discussed by Zellner (1971) and Zaman (1981b); 3. In estimation of structural parameters of a simultaneous equation as recognized in Zellner (1978) and described in Zaman (1981a); 4. In estimating various parameters of economic importance: for example, the investment multiplier, given an estimate of the marginal propensity to consume in a simple Keynesian model; or the long-run supply elasticity from Nerlove’s supply response model (see, for example, Braulke, 1982, Eq. (5)).
A related problem is the estimation of the inverse of the coefficient of variation, that is requiring this estimate:
pffiffiffi v=m. There are many situations
1. When dealing with the coefficient of variation of the normal distribution directly, the expected value of s9x, is infinite (see Johnson and Kotz, 1970, p. 75). This difficulty can be removed by estimating the inverse of the coefficient of variation (Chaturvedi and Rani, 1996); n
Corresponding author. E-mail address: [email protected] (S. Nadarajah).
0378-3758/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jspi.2012.06.018
442
C.S. Withers, S. Nadarajah / Journal of Statistical Planning and Inference 143 (2013) 441–455
2. The exact sampling distribution of the coefficient of variation is quite difficult to obtain for non-normal distributions. In some cases, it is easier to work with the inverse of the coefficient of variation (Sharma and Krishna, 1994; Ng, 2005); 3. In medical imaging, the inverse of the coefficient of variation is used to discriminate rolling leukocytes from cluttered environments (Dong et al., 2005; Sahoo et al., 2006); 4. In finance, the inverse of the coefficient of variation measures the performance index of banks or firms (Baker and Edelman, 1991; Turen, 1996), the risk/return ratio (Powers and Powers, 2009) and the risk-adjusted return (McAuliffe, 2010). The inverse of the coefficient of variation is also known as the Sharp ratio in finance (Kaluszka, 2003); 5. In electrical and electronic engineering, the inverse of the coefficient of variation is used to estimate signal reliability or signal-to-noise ratios (Brown et al., 2001; Bergemann and Zhao, in press); 6. In Physics, the inverse of the coefficient of variation is used to characterize the mixing efficiency of a flat cylinder that is driven by two surface acoustic waves (Frommelt et al., 2008); 7. Chauvet and Potter (2001) use the inverse of the coefficient of variation to examine whether the US business cycle expansion that started in March 1991 is a onetime unique event or whether its length is a result of a change in the stability of the US economy; 8. In structural design and construction, the inverse of the coefficient of variation is often used a reliability index (Zubeck and Kvinson, 1996; Duerr, 2008); 9. Gauthier and Horne (2004) find that the inverse of the coefficient of variation emphasizes the potential for species discrimination; 10. In remote sensing, the inverse of the coefficient of variation provides an indication of the separation between two populations (Ahern, 1988). The inverse of the coefficient of variation is also used as a descriptive parameter in faculty evaluation studies (Rousseau, 1998), to enhance permanent scatterers identification (Refice et al., 2003), to detect robust patterns in the spread of epidemics (Crepey and Barthelemy, 2007), and to model genetic variation in humans (Pavan et al., 2009). Another related problem is the estimation of the ratio of two independent normal means. This problem arises frequently in sample surveys and in biological sciences, for example, with respect to safety assessment of a substance such as a new pharmaceutical compound or a pesticide relative to a vehicle or negative control. Kotz and Johnson (1986, pp. 639–646) provide an excellent discussion of the application areas. There have been only a small number of approaches to estimate 1=m when v is assumed known. The maximum likelihood estimator for 1=m is 1 , x
ð1:1Þ
an estimator with infinite variance. Zellner (1978) provided the following improved estimator of 1=m: x x 2 þ v=n
,
ð1:2Þ
again a biased estimator, in the context of minimizing posterior expected loss. Srivastava and Bhatnagar (1981) considered the class of estimators taking the form of (1.2) x x 2 þ gv=n
,
ð1:3Þ
where g is a constant, and showed that these estimators have bias of the order of Oðn2 Þ. Voinov (1985) provided unbiased estimators for powers of 1=m under the assumption that the sign of m is known. In particular, Voinov (1985) provided the following unbiased estimator of 1=m: ( ) pffiffiffi pffiffiffiffiffiffiffiffiffi nx 2np nðxÞ2 pffiffiffi exp F pffiffiffi , ð1:4Þ 2v v v where FðÞ denotes the standard normal distribution function. pffiffiffi Assuming both m and v are unknown, the estimators for v=m corresponding to (1.1), (1.3) and (1.4) are s , x sx x 2 þ gs2 =n and
,
( ) pffiffiffi pffiffiffiffiffiffiffiffiffi nx nðxÞ2 , F 2np exp s 2s2
respectively.
ð1:5Þ
ð1:6Þ
ð1:7Þ
C.S. Withers, S. Nadarajah / Journal of Statistical Planning and Inference 143 (2013) 441–455
443
Confidence intervals for ratios of normal means have been developed by many authors over the past 50 years. We mention Fieller (1954), Bennett (1963), Chakravarti (1971), Steffens (1971), James et al. (1974), Sevin (1978), Mendoza and Gutierrez-Pena (1999), Tsao and Hwang (1999), Li and Zhang (2003), Diaz-Frances and Sprott (2004), Lee and Lin (2004), Kim (2005), Mendoza (2005), Dilba et al. (2006) and Shi et al. (2008). Confidence intervals for ratios of means from correlated normals have also been studied, see Bose (1942), Roy and Potthoff (1958), Malley (1982), Hannig et al. (2003) and Hannig (2006). The focus of this paper will be with point estimators although the theory developed here can be used to obtain confidence intervals. In this paper, we consider the general problem of estimating vI mJ given a sample from Nðm,vÞ for I real and J an integer. We show how to construct estimators with bias of the order Oðnk Þ for any k 4 1. In particular, we provide unbiased estimators for mJ when J Z0 and estimators of exponentially small bias for mJ when J o0. We believe that this is the first time such general results have been obtained for functions of the normal mean and normal variance. The results are organized as follows. In Section 2, we suppose for given ðn,f Þ we observe independent variables 2
fs vw2f ,
x Nðm,v=nÞ,
ð1:8Þ
where I 4f =2 and means ‘‘distributed as’’. For J Z 0, unbiased estimators are given. For J o0, we give estimators with Q J almost exponentially small bias as n-1. In particular, we estimate 1=m and m2 v. Section 3 gives estimators of pi¼ 1 vIi i mi i based on independent samples of size ni from Nðmi ,vi Þ, i ¼ 1; 2, . . . ,p. For fJ i Z 0g they are unbiased; otherwise their bias is almost exponentially small as min ni -1. Finally, Section 4 performs simulations to show that our estimators for 1=m and pffiffiffi v=m are better than (1.1), (1.3), (1.4), (1.5), (1.6) and (1.7). Throughout this paper, we use the notation a‘ b‘ to mean that a‘ =b‘ -1 as ‘ approaches its limiting value. As noted, we had also used to mean ‘‘distributed as’’. The distinction between these two should be clear from the context. 2. One sample case Suppose (1.8) holds. Then E1=x does not exist though its principal values does. The problem of estimating the latter seems a difficult one—see Withers and Nadarajah (2010). We shall give estimators for t ¼ vI mJ ,
ð2:1Þ
for v m when I 4 f =2 and J is an integer. These are unbiased if J Z 0 and have almost exponentially small bias if J o 0. We assume that if I
J
J o 0,
0 o m0 o 9m9,
ð2:2Þ
where m0 is known. If J o0 we base our estimator on
mn ¼ xð1In Þ þcIn ,
ð2:3Þ
where In ¼ Ið9x9 r m0 Þ, and 0 o 9c9 r 1
is arbitrary,
for example, c ¼ m0 sign x. Set ( x if J Z 0, b¼ h ¼ ðm,vÞ, m mn if J o 0,
ð2:4Þ
b ,s2 Þ, hb ¼ ðm
ð2:5Þ
lI ¼ Eðw2f =f ÞI ¼ ðf =2ÞI Gðf =2þ IÞ=Gðf =2Þ,
ð2:6Þ
J EN2i =lI þ i : aIJi ¼ 2i
ð2:7Þ
Let N Nð0; 1Þ, so EN2i ¼ 1 3 5 ð2i1Þ ¼ ð2iÞ!2i =i!. Theorem 2.1. Suppose (2.1)–(2.7) hold. For k Z1, set X IJ t IJnk ðhÞ ¼ vI mJ ai ðn1 vm2 Þi :
ð2:8Þ
0riok
We have the following:
bÞ estimates t with bias nk as n-1; (a) t IJnk ðh b Þ is unbiased; (b) If J Z0 and k ¼ J=2 þ 1 then t IJnk ðh b Þ estimates vI mJ with bias (c) Suppose J o 0. Fix 0 o A o1 and set kn ¼ 2½nA =2. Then t IJnkn ðh expfð1AÞnA ln ng, as n-1.
ð2:9Þ
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If J o 0 then t IJn1 ðyÞ converges or diverges as vm2 o or 4n=f . The proof of this theorem is given in Appendix A. We now give some examples. For r Z0, set ðJÞr ¼ JðJ1Þ ðJr þ 1Þ, so ( J!=ðJrÞ! if J Z 0, ðJÞr ¼ ð1Þr ðJ þ r1Þr if J o 0:
b Þ, where Example 2.1. If J 4 0 an unbiased estimator of mJ is AJ ðh X i ðJÞ2i ðf =2þ i1Þ1 AJ ¼ AJ ðhÞ ¼ mJ i ðD=4Þ =i!
ð2:10Þ
0 r 2i r J
and D ¼ fn
1
vm2 . The first few are A2 ¼ m2 ð1 þ Df
A1 ¼ m,
A4 ¼ m4 ð1 þ 6Df
1
1
þ 3D2 f
A3 ¼ m3 ð1 þ 3Df
Þ,
1
1
Þ,
ðf þ 2Þ1 Þ,
A5 ¼ m5 ð1 þ 10Df
1
þ 15D2 f
1
ðf þ2Þ1 Þ,
A6 ¼ m6 ð1 þ 15Df
1
þ 45D2 f
1
ðf þ 2Þ1 þ15D3 f
A7 ¼ m7 ð1 þ 21Df
1
þ 105D2 f
1
1
ðf þ 2Þ1 þ 105D3 f
ðf þ 2Þ1 ðf þ4Þ1 Þ, 1
ðf þ2Þ1 ðf þ 4Þ1 Þ:
b Þ, where Example 2.2. If 0 oJ o f an unbiased estimator of ðv1=2 mÞJ is BJ ðh X i ðJÞ2i g 0 g 1 BJ ¼ BJ ðhÞ ¼ DJ1 J2i ðD=4Þ =i!,
ð2:11Þ
0 r 2i r J
where g j ¼ Gððf jÞ=2Þ, D1 ¼ ðvf =2Þ1=2 m, D ¼ fn ( if j Z 0, ðf =21Þj g 0 g 1 2j ¼ ðf =21jÞ1 if j o 0, j
1
vm2 ¼ 2=ðnD21 Þ. Since
we have B1 ¼ D1 g 0 g 1 1 ,
B2 ¼ D21 ðf =21 þ D=2Þ,
1 B3 ¼ D31 g 0 ðg 1 3 þ 3g 1 D=2Þ,
B4 ¼ D41 fðf =21Þ2 þ 3ðf =21ÞD þ3D2 =4g, 1 1 2 B5 ¼ D51 g 0 ðg 1 5 þ 5g 3 D=4 þ15g 1 D =4Þ,
B6 ¼ D6 fðf =21Þ3 þ 15ðf =21Þ2 D=2þ 45ðf =21ÞD2 =4 þ15D3 =8g, and so on. b Þ of (2.10) with 0 r2ir J replaced by 0 r io kn ¼ ½nA , Example 2.3. If J o 0 an estimator of mJ with bias as in (2.9) is AJ ðh where 0 o A o1. In particular, A1 ¼ m1
kX n 1
i ð2iÞ!ðf =2 þ i1Þ1 i ðD=4Þ =i!
ð2:12Þ
i¼0
and A2 ¼ m2
kX n 1
i ð2i þ1Þ!ðf =2þ i1Þ1 i ðD=4Þ =i!,
i¼0 1
where D ¼ fn
vm2 .
b Þ of (2.11) with 0 r2ir J replaced by Example 2.4. If J o 0 an estimator of ðv1=2 mÞJ with bias as in (2.9) is BJ ðh A m 2m 0 ri okn ¼ ½n , where 0 o Ao 1. In particular, for J¼2m, even, v =m is estimated using B2m ¼ D2m 1
kX n 1
i ð2m þ 2i1Þ2i ðf =2 þi þ m1Þ1 i þ m ðD=4Þ =i!,
i¼0
where D1 ¼ ðvf =2Þ1=2 m, D ¼ fn
1
vm2 ¼ 2=ðnD21 Þ.
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445
b Þ is determined by I, J, f and l ¼ ðv=nÞ1=2 9m9, the coefficient of From Theorem 2.1 it follows that the relative bias of t IJnk ðh variation. 3. Multi-sample case Suppose x i Nðmi ,vi =ni Þ and f i s2i =vi w2f for i ¼ 1; 2, . . . ,p with all variables independent. We wish to estimate i Q J tðhÞ ¼ pi¼ 1 vIi i mi i , where Ii 4 f i =2 and Ji is an integer. If Ji o 0 assume 0 o m0i o 9mi 9, where m0i is known. Set ( x i if J i Z 0, bi ¼ h ¼ ðl,vÞ, m mni if Ji o 0,
mni ¼ x i ð1Ini Þ þ ci Ini , Ini ¼ Ið9x i 9 r m0i Þ, vb i ¼ s2i , where 0 r9ci 9r 1 is arbitrary. For k 2 Np , where N ¼ f0; 1,2, . . .g, set t IJnk ðhÞ ¼
p Y i¼1
IJ
t ni ki ðhi Þ, i i
of (2.8) with hi ¼ ðmi ,vi Þ. Then Theorem 2.1 implies. Theorem 3.1. With the notation set as above, we have the following: b Þ estimates tðhÞ with bias Pp nki ; (a) t IJnk ðh i¼1 i b Þ is unbiased; (b) If J Z 0 and k ¼ J=2þ 1 then t IJnk ðh (c) Set ( 2½nAi i =2, where 0 o Ai o 1 if Ji o 0, ki ¼ if Ji Z 0: Ji =2 þ1 b Þ has bias Then t IJnk ðh 8 9 < X = Ai ð1Ai Þni ln ni exp : J o0 ; i
as min ni -1. Example 3.1. An estimator of m1 =m2 with bias expfð1AÞnA2 ln n2 g as n2 -1 is x 1 Af 2 n2 ðmn2 ,s22 Þ, where Afn ðhÞ ¼ A1 of (2.12). 4. A simulation study Here, we perform simulations to compare the estimators proposed in Section 2 with known ones. We conduct two simulation studies. The R (R Development Core Team, 2011) codes used to perform these simulation studies can be obtained from the corresponding author, email: [email protected] Firstly, we compare the estimator for 1=m given by (2.12) assuming that v is unknown versus those given by (1.1), (1.3) and (1.4) assuming that v is known. We use two criteria for comparing the four estimators: mean squared error and bias. We computed (2.12), (1.1), (1.3) and (1.4) by simulating ten thousand replications of samples of size n from N (1, v) for v ¼0.2, 0.5, 2, 10 and n¼ 10, 20, y, 1000. We take f¼n 1, m0 ¼ 0:5 and c ¼ m0 sign x in (2.12). Note that m0 is half the true value of m. This is not an unreasonable lower bound on m. Even smaller values can be taken for m0 , but they would decrease the efficiency of the proposed estimators. In practice, m0 should be chosen as close as possible to m but not too large that m0 is not a lower bound. The constants g in (1.3) and A in (2.12) for a given v are chosen to minimize the respective variances. For g, we follow the procedure:
select a range of values for g 4 0; for each g, compute 0 12 000 10 000 X x 1 10X x 1 j k @ A En ¼ 10 000 k ¼ 1 x 2k þ gv=n 10 000 j ¼ 1 x 2j þ gv=n for all n¼10, 20, y, 1000, where x k denotes the sample mean for the kth replication;
choose the optimal g as the one that minimizes E10 þ E20 þ þ E1000 .
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C.S. Withers, S. Nadarajah / Journal of Statistical Planning and Inference 143 (2013) 441–455
For A, we follow the procedure:
select a range of values for A 2 ð0; 1Þ; for each A, compute 0 12 000 000 1 10X 1 10X ðkÞ ðjÞ A @ A1 A En ¼ 10 000 k ¼ 1 10 000 j ¼ 1 1
for all n ¼10, 20, y, 1000, where AðkÞ 1 is the A1 given by (2.12) for the kth replication; choose the optimal A as the one that minimizes E10 þ E20 þ þ E1000 .
The same ten thousand replications are used for every g and for every A. The optimal choices given by these procedures allow the estimators, (1.3) and (2.12), to realize their greatest potential. The optimal choices for different v are different. However, the optimal choices of g and A appeared to take values around 1 and 1/4, respectively. Hence, for a fair comparison of the simulation results, we shall choose g ¼1 and A ¼1/4 for all v. The plots of the mean squared error and the bias versus n Z 10 for the selected v and for the four estimators are shown in Figs. 1 and 2. The x-axes are plotted on log scale. The following observations can be drawn from the figures:
1. the mean squared errors generally decrease to zero with increasing n; 2. the smallest mean squared errors for either v¼0.2, 0.5 and all n or v¼2, 10 and large n are by the new estimator given by (2.12); 3. the second smallest mean squared errors for either v¼ 0.2, 0.5 and all n or v ¼2, 10 and large n are by (1.3); 4. the third smallest mean squared errors for either v ¼0.2, 0.5 and all n or v¼ 2, 10 and large n are by (1.4); 5. the largest mean squared errors for either v¼0.2, 0.5 and all n or v ¼2, 10 and large n are by (1.1); 6. the smallest mean squared errors for v ¼2, 10 and small n are by (1.3); 7. the second smallest mean squared errors for v¼ 2, 10 and small n are by (1.4) and the new estimator given by (2.12); 8. the largest mean squared errors for v ¼2, 10 and small n are by (1.1); 9. the mean squared errors generally increase with increasing v; 10. the biases generally approach zero with increasing n; 11. the biases are generally positive for (1.1); 12. the biases are generally negative for (1.3) and the new estimator given by (2.12); 13. the smallest biases for all v and large n are by (1.3), (1.4) and the new estimator given by (2.12); 14. the smallest biases are for all v and small n are by (1.4); 15. the second smallest biases for all v and small n are by (1.3); 16. the third smallest biases for all v and small n are by the new estimator given by (2.12); 17. the largest biases for all v and for all n are by (1.1); 18. the biases generally increase with increasing v. The new estimator gives the best performance with respect to mean squared error for all large n. It also gives the best performance with respect to mean squared error for small v and for all n. The best performance with respect to bias is by (1.4), an unbiased estimator. We noted earlier in Section 1 that the estimator (1.1) has infinite variance. So, the mean squared error of (1.1) is also infinite. The estimates reported in Fig. 1 are finite because they are based on a finite number of replications. The estimates appear fairly stable at least for large n. To explain this, we have plotted the mean squared error of (1.1) versus the number of replications, see Fig. 3. We have taken n ¼20, m ¼ 1 and v ¼10. We can see that the divergence of the mean squared error to infinity is very slow even for one hundred thousand replications. Hence, the mean squared errors for (1.1) reported in Fig. 1 should be treated as underestimates. A refined estimator for 1=m based on (2.12) is 8 > < 2 A1 > :2
if A1 o2, if 2r A1 r 2, if A1 42:
This estimator suggested by a referee – for whom we are most grateful – will have smaller mean squared errors than (2.12). We hope to follow up this estimator in a future study. pffiffiffi The second simulation study is to compare the estimators for v=m given by Example 2.4, (1.5), (1.6) and (1.7), assuming that both m and v are unknown. We conducted the simulation as in the first study with f ¼ n1, m0 ¼ 0:5, c ¼ m0
C.S. Withers, S. Nadarajah / Journal of Statistical Planning and Inference 143 (2013) 441–455
v = 0.2
0.025
v = 0.5
0.08
0.015
Mean Squared Error
Mean Squared Error
0.020
0.010
0.06
0.04
0.005
0.02
0.00
0.00 10
20
50
100 n
200
500
1000
10
20
50
v= 2 1.0
0.4
0.8
0.3
0.2
0.0
0.0 50
100 n
500
1000
200
500
1000
0.4
0.2
20
200
0.6
0.1
10
100 n v = 10
0.5
Mean Squared Error
Mean Squared Error
447
200
500
1000
10
20
50
100 n
Fig. 1. Mean squared error of the estimators of 1=m given by (1.1) (in black), (1.3) (in brown), (1.4) (in blue) and (2.12) (in red) for m ¼ 1 and v¼ 0.2, 0.5, 2, 10. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this article.)
sign x, v ¼0.2, 0.5, 2, 10 and n ¼10, 20,y,1000. The constants g in (1.6) and A in Example 2.4 for a given v are chosen similarly to the first simulation study. For g, we follow the procedure:
select a range of values for g 4 0; for each g, compute 0 12 000 10 000 X s x 1 10X s x 1 j j k k @ A En ¼ 10 000 k ¼ 1 x 2k þ gs2 =n 10 000 j ¼ 1 x 2j þ gs2 =n k
j
for all n ¼10, 20,y,1000, where x k denotes the sample mean for the kth replication and sk denotes the sample standard deviation for the kth replication; choose the optimal g as the one that minimizes E10 þ E20 þ þ E1000 .
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C.S. Withers, S. Nadarajah / Journal of Statistical Planning and Inference 143 (2013) 441–455
v = 0.2
v = 0.5
0.06
0.020
0.04 Bias
Bias
0.015
0.010 0.02
0.005 0.00 0.000
10
20
50
100 n
200
500
1000
10
20
50
v= 2
100 n
200
500
1000
v = 10
0
0.4
−2
Bias
Bias
0.2
0.0
−4
−0.2
−6
10
20
50
100 n
200
500
1000
10
20
50
100 n
200
500
1000
Fig. 2. Bias of the estimators of 1=m given by (1.1) (in black), (1.3) (in brown), (1.4) (in blue) and (2.12) (in red) for m ¼ 1 and v¼0.2, 0.5, 2, 10. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this article.)
For A, we follow the procedure:
select a range of values for A 2 ð0; 1Þ; for each A, compute 0 12 000 10 000 X 1 10X 1 ðkÞ ðjÞ @B B A En ¼ 1 10 000 k ¼ 1 10 000 j ¼ 1 1 for all n ¼10, 20, y, 1000, where BðkÞ 1 is the B1 given by Example 2.4 for the kth replication;
choose the optimal A as the one that minimizes E10 þ E20 þ þ E1000 . The optimal choices given by these procedures are also different for different v. Again the optimal choices of g and A appeared to take values around 1 and 1/4, respectively, so we shall take g ¼1 and A¼ 1/4 for all v.
C.S. Withers, S. Nadarajah / Journal of Statistical Planning and Inference 143 (2013) 441–455
449
Mean Squared Error
10000 8000 6000 4000 2000 0 0e+00
2e+04
4e+04 6e+04 Number of Replications
8e+04
1e+05
Fig. 3. Mean squared error of (1.1) versus the number of replications for n ¼20, m ¼ 1 and v¼10.
v = 0.2
v = 0.5
0.015
Mean Squared Error
Mean Squared Error
0.06
0.010
0.005
0.04
0.02
0.00
0.000 10
20
50
100
200
500
1000
10
20
50
0.8
4
0.6
0.4
0.0
0 100 n
1000
200
500
1000
2
1
50
500
3
0.2
20
200
v = 10 5
Mean Squared Error
Mean Squared Error
v= 2 1.0
10
100 n
n
200
500
1000
10
20
50
100 n
pffiffiffi Fig. 4. Mean squared error of the estimators of v=m given by (1.5) (in black), (1.6) (in brown), (1.7) (in blue) and Example 2.4 (in red) for m ¼ 1 and v¼ 0.2, 0.5, 2, 10. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this article.)
450
C.S. Withers, S. Nadarajah / Journal of Statistical Planning and Inference 143 (2013) 441–455
The plots of the mean squared error and the bias versus n Z 10 for the selected v and for the four estimators are shown in Figs. 4 and 5. Again the x-axes are plotted on log scale. The observations from these figures are the same as those from Figs. 1 and 2 except that 1. 2. 3. 4. 5. 6. 7.
the smallest mean squared errors for v ¼0.2 and small n are by (1.6) and (1.7); the second smallest mean squared errors for v ¼0.2 and small n are by the new estimator given by Example 2.4; the smallest mean squared errors for v ¼0.4 and small n are by (1.6) and the new estimator given by Example 2.4; the second smallest mean squared errors for v ¼0.4 and small n are by (1.7); the largest mean squared errors for v ¼0.2, 0.4 and small n are by (1.5); the smallest mean squared errors for v ¼2, 10 and small n are by (1.6); the second smallest mean squared errors for v ¼2, 10 and small n are by (1.7) and the new estimator given by Example 2.4; 8. the largest mean squared errors for v ¼2, 10 and small n are by (1.5); 9. the smallest mean squared errors for all large n are by the new estimator given by Example 2.4; 10. the second smallest mean squared errors for all large n are by (1.6); v = 0.2
v = 0.5 0.02
0.000
0.01
Bias
Bias
−0.004 0.00
−0.01
−0.008
−0.02 −0.012 −0.03 10
20
50
100 n
200
500
1000
10
20
50
v= 2
100 n
200
500
1000
200
500
1000
v = 10 4
0.2 0.1
2
−0.1
Bias
Bias
0.0 0
−0.2 −2 −0.3 −0.4
−4
10
20
50
100 n
200
500
1000
10
20
50
100 n
pffiffiffi Fig. 5. Bias of the estimators of v=m given by (1.5) (in black), (1.6) (in brown), (1.7) (in blue) and Example 2.4 (in red) for m ¼ 1 and v¼ 0.2, 0.5, 2, 10. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this article.)
C.S. Withers, S. Nadarajah / Journal of Statistical Planning and Inference 143 (2013) 441–455
11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23.
the the the the the the the the the the the the the
451
third smallest mean squared errors for all large n are by (1.7); largest mean squared errors for all large n are by (1.5); biases are generally negative for (1.6) and (1.7); smallest biases for v ¼0.2, 0.5 and for all n are by the new estimator given by Example 2.4; second smallest biases for v ¼0.2, 0.5 and for all n are by (1.5); third smallest biases for v ¼0.2, 0.5 and for all n are by (1.7); largest biases for v ¼0.2, 0.5 and for all n are by (1.6); smallest biases for v ¼2, 10 and small n are by (1.7); second smallest biases for v ¼2, 10 and small n are by (1.5); third smallest biases for v ¼2, 10 and small n are by (1.6); largest biases for v ¼2, 10 and small n are by the new estimator given by Example 2.4; smallest biases for v ¼2, 10 and large n are by (1.6) and (1.7), and the new estimator given by Example 2.4; largest biases for v ¼2, 10 and large n are by (1.5).
The new estimator again gives the best performance with respect to mean squared error for large n. The best performance with respect to bias for small v and all n is by the new estimator. The best performance with respect to bias for large v and large n is by (1.6) and (1.7), and the new estimator. In summary, the estimators given by Theorem 2.1 show the best performance with respect to mean squared error for all large n. With respect to bias, the estimators in Theorem 2.1 give the best performance at least for all large n. They also give the smallest biases for some small n. Finally, we like to make a point about the procedures for choosing optimal g and A. Normally, the practitioner will not have the luxury of doing the required simulation for these procedures, even if there is some model that describes the observed data. It would be ideal to have some simple expressions for g or A based only on n, x, and s. Such expressions could be obtained, for example, by minimizing the first few terms of an asymptotic expansion for the mean squared error. In this paper, we have not studied asymptotic properties of the mean squared error of the estimators. Such a study will require substantially more work, so a topic for future work.
Acknowledgments The authors would like to thank the Editor and the two referees for careful reading and for their comments which greatly improved the paper. Appendix A. Proof of Theorem 2.1 p
We precede the proof with some lemmas. Set d ¼ m0 =9m9 2 ð0; 1Þ, d ¼ m1 ðv=nÞ1=2 and 9X9p ¼ ðE9X9 Þ1=p . Lemma A.1. We have PðIn ¼ 1Þ n1=2 expðlnÞ as n-1, where l ¼ ð1dÞ2 m2 =2v. b =m1Þ=d Nð0; 1Þ and a 7 ¼ ð1 7 dÞ=9d9. Then PðIn ¼ 1Þ ¼ Pð91 þ dN9 o dÞ ¼ Fða ÞFða þ Þ fða Þ= Proof. Set N ¼ ðm a n1=2 expðlnÞ, where fðÞ is the derivative of FðÞ. & Lemma A.2. For J and k non-negative integers ð1xÞJ ¼ AJk ðxÞ þBJk ðxÞ, where AJk ðxÞ ¼
k1 X
J j
i¼0
! ðxÞj ,
and BJk ðxÞ ¼
J1 X
J1 þ k
j¼0
jþk
! xj þ k ð1xÞ1j :
Proof. Differentiate ð1xÞ1 ¼
p1 X j¼0
J times.
&
xj þxp ð1xÞ1
452
C.S. Withers, S. Nadarajah / Journal of Statistical Planning and Inference 143 (2013) 441–455 1
Lemma A.3. If x ¼ dN, Q ¼ J1 þk and z ¼ 21=2 d 9ð1In ÞBJk ðxÞ91
¼ 2d
1
9d9 r 1=2 then
1=2 k
9 =ðk=2Þ!:
Q !9d2
ðA:1Þ
Proof. The left hand side of (A.1) is less than or equal to ! J1 X Q jþk 1j E9x9 : d j þk j¼0
Note that Q i i i E9x9 ¼ Q !9d21=2 9 =fðQ iÞ!ði=2Þ!g r Q !9d21=2 9 =ðk=2Þ!, i for iZ k. So, the left hand side of (A.1) is less than or equal to 1
Q !d
ðk=2Þ!1 9d21=2 9
k
J1 X
zj ,
j¼0
which is less than or equal to the right hand side of (A.1).
&
Lemma A.4. Assume (2.5). For j Z 0 set J EN2j : K Jj ¼ vj mJ2j nJj , nJj ¼ 2j Then for J Z 0, k1 X
Ex J ¼
( nj K Jj þ
j¼0
Oðnk Þ 0
if k1 oJ=2, if k1 ZJ=2,
ðA:2Þ
and b Em
J
¼
k1 X
nj K J þ Enk , j
ðA:3Þ
j¼0
where
Enk ¼ mJ Efð1In ÞBJ2k ðxÞ þ In C J2k ðxÞg ¼ Oðnk Þ and C J2k ðxÞ ¼ cJ AJ2k ðxÞ. Proof. Note that x ¼ mð1 þ dNÞ, so (A.2) follows by the binomial expansion. Note that b J ¼ x J ð1In Þ þcJ In ¼ mJ fAJ2k ðxÞ þ ð1In ÞBJ2k ðxÞ þIn C J2k ðxÞg ¼ LA þ LB þ Lc m say, where x ¼ dN. Note that mJ EAJ2k ðxÞ is the first term on the right hand side of (A.3). Also ELB nk by Lemma A.3. Fix 1 o p o1, 1=p þ1=q ¼ 1. Then 9ELC 9 r 9In 9p 9C J2k ðxÞ9q n1=2 expðln=pÞ by Lemma A.1. & Proof of Theorem 2.1. For J Z 0, ( k1 X Oðnk Þ 2I J j I J Es x ¼ n lI v K j þ 0 j¼0
if k1o J=2, if k1Z J=2,
and Es2I x J ¼
k1 X
nj lI vI K J þ Oðnk Þ, j
j¼0
so (a) and (b) follow upon substitution. Put J 0 ¼ J. Then k1 3 X X J bÞ ¼ EtIJnk ðh Akli , EN 2i ðnÞi vI þ i Eðmn ÞJ2i ¼ t 2i i¼0 i¼1 corresponding to 0
ðmn ÞJ ¼ mJ
0
3 X
AJ0li ,
i¼1
equalling the right hand side of (A.3) with ðJ,kÞ ¼ ðJ 0 ,2lÞ.
C.S. Withers, S. Nadarajah / Journal of Statistical Planning and Inference 143 (2013) 441–455
453
2
Choose l ¼ 2½nA =2. Set gm ¼ ðJ0 þ 2m1Þ!ðJ 0 1Þ!1 ðd =2Þm =m!. Then kX þ l2 l1 kX þ l2 X X m Akl1 ¼ gm ð1Þi gm C mk þ gm bmlk , ¼ 1þ i i¼0 k l where the inner sum is over 0 ri r m, i o k and mi o l, k1 X m C mk ¼ ð1Þi i i¼0 is bounded by
m k1
,
k1 X
bmlk ¼
ð1Þi
i ¼ ml þ 1
m i
¼ bm,ml þ 1 þb0m,mk þ 1 ,
and bmN ¼
X0
ð1Þi
N
00 mN X m ð1Þi , b0mN ¼ , i i
m
P P sums to [m/2] halving the last term if m is even, 00 sums from m/2 if m is even, halving the last term, and 00 where sums from (mþ1)/2 if m is odd. m So, b0mN ¼ ð1Þm bmN . For m ¼2M, 0 r bmN r m , so 0 rb r m þ k1 . For m¼2M þ1, 0 rð1ÞM bmN r mNMþ M , so l1 þ MN k1 þ M mlkl1 þl1 bmlk ¼ bm,ml þ 1 bm,mk þ 1 is bounded by þ 2 M M . So, M M kX þ l2 KX þ L2 m 2M l1 þM 9Akl1 19 r gm g2M ¼ S1 þS2 þS3 þ þ2g2M þ 1 k1 l1 M L k P0
say, where K ¼ k=2 and L ¼ l=2. 2 Set r n ¼ d =2. Then gM fr n 4 expð1Þmgm mJ1 gm ðk þ l2ÞJ1 g ¼ rn 4 expð1Þðkþ l2Þ. So, þ l2 k þl2 k X S1 gm ðk þlÞk þ l þ J1:5 k0:5k l0:5l gk T, k1 k
for
m r k þl2
by
Stirling’s
formula,
where
where l2 X
T¼
gi 1,
i¼0
as g-0. So, ln S1 r constant kln nð1AÞð1 þoð1ÞÞ, so S1 behaves as nA1 . Also þ L2 k þl4 K X g2M ðkþ l4ÞJ1 S1 S2 l1 L and S3
KX þ L2
r n g2M ðk þ l4ÞJ lrn S2 S2 ,
L
so Akl1 1 behaves as nA1 . By Lemma A.3, ! k1 X J 0 2 9Akl2 9 ¼ EN2i ðd Þi Eð1In ÞBJ02lþ 2i ðxÞ expfl ln nð1AÞð1 þ oð1ÞÞg, 2i i¼0 by Stirling’s formula. Note that the right hand side of (A.4) behaves (2.9). Also for 1 o p o1 and 1=p þ 1=q ¼ 1, ! ( ) J0 þ 2k1 J 0 J0 J0 2k2 9EIn C 2k ðxÞ9 r 9In 9p 9C 2k ðxÞ9q r9In 9p 9C9 þ 2 9N 9q J0 1 for d r 1=2 since Akl3 ¼
k1 X i¼0
L þ j L
J0 2i
and 9Nj 9q increase with j. So, ! 2
EN 2i ðd Þi EIn C J02lþ 2i ðxÞ
ðA:4Þ
454
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is bounded by 9In 9p
J0 þ 2k2 J 0 1
! EN
( 2k2
29C9
J 0
þ 49N
2l2
9q
J 0 þ 2k þ2l3 J0 1
!) :
2
For d r 1=2 and d rc2 =2, this bound behaves as n1=2 expfnlg by Lemma A.1 using 9Nj 9q fqj expð1Þg5=2 and
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