多个偏正态总体共同位置参数的Bootstrap置信区间

摘要/Abstract

摘要：

该文基于Bootstrap方法研究多个偏正态总体共同位置参数的区间估计和假设检验问题.首先，分别给出未知参数的矩估计和极大似然估计.其次，将徐礼文^[1]对多个正态总体共同均值的探讨推广到多个偏正态总体，进而构造共同位置参数的Bootstrap置信区间和Bootstrap检验统计量. Monte Carlo模拟结果表明，无论是两个总体、三个总体还是五个总体，基于矩估计和惩罚极大似然估计的Bootstrap置信区间在覆盖概率意义下优于其他四种Bootstrap置信区间.最后，将上述方法应用于地区生产总值和生物利用度数据的案例分析，以验证该文所给方法的合理性和有效性.

关键词: 偏正态总体, 共同位置参数, 矩估计, 极大似然估计, Bootstrap置信区间

Abstract:

In this paper, we consider the interval estimation and hypothesis testing problems for the common location parameter of several skew-normal populations when the scale parameters and skewness parameters are unknown. Firstly, we estimate the unknown parameters using the methods of moment estimation and maximum likelihood estimation. Secondly, the Bootstrap confidence intervals and Bootstrap test statistics are constructed, which generalize the results given by Xu [1] under several normal populations. Thirdly, the Monte Carlo simulation results indicate that the Bootstrap confidence intervals based on the methods of moment estimation and maximum penalized likelihood estimation perform better than other four confidence intervals. Finally, the above approaches are applied to the real data examples of regional gross domestic product of China and bioavailability in order to verify the reasonableness and effectiveness of the proposed approaches.

Key words: Skew-normal population, Common location parameter, Moment estimation, Maximum likelihood estimation, Bootstrap confidence interval

中图分类号:

O212

叶仁道,王仲池,罗堃,林雅. 多个偏正态总体共同位置参数的Bootstrap置信区间[J]. 数学物理学报, 2021, 41(1): 194-216.

Rendao Ye,Zhongchi Wang,Kun Luo,Ya Lin. Bootstrap Confidence Intervals for the Common Location Parameter of Several Skew-Normal Populations[J]. Acta mathematica scientia,Series A, 2021, 41(1): 194-216.

图/表 8

表 1

两个总体时六种Bootstrap置信区间的模拟覆盖概率"

(λ₁, λ₂) = (3, 4)
	N1						N2
	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm
$\eta _{1}^{2}$	0.8616	0.8552	0.8740	0.9488	0.9532	0.9460	0.9012	0.8984	0.8980	0.9500	0.9540	0.9548
$\eta _{2}^{2}$	0.8636	0.8564	0.8748	0.9492	0.9528	0.9464	0.9008	0.8968	0.8980	0.9508	0.9536	0.9548
$\eta _{3}^{2}$	0.8636	0.8564	0.8752	0.9492	0.9524	0.9472	0.9012	0.8948	0.8988	0.9508	0.9536	0.9548
$\eta _{4}^{2}$	0.8616	0.8552	0.8740	0.9488	0.9532	0.9460	0.9012	0.8984	0.8980	0.9500	0.9540	0.9548
$\eta _{5}^{2}$	0.8612	0.8552	0.8736	0.9484	0.9532	0.9460	0.9008	0.9000	0.8980	0.9500	0.9548	0.9536
	N3						N4
	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm
$\eta _{1}^{2}$	0.9388	0.9224	0.9248	0.9464	0.9544	0.9588	0.9260	0.9324	0.9396	0.9492	0.9544	0.9532
$\eta _{2}^{2}$	0.9392	0.9196	0.9256	0.9464	0.9536	0.9600	0.9260	0.9292	0.9400	0.9492	0.9532	0.9532
$\eta _{3}^{2}$	0.9396	0.9196	0.9260	0.9464	0.9536	0.9600	0.9260	0.9284	0.9404	0.9488	0.9528	0.9532
$\eta _{4}^{2}$	0.9388	0.9224	0.9248	0.9464	0.9544	0.9588	0.9260	0.9324	0.9396	0.9492	0.9544	0.9532
$\eta _{5}^{2}$	0.9388	0.9224	0.9252	0.9464	0.9548	0.9588	0.9260	0.9332	0.9396	0.9492	0.9544	0.9532
	N5						N6
	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm
$\eta _{1}^{2}$	0.9236	0.9376	0.9404	0.9412	0.9492	0.9520	0.9284	0.9336	0.9400	0.9472	0.9536	0.9576
$\eta _{2}^{2}$	0.9240	0.9348	0.9404	0.9408	0.9472	0.9512	0.9288	0.9300	0.9408	0.9468	0.9536	0.9576
$\eta _{3}^{2}$	0.9244	0.9336	0.9404	0.9408	0.9472	0.9512	0.9288	0.9292	0.9408	0.9472	0.9528	0.9576
$\eta _{4}^{2}$	0.9236	0.9376	0.9404	0.9412	0.9492	0.9520	0.9284	0.9336	0.9400	0.9472	0.9536	0.9576
$\eta _{5}^{2}$	0.9228	0.9392	0.9404	0.9412	0.9496	0.9520	0.9284	0.9348	0.9400	0.9472	0.9540	0.9576
(λ₁, λ₂) = (5, 6)
	N1						N2
	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm
$\eta _{1}^{2}$	0.9312	0.9312	0.9400	0.9652	0.9688	0.9656	0.9548	0.9532	0.9584	0.9656	0.9684	0.9724
$\eta _{2}^{2}$	0.9320	0.9312	0.9408	0.9652	0.9688	0.9660	0.9556	0.9504	0.9592	0.9656	0.9680	0.9724
$\eta _{3}^{2}$	0.9328	0.9304	0.9408	0.9652	0.9688	0.9660	0.9556	0.9496	0.9596	0.9656	0.9680	0.9728
$\eta _{4}^{2}$	0.9312	0.9312	0.9400	0.9652	0.9688	0.9656	0.9548	0.9532	0.9584	0.9656	0.9684	0.9724
$\eta _{5}^{2}$	0.9312	0.9324	0.9400	0.9652	0.9688	0.9656	0.9548	0.9536	0.9580	0.9660	0.9684	0.9720
	N3						N4
	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm
$\eta _{1}^{2}$	0.9680	0.9572	0.9660	0.9612	0.9652	0.9724	0.9584	0.9584	0.9680	0.9648	0.9696	0.9716
$\eta _{2}^{2}$	0.9680	0.9548	0.9660	0.9620	0.9640	0.9724	0.9588	0.9552	0.9680	0.9648	0.9684	0.9716
$\eta _{3}^{2}$	0.9676	0.9544	0.9664	0.9616	0.9640	0.9724	0.9588	0.9544	0.9680	0.9648	0.9684	0.9716
$\eta _{4}^{2}$	0.9680	0.9572	0.9660	0.9612	0.9652	0.9724	0.9584	0.9584	0.9680	0.9648	0.9696	0.9716
$\eta _{5}^{2}$	0.9676	0.9592	0.9656	0.9612	0.9652	0.9728	0.9576	0.9588	0.9680	0.9648	0.9704	0.9716
	N5						N6
	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm
$\eta _{1}^{2}$	0.9532	0.9668	0.9680	0.9620	0.9676	0.9724	0.9564	0.9560	0.9668	0.9588	0.9708	0.9692
$\eta _{2}^{2}$	0.9524	0.9640	0.9680	0.9616	0.9672	0.9724	0.9568	0.9524	0.9664	0.9588	0.9688	0.9692
$\eta _{3}^{2}$	0.9524	0.9636	0.9680	0.9616	0.9672	0.9728	0.9564	0.9520	0.9668	0.9588	0.9680	0.9692
$\eta _{4}^{2}$	0.9532	0.9668	0.9680	0.9620	0.9676	0.9724	0.9564	0.9560	0.9668	0.9588	0.9708	0.9692
$\eta _{5}^{2}$	0.9536	0.9672	0.9676	0.9620	0.9684	0.9720	0.9564	0.9580	0.9672	0.9588	0.9716	0.9692
(λ₁, λ₂) = (8, 9)
	N1						N2
	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm
$\eta _{1}^{2}$	0.9548	0.9564	0.9620	0.9736	0.9752	0.9720	0.9680	0.9664	0.9720	0.9724	0.9744	0.9776
$\eta _{2}^{2}$	0.9548	0.9556	0.9628	0.9736	0.9752	0.9716	0.9684	0.9656	0.9728	0.9724	0.9744	0.9776
$\eta _{3}^{2}$	0.9548	0.9552	0.9628	0.9736	0.9752	0.9716	0.9684	0.9656	0.9736	0.9724	0.9744	0.9780
$\eta _{4}^{2}$	0.9548	0.9564	0.9620	0.9736	0.9752	0.9720	0.9680	0.9664	0.9720	0.9724	0.9744	0.9776
$\eta _{5}^{2}$	0.9548	0.9576	0.9624	0.9736	0.9756	0.9720	0.9680	0.9664	0.9720	0.9724	0.9748	0.9776
	N3						N4
	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm
$\eta _{1}^{2}$	0.9764	0.9676	0.9764	0.9668	0.9708	0.9776	0.9716	0.9680	0.9772	0.9704	0.9764	0.9796
$\eta _{2}^{2}$	0.9764	0.9660	0.9764	0.9672	0.9704	0.9772	0.9720	0.9648	0.9776	0.9700	0.9756	0.9800
$\eta _{3}^{2}$	0.9764	0.9656	0.9764	0.9672	0.9704	0.9772	0.9716	0.9640	0.9776	0.9700	0.9752	0.9800
$\eta _{4}^{2}$	0.9764	0.9676	0.9764	0.9668	0.9708	0.9776	0.9716	0.9680	0.9772	0.9704	0.9764	0.9796
$\eta _{5}^{2}$	0.9760	0.9684	0.9764	0.9668	0.9716	0.9776	0.9704	0.9696	0.9772	0.9704	0.9768	0.9796
	N5						N6
	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm
$\eta _{1}^{2}$	0.9692	0.9748	0.9780	0.9704	0.9748	0.9772	0.9692	0.9772	0.9836	0.9684	0.9788	0.9736
$\eta _{2}^{2}$	0.9696	0.9732	0.9784	0.9700	0.9736	0.9772	0.9656	0.9760	0.9876	0.9680	0.9768	0.9736
$\eta _{3}^{2}$	0.9696	0.9732	0.9784	0.9704	0.9736	0.9772	0.9656	0.9764	0.9809	0.9680	0.9768	0.9736
$\eta _{4}^{2}$	0.9692	0.9748	0.9780	0.9704	0.9748	0.9772	0.9692	0.9772	0.9863	0.9684	0.9788	0.9736
$\eta _{5}^{2}$	0.9696	0.9756	0.9784	0.9704	0.9756	0.9772	0.9700	0.9772	0.9868	0.9684	0.9800	0.9732

表 1

表 2

三个总体时六种Bootstrap置信区间的模拟覆盖概率"

(λ₁, λ₂, λ₃) = (3, 3, 4)
	N1						N2
	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm
$\eta _{1}^{2}$	0.8655	0.8396	0.8668	0.9330	0.9144	0.9428	0.9085	0.8744	0.9080	0.9580	0.9388	0.9620
$\eta _{2}^{2}$	0.8675	0.8380	0.8668	0.9335	0.9180	0.9424	0.9080	0.8724	0.9076	0.9565	0.9432	0.9608
$\eta _{3}^{2}$	0.8685	0.8348	0.8668	0.9310	0.9216	0.9408	0.9065	0.8708	0.9084	0.9560	0.9480	0.9620
$\eta _{4}^{2}$	0.8690	0.8340	0.8672	0.9320	0.9248	0.9396	0.9080	0.8708	0.9104	0.9550	0.9492	0.9620
$\eta _{5}^{2}$	0.8690	0.8336	0.8676	0.9315	0.9228	0.9412	0.9060	0.8712	0.9088	0.9560	0.9484	0.9620
	N3						N4
	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm
$\eta _{1}^{2}$	0.9230	0.8804	0.9324	0.9580	0.9320	0.9624	0.9400	0.8988	0.9416	0.9615	0.9336	0.9664
$\eta _{2}^{2}$	0.9205	0.8736	0.9344	0.9570	0.9380	0.9624	0.9415	0.8976	0.9412	0.9605	0.9352	0.9664
$\eta _{3}^{2}$	0.9275	0.8708	0.9340	0.9560	0.9436	0.9624	0.9420	0.8964	0.9420	0.9605	0.9436	0.9676
$\eta _{4}^{2}$	0.9295	0.8696	0.9340	0.9545	0.9464	0.9620	0.9445	0.8944	0.9432	0.9590	0.9484	0.9672
$\eta _{5}^{2}$	0.9275	0.8720	0.9336	0.9560	0.9448	0.9620	0.9435	0.8964	0.9428	0.9605	0.9452	0.9672
	N5						N6
	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm
$\eta _{1}^{2}$	0.9415	0.8904	0.9448	0.9345	0.9000	0.9456	0.9450	0.8752	0.9420	0.9380	0.8924	0.9448
$\eta _{2}^{2}$	0.9430	0.8844	0.9452	0.9335	0.9088	0.9448	0.9450	0.8684	0.9420	0.9385	0.9016	0.9452
$\eta _{3}^{2}$	0.9450	0.8772	0.9448	0.9335	0.9200	0.9456	0.9465	0.8604	0.9424	0.9400	0.9112	0.9452
$\eta _{4}^{2}$	0.9440	0.8740	0.9448	0.9340	0.9252	0.9452	0.9445	0.8528	0.9424	0.9410	0.9188	0.9448
$\eta _{5}^{2}$	0.9445	0.8768	0.9440	0.9335	0.9228	0.9452	0.9460	0.8588	0.9424	0.9405	0.9136	0.9448
(λ₁, λ₂, λ₃) = (5, 5, 6)
	N1						N2
	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm
$\eta _{1}^{2}$	0.9410	0.9036	0.9444	0.9595	0.9388	0.9664	0.9550	0.9128	0.9588	0.9670	0.9540	0.9756
$\eta _{2}^{2}$	0.9415	0.9004	0.9440	0.9595	0.9436	0.9664	0.9560	0.9072	0.9596	0.9665	0.9576	0.9756
$\eta _{3}^{2}$	0.9415	0.8936	0.9448	0.9590	0.9476	0.9656	0.9580	0.9016	0.9608	0.9670	0.9616	0.9752
$\eta _{4}^{2}$	0.9395	0.8892	0.9452	0.9580	0.9516	0.9644	0.9565	0.8948	0.9616	0.9670	0.9620	0.9744
$\eta _{5}^{2}$	0.9415	0.8920	0.9444	0.9590	0.9484	0.9652	0.9575	0.9016	0.9612	0.9670	0.9616	0.9752
	N3						N4
	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm
$\eta _{1}^{2}$	0.9630	0.8964	0.9704	0.9735	0.9432	0.9796	0.9650	0.8976	0.9704	0.9735	0.9444	0.9796
$\eta _{2}^{2}$	0.9635	0.8852	0.9704	0.9725	0.9492	0.9800	0.9655	0.8904	0.9688	0.9740	0.9516	0.9796
$\eta _{3}^{2}$	0.9650	0.8672	0.9704	0.9730	0.9576	0.9804	0.9655	0.8760	0.9696	0.9745	0.9588	0.9792
$\eta _{4}^{2}$	0.9625	0.8608	0.9692	0.9720	0.9620	0.9800	0.9650	0.8660	0.9708	0.9745	0.9628	0.9784
$\eta _{5}^{2}$	0.9645	0.8652	0.9692	0.9730	0.9604	0.9800	0.9650	0.8744	0.9692	0.9745	0.9604	0.9788
	N5						N6
	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm
$\eta _{1}^{2}$	0.9665	0.8680	0.9716	0.9520	0.9200	0.9696	0.9595	0.8388	0.9672	0.9595	0.9060	0.9660
$\eta _{2}^{2}$	0.9650	0.8536	0.9724	0.9510	0.9292	0.9696	0.9595	0.8196	0.9688	0.9585	0.9192	0.9660
$\eta _{3}^{2}$	0.9650	0.8336	0.9712	0.9515	0.9364	0.9688	0.9620	0.7944	0.9688	0.9585	0.9312	0.9656
$\eta _{4}^{2}$	0.9660	0.8248	0.9688	0.9520	0.9404	0.9688	0.9625	0.7784	0.9676	0.9585	0.9380	0.9668
$\eta _{5}^{2}$	0.9655	0.8312	0.9700	0.9520	0.9380	0.9692	0.9620	0.7900	0.9684	0.9585	0.9320	0.9664
(λ₁, λ₂, λ₃) = (6, 8, 9)
	N1						N2
	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm
$\eta _{1}^{2}$	0.9510	0.9116	0.9568	0.9650	0.9484	0.9728	0.9625	0.9184	0.9656	0.9765	0.9620	0.9816
$\eta _{2}^{2}$	0.9510	0.9104	0.9560	0.9650	0.9524	0.9732	0.9620	0.9108	0.9656	0.9755	0.9628	0.9812
$\eta _{3}^{2}$	0.9525	0.9016	0.9552	0.9650	0.9572	0.9728	0.9640	0.9060	0.9660	0.9735	0.9660	0.9816
$\eta _{4}^{2}$	0.9510	0.8952	0.9548	0.9655	0.9612	0.9724	0.9640	0.9000	0.9664	0.9730	0.9684	0.9800
$\eta _{5}^{2}$	0.9530	0.9008	0.9548	0.9650	0.9588	0.9728	0.9645	0.9048	0.9664	0.9730	0.9660	0.9816
	N3						N4
	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm
$\eta _{1}^{2}$	0.9685	0.8964	0.9748	0.9780	0.9568	0.9820	0.9705	0.9036	0.9744	0.9785	0.9560	0.9828
$\eta _{2}^{2}$	0.9695	0.8868	0.9756	0.9785	0.9604	0.9824	0.9705	0.8928	0.9740	0.9785	0.9616	0.9824
$\eta _{3}^{2}$	0.9695	0.8660	0.9756	0.9775	0.9672	0.9820	0.9705	0.8744	0.9744	0.9780	0.9664	0.9824
$\eta _{4}^{2}$	0.9695	0.8560	0.9760	0.9775	0.9704	0.9820	0.9695	0.8652	0.9756	0.9780	0.9704	0.9824
$\eta _{5}^{2}$	0.9695	0.8644	0.9756	0.9775	0.9668	0.9820	0.9705	0.8728	0.9740	0.9780	0.9672	0.9828
	N5						N6
	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm
$\eta _{1}^{2}$	0.9720	0.8676	0.9776	0.9630	0.9324	0.9768	0.9665	0.8388	0.9736	0.9690	0.9224	0.9716
$\eta _{2}^{2}$	0.9715	0.8516	0.9772	0.9630	0.9372	0.9768	0.9660	0.8068	0.9744	0.9690	0.9320	0.9708
$\eta _{3}^{2}$	0.9705	0.8300	0.9776	0.9610	0.9468	0.9764	0.9680	0.7800	0.9732	0.9695	0.9428	0.9708
$\eta _{4}^{2}$	0.9700	0.8156	0.9752	0.9620	0.9520	0.9768	0.9670	0.7604	0.9724	0.9685	0.9492	0.9696
$\eta _{5}^{2}$	0.9700	0.8272	0.9772	0.9610	0.9476	0.9756	0.9675	0.7748	0.9728	0.9695	0.9452	0.9704

表 2

表 3

五个总体时六种Bootstrap置信区间的模拟覆盖概率"

(λ₁, λ₂, λ₃, λ₄, λ₅) = (3, 3, 4, 4, 5)
	N1						N2
	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm
$\eta _{1}^{2}$	0.8720	0.8332	0.8700	0.9600	0.9280	0.9596	0.9210	0.8760	0.9220	0.9655	0.9220	0.9664
$\eta _{2}^{2}$	0.8690	0.8368	0.8664	0.9545	0.9144	0.9560	0.9190	0.8836	0.9192	0.9615	0.9148	0.9656
$\eta _{3}^{2}$	0.8670	0.8412	0.8644	0.9500	0.9088	0.9544	0.9165	0.8868	0.9188	0.9635	0.9112	0.9660
$\eta _{4}^{2}$	0.8655	0.8440	0.8632	0.9525	0.9196	0.9552	0.9175	0.8900	0.9168	0.9640	0.9228	0.9648
$\eta _{5}^{2}$	0.8595	0.8504	0.8644	0.9500	0.9280	0.9516	0.9135	0.8948	0.9144	0.9645	0.9224	0.9636
	N3						N4
	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm
$\eta _{1}^{2}$	0.9300	0.8784	0.9352	0.9630	0.9132	0.9728	0.9405	0.8888	0.9400	0.9535	0.9016	0.9608
$\eta _{2}^{2}$	0.9290	0.8896	0.9324	0.9650	0.9056	0.9712	0.9390	0.8988	0.9388	0.9545	0.8796	0.9624
$\eta _{3}^{2}$	0.9270	0.8964	0.9316	0.9655	0.8936	0.9700	0.9395	0.9032	0.9396	0.9625	0.8632	0.9644
$\eta _{4}^{2}$	0.9260	0.8996	0.9316	0.9635	0.9152	0.9704	0.9390	0.9084	0.9376	0.9615	0.8916	0.9648
$\eta _{5}^{2}$	0.9240	0.9096	0.9296	0.9640	0.9164	0.9712	0.9380	0.9164	0.9344	0.9615	0.8964	0.9652
	N5						N6
	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm
$\eta _{1}^{2}$	0.9480	0.8772	0.9444	0.9660	0.9040	0.9672	0.9450	0.8688	0.9380	0.9630	0.8732	0.9640
$\eta _{2}^{2}$	0.9480	0.8844	0.9440	0.9670	0.8788	0.9700	0.9445	0.8780	0.9376	0.9580	0.8484	0.9620
$\eta _{3}^{2}$	0.9455	0.8976	0.9420	0.9680	0.8588	0.9692	0.9420	0.8944	0.9376	0.9565	0.8328	0.9572
$\eta _{4}^{2}$	0.9445	0.9048	0.9416	0.9685	0.8896	0.9692	0.9410	0.9016	0.9364	0.9565	0.8708	0.9580
$\eta _{5}^{2}$	0.9450	0.9152	0.9412	0.9640	0.8928	0.9684	0.9410	0.9096	0.9364	0.9590	0.8816	0.9588
(λ₁, λ₂, λ₃, λ₄, λ₅) = (4, 4, 5, 5, 6)
	N1						N2
	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm
$\eta _{1}^{2}$	0.9250	0.8764	0.9248	0.9645	0.9248	0.9668	0.9535	0.8948	0.9568	0.9695	0.9172	0.9740
$\eta _{2}^{2}$	0.9235	0.8808	0.9232	0.9575	0.9096	0.9616	0.9530	0.9016	0.9552	0.9680	0.9088	0.9740
$\eta _{3}^{2}$	0.9220	0.8900	0.9216	0.9565	0.9012	0.9608	0.9505	0.9124	0.9532	0.9710	0.9036	0.9736
$\eta _{4}^{2}$	0.9195	0.8940	0.9212	0.9575	0.9144	0.9604	0.9515	0.9168	0.9516	0.9685	0.9164	0.9720
$\eta _{5}^{2}$	0.9160	0.9016	0.9212	0.9570	0.9188	0.9588	0.9500	0.9252	0.9516	0.9705	0.9188	0.9720
	N3						N4
	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm
$\eta _{1}^{2}$	0.9540	0.8812	0.9608	0.9675	0.9052	0.9756	0.9555	0.8800	0.9596	0.9620	0.8912	0.9720
$\eta _{2}^{2}$	0.9545	0.8968	0.9612	0.9700	0.8960	0.9780	0.9565	0.8924	0.9588	0.9580	0.8700	0.9672
$\eta _{3}^{2}$	0.9525	0.9076	0.9596	0.9710	0.8836	0.9764	0.9545	0.9072	0.9572	0.9645	0.8496	0.9680
$\eta _{4}^{2}$	0.9490	0.9140	0.9568	0.9700	0.9096	0.9772	0.9530	0.9156	0.9568	0.9640	0.8808	0.9704
$\eta _{5}^{2}$	0.9450	0.9288	0.9564	0.9705	0.9076	0.9772	0.9530	0.9272	0.9536	0.9650	0.8880	0.9708
	N5						N6
	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm
$\eta _{1}^{2}$	0.9655	0.8628	0.9640	0.9695	0.8952	0.9692	0.9545	0.8284	0.9536	0.9645	0.8544	0.9684
$\eta _{2}^{2}$	0.9640	0.8712	0.9632	0.9720	0.8648	0.9736	0.9540	0.8524	0.9532	0.9600	0.8264	0.9624
$\eta _{3}^{2}$	0.9625	0.8948	0.9632	0.9715	0.8472	0.9748	0.9535	0.8780	0.9516	0.9580	0.8032	0.9584
$\eta _{4}^{2}$	0.9625	0.9080	0.9620	0.9715	0.8752	0.9748	0.9530	0.8892	0.9528	0.9590	0.8532	0.9596
$\eta _{5}^{2}$	0.9610	0.9256	0.9608	0.9690	0.8792	0.9744	0.9515	0.9064	0.9504	0.9570	0.8632	0.9608
(λ₁, λ₂, λ₃, λ₄, λ₅) = (7, 7, 8, 8, 9)
	N1						N2
	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm
$\eta _{1}^{2}$	0.9615	0.9072	0.9656	0.9710	0.9228	0.9756	0.9740	0.9080	0.9764	0.9770	0.9176	0.9808
$\eta _{2}^{2}$	0.9595	0.9152	0.9644	0.9680	0.9060	0.9712	0.9725	0.9172	0.9760	0.9760	0.9100	0.9804
$\eta _{3}^{2}$	0.9585	0.9236	0.9616	0.9665	0.8972	0.9684	0.9710	0.9308	0.9752	0.9765	0.9048	0.9800
$\eta _{4}^{2}$	0.9575	0.9308	0.9624	0.9660	0.9120	0.9692	0.9690	0.9400	0.9732	0.9770	0.9156	0.9800
$\eta _{5}^{2}$	0.9560	0.9372	0.9620	0.9655	0.9132	0.9700	0.9670	0.9516	0.9716	0.9755	0.9168	0.9792
	N3						N4
	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm
$\eta _{1}^{2}$	0.9750	0.8956	0.9800	0.9755	0.9016	0.9820	0.9735	0.8924	0.9764	0.9675	0.8852	0.9780
$\eta _{2}^{2}$	0.9735	0.9104	0.9792	0.9765	0.8888	0.9828	0.9735	0.9064	0.9768	0.9670	0.8664	0.9792
$\eta _{3}^{2}$	0.9705	0.9268	0.9796	0.9750	0.8796	0.9820	0.9725	0.9252	0.9756	0.9725	0.8432	0.9780
$\eta _{4}^{2}$	0.9695	0.9356	0.9800	0.9745	0.9040	0.9816	0.9715	0.9328	0.9740	0.9725	0.8752	0.9788
$\eta _{5}^{2}$	0.9670	0.9532	0.9780	0.9750	0.9048	0.9816	0.9695	0.9436	0.9732	0.9725	0.8796	0.9808
	N5						N6
	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm
$\eta _{1}^{2}$	0.9780	0.8640	0.9800	0.9735	0.8916	0.9748	0.9720	0.8320	0.9764	0.9635	0.8320	0.9732
$\eta _{1}^{2}$	0.9775	0.8808	0.9800	0.9735	0.8584	0.9768	0.9715	0.8580	0.9756	0.9660	0.8120	0.9696
$\eta _{1}^{2}$	0.9770	0.9100	0.9792	0.9760	0.8352	0.9800	0.9705	0.8940	0.9748	0.9630	0.7788	0.9680
$\eta _{1}^{2}$	0.9760	0.9244	0.9792	0.9765	0.8652	0.9812	0.9700	0.9148	0.9736	0.9635	0.8380	0.9692
$\eta _{1}^{2}$	0.9760	0.9448	0.9776	0.9760	0.8648	0.9800	0.9705	0.9316	0.9728	0.9620	0.8496	0.9692

表 3

表 4

两个总体时六种Bootstrap置信区间的模拟区间长度"

(λ₁, λ₂) = (3, 4)
	N1						N2
	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm
$\eta _{1}^{2}$	0.0411	0.0221	0.0154	1.2987	1.0654	0.6986	0.0275	0.0201	0.0137	1.0710	0.9018	0.5738
$\eta _{2}^{2}$	0.1802	0.1991	0.1384	4.9169	6.6592	4.3665	0.1553	0.1808	0.1236	4.1442	5.6363	3.5863
$\eta _{3}^{2}$	0.4284	0.5529	0.3845	13.9794	17.0478	11.1784	0.4068	0.5022	0.3360	10.8372	14.4292	9.1810
$\eta _{4}^{2}$	0.8817	1.0838	0.7537	36.8349	52.2064	34.2320	0.8903	0.9845	0.6729	32.5810	44.1873	28.1151
$\eta _{5}^{2}$	1.3595	1.7916	1.2460	74.1995	106.5424	69.8603	1.4416	1.6275	1.1124	61.4367	90.1772	57.3771
	N3						N4
	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm
$\eta _{1}^{2}$	0.0252	0.0171	0.0112	0.8519	0.7178	0.4647	0.0383	0.0129	0.0083	0.5550	0.5160	0.3071
$\eta _{2}^{2}$	0.1485	0.1539	0.1009	3.2747	4.4861	2.9044	0.1147	0.1164	0.0746	2.5189	3.2254	1.9195
$\eta _{3}^{2}$	0.3626	0.4274	0.2802	9.6313	11.4847	7.4355	0.2774	0.3235	0.2072	6.4404	8.2571	4.9140
$\eta _{4}^{2}$	0.5683	0.8377	0.5493	26.9447	35.1702	22.7699	0.5835	0.6340	0.4062	21.0352	25.2859	15.0482
$\eta _{5}^{2}$	1.2108	1.3848	0.9080	48.5261	71.7750	46.4686	0.8832	1.0481	0.6715	32.1665	51.6033	30.7102
	N5						N6
	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm
$\eta _{1}^{2}$	0.0336	0.0110	0.0068	0.6014	0.3940	0.2379	0.0345	0.0092	0.0056	0.4908	0.3422	0.2053
$\eta _{2}^{2}$	0.0935	0.0986	0.0615	1.8589	2.4626	1.4867	0.1076	0.0826	0.0503	1.7424	2.1391	1.2834
$\eta _{3}^{2}$	0.2080	0.2740	0.1708	5.7627	6.3043	3.8061	0.2012	0.2295	0.1397	5.5726	5.4763	3.2856
$\eta _{4}^{2}$	0.4737	0.5371	0.3349	13.8291	19.3065	11.6555	0.3902	0.4499	0.2739	13.5280	16.7701	10.0614
$\eta _{5}^{2}$	0.8088	0.8878	0.5536	27.4740	39.4007	23.7863	0.6367	0.7437	0.4527	21.7438	34.2241	20.5332
(λ₁, λ₂) = (5, 6)
	N1						N2
	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm
$\eta _{1}^{2}$	0.0394	0.0210	0.0138	1.2360	1.0023	0.6375	0.0255	0.0183	0.0118	0.9932	0.8023	0.5009
$\eta _{2}^{2}$	0.1646	0.1887	0.1243	4.5252	6.2645	3.9848	0.1372	0.1645	0.1064	3.6578	5.0143	3.1310
$\eta _{3}^{2}$	0.3848	0.5242	0.3452	12.9767	16.0372	10.2012	0.3566	0.4568	0.2956	9.5920	12.8368	8.0154
$\eta _{4}^{2}$	0.7964	1.0275	0.6766	33.7643	49.1118	31.2397	0.7917	0.8955	0.5795	28.7679	39.3111	24.5460
$\eta _{5}^{2}$	1.2185	1.6985	1.1185	67.9331	100.2271	63.7537	1.2786	1.4804	0.9579	53.6549	80.2260	50.0932
	N3						N4
	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm
$\eta _{1}^{2}$	0.0228	0.0143	0.0089	0.7857	0.6215	0.3957	0.0361	0.0098	0.0061	0.5066	0.4393	0.2632
$\eta _{2}^{2}$	0.1275	0.1287	0.0803	2.8605	3.8848	2.4733	0.0952	0.0879	0.0545	2.2166	2.7457	1.6448
$\eta _{3}^{2}$	0.3042	0.3575	0.2231	8.5709	9.9451	6.3316	0.2233	0.2441	0.1515	5.6665	7.0291	4.2107
$\eta _{4}^{2}$	0.4537	0.7008	0.4373	23.6974	30.4554	19.3897	0.4774	0.4785	0.2969	18.6652	21.5255	12.8946
$\eta _{5}^{2}$	1.0214	1.1585	0.7229	41.8990	62.1532	39.5704	0.7079	0.7911	0.4908	27.3300	43.9289	26.3153
	N5						N6
	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm
$\eta _{1}^{2}$	0.0319	0.0080	0.0049	0.5740	0.3437	0.2114	0.0331	0.0068	0.0042	0.4703	0.3005	0.1874
$\eta _{2}^{2}$	0.0777	0.0724	0.0444	1.6875	2.1483	1.3215	0.0953	0.0616	0.0375	1.6146	1.8781	1.1713
$\eta _{3}^{2}$	0.1642	0.2010	0.1233	5.3241	5.4997	3.3830	0.1670	0.1712	0.1042	5.2454	4.8079	2.9985
$\eta _{4}^{2}$	0.3879	0.3939	0.2416	12.4861	16.8425	10.3598	0.3232	0.3355	0.2043	12.5257	14.7231	9.1822
$\eta _{5}^{2}$	0.6670	0.6512	0.3994	24.7334	34.3722	21.1422	0.5259	0.5546	0.3377	19.6984	30.0466	18.7387
(λ₁, λ₂) = (8, 9)
	N1						N2
	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm
$\eta _{1}^{2}$	0.0386	0.0204	0.0131	1.2039	0.9676	0.6056	0.0246	0.0174	0.0109	0.9618	0.7591	0.4681
$\eta _{2}^{2}$	0.1578	0.1837	0.1180	4.3248	6.0474	3.7849	0.1297	0.1563	0.0984	3.4612	4.7443	2.9255
$\eta _{3}^{2}$	0.3660	0.5103	0.3279	12.4635	15.4815	9.6895	0.3357	0.4342	0.2733	9.0888	12.1456	7.4894
$\eta _{4}^{2}$	0.7595	1.0002	0.6427	32.1929	47.4100	29.6727	0.7508	0.8511	0.5357	27.2271	37.1946	22.9350
$\eta _{5}^{2}$	1.1574	1.6535	1.0625	64.7261	96.7542	60.5558	1.2110	1.4069	0.8856	50.5105	75.9066	46.8056
	N3						N4
	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm
$\eta _{1}^{2}$	0.0219	0.0131	0.0080	0.7570	0.5777	0.3643	0.0355	0.0088	0.0054	0.4892	0.4095	0.2485
$\eta _{2}^{2}$	0.1190	0.1178	0.0721	2.6816	3.6109	2.2768	0.0897	0.0788	0.0486	2.1073	2.5595	1.5533
$\eta _{3}^{2}$	0.2806	0.3272	0.2004	8.1129	9.2440	5.8287	0.2080	0.2188	0.1351	5.3869	6.5525	3.9764
$\eta _{4}^{2}$	0.4076	0.6413	0.3927	22.2950	28.3083	17.8497	0.4473	0.4288	0.2648	17.8090	20.0658	12.1772
$\eta _{5}^{2}$	0.9453	1.0601	0.6492	39.0370	57.7713	36.4275	0.6581	0.7089	0.4378	25.5826	40.9500	24.8510
	N5						N6
	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm
$\eta _{1}^{2}$	0.0314	0.0072	0.0045	0.5646	0.3261	0.2034	0.0329	0.0063	0.0039	0.4632	0.2855	0.1814
$\eta _{2}^{2}$	0.0738	0.0651	0.0404	1.6289	2.0381	1.2713	0.0928	0.0564	0.0347	1.5702	1.7846	1.1336
$\eta _{3}^{2}$	0.1535	0.1809	0.1123	5.1740	5.2176	3.2545	0.1599	0.1566	0.0965	5.1318	4.5686	2.9020
$\eta _{4}^{2}$	0.3669	0.3546	0.2201	12.0263	15.9787	9.9664	0.3092	0.3069	0.1891	12.1779	13.9902	8.8866
$\eta _{5}^{2}$	0.6322	0.5862	0.3639	23.7951	32.6094	20.3393	0.5028	0.5073	0.3126	18.9885	28.5507	18.1355

表 4

表 5

三个总体时六种Bootstrap置信区间的模拟区间长度"

(λ₁, λ₂, λ₃) = (3, 3, 4)
	N1						N2
	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm
$\eta _{1}^{2}$	0.1558	0.1979	0.1365	4.3721	6.2373	4.0877	0.1325	0.1794	0.1215	3.7023	5.2625	3.3734
$\eta _{2}^{2}$	0.3006	0.3512	0.2418	7.7139	11.0764	7.2551	0.2495	0.3182	0.2154	6.4917	9.3478	5.9910
$\eta _{3}^{2}$	0.4433	0.5465	0.3751	13.3325	17.2881	11.3168	0.4046	0.4949	0.3343	12.2296	14.5935	9.3507
$\eta _{4}^{2}$	1.0154	1.0665	0.7290	26.7962	33.8652	22.1617	0.7806	0.9649	0.6497	24.4722	28.5907	18.3170
$\eta _{5}^{2}$	1.5271	1.7688	1.2130	40.0428	56.0040	36.6571	1.3446	1.6015	1.0812	35.8804	47.2769	30.2912
	N3						N4
	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm
$\eta _{1}^{2}$	0.1100	0.1527	0.0994	3.0012	3.9181	2.4516	0.0906	0.1159	0.0733	2.3937	3.0532	1.8877
$\eta _{2}^{2}$	0.2372	0.2708	0.1761	4.7556	6.9595	4.3525	0.1882	0.2057	0.1300	3.8212	5.4245	3.3539
$\eta _{3}^{2}$	0.3379	0.4209	0.2729	9.1752	10.8646	6.7914	0.2753	0.3204	0.2019	6.8204	8.4705	5.2370
$\eta _{4}^{2}$	0.5451	0.8201	0.5295	16.6665	21.2852	13.3020	0.5376	0.6261	0.3929	12.8209	16.5977	10.2607
$\eta _{5}^{2}$	1.2405	1.3619	0.8823	25.6048	35.1967	21.9996	0.8750	1.0374	0.6532	19.8941	27.4422	16.9660
	N5						N6
	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm
$\eta _{1}^{2}$	0.0770	0.0981	0.0608	1.9892	2.3561	1.4262	0.0687	0.0822	0.0494	1.8176	2.0390	1.2296
$\eta _{2}^{2}$	0.1658	0.1740	0.1077	2.8175	4.1868	2.5327	0.1298	0.1459	0.0875	2.7334	3.6225	2.1837
$\eta _{3}^{2}$	0.2416	0.2708	0.1668	5.6525	6.5386	3.9525	0.1859	0.2270	0.1356	5.1862	5.6567	3.4092
$\eta _{4}^{2}$	0.4589	0.5281	0.3235	10.3481	12.8113	7.7421	0.4102	0.4424	0.2631	10.0297	11.0841	6.6801
$\eta _{5}^{2}$	0.8659	0.8764	0.5392	13.4808	21.1830	12.8039	0.6102	0.7345	0.4385	14.6270	18.3261	11.0449
(λ₁, λ₂, λ₃) = (5, 5, 6)
	N1						N2
	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm
$\eta _{1}^{2}$	0.1407	0.1878	0.1222	4.0248	5.8983	3.7440	0.1151	0.1632	0.1041	3.2368	4.6661	2.9283
$\eta _{2}^{2}$	0.2739	0.3332	0.2165	7.0974	10.4754	6.6448	0.2185	0.2894	0.1845	5.6653	8.2891	5.2006
$\eta _{3}^{2}$	0.4017	0.5182	0.3357	12.3715	16.3519	10.3646	0.3563	0.4498	0.2864	10.9406	12.9415	8.1174
$\eta _{4}^{2}$	0.9341	1.0101	0.6519	24.9148	32.0333	20.2967	0.6866	0.8763	0.5565	21.9479	25.3549	15.9014
$\eta _{5}^{2}$	1.3924	1.6771	1.0854	36.9300	52.9723	33.5725	1.1886	1.4554	0.9262	31.7051	41.9254	26.2962
	N3						N4
	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm
$\eta _{1}^{2}$	0.0887	0.1279	0.0788	2.6180	3.3553	2.0751	0.0700	0.0875	0.0532	2.1090	2.6020	1.6216
$\eta _{2}^{2}$	0.1993	0.2269	0.1396	4.0753	5.9608	3.6841	0.1517	0.1552	0.0943	3.3151	4.6228	2.8811
$\eta _{3}^{2}$	0.2792	0.3524	0.2163	8.1131	9.3072	5.7488	0.2185	0.2419	0.1463	6.0298	7.2186	4.4990
$\eta _{4}^{2}$	0.4309	0.6860	0.4192	14.5853	18.2361	11.2604	0.4269	0.4728	0.2844	11.2715	14.1454	8.8144
$\eta _{5}^{2}$	1.0506	1.1400	0.6993	22.1638	30.1523	18.6226	0.6914	0.7832	0.4732	17.3326	23.3868	14.5749
	N5						N6
	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm
$\eta _{1}^{2}$	0.0593	0.0720	0.0437	1.8513	2.0602	1.2781	0.0558	0.0614	0.0369	1.7059	1.8071	1.1257
$\eta _{2}^{2}$	0.1345	0.1277	0.0774	2.5730	3.6615	2.2697	0.1069	0.1089	0.0654	2.5355	3.2111	1.9992
$\eta _{3}^{2}$	0.1932	0.1988	0.1197	5.2713	5.7193	3.5423	0.1503	0.1695	0.1013	4.8778	5.0149	3.1219
$\eta _{4}^{2}$	0.3651	0.3879	0.2319	9.6019	11.2065	6.9387	0.3410	0.3304	0.1963	9.4261	9.8271	6.1179
$\eta _{5}^{2}$	0.7096	0.6436	0.3869	12.2461	18.5292	11.4749	0.4951	0.5484	0.3276	13.6283	16.2471	10.1146
(λ₁, λ₂, λ₃) = (6, 8, 9)
	N1						N2
	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm
$\eta _{1}^{2}$	0.1374	0.1851	0.1190	3.8328	5.6934	3.5617	0.1115	0.1595	0.1006	3.0461	4.4063	2.7456
$\eta _{2}^{2}$	0.2679	0.3283	0.2107	6.7565	10.1120	6.3209	0.2122	0.2828	0.1783	5.3267	7.8279	4.8761
$\eta _{3}^{2}$	0.3922	0.5104	0.3265	11.8395	15.7855	9.8587	0.3464	0.4395	0.2767	10.4123	12.2221	7.6110
$\eta _{4}^{2}$	0.9154	0.9943	0.6336	23.8733	30.9242	19.3057	0.6669	0.8558	0.5375	20.9130	23.9460	14.9098
$\eta _{5}^{2}$	1.3617	1.6516	1.0557	35.2072	51.1376	31.9339	1.1564	1.4220	0.8948	29.9936	39.5952	24.6560
	N3						N4
	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm
$\eta _{1}^{2}$	0.0846	0.1226	0.0748	2.4665	3.1207	1.9302	0.0671	0.0827	0.0503	2.0203	2.4502	1.5399
$\eta _{1}^{2}$	0.1922	0.2175	0.1324	3.8063	5.5448	3.4270	0.1465	0.1468	0.0890	3.1575	4.3532	2.7361
$\eta _{1}^{2}$	0.2680	0.3377	0.2051	7.6932	8.6588	5.3476	0.2104	0.2286	0.1381	5.7836	6.7979	4.2725
$\eta _{1}^{2}$	0.4091	0.6571	0.3973	13.7625	16.9670	10.4745	0.4109	0.4466	0.2685	10.7888	13.3214	8.3705
$\eta _{1}^{2}$	1.0145	1.0923	0.6630	20.8033	28.0523	17.3230	0.6651	0.7403	0.4467	16.5347	22.0240	13.8411
	N5						N6
	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm
$\eta _{1}^{2}$	0.0571	0.0680	0.0415	1.8026	1.9617	1.2330	0.0543	0.0586	0.0355	1.6661	1.7271	1.0921
$\eta _{2}^{2}$	0.1306	0.1207	0.0734	2.4866	3.4866	2.1896	0.1043	0.1040	0.0630	2.4648	3.0692	1.9394
$\eta _{3}^{2}$	0.1871	0.1879	0.1136	5.1367	5.4466	3.4172	0.1462	0.1618	0.0976	4.7677	4.7936	3.0287
$\eta _{4}^{2}$	0.3531	0.3666	0.2198	9.3385	10.6722	6.6937	0.3329	0.3155	0.1889	9.2105	9.3939	5.9356
$\eta _{5}^{2}$	0.6897	0.6083	0.3669	11.8101	17.6458	11.0699	0.4818	0.5237	0.3154	13.2715	15.5304	9.8127

表 5

表 6

五个总体时六种Bootstrap置信区间的模拟区间长度"

(λ₁, λ₂, λ₃, λ₄, λ₅) = (3, 3, 4, 4, 5)
	N1						N2
	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm
$\eta _{1}^{2}$	0.1557	0.1978	0.1363	3.4939	4.6909	3.2412	0.1216	0.1617	0.1078	2.8773	3.4579	2.3837
$\eta _{2}^{2}$	0.2734	0.3518	0.2424	8.2839	11.5465	7.9518	0.2131	0.2876	0.1917	6.2643	8.7179	5.9407
$\eta _{3}^{2}$	0.4387	0.5513	0.3804	17.0336	22.4545	15.4017	0.3928	0.4504	0.3005	13.6951	17.0843	11.6810
$\eta _{4}^{2}$	1.0438	1.0821	0.7472	29.4770	38.7087	26.8397	0.7844	0.8838	0.5901	25.7350	29.3646	20.3567
$\eta _{5}^{2}$	1.4713	1.7906	1.2372	44.0148	60.0451	41.9806	1.2714	1.4625	0.9766	37.1142	45.4880	31.8257
	N3						N4
	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm
$\eta _{1}^{2}$	0.1027	0.1398	0.0902	2.5296	2.8396	1.9479	0.0935	0.1159	0.0733	1.8318	2.4289	1.6117
$\eta _{2}^{2}$	0.2213	0.2485	0.1604	5.2675	7.0858	4.7783	0.1706	0.2060	0.1304	4.2271	5.9759	3.9423
$\eta _{3}^{2}$	0.3330	0.3895	0.2515	10.8420	13.8792	9.3431	0.2479	0.3226	0.2044	9.5049	11.5423	7.6441
$\eta _{4}^{2}$	0.5100	0.7648	0.4939	21.4102	24.0453	16.3943	0.5928	0.6331	0.4013	17.7956	20.1387	13.4950
$\eta _{5}^{2}$	0.9985	1.2659	0.8175	26.7238	37.3193	25.6653	0.8089	1.0476	0.6642	22.7187	31.3854	21.2205
	N5						N6
	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm
$\eta _{1}^{2}$	0.0822	0.1093	0.0687	1.7325	2.0830	1.3409	0.0628	0.0822	0.0494	1.6502	1.6998	1.0693
$\eta _{2}^{2}$	0.1728	0.1943	0.1221	3.6881	5.0325	3.3183	0.1143	0.1462	0.0879	3.3868	4.1695	2.6306
$\eta _{3}^{2}$	0.2729	0.3044	0.1915	8.6829	9.8244	6.4719	0.1815	0.2290	0.1378	8.1659	8.2079	5.2124
$\eta _{4}^{2}$	0.5130	0.5974	0.3761	16.8130	17.2494	11.4077	0.4510	0.4493	0.2705	14.5028	14.2889	9.0520
$\eta _{5}^{2}$	0.7733	0.9888	0.6224	24.5885	26.9702	17.9168	0.5895	0.7434	0.4476	16.8295	22.2257	14.1171
(λ₁, λ₂, λ₃, λ₄, λ₅) = (4, 4, 5, 5, 6)
	N1						N2
	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm
$\eta _{1}^{2}$	0.1462	0.1915	0.1272	3.2911	4.4496	3.0456	0.1088	0.1479	0.0948	2.7034	3.2557	2.2115
$\eta _{2}^{2}$	0.2565	0.3406	0.2263	7.8440	11.0346	7.5213	0.1904	0.2630	0.1686	5.8552	8.2163	5.5371
$\eta _{3}^{2}$	0.4124	0.5337	0.3552	16.2008	21.4728	14.5817	0.3571	0.4118	0.2644	12.8877	16.0774	10.8849
$\eta _{4}^{2}$	0.9921	1.0475	0.6977	27.9733	36.9794	25.3735	0.7145	0.8081	0.5192	24.2983	27.6426	18.9434
$\eta _{5}^{2}$	1.3857	1.7332	1.1552	41.6062	57.3327	39.6393	1.1556	1.3371	0.8594	34.8158	42.8044	29.5654
	N3						N4
	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm
$\eta _{1}^{2}$	0.0880	0.1214	0.0758	2.3754	2.6513	1.7957	0.0791	0.0964	0.0591	1.7105	2.2721	1.4887
$\eta _{2}^{2}$	0.1951	0.2159	0.1348	4.9387	6.6647	4.4483	0.1449	0.1714	0.1051	3.9676	5.6446	3.6908
$\eta _{3}^{2}$	0.2919	0.3383	0.2114	10.1844	13.0444	8.6890	0.2077	0.2684	0.1648	8.9952	10.9070	7.1541
$\eta _{4}^{2}$	0.4294	0.6643	0.4151	20.2033	22.5874	15.1945	0.5139	0.5266	0.3234	16.8518	19.0010	12.5837
$\eta _{5}^{2}$	0.8652	1.0995	0.6871	24.7762	35.0130	23.7310	0.6784	0.8714	0.5352	21.1760	29.5517	19.7318
	N5						N6
	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm
$\eta _{1}^{2}$	0.0696	0.0912	0.0561	1.6539	1.9649	1.2599	0.0532	0.0672	0.0400	1.6002	1.6103	1.0197
$\eta _{2}^{2}$	0.1504	0.1622	0.0998	3.5044	4.7685	3.1352	0.0973	0.1195	0.0712	3.2736	3.9771	2.5161
$\eta _{3}^{2}$	0.2378	0.2540	0.1565	8.3027	9.2860	6.0901	0.1548	0.1872	0.1116	7.9416	7.8271	4.9874
$\eta _{4}^{2}$	0.4440	0.4984	0.3073	16.1106	16.2937	10.7029	0.3986	0.3673	0.2191	14.0896	13.6050	8.6378
$\eta _{5}^{2}$	0.6591	0.8249	0.5084	23.4391	25.4566	16.7651	0.5028	0.6077	0.3626	16.1526	21.1227	13.4314
(λ₁, λ₂, λ₃, λ₄, λ₅) = (7, 7, 8, 8, 9)
	N1						N2
	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm
$\eta _{1}^{2}$	0.1355	0.1836	0.1170	3.0581	4.1632	2.8178	0.0949	0.1315	0.0811	2.5129	3.0279	2.0208
$\eta _{2}^{2}$	0.2374	0.3266	0.2081	7.3303	10.4187	7.0135	0.1657	0.2338	0.1442	5.4125	7.6619	5.1002
$\eta _{3}^{2}$	0.3824	0.5116	0.3267	15.2237	20.2877	13.6218	0.3185	0.3661	0.2264	11.9978	14.9555	10.0096
$\eta _{4}^{2}$	0.9334	1.0040	0.6417	26.2275	34.9196	23.6569	0.6387	0.7184	0.4446	22.7218	25.7450	17.3886
$\eta _{5}^{2}$	1.2886	1.6610	1.0626	38.8013	54.0965	36.9026	1.0302	1.1885	0.7359	32.3204	39.8907	27.1060
	N3						N4
	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm
$\eta _{1}^{2}$	0.0747	0.1028	0.0627	2.2214	2.4616	1.6481	0.0685	0.0801	0.0486	1.5998	2.1215	1.3816
$\eta _{2}^{2}$	0.1715	0.1827	0.1114	4.6076	6.2369	4.1287	0.1260	0.1424	0.0865	3.7212	5.3159	3.4558
$\eta _{3}^{2}$	0.2548	0.2864	0.1748	9.5321	12.1893	8.0624	0.1780	0.2230	0.1356	8.5029	10.2646	6.6820
$\eta _{4}^{2}$	0.3566	0.5624	0.3432	18.9937	21.1033	14.0387	0.4556	0.4376	0.2662	15.9378	17.8566	11.7123
$\eta _{5}^{2}$	0.7446	0.9308	0.5684	22.8372	32.6800	21.8718	0.5819	0.7240	0.4405	19.6857	27.7385	18.3212
	N5						N6
	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm	Zmple	Zmle	Zmm	Tmple	Tmle	Tmm
$\eta _{1}^{2}$	0.0593	0.0757	0.0464	1.5813	1.8492	1.1884	0.0476	0.0571	0.0347	1.5523	1.5169	0.9747
$\eta _{2}^{2}$	0.1321	0.1346	0.0825	3.3250	4.5016	2.9623	0.0873	0.1015	0.0617	3.1667	3.7736	2.4147
$\eta _{3}^{2}$	0.2092	0.2109	0.1294	7.9352	8.7334	5.7341	0.1391	0.1589	0.0967	7.7241	7.4248	4.7839
$\eta _{4}^{2}$	0.3879	0.4137	0.2540	15.4452	15.3345	10.0587	0.3678	0.3118	0.1899	13.7053	12.8871	8.2708
$\eta _{5}^{2}$	0.5663	0.6848	0.4202	22.3662	23.9540	15.7304	0.4518	0.5158	0.3143	15.5373	19.9767	12.8403

表 6

图 1

图 2

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