
Citation: Gérald Gaibelet, François Tercé, Sophie Allart, Chantal Lebrun, Xavier Collet, Nadège Jamin, Stéphane Orlowski. Fluorescent probes for detecting cholesterol-rich ordered membrane microdomains: entangled relationships between structural analogies in the membrane and functional homologies in the cell[J]. AIMS Biophysics, 2017, 4(1): 121-151. doi: 10.3934/biophy.2017.1.121
[1] | M. Nagy, H. M. Barakat, M. A. Alawady, I. A. Husseiny, A. F. Alrasheedi, T. S. Taher, A. H. Mansi, M. O. Mohamed . Inference and other aspects for q−Weibull distribution via generalized order statistics with applications to medical datasets. AIMS Mathematics, 2024, 9(4): 8311-8338. doi: 10.3934/math.2024404 |
[2] | Rasha Abd-Elwahaab Attwa, Taha Radwan, Esraa Osama Abo Zaid . Bivariate q-extended Weibull morgenstern family and correlation coefficient formulas for some of its sub-models. AIMS Mathematics, 2023, 8(11): 25325-25342. doi: 10.3934/math.20231292 |
[3] | H. M. Barakat, M. A. Alawady, I. A. Husseiny, M. Nagy, A. H. Mansi, M. O. Mohamed . Bivariate Epanechnikov-exponential distribution: statistical properties, reliability measures, and applications to computer science data. AIMS Mathematics, 2024, 9(11): 32299-32327. doi: 10.3934/math.20241550 |
[4] | Aisha Fayomi, Ehab M. Almetwally, Maha E. Qura . A novel bivariate Lomax-G family of distributions: Properties, inference, and applications to environmental, medical, and computer science data. AIMS Mathematics, 2023, 8(8): 17539-17584. doi: 10.3934/math.2023896 |
[5] | Areej M. AL-Zaydi . On concomitants of generalized order statistics arising from bivariate generalized Weibull distribution and its application in estimation. AIMS Mathematics, 2024, 9(8): 22002-22021. doi: 10.3934/math.20241069 |
[6] | I. A. Husseiny, M. Nagy, A. H. Mansi, M. A. Alawady . Some Tsallis entropy measures in concomitants of generalized order statistics under iterated FGM bivariate distribution. AIMS Mathematics, 2024, 9(9): 23268-23290. doi: 10.3934/math.20241131 |
[7] | Zhipeng Liu, Cailing Li, Zhenhua Bao . On a dependent risk model perturbed by mixed-exponential jump-diffusion processes. AIMS Mathematics, 2025, 10(4): 9882-9899. doi: 10.3934/math.2025452 |
[8] | Fiaz Ahmad Bhatti, Azeem Ali, G. G. Hamedani, Mustafa Ç. Korkmaz, Munir Ahmad . The unit generalized log Burr XII distribution: properties and application. AIMS Mathematics, 2021, 6(9): 10222-10252. doi: 10.3934/math.2021592 |
[9] | Salem A. Alyami, Amal S. Hassan, Ibrahim Elbatal, Naif Alotaibi, Ahmed M. Gemeay, Mohammed Elgarhy . Estimation methods based on ranked set sampling for the arctan uniform distribution with application. AIMS Mathematics, 2024, 9(4): 10304-10332. doi: 10.3934/math.2024504 |
[10] | Haidy A. Newer, Mostafa M. Mohie El-Din, Hend S. Ali, Isra Al-Shbeil, Walid Emam . Statistical inference for the Nadarajah-Haghighi distribution based on ranked set sampling with applications. AIMS Mathematics, 2023, 8(9): 21572-21590. doi: 10.3934/math.20231099 |
In 2013, [1] introduced a new lifetime distribution of one parameter, known as the Bilal distribution. The Bilal distribution is derived as a member of the family of distributions for the median of a random sample drawn from an exponential distribution. The cumulative distribution function (cdf) of the Bilal distribution with the scale parameter σ is given by:
F(x;σ)=1−e−2xσ(3−2e−xσ);σ>0,x≥0. | (1.1) |
The corresponding probability density function (pdf) is defined by:
f(x;σ)=6σe−2xσ(1−e−xσ);σ>0,x≥0, | (1.2) |
with survival function given by:
S(x;σ)=e−2xσ(3−2e−xσ);σ>0,x≥0. |
The Bilal distribution exhibits lower skewness and kurtosis compared to the exponential distribution and belongs to the class of distributions characterized as new, better than used in failure rates. Despite having only one parameter, the Bilal distribution demonstrates a strong capacity to fit two distinct real-world data sets: the first comprising 30 consecutive precipitation measurements (in inches) provided by [2], and the second consisting of waiting times before service for 100 bank customers, as reported by [3]. The closed-form expressions for all its statistical properties, includes key functions such as the quantile function, the hazard rate function (HRF), and a simple expression for the moments attracted several researchers and developed its various extensions. Abd-Elrahman and Niazi [4] proposed various estimators of the parameter for the Bilal distribution based on using Type-2 censored sample. To address the unimodal HRF of the Bilal distribution, Abd-Elrahman [5] proposed a two-parameter generalization referred to as the general Bilal distribution. A three-parameter generalization, called the Harris extended Bilal distribution was introduced, by [6], and its various properties have been discussed. Irshad et al. [7] proposed the Marshal–Olkin Bilal distribution, its applications in statistical process control, and the associated minification process. The proficiency of the univariate Bilal distribution has been firmly established in the literature, demonstrating its superiority over competing models in both theoretical and applied perspectives.
Maya et al. [8] introduced a bivariate version of the one-parameter Bilal distribution using the Morgenstern framework, called the Farlie–Gumbel–Morgenstern bivariate Bilal distribution (FGMBBD), and studied its inferential aspects based on concomitants of order statistics (COS). The pdf of FGMBBD is expressed as follows:
f(w,z)={36σ1σ2e−2wσ1(1−e−wσ1)e−2zσ2(1−e−zσ2)×[1+ϕ(2e−2wσ1{3−2e−wσ1}−1)(2e−2zσ2{3−2e−zσ2}−1)],w>0,z>0;σ1>0,σ2>0;−1≤ϕ≤1.0, otherwise. | (1.3) |
Figure 1 presents 3D plots of the FGMBBD pdf for various parameter values. We can see that the distribution has many shapes depending on the parameters values.
Arun et al. [9] evaluated the performance of the FGMBBD in comparison to the well-known FGM bivariate exponential distribution using two real-world data sets. The first data set, sourced from [10], consists of mineral content measurements in the dominant ulna (X) and non-dominant ulna (Y) of 25 elderly women. The second data set, also from [10], includes tail length (X) and wing length (Y) measurements in millimeters for a sample of 45 female hook-billed kites.
The marginal distributions of the variables W and Z are univariate Bilal distributions, with their respective pdf's given by:
fW(w)=6σ1e−2wσ1(1−e−wσ1);ifσ1>0,w≥0, |
and
fZ(z)=6σ2e−2zσ2(1−e−zσ2);ifσ2>0,z≥0. | (1.4) |
Clearly, E(W)=56σ1,Var(W)=1336σ21,
E(Z)=56σ2, | (1.5) |
and
Var(Z)=1336σ22. | (1.6) |
Note that the mean and variance values of Z depend on σ2. The ranked set sampling (RSS) method, originally developed by [11], is designed to improve the precision of the sample mean as an estimator of the population mean. In this approach, a total of n sets of units, each containing n units, are selected. The units within each set are ordered using a judgmental method or a cost-effective technique that does not involve actual measurements of the selected observations. Subsequently, the unit ranked first in the first set is measured, followed by the unit ranked second in the second set, and this process continues until the unit ranked n largest in the n-th set is measured. The observations obtained through this process form a ranked set sample (rss) of size n. Hence, we only measured n rss units: X1(1:n),X2(2:n),…,Xn(n:n), which are obtained from n2 units. For simplicity, we will use X(i:n) to denote the rss units.
Let the random variable (rv) X have a pdf f(x) and cdf F(x), with mean μ and variance σ2. Consider a simple random sample (srs) of size n, denoted as X1, X2, …, Xn. Let Xj(i:n) represent the ith largest unit in a set of size n in the jth sample. Takahasi and Wakimoto [12] demonstrated that the rss estimator of the population mean, when based on perfect ranking, is unbiased and outperforms the estimator based on srs because it has a lower variance. The mean estimators based on srs and rss are, respectively given by: ˉXSRS=1nn∑i=1Xi, and ˉXRSS=1nn∑i=1X(i:n), with respective variances given by Var(ˉXSRS)=σ2n, and Var(ˉXRSS)=σ2n−1n2n∑i=1(μ(i:n)−μ)2, where μ(i:k)=∞∫−∞xf(i:k)(x)dx. Clearly, Var(ˉXRSS) ≤ Var(ˉXSRS).
Stokes [13] introduced an alternative RSS scheme designed for situations where the primary variable of interest, denoted as Z, is challenging to measure directly, while an auxiliary variable, W, which is correlated with Z can be easily measured. The process involves selecting n independent bivariate sets, each containing n units. In the first set, the Z variable corresponding to the smallest ordered W is measured, followed by the Z variable associated with the second smallest W in the second set, and so on, until the Z variable corresponding to the largest W in the n-th set is measured. The resulting measurements on the Z variable from this new set of n units, selected according to the described method, constitute a rss, as proposed by [13]. Let W(i:n)i represent the observation on the auxiliary variable W from the unit obtained from the i-th set represent Z[i:n]i is used to denote the corresponding measurement of the study variable Z for that unit. Thus, Z[i:n]i for i=1,2,…,n, collectively constitutes the rss of size n. David and Nagaraja [14] mentioned Z[i:n]i as the concomitant variable of the i-th order statistic obtain from the i-th sample. Using Stoke's procedure of RSS, [15] proposed estimators of the scale parameter associated with the Z variable, when (W,Z) follows the FGMBBD.
Suppose that the bivariate random vector (W,Z) follows the FGMBBD with pdf given in (1.3). Draw a rss using [13] scheme. Let W(i:n)i represent the observation obtained on the auxiliary variate W from the ith unit of the rss, and let Z[i:n]i denote the corresponding measurement of the variable related to W(i:n)i,i=1,2,⋯,n. Hence, Z[i:n]i is the ith COS of a random sample of size n drawn from the FGMBBD, with pdf given by (see, [8]) as:
f[i:n](z)=6σ2e−2zσ2(1−e−zσ2)[1+ϕ(n−2i+1)(n+1)(2e−2zσ2{3−2e−zσ2}−1)]. | (1.7) |
Figure 2 shows 3D plots of the f[i:n](z) for some choices of n=10,i=1,10,ϕ=−0.9,0.9. The figure reflects the effect of these selections on the shape of this function.
The mean and variance of Z[i:n]i for i=1,2,⋯,n, are obtained as
E[Z[i:n]i]=σ2[56−1960ϕ(n−2i+1)(n+1)], | (1.8) |
and
Var[Z[i:n]i]=σ22[1336−2531800ϕ(n−2i+1)(n+1)−3613600ϕ2(n−2i+1)2(n+1)2]. | (1.9) |
Since, Z[i:n]i and Z[j:n]j for i≠j are arising from two independent samples, we obtain
Cov[Z[i:n]i,Z[j:n]j]=0,i≠j. |
Al-Saleh and Al-kadiri [16] extended McIntyre's [11] concept of RSS, by introducing the double-stage ranked set sampling (DSRSS). Their findings affirmed that estimators based on the DSRSS scheme exhibit greater efficiency in estimating population parameters compared to those based on both RSS and simple random sampling (SRS) schemes based on the same number of measured units. In a subsequent work, Al-Saleh and Al-Omari [17] extended the DSRSS to the multistage ranked set sampling (MSRSS) and demonstrated an increase in the precision of MSRSS estimators compared to both DSRSS and RSS estimators without needing to increase the sample size. Al-Saleh [18] introduced the concept of steady-state RSS and its corresponding parametric inference. Jemain and Al-Omari [19,20] proposed multistage percentile and quartile RSS methods for estimating the population mean. Jemain [21] suggested multistage median RSS for estimating the population median. For a deeper exploration of RSS and its variations, refer to [22], [23] for the varied L RSS method, [24] for an analysis of generalized robust-regression-type estimators under different RSS methods, and [25] for research on estimating a decreasing mean residual life based on RSS, with applications to survival analysis. Zamanzade et al. [26] considered nonparametric estimation of the mean residual lifetime in RSS with a concomitant variable. Zamanzade et al. [27] considered a ranked-based estimator of the mean past lifetime.
The aforementioned generalizations of RSS play a crucial role in developing inferential aspects of parent bivariate distributions in both theoretical and applied perspectives. Consequently, the objectives of this study are as follows:
(1) To identify the unit that has maximum Fisher information (FI) when ϕ is positive and negative.
(2) Building on the insights from the FI, we define the MSRSS and a steady-state RSS. Using these methodologies, we estimate the scale parameter associated with the study variable Z.
(3) To compare the efficiency of the proposed estimator based on MSRSS and steady-state RSS with the maximum likelihood estimator (MLE).
The rest of this paper is structured as follows. The FI included in the concomitant of a specific order statistic from a random sample arising from a distribution provides essential insight for selecting the most warranted unit(s) from a group of units to establish an appropriate RSS. Thus, the FI about the parameter σ2, included in the concomitant of the ith order statistic from a random sample of size n drawn from the FGMBBD, has been derived and is presented in Section 2. As per the knowledge disseminated from the FI, we have identified that for the FGMBBD, the maximum amount of information about the parameter σ2 is contained in the concomitant of the largest order statistic or the concomitant of the smallest order statistic, according as ϕ is positive or negative. Accordingly, we define an MSRSS in Section 2 and used its observations to introduce the best linear unbiased estimator (BLUE) of σ2 when the dependence parameter ϕ is positive and negative. Here, we have additionally obtained the efficiency of the estimator of σ2 based on the MSRSS and compared it with the MLE of σ2. In Section 3, we proposed BLUE of σ2 based on steady-state RSS and compared the efficiency of the same with respect to the MLE of σ2. The article concludes in Section 4 with some suggestions for future works.
Here, we define the MSRSS and propose the BLUE of σ2 when ϕ is positive and negative based on observations generated by the MSRSS. The MSRSS at r stages (see, [17]) is defined as follows:
(1) Randomly select nr+1 sample units from the target population, where r denotes the number of stages in the MSRSS process. These selected units are then randomly divided into nr−1 sets, each containing n2 units.
(2) For each set from Step 1, adopt Stoke's RSS scheme, to obtain an RSS of size n. This step results in nr−1 ranked sets, each of size n.
(3) Randomly organize the nr−1 ranked sets, each of size n obtained from Step 2, into nr−2 sets, each of size n2. Without performing any actual measurements, apply the RSS scheme to each of the nr−2 sets to produce nr−2 second-stage ranked sets, each of size n.
(4) The process continues, without conducting any actual measurements, until we obtain an rth stage ranked set of size n.
(5) Finally, the n identified units from Step 4 are actually quantified for the variable of interest.
To identify the most suitable unit from a group of units for measurements, we derive the FI regarding the scale parameter σ2 found in the concomitant of the ith order statistic from the FGMBBD.
The pdf of the concomitant of the ith order statistic given in (1.7) can be written as:
f[i:n](z)=6σ2e−2zσ2(1−e−zσ2)[d+2(1−d)e−2zσ2(3−2e−zσ2)], |
where d=d(ϕ,i,n)=1−ϕ(n−2i+1)(n+1). By taking the logarithm, and then the first derivative partially with respect to σ2, we obtain:
∂logf[i:n](z)∂σ2=1σ2[−1+2zσ2−ze−zσ2σ2(1−e−zσ2)+12(1−d)ze−2zσ2(1−e−zσ2)σ2{d+2(1−d)e−2zσ2(3−2e−zσ2)}]. |
Thus,
(∂logf[i:n](z)∂σ2)2=1σ22[−1+2u−ue−u(1−e−u)+12(1−d)ue−2u(1−e−u){d+2(1−d)e−2u(3−2e−u)}]2, |
where u=zσ2. The pdf of the transformed rv U=Zσ2 is obtained by
g(u)=6e−2u(1−e−u)[d+2(1−d)e−2u(3−2e−u)]. |
Then, the FI about the parameter σ2 contained in the concomitant of the ith order statistic is given by:
Jσ2(Z[i:n],ϕ)=E(∂logf[i:n](Z)∂σ2)2=σ2−2q(d), |
where
q(d)=∞∫0[−1+2u−ue−u(1−e−u)+12(1−d)ue−2u(1−e−u){d+2(1−d)e−2u(3−2e−u)}]2×6e−2u(1−e−u)[d+2(1−d)e−2u(3−2e−u)]du. | (2.1) |
Now, fix ϕ>0. Since, d=1−ϕ(n−2i+1)(n+1), i=1,2,⋯,n, the values of d lie in the interval 0≤1−ϕ≤d≤1+ϕ≤2. A graph of the q(d) on the interval 0≤1−ϕ≤d≤1+ϕ≤2 is presented in Figure 3.
From the graph, it is easily observed that the maximum value of the q(d) reaches at the upper extreme point of d, that is when i=n. Thus, one can conclude that when ϕ>0, the maximum FI is obtained in the concomitants of the largest order statistic. Hence, we consider the estimator based on the concomitants of the largest order statistic. Since d(ϕ,i,n)=d(−ϕ,n−i+1,n), we have Jσ2(Z[i:n],ϕ)=Jσ2(Z[n−i+1:n],−ϕ). Hence ϕ<0, the maximum FI is obtained in the concomitants of the smallest order statistic.
By assuming that the rv (W,Z) has a FGMBBD with pdf given in (1.3), where Z is the variable of primary interest and W is an auxiliary variable, in this section first we consider ϕ>0 and carry out a MSRSS based on measurements made on an auxiliary variate to choose the rss to estimate σ2 contained in FGMBBD based on the measurements made on the variable of primary interest. At each stage, from each set, we select the unit with the highest value on the auxiliary variable as the units of the ranked sets, aiming to maximize the FI about σ2 for the final chosen rss.
Let X(r)i,i=1,2,⋯,n be the MSRSS units chosen at the rth stage. Since, the actual measurement of the auxiliary variable on each unit X(r)i,i=1,2,⋯,n has the largest value, we can write Z(r)[n:n]i as the value measured on the variable of the primary interest to X(r)i,i=1,2,⋯,n. Thus, easily one can verify that Z(r)[n:n]i is the same as that of Z[nr:nr], the concomitant of the largest order statistic of nr independent and identically distributed bivariate rvs from the FGMBBD. Also, Z(r)[n:n]i,i=1,2,⋯,n are independently distributed with a pdf given by:
f(r)[n:n]i(z;ϕ)=6σ2e−2zσ2(1−e−zσ2)[1+ϕ(nr−1nr+1)(1−2e−2zσ2{3−2e−zσ2})]. | (2.2) |
Thus, the mean and variance of Z(r)[n:n]i,i=1,2,⋯,n are obtained as:
E[Z(r)[n:n]i]=σ2[56+1960ϕ(nr−1nr+1)], | (2.3) |
and
Var[Z(r)[n:n]i]=σ22[1336+2531800ϕ(nr−1nr+1)−3613600ϕ2(nr−1nr+1)2]. | (2.4) |
If we write
ζnr=56+1960ϕ(nr−1nr+1), | (2.5) |
and
ψnr=1336+2531800ϕ(nr−1nr+1)−3613600ϕ2(nr−1nr+1)2. | (2.6) |
Then, for i≤i≤n, (2.3) and (2.4) can be written as:
E[Z(r)[n:n]i]=σ2ζnr, | (2.7) |
and
Var[Z(r)[n:n]i]=σ22ψnr. | (2.8) |
Since Z(r)[n:n]i and Z(r)[n:n]j (for i≠j) are measurements on Z made from two units involved in two independent samples, we have
Cov(Z(r)[n:n]i,Z(r)[n:n]j)=0,fori≠j. | (2.9) |
Assume that Z(r)[n,n]=(Z(r)[n:n]1,Z(r)[n:n]2,⋯,Z(r)[n:n]n)′. Then, by using (2.7), we obtain the mean vector of Z(r)[n,n] as:
E[Z(r)[n,n]]=σ2ζnr1, | (2.10) |
and by using (2.8) and (2.9), the dispersion matrix of Z(r)[n,n] can be obtained as:
D[Z(r)[n,n]]=σ22ψnrI, | (2.11) |
where 1 is a column vector of n ones and I is a unit matrix of order n. If ϕ>0 is involved in ζnr and ψnr, then (2.10) and (2.11) together defines a generalized Gauss–Markov setup, and hence the BLUE of σ2 is obtained as:
˜σn(r)2=1nζnrn∑i=1Z(r)[n:n]i, | (2.12) |
and its variance is obtained as
Var(˜σn(r)2)=ψnrn(ζnr)2σ22. | (2.13) |
If we take r=1 in the MSRSS method elucidated above, in this case, we get the usual single-stage RSS. Hence, the BLUE ˜σn(1)2 of σ2 is given by
˜σn(1)2=1nζnn∑i=1Z[n:n]i, | (2.14) |
with variance given by
Var(˜σn(1)2)=ψnn(ζn)2σ22, | (2.15) |
where we write Z[n:n]i instead of Z(1)[n:n]i that represents the measurement of the variable of primary interest of the unit selected by the RSS. Also, ζn and ψn obtained by putting r=1 in (2.5) and (2.6), respectively. Irshad et al. [15] computed the asymptotic variance of the MLE ^σ2 of σ2. Using those values, we have evaluated the ratio e(˜σn(1)2|^σ2)=Var(^σ2)Var(˜σn(1)2) for ϕ=0.25(0.25)1; n=2(2)20 as a measure of the efficiency of our estimator ˜σn(1)2 compared to the MLE ^σ2 based on n observations and the results are given in Table 1. From the table one can see that the efficiency increases with increases in ϕ and n. For example, when ϕ=0.5 and n=2,8, we have ˜σn(1)2=1.0596 and 1.1773, respectively. Also, for n=6 and ϕ=0.25,1, the efficiency values are ˜σn(1)2=1.0720 and 1.3580, respectively. However, it is clear that there is not much difference in the efficiency values for negative and positive values of ϕ.
n | ϕ | e(˜σn(1)2|ˆσ2) | ϕ | e(˜σ1(1)2|ˆσ2) |
0.25 | 1.0302 | -0.25 | 1.0302 | |
2 | 0.50 | 1.0596 | -0.50 | 1.0596 |
0.75 | 1.0863 | -0.75 | 1.0863 | |
1.00 | 1.1026 | -1.00 | 1.1030 | |
0.25 | 1.0588 | -0.25 | 1.0588 | |
4 | 0.50 | 1.1259 | -0.50 | 1.1259 |
0.75 | 1.1989 | -0.75 | 1.1989 | |
1.00 | 1.2702 | -1.00 | 1.2710 | |
0.25 | 1.0720 | -0.25 | 1.0720 | |
6 | 0.50 | 1.1577 | -0.50 | 1.1577 |
0.75 | 1.2563 | -0.75 | 1.2563 | |
1.00 | 1.3580 | -1.00 | 1.3600 | |
0.25 | 1.0800 | -0.25 | 1.0800 | |
8 | 0.50 | 1.1773 | -0.50 | 1.1773 |
0.75 | 1.2901 | -0.75 | 1.2901 | |
1.00 | 1.4123 | -1.00 | 1.4150 | |
0.25 | 1.0858 | -0.25 | 1.0858 | |
10 | 0.50 | 1.3273 | -0.50 | 1.3273 |
0.75 | 1.3119 | -0.75 | 1.3119 | |
1.00 | 1.4461 | -1.00 | 1.4490 | |
0.25 | 1.0882 | -0.25 | 1.0882 | |
12 | 0.50 | 1.1950 | -0.50 | 1.1950 |
0.75 | 1.3292 | -0.75 | 1.3292 | |
1.00 | 1.4733 | -1.00 | 1.4733 | |
0.25 | 1.0882 | -0.25 | 1.0882 | |
14 | 0.50 | 1.2026 | -0.50 | 1.2026 |
0.75 | 1.3395 | -0.75 | 1.3395 | |
1.00 | 1.4979 | -1.00 | 1.4979 | |
0.25 | 1.0909 | -0.25 | 1.0909 | |
16 | 0.50 | 1.2105 | -0.50 | 1.2105 |
0.75 | 1.3475 | -0.75 | 1.3475 | |
1.00 | 1.5122 | -1.00 | 1.5122 | |
0.25 | 1.0951 | -0.25 | 1.0951 | |
18 | 0.50 | 1.2119 | -0.50 | 1.2119 |
0.75 | 1.3606 | -0.75 | 1.3606 | |
1.00 | 1.5249 | -1.00 | 1.5249 | |
0.25 | 1.0928 | -0.25 | 1.0928 | |
20 | 0.50 | 1.2170 | -0.50 | 1.2170 |
0.75 | 1.3583 | -0.75 | 1.3583 | |
1.00 | 1.5309 | -1.00 | 1.5309 |
Al-Saleh [18] has proposed the steady-state RSS by letting r go to +∞. If we apply the steady state RSS, then the asymptotic distribution of Z(r)[n:n]i is obtained from (2.2) with a pdf given by
f(∞)[n:n]i(z;ϕ)=6σ2e−2zσ2(1−e−zσ2)[1+ϕ(1−2e−2zσ2{3−2e−zσ2})]. | (3.1) |
Clearly, Z(∞)[n:n]i,i=1,2,⋯,n are independent and identically distributed (iid) rvs, each with pdf defined in (3.1). Then, Z(∞)[n:n]i,i=1,2,⋯,n, may be considered as steady-state rss of size n. Therefore, from (2.3) and (2.4), the mean and variance of Z(∞)[n:n]i for i=1,2,⋯,n are obtained as:
E[Z(∞)[n:n]i]=σ2[56+1960ϕ], | (3.2) |
and
Var[Z(∞)[n:n]i]=σ22[1336+2531800ϕ−3613600ϕ2]. | (3.3) |
Let Z(∞)[n:n]=(Z(∞)[n:n]1,Z(∞)[n:n]2,⋯,Z(∞)[n:n]n)′. Then, the BLUE ˜σn(∞)2 based on Z(∞)[n:n] and the variance of ˜σn(∞)2 is obtained by taking the limit as r→∞ in (2.12) and (2.13), respectively, and are given by
˜σn(∞)2=1n[56+1960ϕ]n∑i=1Z(∞)[n:n]i, | (3.4) |
and
Var(˜σn(∞)2)=[1336+2531800ϕ−3613600ϕ2]n[56+1960ϕ]2σ22. | (3.5) |
Using the results of [15], we obtain the efficiency of ˜σn(∞)2 relative to ^σ2 by taking the ratio of Var(^σ2) with respect to Var(˜σn(∞)2) and is given by:
E(˜σn(∞)2|^σ2)=Var(^σ2)Var(˜σn(∞)2)=I(−1)σ2(ϕ)[56+1960ϕ]2[1336+2531800ϕ−3613600ϕ2]. |
Thus, the efficiency e(˜σn(∞)2|^σ2) is independent of the sample size n, and that is for a fixed ϕ, the e(˜σn(∞)2|^σ2) has the same values for all n.
Additionally, we have computed the value of e(˜σn(∞)2|^σ2) for ϕ=0.25(0.25)1, and the results are given in Table 2. From the table one can observe that the efficiency of ˜σn(∞)2 increases as ϕ increases. Moreover, the estimator ˜σn(∞)2 possesses the highest efficiency among other estimators of σ2 considered in this study, and the value of the efficiencies varies from 1.1073 to 1.6352.
ϕ | e(˜σn(∞)2|^σ2) | ϕ | e(˜σ1(∞)2|^σ2) |
0.25 | 1.1073 | -0.25 | 1.1073 |
0.50 | 1.2472 | -0.50 | 1.2472 |
0.75 | 1.4223 | -0.75 | 1.4226 |
1.00 | 1.6352 | -1.00 | 1.6365 |
As mentioned earlier in the case of FGMBBD, when ϕ<0, the concomitant of the smallest order statistic contains the maximum FI about σ2. Consequently, we explore an MSRSS in this scenario, where we select a unit of a sample with the smallest value on the auxiliary variable as the units of ranked sets at each step and from each set, with the goal of maximizing FI on the final selected rss.
Let Z(r)[1:n]i,i=1,2,⋯,n, represents the value measured on the primary variable of interest for the units selected at the rth stage of the MSRSS. Then, it is simple to see that Z(r)[1:n]i, the concomitant of the least order statistic of nr iid bivariate random variables with FGMBBD, has the same distribution as Z[1:nr]. Moreover, Z(r)[1:n]i,i=1,2,⋯,n, are also independently distributed with pdf given by:
f(r)[1:n]i(z;ϕ)=6σ2e−2zσ2(1−e−zσ2)[1−ϕ(nr−1nr+1)(1−2e−2zσ2{3−2e−zσ2})]. | (3.6) |
Clearly, from (2.2) and (3.6), we have:
f(r)[1:n]i(z;ϕ)=f(r)[n:n]i(z;−ϕ). | (3.7) |
Hence, it is obvious that E(Z(r)[n:n]i) for ϕ>0 and E(Z(r)[1:n]i) for ϕ<0 are the same. Similarly, Var(Z(r)[n:n]i) for ϕ>0 and Var(Z(r)[1:n]i) for ϕ<0 are the same. Consequently, if ˜σ1(r)2 is the BLUE of σ2 involved in the FGMBBD for ϕ<0 based on the MSRSS observations Z(r)[1:n]i,i=1,2,⋯,n, then the coefficients of Z(r)[1:n]i,i=1,2,⋯,n in the BLUE ˜σ1(r)2 for ϕ<0 the same as the coefficients of Z(r)[n:n]i,i=1,2,⋯,n in the BLUE ˜σn(r)2 for ϕ>0.
Further, we have Var(˜σ1(r)2)=Var(˜σn(r)2). Hence, Var(˜σ1(1)2)=Var(˜σn(1)2) and Var(˜σ1(∞)2)=Var(˜σn(∞)2), where ˜σ1(1)2 is the BLUE of σ2 for ϕ<0 is the usual single-stage rss observations Z(r)[1:n]i,i=1,2,⋯,n and ˜σ1(∞)2 is the BLUE of σ2 for ϕ<0 based on the steady-state rss observations Z(∞)[1:n]i,i=1,2,⋯,n. We have computed the efficiency e(˜σ1(r)2|ˆσ2) of the BLUE ˜σ1(r)2 relative to ˆσ2, the MLE of σ2 for ϕ=−1,−0.75,−0.50,−0.25; n=2(2)20 and the results are included in Table 1. Also, we have computed e(˜σ1(∞)2|ˆσ2) for ϕ=−1,−0.75,−0.50,and−0.25, and the results are given in Table 2. Based on Table 2, one can infer that the efficiency increases as |ϕ| increases, and the value of the efficiency varies from 1.1073 to 1.6365.
Remark 3.1. If (W,Z) follows a FGMBBD with pdf given in (1.3), then the correlation coefficient between W and Z is given by
Corr(W,Z)=3611300ϕ,−1≤ϕ≤1. |
Obviously, the correlation coefficient is maximum when |ϕ| is as large as 1. From the tables, we can observe that the efficiencies of all estimators increase as |ϕ| increases for a given sample size. Consequently, we observe that more information about σ2 can be extracted from the RSS when |ϕ| is large. Therefore, we conclude that concomitant ranking is more effective for estimating σ2 when the modulus value of the dependence parameter ϕ is large (ϕ approaches to ±1).
Remark 3.2. The assumption that ϕ is known may be considered unrealistic in certain real-life situations. [8] addressed this issue by proposing a moment-type estimator for ϕ, which is expressed as follows:
ˆϕ={−1,ifϰ<−36113001300361ϰ,if−3611300≤ϰ≤36113001,ifϰ>3611300, |
where ϰ is the sample correlation from (W(i:n)i,Z[i:n]i) for i=1,2,⋯,n.
This study focuses on estimating the scale parameter of the primary variable Z using the MSRSS, where the ranking is based on the marginal observations of an auxiliary variable W. It is assumed that the joint distribution of (W,Z) follows the FGMBBD. Considering the dependence parameter ϕ is known, the BLUE for the scale parameter of Z under MSRSS as well as steady-state RSS schemes is presented. The efficiency of the suggested estimators is compared to the MLE based on the same number of measured units. The results indicate that the proposed estimators are more efficient than the classical estimators considered in this study. For further studies, one can estimate other parameters or use different modifications of RSS to estimate the same parameter.
Conceptualization, S.P.A., M.R.I., R.M., S.S.A., and A.I.A; methodology, S.P.A., M.R.I., and R.M.; software, S.P.A., M.R.I., A.I.A., and R.M.; validation, S.P.A., M.R.I., R.M., S.S.A., and A.I.A.; formal analysis, M.R.I., S.P.A., R.M., and A.I.A; writing-original draft preparation, M.R.I., S.P.A., R.M., A.I.A., and S.S.A.; writing-review and editing, M.R.I., S.P.A., R.M., A.I.A., and S.S.A.; visualization, M.R.I., S.P.A., R.M., A.I.A., and S.S.A. All authors have read and approved the final version of the manuscript for publication.
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.
There is no conflict of interest with the publication of this work.
[1] |
Klymchenko AS, Kreder R (2014) Fluorescent probes for lipid rafts: from model membranes to living cells. Chem Biol 21: 97-113. doi: 10.1016/j.chembiol.2013.11.009
![]() |
[2] |
Sankaram MB, Thompson TE (1991) Cholesterol-induced fluid-phase immiscibility in membranes. Proc Natl Acad Sci USA 88: 8686-8690. doi: 10.1073/pnas.88.19.8686
![]() |
[3] |
Veatch SL, Keller SL (2005) Seeing spots: complex phase behavior in simple membranes. Biochim Biophys Acta 1746: 172-185. doi: 10.1016/j.bbamcr.2005.06.010
![]() |
[4] |
Daly TA, Wang M, Regen SL (2011) The origin of cholesterol's condensing effect. Langmuir 27: 2159-2161. doi: 10.1021/la105039q
![]() |
[5] |
Hakobyan D, Heuer A (2014) Key molecular requirements for raft formation in lipid/cholesterol membranes. PLoS One 9: e87369. doi: 10.1371/journal.pone.0087369
![]() |
[6] |
Krause MR, Regen SL (2014) The structural role of cholesterol in cell membranes: from condensed bilayers to lipid rafts. Acc Chem Res 47: 3512-3521. doi: 10.1021/ar500260t
![]() |
[7] |
Feigenson GW (2006) Phase behavior of lipid mixtures. Nat Chem Biol 2: 560-563. doi: 10.1038/nchembio1106-560
![]() |
[8] |
Rog T, Pasenkiewicz-Gierula M, Vattulainen I, et al. (2009) Ordering effects of cholesterol and its analogues. Biochim Biophys Acta 1788: 97-121. doi: 10.1016/j.bbamem.2008.08.022
![]() |
[9] |
Garcia-Saez AJ, Schwille P (2010) Stability of lipid domains. FEBS Lett 584: 1653-1658. doi: 10.1016/j.febslet.2009.12.036
![]() |
[10] |
Shaw JE, Epand RF, Epand RM, et al. (2006) Correlated fluorescence-atomic force microscopy of membrane domains: structure of fluorescence probes determines lipid localization. Biophys J 90: 2170-2178. doi: 10.1529/biophysj.105.073510
![]() |
[11] |
Mouritsen OG, Zuckermann MJ (2004) What's so special about cholesterol? Lipids 39: 1101-1113. doi: 10.1007/s11745-004-1336-x
![]() |
[12] |
Wang J, Megha, London E (2004) Relationship between sterol/steroid structure and participation in ordered lipid domains (lipid rafts): implications for lipid raft structure and function. Biochemistry 43: 1010-1018. doi: 10.1021/bi035696y
![]() |
[13] |
Wenz JJ, Barrantes FJ (2003) Steroid structural requirements for stabilizing or disrupting lipid domains. Biochemistry 42: 14267-14276. doi: 10.1021/bi035759c
![]() |
[14] |
Xu X, Bittman R, Duportail G, et al. (2001) Effect of the structure of natural sterols and sphingolipids on the formation of ordered sphingolipid/sterol domains (rafts) Comparison of cholesterol to plant, fungal, and disease-associated sterols and comparison of sphingomyelin, cerebrosides, and ceramide. J Biol Chem 276: 33540-33546. doi: 10.1074/jbc.M104776200
![]() |
[15] |
Xu X, London E (2000) The effect of sterol structure on membrane lipid domains reveals how cholesterol can induce lipid domain formation. Biochemistry 39: 843-849. doi: 10.1021/bi992543v
![]() |
[16] | Bernardino de la Serna J, Schutz GJ, Eggeling C, et al. (2016) There is no simple model of the plasma membrane organization. Front Cell Dev Biol 4: art106. |
[17] |
van Meer G, Stelzer EH, Wijnaendts-van-Resandt RW, et al. (1987) Sorting of sphingolipids in epithelial (Madin-Darby canine kidney) cells. J Cell Biol 105: 1623-1635. doi: 10.1083/jcb.105.4.1623
![]() |
[18] |
Simons K, Ikonen E (1997) Functional rafts in cell membranes. Nature 387: 569-572. doi: 10.1038/42408
![]() |
[19] |
Gupta N, DeFranco AL (2007) Lipid rafts and B cell signaling. Semin Cell Dev Biol 18: 616-626. doi: 10.1016/j.semcdb.2007.07.009
![]() |
[20] |
Ikonen E (2001) Roles of lipid rafts in membrane transport. Curr Opin Cell Biol 13: 470-477. doi: 10.1016/S0955-0674(00)00238-6
![]() |
[21] |
Lasserre R, Guo XJ, Conchonaud F, et al. (2008) Raft nanodomains contribute to Akt/PKB plasma membrane recruitment and activation. Nat Chem Biol 4: 538-547. doi: 10.1038/nchembio.103
![]() |
[22] |
Orlowski S, Comera C, Terce F, et al. (2007) Lipid rafts: dream or reality for cholesterol transporters? Eur Biophys J 36: 869-885. doi: 10.1007/s00249-007-0193-8
![]() |
[23] |
Orlowski S, Martin S, Escargueil A (2006) P-glycoprotein and 'lipid rafts': some ambiguous mutual relationships (floating on them, building them or meeting them by chance?). Cell Mol Life Sci 63: 1038-1059. doi: 10.1007/s00018-005-5554-9
![]() |
[24] |
Owen DM, Magenau A, Williamson D, et al. (2012) The lipid raft hypothesis revisited—new insights on raft composition and function from super-resolution fluorescence microscopy. Bioessays 34: 739-747. doi: 10.1002/bies.201200044
![]() |
[25] | Simons K, Toomre D (2000) Lipid rafts and signal transduction. Nat Rev Mol Cell Biol 1: 31-39. |
[26] |
Carquin M, D'Auria L, Pollet H, et al. (2016) Recent progress on lipid lateral heterogeneity in plasma membranes: From rafts to submicrometric domains. Prog Lipid Res 62: 1-24. doi: 10.1016/j.plipres.2015.12.004
![]() |
[27] |
Jacobson K, Mouritsen OG, Anderson RG (2007) Lipid rafts: at a crossroad between cell biology and physics. Nat Cell Biol 9: 7-14. doi: 10.1038/ncb0107-7
![]() |
[28] |
Mayor S, Rao M (2004) Rafts: scale-dependent, active lipid organization at the cell surface. Traffic 5: 231-240. doi: 10.1111/j.1600-0854.2004.00172.x
![]() |
[29] |
Pike LJ (2004) Lipid rafts: heterogeneity on the high seas. Biochem J 378: 281-292. doi: 10.1042/bj20031672
![]() |
[30] |
Simons K, Gerl MJ (2010) Revitalizing membrane rafts: new tools and insights. Nat Rev Mol Cell Biol 11: 688-699. doi: 10.1038/nrm2977
![]() |
[31] |
Wustner D (2007) Fluorescent sterols as tools in membrane biophysics and cell biology. Chem Phys Lipids 146: 1-25. doi: 10.1016/j.chemphyslip.2006.12.004
![]() |
[32] |
Baker BY, Epand RF, Epand RM, et al. (2007) Cholesterol binding does not predict activity of the steroidogenic acute regulatory protein, StAR. J Biol Chem 282: 10223-10232. doi: 10.1074/jbc.M611221200
![]() |
[33] |
Honigmann A, Pralle A (2016) Compartmentalization of the Cell Membrane. J Mol Biol 428: 4739-4748. doi: 10.1016/j.jmb.2016.09.022
![]() |
[34] |
Lenne PF, Wawrezinieck L, Conchonaud F, et al. (2006) Dynamic molecular confinement in the plasma membrane by microdomains and the cytoskeleton meshwork. EMBO J 25: 3245-3256. doi: 10.1038/sj.emboj.7601214
![]() |
[35] | van Zanten TS, Mayor S (2015) Current approaches to studying membrane organization. F1000Research: 6868. |
[36] |
Lingwood D, Simons K (2010) Lipid rafts as a membrane-organizing principle. Science 327: 46-50. doi: 10.1126/science.1174621
![]() |
[37] |
Devaux PF, Morris R (2004) Transmembrane asymmetry and lateral domains in biological membranes. Traffic 5: 241-246. doi: 10.1111/j.1600-0854.2004.0170.x
![]() |
[38] |
Lorent JH, Levental I (2015) Structural determinants of protein partitioning into ordered membrane domains and lipid rafts. Chem Phys Lipids 192: 23-32. doi: 10.1016/j.chemphyslip.2015.07.022
![]() |
[39] |
Chum T, Glatzova D, Kvicalova Z, et al. (2016) The role of palmitoylation and transmembrane domain in sorting of transmembrane adaptor proteins. J Cell Sci 129: 95-107. doi: 10.1242/jcs.175190
![]() |
[40] |
Levental I, Grzybek M, Simons K (2010) Greasing their way: lipid modifications determine protein association with membrane rafts. Biochemistry 49: 6305-6316. doi: 10.1021/bi100882y
![]() |
[41] |
Anderson RG, Jacobson K (2002) A role for lipid shells in targeting proteins to caveolae, rafts, and other lipid domains. Science 296: 1821-1825. doi: 10.1126/science.1068886
![]() |
[42] |
Kaiser HJ, Orlowski A, Rog T, et al. (2011) Lateral sorting in model membranes by cholesterol-mediated hydrophobic matching. Proc Natl Acad Sci USA 108: 16628-16633. doi: 10.1073/pnas.1103742108
![]() |
[43] |
Nyholm TK (2015) Lipid-protein interplay and lateral organization in biomembranes. Chem Phys Lipids 189: 48-55. doi: 10.1016/j.chemphyslip.2015.05.008
![]() |
[44] |
Fujiwara T, Ritchie K, Murakoshi H, et al. (2002) Phospholipids undergo hop diffusion in compartmentalized cell membrane. J Cell Biol 157: 1071-1081. doi: 10.1083/jcb.200202050
![]() |
[45] | Braccia A, Villani M, Immerdal L, et al. (2003) Microvillar membrane microdomains exist at physiological temperature. Role of galectin-4 as lipid raft stabilizer revealed by "superrafts". J Biol Chem 278: 15679-15684. |
[46] |
Hansen GH, Immerdal L, Thorsen E, et al. (2001) Lipid rafts exist as stable cholesterol-independent microdomains in the brush border membrane of enterocytes. J Biol Chem 276: 32338-32344. doi: 10.1074/jbc.M102667200
![]() |
[47] |
Danielsen EM, Hansen GH (2003) Lipid rafts in epithelial brush borders: atypical membrane microdomains with specialized functions. Biochim Biophys Acta 1617: 1-9. doi: 10.1016/j.bbamem.2003.09.005
![]() |
[48] |
Meder D, Moreno MJ, Verkade P, et al. (2006) Phase coexistence and connectivity in the apical membrane of polarized epithelial cells. Proc Natl Acad Sci USA 103: 329-334. doi: 10.1073/pnas.0509885103
![]() |
[49] |
Lingwood D, Ries J, Schwille P, et al. (2008) Plasma membranes are poised for activation of raft phase coalescence at physiological temperature. Proc Natl Acad Sci USA 105: 10005-10010. doi: 10.1073/pnas.0804374105
![]() |
[50] |
Mizuno H, Abe M, Dedecker P, et al. (2011) Fluorescent probes for superresolution imaging of lipid domains on the plasma membrane. Chem Sci 2: 1548-1553. doi: 10.1039/c1sc00169h
![]() |
[51] |
Carquin M, Conrard L, Pollet H, et al. (2015) Cholesterol segregates into submicrometric domains at the living erythrocyte membrane: evidence and regulation. Cell Mol Life Sci 72: 4633-4651. doi: 10.1007/s00018-015-1951-x
![]() |
[52] |
Carquin M, Pollet H, Veiga-da-Cunha M, et al. (2014) Endogenous sphingomyelin segregates into submicrometric domains in the living erythrocyte membrane. J Lipid Res 55: 1331-1342. doi: 10.1194/jlr.M048538
![]() |
[53] |
D'Auria L, Van der Smissen P, Bruyneel F, et al. (2011) Segregation of fluorescent membrane lipids into distinct micrometric domains: evidence for phase compartmentation of natural lipids? PLoS One 6: e17021. doi: 10.1371/journal.pone.0017021
![]() |
[54] |
Frisz JF, Klitzing HA, Lou K, et al. (2013) Sphingolipid domains in the plasma membranes of fibroblasts are not enriched with cholesterol. J Biol Chem 288: 16855-16861. doi: 10.1074/jbc.M113.473207
![]() |
[55] |
Frisz JF, Lou K, Klitzing HA, et al. (2013) Direct chemical evidence for sphingolipid domains in the plasma membranes of fibroblasts. Proc Natl Acad Sci USA 110: E613-622. doi: 10.1073/pnas.1216585110
![]() |
[56] |
Magenau A, Benzing C, Proschogo N, et al. (2011) Phagocytosis of IgG-coated polystyrene beads by macrophages induces and requires high membrane order. Traffic 12: 1730-1743. doi: 10.1111/j.1600-0854.2011.01272.x
![]() |
[57] |
Stancevic B, Kolesnick R (2010) Ceramide-rich platforms in transmembrane signaling. FEBS Lett 584: 1728-1740. doi: 10.1016/j.febslet.2010.02.026
![]() |
[58] |
Garcia-Ruiz C, Morales A, Fernandez-Checa JC (2015) Glycosphingolipids and cell death: one aim, many ways. Apoptosis 20: 607-620. doi: 10.1007/s10495-015-1092-6
![]() |
[59] |
Contreras FX, Sot J, Ruiz-Arguello MB, et al. (2004) Cholesterol modulation of sphingomyelinase activity at physiological temperatures. Chem Phys Lipids 130: 127-134. doi: 10.1016/j.chemphyslip.2004.02.003
![]() |
[60] |
Cheng JP, Nichols BJ (2016) Caveolae: One Function or Many? Trends Cell Biol 26: 177-189. doi: 10.1016/j.tcb.2015.10.010
![]() |
[61] |
Parton RG, del Pozo MA (2013) Caveolae as plasma membrane sensors, protectors and organizers. Nat Rev Mol Cell Biol 14: 98-112. doi: 10.1038/nrm3512
![]() |
[62] |
Parton RG, Simons K (2007) The multiple faces of caveolae. Nat Rev Mol Cell Biol 8: 185-194. doi: 10.1038/nrm2122
![]() |
[63] |
Shvets E, Ludwig A, Nichols BJ (2014) News from the caves: update on the structure and function of caveolae. Curr Opin Cell Biol 29: 99-106. doi: 10.1016/j.ceb.2014.04.011
![]() |
[64] |
Fridolfsson HN, Roth DM, Insel PA, et al. (2014) Regulation of intracellular signaling and function by caveolin. FASEB J 28: 3823-3831. doi: 10.1096/fj.14-252320
![]() |
[65] |
Banning A, Tomasovic A, Tikkanen R (2011) Functional aspects of membrane association of reggie/flotillin proteins. Curr Protein Pept Sci 12: 725-735. doi: 10.2174/138920311798841708
![]() |
[66] |
Ortegren U, Yin L, Ost A, et al. (2006) Separation and characterization of caveolae subclasses in the plasma membrane of primary adipocytes; segregation of specific proteins and functions. FEBS J 273: 3381-3392. doi: 10.1111/j.1742-4658.2006.05345.x
![]() |
[67] |
Sinha B, Koster D, Ruez R, et al. (2011) Cells respond to mechanical stress by rapid disassembly of caveolae. Cell 144: 402-413. doi: 10.1016/j.cell.2010.12.031
![]() |
[68] |
Roper K, Corbeil D, Huttner WB (2000) Retention of prominin in microvilli reveals distinct cholesterol-based lipid micro-domains in the apical plasma membrane. Nat Cell Biol 2: 582-592. doi: 10.1038/35023524
![]() |
[69] |
Zimmerman B, Kelly B, McMillan BJ, et al. (2016) Crystal structure of a full-length human tetraspanin reveals a cholesterol-binding pocket. Cell 167: 1041-1051. doi: 10.1016/j.cell.2016.09.056
![]() |
[70] |
Hemler ME (2005) Tetraspanin functions and associated microdomains. Nat Rev Mol Cell Biol 6: 801-811. doi: 10.1038/nrm1736
![]() |
[71] |
Yanez-Mo M, Barreiro O, Gordon-Alonso M, et al. (2009) Tetraspanin-enriched microdomains: a functional unit in cell plasma membranes. Trends Cell Biol 19: 434-446. doi: 10.1016/j.tcb.2009.06.004
![]() |
[72] |
Zhang XA, Huang C (2012) Tetraspanins and cell membrane tubular structures. Cell Mol Life Sci 69: 2843-2852. doi: 10.1007/s00018-012-0954-0
![]() |
[73] |
Hugel B, Martinez MC, Kunzelmann C, et al. (2005) Membrane microparticles: two sides of the coin. Physiology (Bethesda) 20: 22-27. doi: 10.1152/physiol.00029.2004
![]() |
[74] |
Raposo G, Stoorvogel W (2013) Extracellular vesicles: exosomes, microvesicles, and friends. J Cell Biol 200: 373-383. doi: 10.1083/jcb.201211138
![]() |
[75] |
van der Pol E, Boing AN, Harrison P, et al. (2012) Classification, functions, and clinical relevance of extracellular vesicles. Pharmacol Rev 64: 676-705. doi: 10.1124/pr.112.005983
![]() |
[76] | Kravets FG, Lee J, Singh B, et al. (2000) Prostasomes: current concepts. Prostate 43: 169-174. |
[77] |
Llorente A, van Deurs B, Sandvig K (2007) Cholesterol regulates prostasome release from secretory lysosomes in PC-3 human prostate cancer cells. Eur J Cell Biol 86: 405-415. doi: 10.1016/j.ejcb.2007.05.001
![]() |
[78] |
Arvidson G, Ronquist G, Wikander G, et al. (1989) Human prostasome membranes exhibit very high cholesterol/phospholipid ratios yielding high molecular ordering. Biochim Biophys Acta 984: 167-173. doi: 10.1016/0005-2736(89)90212-5
![]() |
[79] |
Llorente A, de Marco MC, Alonso MA (2004) Caveolin-1 and MAL are located on prostasomes secreted by the prostate cancer PC-3 cell line. J Cell Sci 117: 5343-5351. doi: 10.1242/jcs.01420
![]() |
[80] |
Kastelowitz N, Yin H (2014) Exosomes and microvesicles: identification and targeting by particle size and lipid chemical probes. ChemBioChem 15: 923-928. doi: 10.1002/cbic.201400043
![]() |
[81] |
Bagnat M, Keranen S, Shevchenko A, et al. (2000) Lipid rafts function in biosynthetic delivery of proteins to the cell surface in yeast. Proc Natl Acad Sci USA 97: 3254-3259. doi: 10.1073/pnas.97.7.3254
![]() |
[82] |
Dupre S, Haguenauer-Tsapis R (2003) Raft partitioning of the yeast uracil permease during trafficking along the endocytic pathway. Traffic 4: 83-96. doi: 10.1034/j.1600-0854.2003.40204.x
![]() |
[83] |
Lauwers E, Andre B (2006) Association of yeast transporters with detergent-resistant membranes correlates with their cell-surface location. Traffic 7: 1045-1059. doi: 10.1111/j.1600-0854.2006.00445.x
![]() |
[84] |
Malinska K, Malinsky J, Opekarova M, et al. (2003) Visualization of protein compartmentation within the plasma membrane of living yeast cells. Mol Biol Cell 14: 4427-4436. doi: 10.1091/mbc.E03-04-0221
![]() |
[85] |
Malinska K, Malinsky J, Opekarova M, et al. (2004) Distribution of Can1p into stable domains reflects lateral protein segregation within the plasma membrane of living S. cerevisiae cells. J Cell Sci 117: 6031-6041. doi: 10.1242/jcs.01493
![]() |
[86] |
Spira F, Mueller NS, Beck G, et al. (2012) Patchwork organization of the yeast plasma membrane into numerous coexisting domains. Nat Cell Biol 14: 640-648. doi: 10.1038/ncb2487
![]() |
[87] | Bagnat M, Simons K (2002) Lipid rafts in protein sorting and cell polarity in budding yeast Saccharomyces cerevisiae. Biol Chem 383: 1475-1480. |
[88] | Wachtler V, Balasubramanian MK (2006) Yeast lipid rafts?—an emerging view. Trends Cell Biol 16: 1-4. |
[89] |
Klose C, Ejsing CS, Garcia-Saez AJ, et al. (2010) Yeast lipids can phase-separate into micrometer-scale membrane domains. J Biol Chem 285: 30224-30232. doi: 10.1074/jbc.M110.123554
![]() |
[90] |
Aresta-Branco F, Cordeiro AM, Marinho HS, et al. (2011) Gel domains in the plasma membrane of Saccharomyces cerevisiae: highly ordered, ergosterol-free, and sphingolipid-enriched lipid rafts. J Biol Chem 286: 5043-5054. doi: 10.1074/jbc.M110.154435
![]() |
[91] |
Mongrand S, Stanislas T, Bayer EM, et al. (2010) Membrane rafts in plant cells. Trends Plant Sci 15: 656-663. doi: 10.1016/j.tplants.2010.09.003
![]() |
[92] |
Tapken W, Murphy AS (2015) Membrane nanodomains in plants: capturing form, function, and movement. J Exp Bot 66: 1573-1586. doi: 10.1093/jxb/erv054
![]() |
[93] |
Malinsky J, Opekarova M, Grossmann G, et al. (2013) Membrane microdomains, rafts, and detergent-resistant membranes in plants and fungi. Annu Rev Plant Biol 64: 501-529. doi: 10.1146/annurev-arplant-050312-120103
![]() |
[94] |
Lopez D, Kolter R (2010) Functional microdomains in bacterial membranes. Genes Dev 24: 1893-1902. doi: 10.1101/gad.1945010
![]() |
[95] |
Lopez D (2015) Molecular composition of functional microdomains in bacterial membranes. Chem Phys Lipids 192: 3-11. doi: 10.1016/j.chemphyslip.2015.08.015
![]() |
[96] |
Saenz JP, Grosser D, Bradley AS, et al. (2015) Hopanoids as functional analogues of cholesterol in bacterial membranes. Proc Natl Acad Sci USA 112: 11971-11976. doi: 10.1073/pnas.1515607112
![]() |
[97] |
Huang Z, London E (2016) Cholesterol lipids and cholesterol-containing lipid rafts in bacteria. Chem Phys Lipids 199: 11-16. doi: 10.1016/j.chemphyslip.2016.03.002
![]() |
[98] |
Haidekker MA, Theodorakis EA (2007) Molecular rotors--fluorescent biosensors for viscosity and flow. Org Biomol Chem 5: 1669-1678. doi: 10.1039/B618415D
![]() |
[99] |
Demchenko AP, Mely Y, Duportail G, et al. (2009) Monitoring biophysical properties of lipid membranes by environment-sensitive fluorescent probes. Biophys J 96: 3461-3470. doi: 10.1016/j.bpj.2009.02.012
![]() |
[100] |
Bagatolli LA (2006) To see or not to see: lateral organization of biological membranes and fluorescence microscopy. Biochim Biophys Acta 1758: 1541-1556. doi: 10.1016/j.bbamem.2006.05.019
![]() |
[101] |
Gaus K, Zech T, Harder T (2006) Visualizing membrane microdomains by Laurdan 2-photon microscopy. Mol Membr Biol 23: 41-48. doi: 10.1080/09687860500466857
![]() |
[102] |
Gaus K, Gratton E, Kable EP, et al. (2003) Visualizing lipid structure and raft domains in living cells with two-photon microscopy. Proc Natl Acad Sci USA 100: 15554-15559. doi: 10.1073/pnas.2534386100
![]() |
[103] |
Sezgin E, Sadowski T, Simons K (2014) Measuring lipid packing of model and cellular membranes with environment sensitive probes. Langmuir 30: 8160-8166. doi: 10.1021/la501226v
![]() |
[104] |
Kucherak OA, Oncul S, Darwich Z, et al. (2010) Switchable nile red-based probe for cholesterol and lipid order at the outer leaflet of biomembranes. J Am Chem Soc 132: 4907-4916. doi: 10.1021/ja100351w
![]() |
[105] |
Baumgart T, Hunt G, Farkas ER, et al. (2007) Fluorescence probe partitioning between Lo/Ld phases in lipid membranes. Biochim Biophys Acta 1768: 2182-2194. doi: 10.1016/j.bbamem.2007.05.012
![]() |
[106] | Solanko KA, Modzel M, Solanko LM, et al. (2015) Fluorescent sterols and cholesteryl esters as probes for intracellular cholesterol transport. Lipid Insights 8: 95-114. |
[107] |
Gimpl G, Gehrig-Burger K (2007) Cholesterol reporter molecules. Biosci Rep 27: 335-358. doi: 10.1007/s10540-007-9060-1
![]() |
[108] |
Fujimoto T, Hayashi M, Iwamoto M, et al. (1997) Crosslinked plasmalemmal cholesterol is sequestered to caveolae: analysis with a new cytochemical probe. J Histochem Cytochem 45: 1197-1205. doi: 10.1177/002215549704500903
![]() |
[109] |
Shimada Y, Maruya M, Iwashita S, et al. (2002) The C-terminal domain of perfringolysin O is an essential cholesterol-binding unit targeting to cholesterol-rich microdomains. Eur J Biochem 269: 6195-6203. doi: 10.1046/j.1432-1033.2002.03338.x
![]() |
[110] | Liu SL, Sheng R, Jung JH, et al. (2017) Orthogonal lipid sensors identify transbilayer asymmetry of plasma membrane cholesterol. Nat Chem Biol 13: 268-274. |
[111] |
Koivusalo M, Alvesalo J, Virtanen JA, et al. (2004) Partitioning of pyrene-labeled phospho- and sphingolipids between ordered and disordered bilayer domains. Biophys J 86: 923-935. doi: 10.1016/S0006-3495(04)74168-5
![]() |
[112] |
Koivusalo M, Jansen M, Somerharju P, et al. (2007) Endocytic trafficking of sphingomyelin depends on its acyl chain length. Mol Biol Cell 18: 5113-5123. doi: 10.1091/mbc.E07-04-0330
![]() |
[113] |
Pagano RE, Watanabe R, Wheatley C, et al. (1999) Use of N-[5-(5,7-dimethyl boron dipyrromethene difluoride-sphingomyelin to study membrane traffic along the endocytic pathway. Chem Phys Lipids 102: 55-63. doi: 10.1016/S0009-3084(99)00075-4
![]() |
[114] |
Scheidt HA, Muller P, Herrmann A, et al. (2003) The potential of fluorescent and spin-labeled steroid analogs to mimic natural cholesterol. J Biol Chem 278: 45563-45569. doi: 10.1074/jbc.M303567200
![]() |
[115] |
Smutzer G, Crawford BF, Yeagle PL (1986) Physical properties of the fluorescent sterol probe dehydroergosterol. Biochim Biophys Acta 862: 361-371. doi: 10.1016/0005-2736(86)90239-7
![]() |
[116] |
Ohvo-Rekila H, Akerlund B, Slotte JP (2000) Cyclodextrin-catalyzed extraction of fluorescent sterols from monolayer membranes and small unilamellar vesicles. Chem Phys Lipids 105: 167-178. doi: 10.1016/S0009-3084(00)00122-5
![]() |
[117] |
Hao M, Lin SX, Karylowski OJ, et al. (2002) Vesicular and non-vesicular sterol transport in living cells The endocytic recycling compartment is a major sterol storage organelle. J Biol Chem 277: 609-617. doi: 10.1074/jbc.M108861200
![]() |
[118] |
Wustner D, Herrmann A, Hao M, et al. (2002) Rapid nonvesicular transport of sterol between the plasma membrane domains of polarized hepatic cells. J Biol Chem 277: 30325-30336. doi: 10.1074/jbc.M202626200
![]() |
[119] |
Wustner D, Mondal M, Tabas I, et al. (2005) Direct observation of rapid internalization and intracellular transport of sterol by macrophage foam cells. Traffic 6: 396-412. doi: 10.1111/j.1600-0854.2005.00285.x
![]() |
[120] |
Frolov A, Petrescu A, Atshaves BP, et al. (2000) High density lipoprotein-mediated cholesterol uptake and targeting to lipid droplets in intact L-cell fibroblasts A single- and multiphoton fluorescence approach. J Biol Chem 275: 12769-12780. doi: 10.1074/jbc.275.17.12769
![]() |
[121] | Wustner D, Mondal M, Huang A, et al. (2004) Different transport routes for high density lipoprotein and its associated free sterol in polarized hepatic cells. J Lipid Res 45: 427-437. |
[122] |
Liu J, Chang CC, Westover EJ, et al. (2005) Investigating the allosterism of acyl-CoA:cholesterol acyltransferase (ACAT) by using various sterols: in vitro and intact cell studies. Biochem J 391: 389-397. doi: 10.1042/BJ20050428
![]() |
[123] |
Mukherjee S, Zha X, Tabas I, et al. (1998) Cholesterol distribution in living cells: fluorescence imaging using dehydroergosterol as a fluorescent cholesterol analog. Biophys J 75: 1915-1925. doi: 10.1016/S0006-3495(98)77632-5
![]() |
[124] |
Sato SB, Ishii K, Makino A, et al. (2004) Distribution and transport of cholesterol-rich membrane domains monitored by a membrane-impermeant fluorescent polyethylene glycol-derivatized cholesterol. J Biol Chem 279: 23790-23796. doi: 10.1074/jbc.M313568200
![]() |
[125] |
Mazeres S, Schram V, Tocanne JF, et al. (1996) 7-nitrobenz-2-oxa-1,3-diazole-4-yl-labeled phospholipids in lipid membranes: differences in fluorescence behavior. Biophys J 71: 327-335. doi: 10.1016/S0006-3495(96)79228-7
![]() |
[126] |
Loura LM, Fedorov A, Prieto M (2001) Exclusion of a cholesterol analog from the cholesterol-rich phase in model membranes. Biochim Biophys Acta 1511: 236-243. doi: 10.1016/S0005-2736(01)00269-3
![]() |
[127] | Sezgin E, Can FB, Schneider F, et al. (2016) A comparative study on fluorescent cholesterol analogs as versatile cellular reporters. J Lipid Res 57: 299-309. |
[128] |
Gaibelet G, Allart S, Terce F, et al. (2015) Specific cellular incorporation of a pyrene-labelled cholesterol: lipoprotein-mediated delivery toward ordered intracellular membranes. PLoS One 10: e0121563. doi: 10.1371/journal.pone.0121563
![]() |
[129] |
Dagher G, Donne N, Klein C, et al. (2003) HDL-mediated cholesterol uptake and targeting to lipid droplets in adipocytes. J Lipid Res 44: 1811-1820. doi: 10.1194/jlr.M300267-JLR200
![]() |
[130] | Sparrow CP, Patel S, Baffic J, et al. (1999) A fluorescent cholesterol analog traces cholesterol absorption in hamsters and is esterified in vivo and in vitro. J Lipid Res 40: 1747-1757. |
[131] | Wiegand V, Chang TY, Strauss JF, 3rd, et al. (2003) Transport of plasma membrane-derived cholesterol and the function of Niemann-Pick C1 Protein. FASEB J 17: 782-784. |
[132] |
Shrivastava S, Haldar S, Gimpl G, et al. (2009) Orientation and dynamics of a novel fluorescent cholesterol analogue in membranes of varying phase. J Phys Chem B 113: 4475-4481. doi: 10.1021/jp808309u
![]() |
[133] |
Huang H, McIntosh AL, Atshaves BP, et al. (2010) Use of dansyl-cholestanol as a probe of cholesterol behavior in membranes of living cells. J Lipid Res 51: 1157-1172. doi: 10.1194/jlr.M003244
![]() |
[134] |
Li Z, Mintzer E, Bittman R (2006) First synthesis of free cholesterol-BODIPY conjugates. J Org Chem 71: 1718-1721. doi: 10.1021/jo052029x
![]() |
[135] |
Bergstrom F, Mikhalyov I, Hagglof P, et al. (2002) Dimers of dipyrrometheneboron difluoride (BODIPY) with light spectroscopic applications in chemistry and biology. J Am Chem Soc 124: 196-204. doi: 10.1021/ja010983f
![]() |
[136] |
Marks DL, Bittman R, Pagano RE (2008) Use of Bodipy-labeled sphingolipid and cholesterol analogs to examine membrane microdomains in cells. Histochem Cell Biol 130: 819-832. doi: 10.1007/s00418-008-0509-5
![]() |
[137] |
Ariola FS, Li Z, Cornejo C, et al. (2009) Membrane fluidity and lipid order in ternary giant unilamellar vesicles using a new bodipy-cholesterol derivative. Biophys J 96: 2696-2708. doi: 10.1016/j.bpj.2008.12.3922
![]() |
[138] |
Wustner D, Solanko L, Sokol E, et al. (2011) Quantitative assessment of sterol traffic in living cells by dual labeling with dehydroergosterol and BODIPY-cholesterol. Chem Phys Lipids 164: 221-235. doi: 10.1016/j.chemphyslip.2011.01.004
![]() |
[139] |
Solanko LM, Honigmann A, Midtiby HS, et al. (2013) Membrane orientation and lateral diffusion of BODIPY-cholesterol as a function of probe structure. Biophys J 105: 2082-2092. doi: 10.1016/j.bpj.2013.09.031
![]() |
[140] |
Holtta-Vuori M, Uronen RL, Repakova J, et al. (2008) BODIPY-cholesterol: a new tool to visualize sterol trafficking in living cells and organisms. Traffic 9: 1839-1849. doi: 10.1111/j.1600-0854.2008.00801.x
![]() |
[141] |
Mazeres S, Lagane B, Welby M, et al. (2001) Probing the lateral organization of membranes: fluorescence repercussions of pyrene probe distribution. Spectrochim Acta A Mol Biomol Spectrosc 57: 2297-2311. doi: 10.1016/S1386-1425(01)00486-3
![]() |
[142] |
Le Guyader L, Le Roux C, Mazeres S, et al. (2007) Changes of the membrane lipid organization characterized by means of a new cholesterol-pyrene probe. Biophys J 93: 4462-4473. doi: 10.1529/biophysj.107.112821
![]() |
[143] |
Lagane B, Mazeres S, Le Grimellec C, et al. (2002) Lateral distribution of cholesterol in membranes probed by means of a pyrene-labelled cholesterol: effects of acyl chain unsaturation. Biophys Chem 95: 7-22. doi: 10.1016/S0301-4622(01)00235-6
![]() |
[144] |
Lecompte MF, Gaibelet G, Lebrun C, et al. (2015) Cholesterol and sphingomyelin-containing model condensed lipid monolayers: heterogeneities involving ordered microdomains assessed by two cholesterol derivatives. Langmuir 31: 11921-11931. doi: 10.1021/acs.langmuir.5b02646
![]() |
[145] | Astarie-Dequeker C, Le Guyader L, Malaga W, et al. (2009) Phthiocerol dimycocerosates of M. tuberculosis participate in macrophage invasion by inducing changes in the organization of plasma membrane lipids. PLoS Pathog 5: e1000289. |
[146] |
Rohrl C, Meisslitzer-Ruppitsch C, Bittman R, et al. (2012) Combined light and electron microscopy using diaminobenzidine photooxidation to monitor trafficking of lipids derived from lipoprotein particles. Curr Pharm Biotechnol 13: 331-340. doi: 10.2174/138920112799095338
![]() |
[147] |
Holtta-Vuori M, Sezgin E, Eggeling C, et al. (2016) Use of BODIPY-Cholesterol (TF-Chol) for Visualizing Lysosomal Cholesterol Accumulation. Traffic 17: 1054-1057. doi: 10.1111/tra.12414
![]() |
[148] |
Gaibelet G, Terce F, Bertrand-Michel J, et al. (2013) 21-Methylpyrenyl-cholesterol stably and specifically associates with lipoprotein peripheral hemi-membrane: a new labelling tool. Biochem Biophys Res Commun 440: 533-538. doi: 10.1016/j.bbrc.2013.09.101
![]() |
[149] |
Storey SM, Gallegos AM, Atshaves BP, et al. (2007) Selective cholesterol dynamics between lipoproteins and caveolae/lipid rafts. Biochemistry 46: 13891-13906. doi: 10.1021/bi700690s
![]() |
[150] |
Ohsaki Y, Cheng J, Suzuki M, et al. (2009) Biogenesis of cytoplasmic lipid droplets: from the lipid ester globule in the membrane to the visible structure. Biochim Biophys Acta 1791: 399-407. doi: 10.1016/j.bbalip.2008.10.002
![]() |
[151] |
Mukherjee S, Soe TT, Maxfield FR (1999) Endocytic sorting of lipid analogues differing solely in the chemistry of their hydrophobic tails. J Cell Biol 144: 1271-1284. doi: 10.1083/jcb.144.6.1271
![]() |
[152] |
Prinz WA (2007) Non-vesicular sterol transport in cells. Prog Lipid Res 46: 297-314. doi: 10.1016/j.plipres.2007.06.002
![]() |
n | ϕ | e(˜σn(1)2|ˆσ2) | ϕ | e(˜σ1(1)2|ˆσ2) |
0.25 | 1.0302 | -0.25 | 1.0302 | |
2 | 0.50 | 1.0596 | -0.50 | 1.0596 |
0.75 | 1.0863 | -0.75 | 1.0863 | |
1.00 | 1.1026 | -1.00 | 1.1030 | |
0.25 | 1.0588 | -0.25 | 1.0588 | |
4 | 0.50 | 1.1259 | -0.50 | 1.1259 |
0.75 | 1.1989 | -0.75 | 1.1989 | |
1.00 | 1.2702 | -1.00 | 1.2710 | |
0.25 | 1.0720 | -0.25 | 1.0720 | |
6 | 0.50 | 1.1577 | -0.50 | 1.1577 |
0.75 | 1.2563 | -0.75 | 1.2563 | |
1.00 | 1.3580 | -1.00 | 1.3600 | |
0.25 | 1.0800 | -0.25 | 1.0800 | |
8 | 0.50 | 1.1773 | -0.50 | 1.1773 |
0.75 | 1.2901 | -0.75 | 1.2901 | |
1.00 | 1.4123 | -1.00 | 1.4150 | |
0.25 | 1.0858 | -0.25 | 1.0858 | |
10 | 0.50 | 1.3273 | -0.50 | 1.3273 |
0.75 | 1.3119 | -0.75 | 1.3119 | |
1.00 | 1.4461 | -1.00 | 1.4490 | |
0.25 | 1.0882 | -0.25 | 1.0882 | |
12 | 0.50 | 1.1950 | -0.50 | 1.1950 |
0.75 | 1.3292 | -0.75 | 1.3292 | |
1.00 | 1.4733 | -1.00 | 1.4733 | |
0.25 | 1.0882 | -0.25 | 1.0882 | |
14 | 0.50 | 1.2026 | -0.50 | 1.2026 |
0.75 | 1.3395 | -0.75 | 1.3395 | |
1.00 | 1.4979 | -1.00 | 1.4979 | |
0.25 | 1.0909 | -0.25 | 1.0909 | |
16 | 0.50 | 1.2105 | -0.50 | 1.2105 |
0.75 | 1.3475 | -0.75 | 1.3475 | |
1.00 | 1.5122 | -1.00 | 1.5122 | |
0.25 | 1.0951 | -0.25 | 1.0951 | |
18 | 0.50 | 1.2119 | -0.50 | 1.2119 |
0.75 | 1.3606 | -0.75 | 1.3606 | |
1.00 | 1.5249 | -1.00 | 1.5249 | |
0.25 | 1.0928 | -0.25 | 1.0928 | |
20 | 0.50 | 1.2170 | -0.50 | 1.2170 |
0.75 | 1.3583 | -0.75 | 1.3583 | |
1.00 | 1.5309 | -1.00 | 1.5309 |
ϕ | e(˜σn(∞)2|^σ2) | ϕ | e(˜σ1(∞)2|^σ2) |
0.25 | 1.1073 | -0.25 | 1.1073 |
0.50 | 1.2472 | -0.50 | 1.2472 |
0.75 | 1.4223 | -0.75 | 1.4226 |
1.00 | 1.6352 | -1.00 | 1.6365 |
n | ϕ | e(˜σn(1)2|ˆσ2) | ϕ | e(˜σ1(1)2|ˆσ2) |
0.25 | 1.0302 | -0.25 | 1.0302 | |
2 | 0.50 | 1.0596 | -0.50 | 1.0596 |
0.75 | 1.0863 | -0.75 | 1.0863 | |
1.00 | 1.1026 | -1.00 | 1.1030 | |
0.25 | 1.0588 | -0.25 | 1.0588 | |
4 | 0.50 | 1.1259 | -0.50 | 1.1259 |
0.75 | 1.1989 | -0.75 | 1.1989 | |
1.00 | 1.2702 | -1.00 | 1.2710 | |
0.25 | 1.0720 | -0.25 | 1.0720 | |
6 | 0.50 | 1.1577 | -0.50 | 1.1577 |
0.75 | 1.2563 | -0.75 | 1.2563 | |
1.00 | 1.3580 | -1.00 | 1.3600 | |
0.25 | 1.0800 | -0.25 | 1.0800 | |
8 | 0.50 | 1.1773 | -0.50 | 1.1773 |
0.75 | 1.2901 | -0.75 | 1.2901 | |
1.00 | 1.4123 | -1.00 | 1.4150 | |
0.25 | 1.0858 | -0.25 | 1.0858 | |
10 | 0.50 | 1.3273 | -0.50 | 1.3273 |
0.75 | 1.3119 | -0.75 | 1.3119 | |
1.00 | 1.4461 | -1.00 | 1.4490 | |
0.25 | 1.0882 | -0.25 | 1.0882 | |
12 | 0.50 | 1.1950 | -0.50 | 1.1950 |
0.75 | 1.3292 | -0.75 | 1.3292 | |
1.00 | 1.4733 | -1.00 | 1.4733 | |
0.25 | 1.0882 | -0.25 | 1.0882 | |
14 | 0.50 | 1.2026 | -0.50 | 1.2026 |
0.75 | 1.3395 | -0.75 | 1.3395 | |
1.00 | 1.4979 | -1.00 | 1.4979 | |
0.25 | 1.0909 | -0.25 | 1.0909 | |
16 | 0.50 | 1.2105 | -0.50 | 1.2105 |
0.75 | 1.3475 | -0.75 | 1.3475 | |
1.00 | 1.5122 | -1.00 | 1.5122 | |
0.25 | 1.0951 | -0.25 | 1.0951 | |
18 | 0.50 | 1.2119 | -0.50 | 1.2119 |
0.75 | 1.3606 | -0.75 | 1.3606 | |
1.00 | 1.5249 | -1.00 | 1.5249 | |
0.25 | 1.0928 | -0.25 | 1.0928 | |
20 | 0.50 | 1.2170 | -0.50 | 1.2170 |
0.75 | 1.3583 | -0.75 | 1.3583 | |
1.00 | 1.5309 | -1.00 | 1.5309 |
ϕ | e(˜σn(∞)2|^σ2) | ϕ | e(˜σ1(∞)2|^σ2) |
0.25 | 1.1073 | -0.25 | 1.1073 |
0.50 | 1.2472 | -0.50 | 1.2472 |
0.75 | 1.4223 | -0.75 | 1.4226 |
1.00 | 1.6352 | -1.00 | 1.6365 |