Improved Estimation of the Scale Parameter for Log-Logistic Distribution Using Balanced Ranked Set Sampling

Housila P. Singh; Vishal Mehta

doi:10.21307/stattrans-2016-057

VOLUME 18 , ISSUE 1 (March 2017) > List of articles

Improved Estimation of the Scale Parameter for Log-Logistic Distribution Using Balanced Ranked Set Sampling

Keywords : minimum mean squared error estimator, shrinkage estimator, log-logistic distribution, best linear unbiased estimator, median ranked set sample, AMS Subject Classification: 62G30; 62H12

Citation Information : Statistics in Transition New Series. Volume 18, Issue 1, Pages 53-74, DOI: https://doi.org/10.21307/stattrans-2016-057

License : (CC BY 4.0)

Published Online: 11-July-2017

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ABSTRACT

In this article we have suggested some improved estimators of a scale parameter of log-logistic distribution (LLD) under a situation where the units in a sample can be ordered by judgement method without any error. We have also suggested some linear shrinkage estimator of a scale parameter of LLD. Efficiency comparisons are also made in this work.

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1. Introduction

Ranked set sampling (RSS) is a method of sampling that can be advantageous when quantification of all sampling units is costly but a small set of units can be easily ranked, according to the character under investigation, without actual quantification. The technique was first introduced by McIntyre (1952) for estimating means pasture and forage yields. The theory and application of ranked set sampling given by Chen et al. (2004).

A random variable X is said to have a log-logistic distribution with the scale parameter α and the shape parameter β if its cumulative distribution function (CDF) and probability density function (PDF) are respectively given as (see, Lesitha and Thomas (2012))

(1)

F (x; α, β) = \frac{x^{β}}{α^{β} + x^{β}}, x > 0, α > 0, β > 1

and

(2)

f (x; α, β) = \frac{β α^{β} x^{β - 1}}{{(α^{β} + x^{β})}^{2}}, x > 0, α > 0, β > 1 .

Also, the kth moment of (2) exists only when k < β and is given by

(3)

E (X^{k}) = α^{k} B (1 - \frac{k}{β}, 1 + \frac{k}{β}),

where B denotes beta function.

The applications of log-logistic distribution are well known in a survival analysis of data sets such as survival times of cancer patients in which the hazard rate increases initially and decreases later (for example, see Bennett (1983)). In economic studies of distributions of wealth or income, it is known as Fisk distribution (see Fisk (1961)) and is considered as an equivalent alternative to a lognormal distribution. For further details on the importance and applications of a log-logistic distribution one may refer to Shoukri et al. (1988), Geskus (2001), Robson and Reed (1999) and Ahmad et al. (1988). For current reference in this context the reader is referred to Singh and Mehta (2013; 2014, a, b, 2015, 2016 a, b, c), Mehta and Singh (2014) and Mehta (2015).

If X_1:n, X_2:n, …, X_n:n are the order statistics of a random sample of size n drawn from (1) then

(4)

Y_{r:n} = \frac{X_{r:n}}{α}, r = 1, 2, \dots, n,

are distributed as order statistics of the same sample size drawn from a LLD(1, β) with PDF given by

(5)

g (y, β) = \frac{β y^{β - 1}}{{(1 + y^{β})}^{2}}, y > 0, β > 1 .

For a detailed description of various properties of order statistics arising from a LLD(1, β) one may refer to Ragab and Green (1984). Balakrishnan and Malik (1987) have given some recurrence relations on the single and product moments of order statistics arising from a LLD(1, β). Suppose

(6)

γ_{r:n} = E (Y_{r:n}), r = 1, 2, \dots, n,

(7)

σ_{r,s:n} = Cov (Y_{r:n}, Y_{s:n}), 1 \leq r < s \leq n

and

(8)

σ_{r,r:n} = Var (Y_{r:n}), 1 \leq r \leq n .

By using (4) in (6)-(8) we have

(9)

E (X_{r:n}) = α γ_{r:n}, 1 \leq r \leq n

(10)

Cov (X_{r:n}, X_{s:n}) = α^{2} σ_{r,s:n}, 1 \leq r < s \leq n

(11)

Var (X_{r:n}) = α^{2} σ_{r,r:n}, 1 \leq r \leq n .

Lesitha and Thomas (2012) have computed the values of γ_r:n and σ_r,s:n,1 ≤ r,s ≤ n independently for n = 2(1)8 and for β = 2.5(0.5)5.0 using Mathcad software so as to use those values for the computation of BLUE of α based on order statistics. If X = (X_l:n, X_2:n, …, X_n:n)′ then the mean vector E(X) and dispersion matrix D(X) of X are

E (X) = γ α

and

D (X) = α^{2} G,

where γ = (γ_1:n, γ_2:n, …, γ_n:n)′ and G = ((σ_r,s:n)).

Thus, by Gauss-Markov theorem Lesitha and Thomas (2012) gives the BLUE $\hat{α}$ based on order statistics of a random sample of size n as:

t_{1} = \hat{α} = {(γ^{'} G^{- 1} γ)}^{- 1} γ^{'} G^{- 1} X

and

(12)

Var (t_{1}) = {(γ^{'} G^{- 1} γ)}^{- 1} α^{2} = α^{2} V_{1},

where V₁ = (γ′G^–1γ)^–1.

Lesitha and Thomas (2012) further estimate α based on the mean of unbiased estimators of α defined from each individual observations in the balanced ranked set sampling as:

t_{2} = α^{*} = \frac{1}{n} \sum_{r = 1}^{n} [\frac{X_{(r : n) r}}{γ_{r : n}}],

with

(13)

Var (t_{2}) = \frac{1}{n^{2}} \sum_{r = 1}^{n} [\frac{σ_{r, r : n}}{γ_{r : n}^{2}}] α^{2} = α^{2} V_{2},

where

V_{2} = \frac{1}{n^{2}} \sum_{r = 1}^{n} [\frac{σ_{r,r:n}}{γ_{r:n}^{2}}]

Lesitha and Thomas (2012) also estimate α based on BLUE in the balanced ranked set sampling as:

t_{3} = α^{**} = {(γ^{'} G_{1}^{- 1} γ)}^{- 1} γ^{'} G_{1}^{- 1} X_{rss}

and

(14)

Var (t_{3}) = {(γ^{'} G_{1}^{- 1} γ)}^{- 1} α^{2} = α^{2} V_{3} .

where γ = (γ_1:n, γ_2:n, …, γ_n:n)′, G₁ = diag(σ_1,1:n, σ_2,2:n, …, σ_n,n:n) and

V_{3} = {(γ^{'} G_{1}^{- 1} γ)}^{- 1}

When n is small the estimators α^* and α^** may not be acceptable for the expected level of precision. In such situations Lesitha and Thomas (2012) makes N cycles of RSS. For details see Chen et al. (2004). Suppose $α_{i}^{*}$ and $α_{i}^{* *}$ denote the estimators of α corresponding to α^* and α^** respectively, based on the ith cycle. Then, estimators of α based on N cycles are given by:

\bar{α^{*}} = \frac{1}{N} \sum_{i = 1}^{N} α_{i}^{*},

and

\bar{α^{* *}} = \frac{1}{N} \sum_{i = 1}^{N} α_{i}^{* *}

with

Var (\bar{α^{*}}) = \frac{α^{2}}{N n^{2}} \sum_{r = 1}^{N} [\frac{σ_{r, r : n}}{γ_{r : n}^{2}}],

and

Var (\bar{α^{* *}}) = \frac{{(γ^{'} G_{1}^{- 1} γ)}^{- 1} α^{2}}{N} .

Median ranked set sampling (MRSS) was first introduced by Muttlak (1997) to estimate the mean of a normal distribution. In general, MRSS is applied as a modification of RSS when one is interested in estimating a parameter associated with the central tendency of a distribution. The procedures of MRSS are given as: Select n independent samples each with n units as in the case of RSS. Then rank the units in each sample either by judgement method or by using some inexpensive means without having actual measurement on the unit. Lesitha and Thomas (2012) used MRSS method to estimate α as:

t_{4} = \tilde{α} = {\begin{array}{l} \begin{array}{l} \frac{1}{n} \sum_{r = 1}^{n} [\frac{X_{(m : n) r}}{γ_{m : n}}] & ; w h e n m = \frac{n + 1}{2} a n d n i s o d d \end{array} \\ {(γ_{1}^{'} G_{2}^{- 1} γ_{1})}^{- 1} γ_{1}^{'} G_{2}^{- 1} X_{mrss}; w h e n m = \frac{n}{2} a n d n i s e v e n \end{array}

with

(15)

Var (t_{4}) = {\begin{array}{l} \begin{array}{l} \frac{1}{n} [\frac{σ_{m, m : n}}{γ_{m : n}^{2}}] α^{2} & ; w h e n m = \frac{n + 1}{2} a n d n i s o d d \end{array} \\ {(γ_{1}^{'} G_{2}^{- 1} γ_{1})}^{- 1} α^{2}; w h e n m = \frac{n}{2} a n d n i s e v e n \end{array} = a^{2} V_{4},

where

\begin{array}{l} X_{m r s s} = {(X_{(m:n) 1}, X_{(m + 1:n) 2}, X_{(m:n) 3}, X_{(m + 1:n) 4}, \dots, X_{(m + 1:n) n})}^{'} \\ γ_{1} = {(γ_{m:n}, γ_{m + 1:n}, γ_{m:n}, γ_{m + 1:n}, \dots, γ_{m + 1:n})}^{'} \\ G_{2} = d i a g (σ_{m, m:n}, σ_{m + 1, m + 1:n}, σ_{m, m:n}, σ_{m + 1, m + 1:n}, \dots, σ_{m + 1, m + 1:n}) and \\ V_{4} = {\begin{array}{l} \begin{array}{l} \frac{1}{n} [\frac{σ_{m, m:n}}{γ_{m:n}^{2}}] & ; w h e n m = \frac{n + 1}{2} a n d n i s o d d \end{array} \\ {(γ_{1}^{'} G_{2}^{- 1} γ_{1})}^{- 1}; w h e n m = \frac{n}{2} a n d n i s e v e n \end{array} \end{array}

2. Improved estimation of the scale parameter α

Let t_i, i = 1,2,3,4 be an unbiased estimator of the parameter α, then we define a class of estimators for α as

T_{i} = A_{i} t_{i}, i = 1, 2, 3, 4,

where A_i′s,i = 1,2,3,4 are suitably chosen constants such that mean squared error of the estimators T_i′s,i = 1,2,3,4 is minimum.

The biases and mean squared errors (MSEs) of T_i,i = 1,2,3,4 are respectively given by

B (T_{i}) = α (A_{i} - 1),

and

M S E (T_{i}) = A_{i}^{2} V a r (t_{i}) + {(A_{i} - 1)}^{2} α^{2} = α^{2} [A_{i}^{2} (1 + V_{i}) - 2 A_{i} + 1] .

The MSE(T_i), i = 1,2,3,4 is minimized for

A_{i} = {(1 + V_{i})}^{- 1}, i = 1, 2, 3, 4.

Thus, the resulting minimum MSE estimator of α is given by

T_{0 i} = t_{i} {(1 + V_{i})}^{- 1}, i = 1, 2, 3, 4.

The biases and MSEs of T_0i,i = 1,2,3,4 are respectively given as

(16)

B (T_{0 i}) = - α (\frac{V_{i}}{1 + V_{i}}),

and

(17)

M S E (T_{0 i}) = α^{2} (\frac{V_{i}}{1 + V_{i}}) .

We have from (12) - (15) and (17) that

(18)

V a r (t_{i}) - M S E (T_{0 i}) = α^{2} \frac{V_{i}^{2}}{(1 + V_{i})} > 0, i = 1, 2, 3, 4 .

It follows from (18) that the proposed MMSE estimators T_0i′ s,i = 1,2,3,4 are better than the corresponding usual unbiased estimators t_i′ s,i = 1,2,3,4.

3. Improved estimation of the scale parameter α with prior information

Let t_i,i = 1,2,3,4 be an unbiased estimator of the parameter α, then we define a class of estimators of α using the prior point estimate α₀ of α as

(19)

T_{1 i} = α_{0} + B_{i} t_{i}, i = 1, 2, 3, 4,

where B_i′ s,i = 1,2,3,4 are suitably chosen constants such that mean squared error of the estimators T_1i′ s,i = 1,2,3,4 are minimum.

The biases and mean squared errors (MSEs) of T_1i, i = 1,2,3,4 are respectively given by

B (T_{1 i}) = α (ϕ + B_{i}),

and

M S E (T_{1 i}) = α^{2} [ϕ^{2} + B_{i}^{2} (1 + V_{i}) + 2 ϕ B_{i}] .

where

ϕ = (\frac{α_{0}}{α} - 1) = (λ - 1)

with

λ = \frac{α_{0}}{α}

The MSE(T_1i),i = 1,2,3,4 is minimized for

(20)

B_{i} = ϕ {(1 + V_{i})}^{- 1}, i = 1, 2, 3, 4 .

The value of B_i,i = 1,2,3,4 at (20) depends on the unknown parameter α, so an estimate of B_i,i = 1,2,3,4 based on sample data is given by

B_{i}^{*} = - \frac{ϕ^{*}}{(1 + V_{i})} = - \frac{(θ_{20} - t_{i})}{t_{i} (1 + V_{i})}, i = 1, 2, 3, 4.

Putting $B_{i}^{*}$ ,i = 1,2,3,4 in (19), we get a shrinkage estimator of t_i′ s,i = 1,2,3,4 as

(21)

T_{1 i}^{*} = α_{0} - {(1 + V_{i})}^{- 1} (α_{0} - t_{i}), i = 1, 2, 3, 4 .

The biases and mean squared errors (MSEs) of the estimators $T_{1 i}^{*}' s, i = 1, 2, 3, 4$ are respectively given by

(22)

B (T_{1 i}^{*}) = α ϕ (\frac{V_{i}}{1 + V_{i}}),

and

(23)

M S E (T_{1 i}^{*}) = α^{2} \frac{V_{i} (ϕ^{2} V_{i} + 1)}{{(1 + V_{i})}^{2}} .

Comparisons of the proposed shrinkage estimators $T_{1 i}^{*}' s, i = 1, 2, 3, 4$ with that of corresponding usual unbiased estimators t_i′ s,i = 1,2,3,4 are given in the following Theorem 1.

Theorem 1: The proposed shrinkage estimators $T_{1 i}^{*}' s, i = 1, 2, 3, 4$ are better than the corresponding usual unbiased estimators t_i′ s,i = 1,2,3,4 if

\begin{matrix} λ \in (0, (1 + \sqrt{(2 + V_{i})})), \\ i . e . i f α_{0} \in (0, α {1 + \sqrt{(2 + V_{i})}}), \\ i . e . i f α \in (\frac{α_{0}}{{1 + \sqrt{(2 + V_{i})}}}, \infty) . \end{matrix}

Proof: From (12) - (15) and (23), we have that

(24)

\begin{array}{l} MSE (T_{1 i}^{*}) < V a r (t_{i}), i = 1, 2, 3, 4 if \\ α^{2} \frac{V_{i} (ϕ^{2} V_{i} + 1)}{{(1 + V_{i})}^{2}} < α^{2} V_{i}, \\ i . e . i f \frac{(ϕ^{2} V_{i} + 1)}{{(1 + V_{i})}^{2}} < 1 \\ i . e . i f ϕ^{2} < \frac{({(1 + V_{i})}^{2} - 1)}{V_{i}}, \\ i . e . i f ϕ^{2} < \frac{V_{i} (2 + V_{i})}{V_{i}}, \\ i . e . i f ϕ^{2} < 2 + V_{i}, \\ i . e . i f {(λ - 1)}^{2} < 2 + V_{i}, \\ i . e . i f {1 - \sqrt{(2 + V_{i})}} < λ < {1 + \sqrt{(2 + V_{i})}} . \end{array}

Since ${1 - \sqrt{(2 + V_{i})}} < 0$ and λ(= α₀ / α) cannot be negative therefore (24) reduces to

0 < λ < {1 + \sqrt{(2 + V_{i})}},

0 < α_{0} < α {1 + \sqrt{(2 + V_{i})}},

\frac{α_{0}}{{1 + \sqrt{(2 + V_{i})}}} < α < \infty .

Hence the theorem. ♦

Further, we have compared the proposed shrinkage estimators $T_{1 i}^{*}' s, i = 1, 2, 3, 4$ with that of corresponding MMSE estimators T_0i′ s,i = 1,2,3,4 and the results are presented in Theorem 2.

Theorem 2: The proposed shrinkage estimators $T_{1 i}^{*}' s, i = 1, 2, 3, 4$ are better than the corresponding MMSE estimators T_0i′ s,i = 1,2,3,4 if

\begin{array}{l} λ \in (0, 2), \\ i . e . i f α_{0} \in (0, 2 α), \\ i . e . i f α \in (\frac{α_{0}}{2}, \infty) . \end{array}

Proof: From (17) and (23) we have that

\begin{array}{l} M S E (T_{1 i}^{*}) < M S E (T_{0 i}, i = 1, 2, \dots, 7 if \\ α^{2} \frac{V_{i} (ϕ^{2} V_{i} + 1)}{{(1 + V_{i})}^{2}} < α^{2} \frac{V_{i}}{1 + V_{i}}, \\ \begin{array}{l} i . e . i f & \frac{(ϕ^{2} V_{i} + 1)}{(1 + V_{i})} < 1, \\ i . e . i f & ϕ^{2} < 1, \\ i . e . i f & - 1 < ϕ < 1, \\ i . e . i f & 0 < λ < 2, \\ o r & 0 < α_{0} < 2 α, \\ o r & \frac{α_{0}}{2} < α < \infty . \end{array} \end{array}

Hence the theorem. ♦

4. Relative efficiencies

We have computed the relative efficiencies of various suggested estimators to usual estimators by using the formulae:

\begin{array}{l} e_{1} = R E (T_{01}, t_{1}) = 1 + V_{1}; \\ e_{2} = R E (T_{02}, t_{2}) = 1 + V_{2}; \\ e_{3} = R E (T_{03}, t_{3}) = 1 + V_{3}; \\ e_{4} = R E (T_{04}, t_{4}) = 1 + V_{4}; \\ e_{5} = R E (T_{04}, T_{01}) = \frac{V_{1} (1 + V_{4})}{V_{4} (1 + V_{1})}; \\ e_{6} = R E (T_{04}, T_{02}) = \frac{V_{2} (1 + V_{4})}{V_{4} (1 + V_{2})}; \\ e_{7} = R E (T_{04}, T_{03}) = \frac{V_{3} (1 + V_{4})}{V_{4} (1 + V_{3})}; \\ e_{8} = R E (T_{11}^{*}, t_{1}) = \frac{{(1 + V_{1})}^{2}}{(ϕ^{2} V_{1} + 1)}; \\ e_{9} = R E (T_{11}^{*}, T_{01}) = \frac{(1 + V_{1})}{(ϕ^{2} V_{1} + 1)}; \end{array}

\begin{array}{l} e_{10} = R E (T_{12}^{*}, t_{2}) = \frac{{(1 + V_{2})}^{2}}{(ϕ^{2} V_{2} + 1)}; \\ e_{11} = R E (T_{12}^{*}, T_{02}) = \frac{(1 + V_{2})}{(ϕ^{2} V_{2} + 1)}; \\ e_{12} = R E (T_{13}^{*}, t_{3}) = \frac{{(1 + V_{3})}^{2}}{(ϕ^{2} V_{3} + 1)}; \\ e_{13} = R E (T_{13}^{*}, T_{03}) = \frac{(1 + V_{3})}{(ϕ^{2} V_{3} + 1)}; \\ e_{14} = R E (T_{14}^{*}, t_{4}) = \frac{{(1 + V_{4})}^{2}}{(ϕ^{2} V_{4} + 1)}; \end{array}

and

e_{15} = R E (T_{14}^{*}, T_{04}) = \frac{(1 + V_{4})}{(ϕ^{2} V_{4} + 1)} .

The values of e_i,i = 1,2,…,7 are shown in Table 1 for n = 2(1)8 and β = 2.5(0.5)5.

Table 1.

The values of e_i′ s,i = 1,2,…,7.

N	ß	e₁	e₂	e₃	e₄	e₅	e₆	e₇
2	2.5	1.3371	1.4025	1.2799	1.2799	1.1530	1.3124	1.0000
	3.0	1.2158	1.1978	1.1674	1.1674	1.2381	1.1516	1.0000
	3.5	1.1514	1.1254	1.1135	1.1135	1.2901	1.0932	1.0000
	4.0	1.1126	1.0885	1.0827	1.0827	1.3242	1.0643	1.0000
	4.5	1.0872	1.0665	1.0633	1.0633	1.3476	1.0474	1.0000
	5.0	1.0697	1.0521	1.0501	1.0501	1.3644	1.0368	1.0000
3	2.5	1.2011	1.1904	1.1180	1.0832	2.1798	2.0828	1.3740
	3.0	1.1308	1.0970	1.0752	1.0543	2.2446	1.7159	1.3572
	3.5	1.0937	1.0626	1.0526	1.0385	2.3093	1.5893	1.3475
	4.0	1.0706	1.0447	1.0391	1.0288	2.3527	1.5272	1.3422
	4.5	1.0552	1.0338	1.0303	1.0224	2.3815	1.4910	1.3382
	5.0	1.0443	1.0266	1.0242	1.0180	2.4033	1.4675	1.3356
4	2.5	1.1392	1.1118	1.0648	1.0477	2.6846	2.2089	1.3376
	3.0	1.0939	1.0582	1.0424	1.0317	2.7929	1.7903	1.3246
	3.5	1.0678	1.0380	1.0301	1.0227	2.8598	1.6495	1.3176
	4.0	1.0514	1.0273	1.0226	1.0171	2.9034	1.5803	1.3126
	4.5	1.0413	1.0208	1.0176	1.0134	3.0062	1.5408	1.3102
	5.0	1.0324	1.0164	1.0141	1.0107	2.9570	1.5158	1.3085
5	2.5	1.1078	1.0740	1.0409	1.0278	3.5999	2.5482	1.4547
	3.0	1.0733	1.0391	1.0272	1.0187	3.7126	2.0476	1.4397
	3.5	1.0531	1.0257	1.0195	1.0135	3.7819	1.8805	1.4317
	4.0	1.0403	1.0186	1.0147	1.0101	3.8704	1.8188	1.4421
	4.5	1.0316	1.0141	1.0115	1.0080	3.8489	1.7503	1.4223
	5.0	1.0256	1.0112	1.0092	1.0065	3.8785	1.7199	1.4196
6	2.5	1.0880	1.0528	1.0282	1.0196	4.2168	2.6150	1.4288
	3.0	1.0600	1.0282	1.0189	1.0133	4.3313	2.0988	1.4170
	3.5	1.0437	1.0187	1.0136	1.0096	4.3995	1.9265	1.4101
	4.0	1.0332	1.0135	1.0103	1.0073	4.4473	1.8430	1.4065
	4.5	1.0261	1.0103	1.0080	1.0057	4.4756	1.7943	1.4024
	5.0	1.0211	1.0082	1.0065	1.0046	4.5010	1.7638	1.4009
7	2.5	1.0735	1.0397	1.0206	1.0138	5.0286	2.8038	1.4800
	3.0	1.0509	1.0214	1.0139	1.0094	5.1991	2.2498	1.4690
	3.5	1.0372	1.0147	1.0106	1.0068	5.2926	2.1298	1.5419
	4.0	1.0282	1.0103	1.0076	1.0052	5.3054	1.9708	1.4542
	4.5	1.0222	1.0079	1.0060	1.0041	5.3447	1.9217	1.4556
	5.0	1.0179	1.0062	1.0048	1.0033	5.3582	1.8854	1.4494
8	2.5	1.0643	1.0310	1.0157	1.0107	5.7315	2.8525	1.4632
	3.0	1.0444	1.0168	1.0106	1.0073	5.8758	2.2888	1.4526
	3.5	1.0329	1.0112	1.0077	1.0053	6.0530	2.1067	1.4484
	4.0	1.0245	1.0081	1.0058	1.0040	5.9651	2.0116	1.4441
	4.5	1.0190	1.0062	1.0046	1.0032	5.9067	1.9593	1.4396
	5.0	1.0156	1.0049	1.0037	1.0025	6.0905	1.9479	1.4568

The values of e_i,i = 8,9,…,15 are shown in Tables 2 to 5 for n = 2(1)8; β = 2.5(0.5)5 and different values of $λ = \frac{α_{0}}{α}$ .

Table 2.

The values of e_i for i = 8 and 9

		$e_{8} = R E (T_{11}^{*}, t_{1})$						Range of λ in which $T_{11}^{*}$ is efficient to t₁
n	ß	λ = 1.00	λ = 1.25 and λ = 0.75	λ = 1.50 and λ = 0.50	λ = 1.75 and λ = 0.25	λ = 2.00	λ = 2.25	Range of λ in which $T_{11}^{*}$ is efficient to t₁
2	2.5	1.7878	1.7509	1.6489	1.5028	1.3371	1.1710	(0,2.53)
	3.0	1.4782	1.4586	1.4026	1.3182	1.2158	1.1054	(0,2.49)
	3.5	1.3257	1.3133	1.2773	1.2217	1.1514	1.0721	(0,2.47)
	4.0	1.2378	1.2292	1.2039	1.1641	1.1126	1.0527	(0,2.45)
	4.5	1.1820	1.1756	1.1568	1.1268	1.0872	1.0403	(0,2.44)
	5.0	1.1442	1.1392	1.1246	1.1010	1.0697	1.0319	(0,2.44)
3	2.5	1.4426	1.4247	1.3735	1.2960	1.2011	1.0977	(0,2.48)
	3.0	1.2786	1.2683	1.2382	1.1910	1.1308	1.0617	(0,2.46)
	3.5	1.1961	1.1892	1.1688	1.1363	1.0937	1.0434	(0,2.45)
	4.0	1.1461	1.1411	1.1263	1.1024	1.0706	1.0323	(0,2.44)
	4.5	1.1133	1.1095	1.0982	1.0798	1.0552	1.0250	(0,2.43)
	5.0	1.0906	1.0876	1.0787	1.0641	1.0443	1.0200	(0,2.43)
4	2.5	1.2977	1.2865	1.2541	1.2035	1.1392	1.0659	(0,2.46)
	3.0	1.1966	1.1896	1.1692	1.1366	1.0939	1.0435	(0,2.45)
	3.5	1.1402	1.1354	1.1212	1.0983	1.0678	1.0310	(0,2.44)
	4.0	1.1053	1.1018	1.0913	1.0743	1.0514	1.0232	(0,2.43)
	4.5	1.0843	1.0815	1.0732	1.0597	1.0413	1.0186	(0,2.43)
	5.0	1.0659	1.0638	1.0574	1.0468	1.0324	1.0145	(0,2.43)
5	2.5	1.2272	1.2190	1.1950	1.1570	1.1078	1.0503	(0,2.45)
	3.0	1.1519	1.1466	1.1312	1.1063	1.0733	1.0336	(0,2.44)
	3.5	1.1091	1.1054	1.0945	1.0769	1.0531	1.0241	(0,2.43)
	4.0	1.0823	1.0796	1.0715	1.0583	1.0403	1.0181	(0,2.43)
	4.5	1.0643	1.0622	1.0559	1.0457	1.0316	1.0141	(0,2.43)
	5.0	1.0518	1.0501	1.0451	1.0369	1.0256	1.0114	(0,2.42)
6	2.5	1.1837	1.1772	1.1582	1.1279	1.0880	1.0406	(0,2.44)
	3.0	1.1237	1.1195	1.1071	1.0870	1.0600	1.0273	(0,2.44)
	3.5	1.0892	1.0863	1.0775	1.0631	1.0437	1.0197	(0,2.43)
	4.0	1.0675	1.0653	1.0587	1.0479	1.0332	1.0149	(0,2.43)
	4.5	1.0529	1.0512	1.0461	1.0377	1.0261	1.0116	(0,2.42)
	5.0	1.0426	1.0413	1.0372	1.0304	1.0211	1.0094	(0,2.42)
7	2.5	1.1524	1.1471	1.1316	1.1066	1.0735	1.0337	(0,2.44)
	3.0	1.1043	1.1008	1.0905	1.0736	1.0509	1.0230	(0,2.43)
	3.5	1.0759	1.0734	1.0659	1.0538	1.0372	1.0167	(0,2.43)
	4.0	1.0572	1.0554	1.0498	1.0407	1.0282	1.0126	(0,2.42)
	4.5	1.0449	1.0434	1.0391	1.0320	1.0222	1.0099	(0,2.42)
	5.0	1.0362	1.0350	1.0316	1.0258	1.0179	1.0079	(0,2.42)
8	2.5	1.1327	1.1282	1.1148	1.0932	1.0643	1.0293	(0,2.44)
	3.0	1.0907	1.0877	1.0787	1.0641	1.0444	1.0200	(0,2.43)
	3.5	1.0669	1.0647	1.0582	1.0475	1.0329	1.0147	(0,2.43)
	4.0	1.0497	1.0481	1.0433	1.0354	1.0245	1.0109	(0,2.42)
	4.5	1.0384	1.0372	1.0335	1.0274	1.0190	1.0084	(0,2.42)
	5.0	1.0315	1.0305	1.0275	1.0225	1.0156	1.0069	(0,2.42)

		$e_{9} = R E (T_{11}^{*}, T_{01})$					Range of λ in which $T_{11}^{}$ is efficient to T₀₁*
n	ß	λ = 1.00	λ = 1.25 and λ = 0.75	λ = 1.50 and λ = 0.50	λ = 1.75 and λ = 0.25	λ = 2.00 and λ = 0.00	Range of λ in which $T_{11}^{}$ is efficient to T₀₁*
2	2.5	1.3371	1.3095	1.2332	1.1240	1.0000	[0,2]
	3.0	1.2158	1.1996	1.1536	1.0842	1.0000	[0,2]
	3.5	1.1514	1.1406	1.1094	1.0610	1.0000	[0,2]
	4.0	1.1126	1.1048	1.0821	1.0463	1.0000	[0,2]
	4.5	1.0872	1.0813	1.0640	1.0364	1.0000	[0,2]
	5.0	1.0697	1.0650	1.0514	1.0293	1.0000	[0,2]
3	2.5	1.2011	1.1862	1.1436	1.0790	1.0000	[0,2]
	3.0	1.1308	1.1216	1.0950	1.0533	1.0000	[0,2]
	3.5	1.0937	1.0873	1.0687	1.0389	1.0000	[0,2]
	4.0	1.0706	1.0659	1.0520	1.0297	1.0000	[0,2]
	4.5	1.0552	1.0515	1.0408	1.0234	1.0000	[0,2]
	5.0	1.0443	1.0414	1.0329	1.0189	1.0000	[0,2]
4	2.5	1.1392	1.1294	1.1009	1.0565	1.0000	[0,2]
	3.0	1.0939	1.0875	1.0688	1.0390	1.0000	[0,2]
	3.5	1.0678	1.0633	1.0500	1.0286	1.0000	[0,2]
	4.0	1.0514	1.0480	1.0380	1.0218	1.0000	[0,2]
	4.5	1.0413	1.0386	1.0307	1.0177	1.0000	[0,2]
	5.0	1.0324	1.0304	1.0241	1.0139	1.0000	[0,2]
5	2.5	1.1078	1.1004	1.0787	1.0445	1.0000	[0,2]
	3.0	1.0733	1.0684	1.0540	1.0308	1.0000	[0,2]
	3.5	1.0531	1.0496	1.0393	1.0226	1.0000	[0,2]
	4.0	1.0403	1.0377	1.0300	1.0173	1.0000	[0,2]
	4.5	1.0316	1.0296	1.0235	1.0136	1.0000	[0,2]
	5.0	1.0256	1.0239	1.0191	1.0110	1.0000	[0,2]
6	2.5	1.0880	1.0820	1.0646	1.0367	1.0000	[0,2]
	3.0	1.0600	1.0561	1.0444	1.0254	1.0000	[0,2]
	3.5	1.0437	1.0408	1.0324	1.0186	1.0000	[0,2]
	4.0	1.0332	1.0311	1.0247	1.0143	1.0000	[0,2]
	4.5	1.0261	1.0244	1.0195	1.0113	1.0000	[0,2]
	5.0	1.0211	1.0197	1.0157	1.0091	1.0000	[0,2]
7	2.5	1.0735	1.0686	1.0541	1.0309	1.0000	[0,2]
	3.0	1.0509	1.0475	1.0377	1.0216	1.0000	[0,2]
	3.5	1.0372	1.0348	1.0277	1.0160	1.0000	[0,2]
	4.0	1.0282	1.0264	1.0210	1.0122	1.0000	[0,2]
	4.5	1.0222	1.0208	1.0166	1.0096	1.0000	[0,2]
	5.0	1.0179	1.0168	1.0134	1.0078	1.0000	[0,2]
8	2.5	1.0643	1.0600	1.0474	1.0271	1.0000	[0,2]
	3.0	1.0444	1.0415	1.0329	1.0189	1.0000	[0,2]
	3.5	1.0329	1.0308	1.0245	1.0141	1.0000	[0,2]
	4.0	1.0245	1.0230	1.0183	1.0106	1.0000	[0,2]
	4.5	1.0190	1.0178	1.0142	1.0082	1.0000	[0,2]
	5.0	1.0156	1.0146	1.0117	1.0068	1.0000	[0,2]

Table 3.

The values of e_i for i = 10 and 11

		$e_{10} = R E (T_{12}^{*}, t_{2})$						Range of λ in which $T_{12}^{*}$ is efficient to t₂
n	ß	λ = 1.00	λ = 1.25 and λ = 0.75	λ = 1.50 and λ = 0.50	λ = 1.75 and λ = 0.25	λ = 2.00	λ = 2.25	Range of λ in which $T_{12}^{*}$ is efficient to t₂
2	2.5	1.9669	1.9186	1.7871	1.6038	1.4025	1.2075	(0,2.55)
	3.0	1.4347	1.4171	1.3671	1.2910	1.1978	1.0960	(0,2.48)
	3.5	1.2665	1.2566	1.2280	1.1830	1.1254	1.0590	(0,2.46)
	4.0	1.1849	1.1784	1.1592	1.1287	1.0885	1.0409	(0,2.45)
	4.5	1.1374	1.1327	1.1188	1.0964	1.0665	1.0304	(0,2.44)
	5.0	1.1069	1.1033	1.0926	1.0754	1.0521	1.0236	(0,2.43)
3	2.5	1.4171	1.4004	1.3527	1.2800	1.1904	1.0922	(0,2.48)
	3.0	1.2034	1.1961	1.1749	1.1411	1.0970	1.0450	(0,2.45)
	3.5	1.1292	1.1248	1.1118	1.0908	1.0626	1.0285	(0,2.44)
	4.0	1.0914	1.0884	1.0794	1.0646	1.0447	1.0202	(0,2.43)
	4.5	1.0688	1.0665	1.0598	1.0488	1.0338	1.0151	(0,2.43)
	5.0	1.0539	1.0522	1.0470	1.0384	1.0266	1.0119	(0,2.42)
4	2.5	1.2360	1.2274	1.2024	1.1629	1.1118	1.0523	(0,2.45)
	3.0	1.1199	1.1158	1.1038	1.0843	1.0582	1.0265	(0,2.43)
	3.5	1.0775	1.0749	1.0673	1.0549	1.0380	1.0171	(0,2.43)
	4.0	1.0554	1.0536	1.0482	1.0394	1.0273	1.0122	(0,2.42)
	4.5	1.0419	1.0406	1.0366	1.0299	1.0208	1.0092	(0,2.42)
	5.0	1.0330	1.0320	1.0288	1.0236	1.0164	1.0072	(0,2.42)
5	2.5	1.1534	1.1481	1.1325	1.1073	1.0740	1.0339	(0,2.44)
	3.0	1.0798	1.0771	1.0693	1.0565	1.0391	1.0176	(0,2.43)
	3.5	1.0521	1.0504	1.0454	1.0371	1.0257	1.0115	(0,2.42)
	4.0	1.0375	1.0363	1.0327	1.0267	1.0186	1.0082	(0,2.42)
	4.5	1.0285	1.0276	1.0249	1.0204	1.0141	1.0062	(0,2.42)
	5.0	1.0225	1.0218	1.0196	1.0161	1.0112	1.0049	(0,2.42)
6	2.5	1.1084	1.1047	1.0939	1.0764	1.0528	1.0239	(0,2.43)
	3.0	1.0572	1.0554	1.0498	1.0407	1.0282	1.0126	(0,2.42)
	3.5	1.0377	1.0365	1.0328	1.0269	1.0187	1.0083	(0,2.42)
	4.0	1.0272	1.0263	1.0237	1.0194	1.0135	1.0060	(0,2.42)
	4.5	1.0207	1.0201	1.0181	1.0148	1.0103	1.0045	(0,2.42)
	5.0	1.0164	1.0159	1.0143	1.0117	1.0082	1.0036	(0,2.42)
7	2.5	1.0809	1.0783	1.0703	1.0573	1.0397	1.0178	(0,2.43)
	3.0	1.0433	1.0419	1.0377	1.0308	1.0214	1.0095	(0,2.42)
	3.5	1.0295	1.0286	1.0258	1.0211	1.0147	1.0065	(0,2.42)
	4.0	1.0207	1.0200	1.0181	1.0148	1.0103	1.0045	(0,2.42)
	4.5	1.0158	1.0153	1.0138	1.0113	1.0079	1.0035	(0,2.42)
	5.0	1.0125	1.0121	1.0109	1.0090	1.0062	1.0027	(0,2.42)
8	2.5	1.0629	1.0609	1.0548	1.0447	1.0310	1.0138	(0,2.43)
	3.0	1.0339	1.0328	1.0296	1.0242	1.0168	1.0074	(0,2.42)
	3.5	1.0225	1.0218	1.0197	1.0161	1.0112	1.0049	(0,2.42)
	4.0	1.0163	1.0158	1.0143	1.0117	1.0081	1.0036	(0,2.42)
	4.5	1.0125	1.0121	1.0109	1.0090	1.0062	1.0027	(0,2.42)
	5.0	1.0099	1.0096	1.0087	1.0071	1.0049	1.0022	(0,2.42)

		$e_{11} = R E (T_{12}^{*}, T_{02})$					Range of λ in which $T_{12}^{}$ is efficient to T₀₂*
n	ß	λ = 1.00	λ = 1.25 and λ = 0.75	λ = 1.50 and λ = 0.50	λ = 1.75 and λ = 0.25	λ = 2.00 and λ = 0.00	Range of λ in which $T_{12}^{}$ is efficient to T₀₂*
2	2.5	1.4025	1.3680	1.2743	1.1436	1.0000	[0,2]
	3.0	1.1978	1.1831	1.1413	1.0779	1.0000	[0,2]
	3.5	1.1254	1.1166	1.0912	1.0512	1.0000	[0,2]
	4.0	1.0885	1.0825	1.0650	1.0369	1.0000	[0,2]
	4.5	1.0665	1.0621	1.0491	1.0280	1.0000	[0,2]
	5.0	1.0521	1.0487	1.0386	1.0221	1.0000	[0,2]
3	2.5	1.1904	1.1764	1.1363	1.0752	1.0000	[0,2]
	3.0	1.0970	1.0904	1.0710	1.0402	1.0000	[0,2]
	3.5	1.0626	1.0585	1.0463	1.0265	1.0000	[0,2]
	4.0	1.0447	1.0418	1.0332	1.0191	1.0000	[0,2]
	4.5	1.0338	1.0316	1.0252	1.0145	1.0000	[0,2]
	5.0	1.0266	1.0249	1.0198	1.0115	1.0000	[0,2]
4	2.5	1.1118	1.1040	1.0815	1.0460	1.0000	[0,2]
	3.0	1.0582	1.0544	1.0430	1.0247	1.0000	[0,2]
	3.5	1.0380	1.0356	1.0282	1.0163	1.0000	[0,2]
	4.0	1.0273	1.0256	1.0203	1.0118	1.0000	[0,2]
	4.5	1.0208	1.0194	1.0155	1.0090	1.0000	[0,2]
	5.0	1.0164	1.0153	1.0122	1.0071	1.0000	[0,2]
5	2.5	1.0740	1.0690	1.0545	1.0311	1.0000	[0,2]
	3.0	1.0391	1.0366	1.0291	1.0167	1.0000	[0,2]
	3.5	1.0257	1.0241	1.0192	1.0111	1.0000	[0,2]
	4.0	1.0186	1.0174	1.0139	1.0080	1.0000	[0,2]
	4.5	1.0141	1.0132	1.0106	1.0061	1.0000	[0,2]
	5.0	1.0112	1.0105	1.0084	1.0049	1.0000	[0,2]
6	2.5	1.0528	1.0493	1.0391	1.0224	1.0000	[0,2]
	3.0	1.0282	1.0264	1.0210	1.0122	1.0000	[0,2]
	3.5	1.0187	1.0175	1.0139	1.0081	1.0000	[0,2]
	4.0	1.0135	1.0126	1.0101	1.0059	1.0000	[0,2]
	4.5	1.0103	1.0097	1.0077	1.0045	1.0000	[0,2]
	5.0	1.0082	1.0076	1.0061	1.0036	1.0000	[0,2]
7	2.5	1.0397	1.0371	1.0295	1.0170	1.0000	[0,2]
	3.0	1.0214	1.0200	1.0160	1.0093	1.0000	[0,2]
	3.5	1.0147	1.0137	1.0110	1.0064	1.0000	[0,2]
	4.0	1.0103	1.0097	1.0077	1.0045	1.0000	[0,2]
	4.5	1.0079	1.0074	1.0059	1.0034	1.0000	[0,2]
	5.0	1.0062	1.0058	1.0047	1.0027	1.0000	[0,2]
8	2.5	1.0310	1.0290	1.0231	1.0133	1.0000	[0,2]
	3.0	1.0168	1.0158	1.0126	1.0073	1.0000	[0,2]
	3.5	1.0112	1.0105	1.0084	1.0049	1.0000	[0,2]
	4.0	1.0081	1.0076	1.0061	1.0035	1.0000	[0,2]
	4.5	1.0062	1.0058	1.0047	1.0027	1.0000	[0,2]
	5.0	1.0049	1.0046	1.0037	1.0022	1.0000	[0,2]

Table 4.

The values of e_i for i = 12 and 13

		$e_{12} = R E (T_{13}^{*}, t_{3})$						Range of λ in which $T_{13}^{*}$ is efficient to t₃
n	ß	λ = 1.00	λ = 1.25 and λ = 0.75	λ = 1.50 and λ = 0.50	λ = 1.75 and λ = 0.25	λ = 2.00	λ = 2.25	Range of λ in which $T_{13}^{*}$ is efficient to t₃
2	2.5	1.6380	1.6099	1.5309	1.4152	1.2799	1.1397	(0,2.51)
	3.0	1.3628	1.3487	1.3080	1.2455	1.1674	1.0803	(0,2.47)
	3.5	1.2398	1.2311	1.2056	1.1654	1.1135	1.0531	(0,2.45)
	4.0	1.1723	1.1663	1.1485	1.1202	1.0827	1.0381	(0,2.44)
	4.5	1.1306	1.1262	1.1130	1.0917	1.0633	1.0288	(0,2.44)
	5.0	1.1028	1.0993	1.0891	1.0725	1.0501	1.0227	(0,2.43)
3	2.5	1.2499	1.2407	1.2141	1.1721	1.1180	1.0553	(0,2.46)
	3.0	1.1560	1.1506	1.1347	1.1091	1.0752	1.0345	(0,2.44)
	3.5	1.1080	1.1044	1.0936	1.0761	1.0526	1.0238	(0,2.43)
	4.0	1.0797	1.0771	1.0692	1.0565	1.0391	1.0176	(0,2.43)
	4.5	1.0614	1.0594	1.0535	1.0437	1.0303	1.0135	(0,2.42)
	5.0	1.0489	1.0473	1.0426	1.0348	1.0242	1.0107	(0,2.42)
4	2.5	1.1338	1.1293	1.1158	1.0940	1.0648	1.0296	(0,2.44)
	3.0	1.0867	1.0838	1.0753	1.0613	1.0424	1.0191	(0,2.43)
	3.5	1.0612	1.0592	1.0533	1.0435	1.0301	1.0135	(0,2.42)
	4.0	1.0457	1.0442	1.0398	1.0326	1.0226	1.0100	(0,2.42)
	4.5	1.0355	1.0344	1.0310	1.0253	1.0176	1.0078	(0,2.42)
	5.0	1.0284	1.0275	1.0248	1.0203	1.0141	1.0062	(0,2.42)
5	2.5	1.0835	1.0808	1.0726	1.0592	1.0409	1.0184	(0,2.43)
	3.0	1.0551	1.0533	1.0480	1.0392	1.0272	1.0121	(0,2.42)
	3.5	1.0393	1.0381	1.0343	1.0281	1.0195	1.0086	(0,2.42)
	4.0	1.0295	1.0286	1.0258	1.0211	1.0147	1.0065	(0,2.42)
	4.5	1.0231	1.0223	1.0201	1.0165	1.0115	1.0051	(0,2.42)
	5.0	1.0185	1.0179	1.0162	1.0133	1.0092	1.0041	(0,2.42)
6	2.5	1.0571	1.0553	1.0497	1.0406	1.0282	1.0126	(0,2.42)
	3.0	1.0381	1.0369	1.0332	1.0272	1.0189	1.0084	(0,2.42)
	3.5	1.0274	1.0265	1.0239	1.0196	1.0136	1.0060	(0,2.42)
	4.0	1.0206	1.0200	1.0180	1.0148	1.0103	1.0045	(0,2.42)
	4.5	1.0161	1.0156	1.0141	1.0116	1.0080	1.0035	(0,2.42)
	5.0	1.0130	1.0126	1.0113	1.0093	1.0065	1.0028	(0,2.42)
7	2.5	1.0415	1.0402	1.0362	1.0296	1.0206	1.0091	(0,2.42)
	3.0	1.0279	1.0270	1.0244	1.0200	1.0139	1.0061	(0,2.42)
	3.5	1.0213	1.0206	1.0186	1.0152	1.0106	1.0047	(0,2.42)
	4.0	1.0152	1.0147	1.0133	1.0109	1.0076	1.0033	(0,2.42)
	4.5	1.0119	1.0116	1.0104	1.0086	1.0060	1.0026	(0,2.42)
	5.0	1.0096	1.0093	1.0084	1.0069	1.0048	1.0021	(0,2.42)
8	2.5	1.0316	1.0306	1.0275	1.0226	1.0157	1.0069	(0,2.42)
	3.0	1.0213	1.0207	1.0186	1.0153	1.0106	1.0047	(0,2.42)
	3.5	1.0154	1.0149	1.0135	1.0111	1.0077	1.0034	(0,2.42)
	4.0	1.0117	1.0113	1.0102	1.0084	1.0058	1.0026	(0,2.42)
	4.5	1.0092	1.0089	1.0080	1.0066	1.0046	1.0020	(0,2.42)
	5.0	1.0074	1.0072	1.0065	1.0053	1.0037	1.0016	(0,2.42)

		$e_{13} = R E (T_{13}^{*}, T_{03})$					Range of λ in which $T_{13}^{}$ is efficient to T₀₃*
n	ß	λ = 1.00	λ = 1.25 and λ = 0.75	λ = 1.50 and λ = 0.50	λ = 1.75 and λ = 0.25	λ = 2.00 and λ = 0.00	Range of λ in which $T_{13}^{}$ is efficient to T₀₃*
2	2.5	1.2799	1.2578	1.1962	1.1058	1.0000	[0,2]
	3.0	1.1674	1.1553	1.1205	1.0669	1.0000	[0,2]
	3.5	1.1135	1.1056	1.0828	1.0467	1.0000	[0,2]
	4.0	1.0827	1.0772	1.0608	1.0346	1.0000	[0,2]
	4.5	1.0633	1.0591	1.0467	1.0267	1.0000	[0,2]
	5.0	1.0501	1.0469	1.0371	1.0213	1.0000	[0,2]
3	2.5	1.1180	1.1098	1.0859	1.0484	1.0000	[0,2]
	3.0	1.0752	1.0702	1.0553	1.0316	1.0000	[0,2]
	3.5	1.0526	1.0492	1.0389	1.0224	1.0000	[0,2]
	4.0	1.0391	1.0365	1.0290	1.0167	1.0000	[0,2]
	4.5	1.0303	1.0283	1.0225	1.0130	1.0000	[0,2]
	5.0	1.0242	1.0226	1.0180	1.0104	1.0000	[0,2]
4	2.5	1.0648	1.0605	1.0478	1.0274	1.0000	[0,2]
	3.0	1.0424	1.0397	1.0315	1.0181	1.0000	[0,2]
	3.5	1.0301	1.0282	1.0224	1.0130	1.0000	[0,2]
	4.0	1.0226	1.0211	1.0168	1.0098	1.0000	[0,2]
	4.5	1.0176	1.0165	1.0131	1.0076	1.0000	[0,2]
	5.0	1.0141	1.0132	1.0105	1.0061	1.0000	[0,2]
5	2.5	1.0409	1.0383	1.0304	1.0175	1.0000	[0,2]
	3.0	1.0272	1.0254	1.0203	1.0117	1.0000	[0,2]
	3.5	1.0195	1.0182	1.0145	1.0084	1.0000	[0,2]
	4.0	1.0147	1.0137	1.0110	1.0064	1.0000	[0,2]
	4.5	1.0115	1.0107	1.0086	1.0050	1.0000	[0,2]
	5.0	1.0092	1.0086	1.0069	1.0040	1.0000	[0,2]
6	2.5	1.0282	1.0264	1.0210	1.0121	1.0000	[0,2]
	3.0	1.0189	1.0177	1.0141	1.0082	1.0000	[0,2]
	3.5	1.0136	1.0127	1.0102	1.0059	1.0000	[0,2]
	4.0	1.0103	1.0096	1.0077	1.0045	1.0000	[0,2]
	4.5	1.0080	1.0075	1.0060	1.0035	1.0000	[0,2]
	5.0	1.0065	1.0061	1.0048	1.0028	1.0000	[0,2]
7	2.5	1.0206	1.0193	1.0153	1.0089	1.0000	[0,2]
	3.0	1.0139	1.0130	1.0104	1.0060	1.0000	[0,2]
	3.5	1.0106	1.0099	1.0079	1.0046	1.0000	[0,2]
	4.0	1.0076	1.0071	1.0057	1.0033	1.0000	[0,2]
	4.5	1.0060	1.0056	1.0045	1.0026	1.0000	[0,2]
	5.0	1.0048	1.0045	1.0036	1.0021	1.0000	[0,2]
8	2.5	1.0157	1.0147	1.0117	1.0068	1.0000	[0,2]
	3.0	1.0106	1.0099	1.0079	1.0046	1.0000	[0,2]
	3.5	1.0077	1.0072	1.0057	1.0033	1.0000	[0,2]
	4.0	1.0058	1.0055	1.0044	1.0025	1.0000	[0,2]
	4.5	1.0046	1.0043	1.0034	1.0020	1.0000	[0,2]
	5.0	1.0037	1.0035	1.0028	1.0016	1.0000	[0,2]

Table 5.

The values of e_i for i = 14 and 15

		$e_{14} = R E (T_{14}^{*}, t_{4})$						Range of λ in which $T_{14}^{*}$ is efficient to t₄
n	ß	λ = 1.00	λ = 1.25 and λ = 0.75	λ = 1.50 and λ = 0.50	λ = 1.75 and λ = 0.25	λ = 2.00	λ = 2.25	Range of λ in which $T_{14}^{*}$ is efficient to t₄
2	2.5	1.6380	1.6099	1.5309	1.4152	1.2799	1.1397	(0,2.51)
	3.0	1.3628	1.3487	1.3080	1.2455	1.1674	1.0803	(0,2.47)
	3.5	1.2398	1.2311	1.2056	1.1654	1.1135	1.0531	(0,2.45)
	4.0	1.1723	1.1663	1.1485	1.1202	1.0827	1.0381	(0,2.44)
	4.5	1.1306	1.1262	1.1130	1.0917	1.0633	1.0288	(0,2.44)
	5.0	1.1028	1.0993	1.0891	1.0725	1.0501	1.0227	(0,2.43)
3	2.5	1.1733	1.1672	1.1494	1.1209	1.0832	1.0383	(0,2.44)
	3.0	1.1116	1.1078	1.0967	1.0786	1.0543	1.0246	(0,2.43)
	3.5	1.0785	1.0759	1.0682	1.0557	1.0385	1.0173	(0,2.43)
	4.0	1.0585	1.0566	1.0509	1.0416	1.0288	1.0129	(0,2.42)
	4.5	1.0454	1.0439	1.0396	1.0324	1.0224	1.0100	(0,2.42)
	5.0	1.0363	1.0351	1.0316	1.0259	1.0180	1.0080	(0,2.42)
4	2.5	1.0976	1.0944	1.0847	1.0690	1.0477	1.0215	(0,2.43)
	3.0	1.0644	1.0623	1.0561	1.0458	1.0317	1.0142	(0,2.43)
	3.5	1.0459	1.0445	1.0400	1.0327	1.0227	1.0101	(0,2.42)
	4.0	1.0345	1.0334	1.0301	1.0247	1.0171	1.0076	(0,2.42)
	4.5	1.0269	1.0261	1.0235	1.0193	1.0134	1.0059	(0,2.42)
	5.0	1.0216	1.0209	1.0189	1.0155	1.0107	1.0047	(0,2.42)
5	2.5	1.0563	1.0545	1.0490	1.0401	1.0278	1.0124	(0,2.42)
	3.0	1.0378	1.0366	1.0330	1.0270	1.0187	1.0083	(0,2.42)
	3.5	1.0272	1.0264	1.0238	1.0195	1.0135	1.0060	(0,2.42)
	4.0	1.0203	1.0197	1.0178	1.0146	1.0101	1.0045	(0,2.42)
	4.5	1.0161	1.0156	1.0141	1.0116	1.0080	1.0035	(0,2.42)
	5.0	1.0130	1.0126	1.0113	1.0093	1.0065	1.0028	(0,2.42)
6	2.5	1.0395	1.0382	1.0344	1.0282	1.0196	1.0087	(0,2.42)
	3.0	1.0267	1.0258	1.0233	1.0191	1.0133	1.0059	(0,2.42)
	3.5	1.0193	1.0187	1.0169	1.0138	1.0096	1.0042	(0,2.42)
	4.0	1.0146	1.0142	1.0128	1.0105	1.0073	1.0032	(0,2.42)
	4.5	1.0115	1.0111	1.0100	1.0082	1.0057	1.0025	(0,2.42)
	5.0	1.0092	1.0090	1.0081	1.0066	1.0046	1.0020	(0,2.42)
7	2.5	1.0278	1.0269	1.0243	1.0199	1.0138	1.0061	(0,2.42)
	3.0	1.0189	1.0183	1.0165	1.0135	1.0094	1.0041	(0,2.42)
	3.5	1.0137	1.0133	1.0120	1.0098	1.0068	1.0030	(0,2.42)
	4.0	1.0104	1.0101	1.0091	1.0075	1.0052	1.0023	(0,2.42)
	4.5	1.0082	1.0079	1.0071	1.0059	1.0041	1.0018	(0,2.42)
	5.0	1.0066	1.0064	1.0058	1.0047	1.0033	1.0014	(0,2.42)
8	2.5	1.0214	1.0207	1.0187	1.0153	1.0107	1.0047	(0,2.42)
	3.0	1.0146	1.0142	1.0128	1.0105	1.0073	1.0032	(0,2.42)
	3.5	1.0106	1.0103	1.0093	1.0076	1.0053	1.0023	(0,2.42)
	4.0	1.0081	1.0078	1.0071	1.0058	1.0040	1.0018	(0,2.42)
	4.5	1.0064	1.0062	1.0056	1.0046	1.0032	1.0014	(0,2.42)
	5.0	1.0051	1.0049	1.0044	1.0036	1.0025	1.0011	(0,2.42)

		$e_{15} = R E (T_{14}^{*}, T_{04})$					Range of λ in which $T_{14}^{}$ is efficient to T₀₄*
n	ß	λ = 1.00	λ = 1.25 and λ = 0.75	λ = 1.50 and λ = 0.50	λ = 1.75 and λ = 0.25	λ = 2.00 and λ = 0.00	Range of λ in which $T_{14}^{}$ is efficient to T₀₄*
2	2.5	1.2799	1.2578	1.1962	1.1058	1.0000	[0,2]
	3.0	1.1674	1.1553	1.1205	1.0669	1.0000	[0,2]
	3.5	1.1135	1.1056	1.0828	1.0467	1.0000	[0,2]
	4.0	1.0827	1.0772	1.0608	1.0346	1.0000	[0,2]
	4.5	1.0633	1.0591	1.0467	1.0267	1.0000	[0,2]
	5.0	1.0501	1.0469	1.0371	1.0213	1.0000	[0,2]
3	2.5	1.0832	1.0776	1.0611	1.0348	1.0000	[0,2]
	3.0	1.0543	1.0508	1.0402	1.0231	1.0000	[0,2]
	3.5	1.0385	1.0360	1.0286	1.0165	1.0000	[0,2]
	4.0	1.0288	1.0270	1.0215	1.0124	1.0000	[0,2]
	4.5	1.0224	1.0210	1.0167	1.0097	1.0000	[0,2]
	5.0	1.0180	1.0168	1.0134	1.0078	1.0000	[0,2]
4	2.5	1.0477	1.0446	1.0353	1.0203	1.0000	[0,2]
	3.0	1.0317	1.0297	1.0236	1.0136	1.0000	[0,2]
	3.5	1.0227	1.0213	1.0169	1.0098	1.0000	[0,2]
	4.0	1.0171	1.0160	1.0128	1.0074	1.0000	[0,2]
	4.5	1.0134	1.0125	1.0100	1.0058	1.0000	[0,2]
	5.0	1.0107	1.0101	1.0080	1.0047	1.0000	[0,2]
5	2.5	1.0278	1.0260	1.0207	1.0120	1.0000	[0,2]
	3.0	1.0187	1.0175	1.0140	1.0081	1.0000	[0,2]
	3.5	1.0135	1.0127	1.0101	1.0059	1.0000	[0,2]
	4.0	1.0101	1.0095	1.0076	1.0044	1.0000	[0,2]
	4.5	1.0080	1.0075	1.0060	1.0035	1.0000	[0,2]
	5.0	1.0065	1.0061	1.0048	1.0028	1.0000	[0,2]
6	2.5	1.0196	1.0183	1.0146	1.0085	1.0000	[0,2]
	3.0	1.0133	1.0124	1.0099	1.0058	1.0000	[0,2]
	3.5	1.0096	1.0090	1.0072	1.0042	1.0000	[0,2]
	4.0	1.0073	1.0068	1.0055	1.0032	1.0000	[0,2]
	4.5	1.0057	1.0054	1.0043	1.0025	1.0000	[0,2]
	5.0	1.0046	1.0043	1.0035	1.0020	1.0000	[0,2]
7	2.5	1.0138	1.0129	1.0103	1.0060	1.0000	[0,2]
	3.0	1.0094	1.0088	1.0070	1.0041	1.0000	[0,2]
	3.5	1.0068	1.0064	1.0051	1.0030	1.0000	[0,2]
	4.0	1.0052	1.0049	1.0039	1.0023	1.0000	[0,2]
	4.5	1.0041	1.0038	1.0031	1.0018	1.0000	[0,2]
	5.0	1.0033	1.0031	1.0025	1.0014	1.0000	[0,2]
8	2.5	1.0107	1.0100	1.0080	1.0046	1.0000	[0,2]
	3.0	1.0073	1.0068	1.0055	1.0032	1.0000	[0,2]
	3.5	1.0053	1.0050	1.0040	1.0023	1.0000	[0,2]
	4.0	1.0040	1.0038	1.0030	1.0018	1.0000	[0,2]
	4.5	1.0032	1.0030	1.0024	1.0014	1.0000	[0,2]
	5.0	1.0025	1.0024	1.0019	1.0011	1.0000	[0,2]

5. Conclusion

It is observed from Table 1 that the values of the relative efficiencies e_i′ s,i = 1,2,…,7 of the proposed minimum mean squared error (MMSE) estimators T_0i, i = 1,2,3,4 with respect to Lesitha and Thomas (2012) estimators t_i′ s,i = 1,2,3,4 respectively are greater than ‘unity’. Thus, the proposed estimators T_0i, i = 1,2,3,4 are more efficient than the corresponding usual estimators t_i′ s,i = 1,2,3,4 respectively.

It is further observed that the values of the relative efficiency e₅ of T₀₄ with respect to T₀₁ are the largest among e_i′ s,i = 1,2,…,7, from which follows that the proposed MMSE estimator T₀₄ is the best estimator among Lesitha and Thomas (2012) estimators t_i′ s,i = 1,2,3,4 and MMSE estimators T_0i,i = 1,2,3,4.

Tables 2 to 5 demonstrate that for fixed (n, β) the values of relative efficiencies e_i′ s,i = 8,9,…,15 increase as λ increases up to 1, while it decreases if λ goes beyond ‘unity’. When the value of λ is unity (i.e. the guessed value α₀ coincides with the true value α) a higher gain in efficiency is seen. For fixed values of (n, λ) the values of e_i′ s,i = 8,9,…,15 decrease as β increases. When (β, λ) are fixed the values of e_i′ s,i = 8,9,…,15 also decrease as sample size n increases. A higher gain in efficiency is obtained when the sample size n is small. In general, the estimators $T_{1 i}^{*}, i = 1, 2, 3, 4$ are more efficient than Lesitha and Thomas (2012) estimators t_i′ s,i = 1,2,3,4 and MMSE estimators T_0i, i = 1,2,3,4 respectively when λ∈(0,2.42). It is further observed that $T_{1 i}^{*}, i = 1, 2, 3, 4$ are respectively better than MMSE estimators T_0i,i = 1,2,3,4 when λ∈[0,2].

Acknowledgement

The authors are highly grateful to the referees for their constructive comments/ suggestions that helped in the improvement of the revised version of the paper.

References

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FIGURES & TABLES

REFERENCES

Ahmad, M. I., Sinclair, C. D., Werritty, A., (1988). Log-logistic flood frequency analysis. J Hydrol, 98, pp. 205–224.
[CROSSREF]
Balakrishnan, N., Malik, H. J., (1987). Best linear unbiased estimation of location and scale parameter of the log-logistic distribution. Commun Stat Theory Methods, 16, pp. 3477–3495.
[CROSSREF]
Bennett, S., (1983). Log-logistic regression models for survival data. J R Stat Soc, Ser C 32, pp. 165–171.
Chen, Z., Bai, Z., Sinha, B. K., (2004). Ranked set sampling, theory and applications. Lecture Notes in Statistics, Springer, New York.
[CROSSREF]
Fisk, P. R., (1961). The graduation of income distributions. Econometrica, 29, pp. 171–185.
[CROSSREF]
Geskus, R. B., (2001). Methods for estimating the AIDS incubation time distribution when data of seroconversion is censored. Stat Med, 20, 795–812.
[CROSSREF]
Lesitha, G., Thomas, P. Y., (2012). Estimation of the scale parameter of A LOG-LOGISTICS DISTRIBUTION. METRIKA, DOI 10.1007/S00184-012-0397-5.
[CROSSREF] [URL]
Mcintyre, G. A., (1952). A method for unbiased selective sampling using ranked sets. Aust J Agric Res, 3, pp. 385–390.
[CROSSREF]
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Publications

Statistics in Transition New Series

Improved Estimation of the Scale Parameter for Log-Logistic Distribution Using Balanced Ranked Set Sampling

ARTICLE

ABSTRACT

1. Introduction

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

(13)

(14)

(15)

2. Improved estimation of the scale parameter α

(16)

(17)

(18)

3. Improved estimation of the scale parameter α with prior information

(19)

(20)

(21)

(22)

(23)

(24)

4. Relative efficiencies

Table 1.

The values of ei′ s,i = 1,2,…,7.

Table 2.

The values of ei for i = 8 and 9

Table 3.

The values of ei for i = 10 and 11

Table 4.

The values of ei for i = 12 and 13

Table 5.

The values of ei for i = 14 and 15

5. Conclusion

Acknowledgement

References

FIGURES & TABLES

REFERENCES

EXTRA FILES

COMMENTS

The values of e_i′ s,i = 1,2,…,7.

The values of e_i for i = 8 and 9

The values of e_i for i = 10 and 11

The values of e_i for i = 12 and 13

The values of e_i for i = 14 and 15