Generalized Inverted Exponential Distribution; Maximum Likelihood Estimation; Triangular Fuzzy Number; Graded Mean Method; Centroid Method.

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ESTIMATING THE PARAMETERS OF GENENERALIZED INVERTED EXPONENTIAL DISTRIBUTION WITH FUZZY DATA BY

DEPENDING ON PROPOSED APPROACH IN GENERALIZATION RANKING METHODS

Dr.Qutaiba N. Nayef Al-Kazaz 1^#1, Hawraa J. Kadhim Al-Saadi 2^#2

#1 Department of Statistics, College of Administration and Economics, Baghdad University, Iraq.

#2 Department of Mathematics, College of Science for Women, Baghdad University, Iraq

ABSTRACT

In this paper, we propose a new ranking function which depends on the generalization and the incorporation between the graded mean method and the centroid method to eliminate the fuzziness of the generalized inverted exponential distribution's data then we estimate the parameters of this distribution by maximum likelihood estimation. Simulation experiments used to compare the results.

Key words:

Generalized Inverted Exponential Distribution; Maximum Likelihood Estimation; Triangular Fuzzy Number; Graded Mean Method; Centroid Method.

1- INTRODUCTION

Statistical analysis, in traditional form, is based on crispness of data, random variable, point estimation, hypotheses, parameters and so on. But in this paper, a statistical analysis is based on the fuzziness of the data where the fuzzy data^[12] is set-valued observations in which data consists of sets rather than points of some domain of interest. Therefore; to treat the fuzzy problem suffered by the data distributed according to the generalized inverted exponential (GIE) distribution, which it parameters will be estimated in this paper we must use the ranking functions, where ranking of fuzzy quantities is based on extracting various features from fuzzy sets. These features may be a center of gravity, an area under the membership function, or various intersection points between fuzzy sets^[13].

There are studies that have estimated the parameters of the (GIE) distribution in classical set theory:

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©2016 RS Publication, [email protected] Page 325 Abouammob and Alshingiti^[1] have proposed the (GIE) distribution by introducing a shape parameter to the inverted exponential distribution. They studied parameters estimation by using maximum likelihood estimation and least square estimation methods. Also they discussed statistical and reliability properties of the distribution .Singh, Singh and Kumar ^[17] estimated the (GIE) distribution parameters by using maximum likelihood estimation and Bayesian estimation methods in case of the progressive type-II censoring scheme with binomial removal. Potdar and Shirke^[14] have introduced a generalized inverted scale parameter family of distributions, the (GIE) distribution is a member of this family. Maximum likelihood estimators of scale and shape parameters were obtained. Umesh, Sanjay and Rajwant ^[16] have proposed the method of maximum product of spacing for parameters point estimation of (GIE) distribution. They compared the proposed estimation and traditional estimation methods (maximum likelihood estimation and least square estimation).

2- THE MODEL

This lifetime distribution is capable of modeling various shapes of failure rates and hence various shapes of ageing criteria^[1].

The probability density function (PDF) and the cumulative distribution function (CDF) of the (GIE) distribution are given by the forms respectively ^[14]:

Where is a shape parameter and is a scale parameter.

Fig (2-1) represents the PDF and the CDF of the (GIE) distribution respectively when and and

The reliability (survival) function and the hazard (failure) rate function of the (GIE) distribution are given by respectively:

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Fig (2-2) represents the reliability function and the hazard function of the (GIE) distribution respectively when and and

3- PARAMETERS ESTIMATION

The main aim of a probability distribution study is to estimate the parameters of those distributions.

The likelihood function of the (GIE) distribution for a sample of size n is given as follows:

By taking the log function to equation (3-1) we get

Differentiating equation (3-2) with respect to and respectively and equating the derivatives to zero gives

The equations (3-3) and (3-4) are nonlinear; therefore we can use any iterative procedure such as Newton-Raphson method to get a solution.

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The fuzzy set is a generalization of a classical set that was introduced by Zadeh in 1965 as a mathematical way to represent many phenomena's in real life when precise measurements are not available such as high temperature, water levels in lakes and river, speed measurements, blood pressure, etc.

We will introduce some concepts of fuzzy set theory as follows

Definition4.1. Let be a nonempty set. A fuzzy set in T is defined as a set of ordered pairs;

where is called the membership function for the fuzzy set it's defined as^{[2, 15]}:-

Definition4.2. The support of a fuzzy set is the set of all points t such that (i.e)^[2]

Definition4.3. The -cut of a fuzzy set is the set of all points t in T such that , where

Where is a nonempty bounded closed interval contained in T and can by denoted by where and are lower and upper bounds.

But the strong -cut is defined as ^{[2, 20]}:-

Definition4.4. The height of a fuzzy set is any element t in T such that this element has the largest membership ^{[4, 15]}

Definition4.5. A fuzzy set is called normal if . Otherwise is called subnormal ^[5,

15].

Definition4.6. A fuzzy set is called convex

^[15].

Definition4.7. A fuzzy set of R (the set of real numbers) is called fuzzy number if it's satisfies the following conditions ^[2]:-

1-Normality 2-Convexity

Definition4.8.Triangular fuzzy number is a fuzzy number represented with three points as follows ^[10]:-

where is left end point, is center point and is right end point. It's represented by triangular membership function

Remark4.9. The -cut of a triangular fuzzy number can be expressed as ^[20]:

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, then

We called equation (4-3) is a triangular fuzzy data this is because represented a data satisfying the (GIE) distribution.

We must use the ranking function methods to convert the formula of the triangular fuzzy data (4- 3) to crisp data.

We will explain that in the following section.

5- Ranking Functions Methods

An efficient approach for comparing the fuzzy numbers is by using of a ranking function

[9]. It's an important aspect of decision making in a fuzzy environment. In fuzzy decision making problems, fuzzy numbers must be ranked before an action is taken by a decision maker ^[11]. To transfer a fuzzy numbers to crisp numbers we must define a ranking function as follows:- , where is a set of all fuzzy numbers, which maps each fuzzy number into the real line , where a natural order exists.

Let and be two triangular fuzzy numbers in then [9]

1) 2) 3)

Now we want to find the relationship between two of ranking functions methods, the generalization of graded mean integration representation method and the generalization of centroid method.

Definition5.1. let be a fuzzy number, and be the lower and upper bounds respectively. Then the graded mean -preference integration representation of is ^[6]

Where .

We generalized equation by putting the -th order ( of cut as follows

The value of depends on the decision maker's preference. Usually we chooses since it does not bias to the left or the right ^[3]then equation (5-2) becomes

We call equation (5-3) as the generalization of graded mean integration representation method.

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Yarer (1981) proposed it as follows

Where is a measure of the importance of the value of t.

Bander and Simonovic (1996- 2000) generalized equation (5-4) by using ( as a power for membership function as follows

Cheng (1998) supposed ,^[13] we gets

Then the Generalization of Centroid Method of triangular fuzzy number is defined as

Remark5.3. The relationship between equation (5-3) and equation (5-7) is

To prove the relation (5-8) notice the following:

Many studies are special cases of equation (5-3) or equation (5-7) If (Wang, 2009) we get

If (Khalaf, 2014) (Tang, 2003) we get

If (Nareshkumar&Ghuru, 2014) we get

If (Hussein& Dheyab, 2015) we get

If we get

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Simulation procedure will be used to study each case and comparison between ranking methods which is resulting from different values of . Simulation procedure is explained in the following steps:

The first step:

In this step determine the default values as following:

1- Choosing the sample sizes n= 14, 25, 50 and100.

2- Choosing the default values of parameters and and and . The Second step:

The formula of a random number generation of (GIE) distribution is

Where is a random variable uniformly distributed on (0, 1).

The third step:

Using formula (4-3) to obtain a fuzzy data set for all sizes The fourth step:

In this step we convert a fuzzy data to crisp data by using the formulas (5-9), (5-10), (5-11), (5- 12) and (5-13).

The fifth step:

Estimating the model parameters by solving equations (3-3) and (3-4) for each data set generated from the previous step.

The Sixth step:

We compare all estimators by using the mean squared error of the model, which it is given by

Where represents the number of experiment replicates. (In this paper =500)

The method with the minimum mean squared error for model becomes the best method.

The mean squared error for and given respectively by

After applying the six steps above, we reached the following results The nine tables below including the following for each sample size:

1- The estimators of and

2- The mean squared error ( ) for the model, . 3- Different default values of and

4- The following values of and

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and

n

Parameters K

MSE

For Model

n MSE

For Model

14

-0.5 0.3437596

(0.0095699)

0.9674095 (0.0122586)

0.001592

25

0.3191451 (0.0028447)

0.9818420 (0.0062885)

0.001360

0 0.3437766

(0.0095692)

0.9713409 (0.0121044)

0.001612 0.3191445

(0.0028447)

0.9858839 (0.0062074)

0.001364

1 0.3436748

(0.0095275)

0.9754310 (0.0119441)

0.001629 0.3191281

(0.0028412)

0.9899790 (0.0061513)

0.001370

2 0.3437637

(0.0095666)

0.9773972 (0.0119373)

0.001640 0.3191415

(0.0028447)

0.9920166 (0.0061489)

0.001372

3 0.3437736

(0.0095684)

0.9785934 (0.011910)

0.001646 0.3191418

(0.0028442)

0.9932466 (0.0061443)

0.001374

50

-0.5 0.312917416 (0.0016815)

0.9853289 (0.0052074)

0.0004277

100

0.309824614 (0.0011190)

0.9751752 (0.0046129)

0.000230

0 0.31291473

(0.0016814)

0.9893899 (0.0051461)

0.0004258 0.30983298

(0.0011188)

0.9791646 (0.0044615)

0.000233

1 0.312917562

(0.0016815)

0.9934716 (0.0051176)

0.0004241 0.309832989

(0.0011188)

0.9832107 (0.0043426)

0.000237

2 0.312917402

(0.0016815)

0.9955290 (0.0051160)

0.0004233 0.309833028

(0.0011188)

0.9852463 (0.0042953)

0.000238

3 0.312908729

(0.0016827)

0.9967942 (0.0051123)

0.0004229 0.309828108

(0.0011192)

0.9864901 (0.0042759)

0.000239

Table (6-2) Simulation results for the parameters estimation (MSE for parameters) when and

N

Parameters K

MSE

For Model

n MSE

For Model

14

-0.5 0.3357381 (0.0082280)

1.4576935 (0.0279388)

0.005223

25

0.3184931 (0.0029346)

1.4730465 (0.0170734)

0.002301

0 0.3357377

(0.0082281)

1.4636946 (0.0276838)

0.005221 0.3184932

(0.0029346)

1.4791086 (0.0169192)

0.002296

1 0.3357376

(0.0082281)

1.4697431 (0.0274982)

0.005218 0.3184938

(0.0029347)

1.4852186 (0.0168375)

0.002292

2 0.3355802

(0.0081516)

1.4729779 (0.0272531)

0.005221 0.3184911

(0.0029344)

1.4883014 (0.0168221)

0.002290

3 0.3356564

(0.0082039)

1.4748001 (0.0273814)

0.005216 0.3184941

(0.0029347)

1.4901435 (0.0168266)

0.002289

50

-0.5 0.3086675 (0.0012862)

1.4752211 (0.0110793)

0.000892

100

0.3068773 (0.0008770)

1.4749317 (0.0082621)

0.000654

0 0.3086674

(0.0012862)

1.4812923 (0.0109017)

0.000908 0.3068773

(0.0008770)

1.4810012 (0.0080576)

0.000664

1 0.3086670

(0.0012861)

1.4874150 (0.0107978)

0.000925 0.3068501

(0.0008697)

1.4871728 (0.0078922)

0.000675

2 0.3086606

(0.0012850)

1.4905154 (0.0107785)

0.000934 0.3068773

(0.0008770)

1.4901998 (0.0078886)

0.000681

3 0.3086590

(0.0012869)

1.4923950 (0.0107934)

0.000939 0.3068759

(0.0008768)

1.4920583 (0.0078759)

0.000686

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and

N

Parameters K

MSE

For Model

n MSE

For Model

14

-0.5 0.3227562 (0.0052749)

1.9485729 (0.0420154)

0.0067742

25

0.3109791 (0.0020854)

1.9857247 (0.0255141)

0.0028007

0 0.3227563

(0.0052749)

1.9565914 (0.0415798)

0.0068592 0.3109789

(0.0020854)

1.9938976 (0.0255563)

0.0028373

1 0.3227006

(0.0052567)

1.9648705 (0.0413347)

0.0069410 0.3109616

(0.0020817)

2.0022086 (0.0257613)

0.0028748

2 0.3227563

(0.0052749)

1.9687442 (0.0411670)

0.0069900 0.3109755

(0.0020853)

2.0062998 (0.0258785)

0.0028953

3 0.3226076

(0.0052031)

1.971619 (0.0412741)

0.0069912 0.3109787

(0.0020859)

2.0087788 (0.0259786)

0.0029073

50

-0.5 0.3073820 (0.0013447)

1.9797943 (0.0197695)

0.0026329

100

0.3071695 (0.0007694)

1.9661815 (0.0130825)

0.0011495

0 0.3073820

(0.0013447)

1.9879416 (0.0196663)

0.0026778 0.3071699

(0.0007694)

1.9742706 (0.0126989)

0.0011717

1 0.3073820

(0.0013447)

1.9961563 (0.0196974)

0.0027240 0.30716967

(0.0007694)

1.9824297 (0.0124450)

0.0011949

2 0.3073751

(0.0013449)

2.0003302 (0.0197657)

0.0027474 0.3071682

(0.0007695)

1.9865438 (0.0123715)

0.0012069

3 0.3073525

(0.0013482)

2.0030328 (0.0200624)

0.0027613 0.3071699

(0.0007694)

1.9890040 (0.0123381)

0.0012142

Fig (6-1) an illustrative overview to curve ofthe (GIE)distribution with real data and fuzzy data when n = 100 and and and

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and

N Parameters K

MSE

For Model

n MSE

For Model

14

-0.5 0.5895954 (0.0241957)

0.957426 (0.0081098)

0.0053863

25

0.571725 (0.0167536)

0.965825 (0.0059769)

0.0039184

0 0.5897896

(0.0243086)

0.961339 (0.0078579)

0.0054530 0.571831

(0.0168069)

0.969732 (0.0057735)

0.0039647

1 0.5897182

(0.0242709)

0.965315 (0.0076101)

0.0055120 0.572140

(0.0169308)

0.973596 (0.0056238)

0.0040158

2 0.5898162

(0.0243059)

0.967273 (0.0075073)

0.0055471 0.572048

(0.0168885)

0.975640 (0.0055371)

0.0040370

3 0.5898148

(0.0243058)

0.9684780 (0.0074459)

0.0055663 0.572142

(0.0169306)

0.976822 (0.0054967)

0.0040537

50

-0.5 0.5406477 (0.0078503)

0.9769376 (0.0040365)

0.0016176

100

0.520651 (0.0066436)

0.979132 (0.0109244)

0.0016306

0 0.5406029

(0.0078324)

0.9809816 (0.0038956)

0.0016365 0.520653

(0.0066410)

0.983160 (0.0106520)

0.0016611

1 0.5406597

(0.0078528)

0.9850046 (0.0037886)

0.0016585 0.520652

(0.0066438)

0.9872233 (0.0104615)

0.0016926

2 0.5406531

(0.0078514)

0.9870471 (0.0037458)

0.0016693 0.520648

(0.0066429)

0.9892709 (0.0103922)

0.0017087

3 0.5406598

(0.0078528)

0.9882715 (0.0037250)

0.0016760 0.520650

(0.0066436)

0.9904986 (0.0103579)

0.0017185

N

Parameters K

MSE

For Model

n MSE

For Model

14

-0.5 0.5675878 (0.0383427)

1.4511404 (0.0265288)

0.0210676

25

0.5444694 (0.0119034)

1.4674738 (0.0168514)

0.0103239

0 0.5674692

(0.0382885)

1.4572572 (0.0261955)

0.0213206 0.5444592

(0.0119023)

1.4735300 (0.0166238)

0.0103417

1 0.5672965

(0.0381444)

1.4634275 (0.0259091)

0.0215676 0.5444680

(0.0119034)

1.4796049 (0.0164722)

0.0103615

2 0.5675856

(0.0383424)

1.4661675 (0.0257879)

0.0217411 0.5444675

(0.0119026)

1.4826664 (0.0164219)

0.0103722

3 0.5675776

(0.0383404)

1.4680015 (0.0257278)

0.0218235 0.5444691

(0.0119033)

1.4845093 (0.0164021)

0.0103789

50

-0.5 0.5341573 (0.0066436)

1.4680817 (0.0109244)

0.0028820

100

0.5115359 (0.0024376)

1.4779618 (0.0067217)

0.0016366

0 0.5341313

(0.0066410)

1.4741695 (0.0106519)

0.0029391 0.5115517

(0.0024372)

1.4840090 (0.0065407)

0.0016306

1 0.5341573

(0.0066437)

1.4802149 (0.0104615)

0.0029987 0.5115227

(0.0024442)

1.4902491 (0.0064958)

0.0016268

2 0.5341538

(0.0066429)

1.4832853 (0.0103922)

0.0030292 0.5115511

(0.0024371)

1.4932278 (0.0064092)

0.0016247

3 0.5341567

(0.0066436)

1.4851253 (0.0103579)

0.0030479 0.5115503

(0.0024374)

1.4950879 (0.0064045)

0.0016240

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and

n

Parameters K

MSE

For Model

n MSE

For Model

14

-0.5 0.5421249 (0.0175025)

1.9517278 (0.0328766)

0.0361638

25

0.5364559 (0.0084539)

1.9504150 (0.0292819)

0.0129918

0 0.5421260

(0.0175029)

1.9597580 (0.0324186)

0.0363833 0.53645572

(0.0084539)

1.9584419 (0.0287710)

0.0131851

1 0.5421241

(0.0175024)

1.9678603 (0.0320874)

0.0366081 0.53642664

(0.0084433)

1.9665934 (0.0284009)

0.0133796

2 0.5421269

(0.0175028)

1.9719278 (0.0319714)

0.0367225 0.53645369

(0.0084544)

1.9706132 (0.0282427)

0.0134843

3 0.5421255

(0.0175026)

1.9743841 (0.0319161)

0.0367919 0.53642019

(0.0084268)

1.9730814 (0.0281476)

0.0135439

50

-0.5 0.5160682 (0.0039317)

1.9661023 (0.0168877)

0.0089240

100

0.50783612 (0.0018726)

1.9701857 (0.0100866)

0.0020353

0 0.5160680

(0.0039317)

1.9741940 (0.0165344)

0.0090779 0.50783691

(0.0018728)

1.9782919 (0.0097452)

0.0020195

1 0.516068

(0.0039317)

1.98235314 (0.0163119)

0.0092361 0.50783722

(0.0018727)

1.9864657 (0.0095340)

0.0020063

2 0.5160670

(0.0039315)

1.98645861 (0.0162502)

0.0093168 0.50783577

(0.0018725)

1.9905818 (0.0094774)

0.0020008

3 0.5160681

(0.0039317)

1.98892633 (0.0162288)

0.0093658 0.50783715

(0.0018728)

1.9930545 (0.0094613)

0.0019978

Fig (6-2) an illustrative overview to curve of the (GIE)distribution with real data and fuzzy data when n = 100 and and and

(12)

and

n

Parameters K

MSE

For Model

n MSE

For Model

14

-0.5 0.78001050 (0.0095404)

0.982607 (0.0045546)

0.010422

25

0.787250614 (0.0071537)

0.98870547 (0.0024831)

0.0094753

0 0.78008813

(0.0095207)

0.986596 (0.0044535)

0.010544 0.787203169

(0.0071449)

0.99276874 (0.0024279)

0.0095688

1 0.78009269

(0.0095199)

0.990672 (0.0043963)

0.010669 0.787249092

(0.00715456)

0.99687844 (0.0024047)

0.0096648

2 0.78009363

(0.0095197)

0.992723 (0.0043801)

0.010732 0.787253828

(0.0071531)

0.99893947 (0.0024056)

0.0097139

3 0.77989830

(0.0096113)

0.994107 (0.0043121)

0.010772 0.787253281

(0.0071532)

1.00018203 (0.0024104)

0.0097437

50

-0.5 0.786487 (0.006625)

0.99212 (0.001931)

0.003189

100

0.79975612 (0.0047254)

0.98850525 (0.0018139)

0.00152670

0 0.786472

(0.006627)

0.996198 (0.001898)

0.003162 0.799754891

(0.0047256)

0.99257336 (0.0017508)

0.00152213

1 0.786487

(0.006625)

1.00032 (0.0019)

0.003137 0.799748909

(0.0047270)

0.99667818 (0.0017209)

0.00151960

2 0.786352

(0.006677)

1.002539 (0.001947)

0.003127 0.799754158

(0.0047257)

0.99873858 (0.0017183)

0.001519083

3 0.786453

(0.006636)

1.003664 (0.001929)

0.003118 0.799750808

(0.0047263)

0.99998176 (0.0017209)

0.001519089

n

Parameters K

MSE

For Model

n MSE

For Model

14

-0.5 0.877256 (0.0264558)

1.457348 (0.0096387)

0.01394

25

0.861408 (0.0174345)

1.460413 (0.0064174)

0.00880

0 0.877265

(0.0264611)

1.463342 (0.0092270)

0.01409 0.861478

(0.0174369)

1.466366 (0.0060097)

0.00892

1 0.877271

(0.0264608)

1.469384 (0.00888610)

0.01424 0.861474

(0.0174360)

1.472427 (0.0056789)

0.00905

2 0.877261

(0.0264577)

1.472430 (0.0087419)

0.01431 0.861480

(0.0174371)

1.475473 (0.0055407)

0.00911

3 0.877181

(0.0264797)

1.474396 (0.0087192)

0.01436 0.861481

(0.0174375)

1.477307 (0.0054663)

0.00915

50

-0.5 0.841824 (0.014915)

1.461454 (0.006923)

0.00677

100

0.829853 (0.0081429)

1.470215 (0.0049349)

0.01015

0 0.841803

(0.014911)

1.467483 (0.006542)

0.0068 0.829811

(0.0081229)

1.476275 (0.0046424)

0.01018

1 0.841821

(0.014916)

1.473535 (0.006228)

0.00683 0.829779

(0.0081139)

1.482390 (0.0044222)

0.01021

2 0.841808

(0.014907)

1.476583 (0.006097)

0.00685 0.829802

(0.0081245)

1.485456 (0.0043442)

0.01023

3 0.84179

(0.014932)

1.478487 (0.006057)

0.00686 0.829822

(0.0081271)

1.487289 (0.0043027)

0.01024

(13)

and

n

Parameters K

MSE

For Model

N MSE

For Model

14

-0.5 0.985471

(0.1368017)

1.903517 (0.0479065)

0.03540

25

0.88753 (0.0365993)

1.94339 (0.023152)

0.02434

0 0.985466

(0.1368001)

1.911356 (0.0467732)

0.03553 0.88757

(0.0366068)

1.95134 (0.0224826)

0.02460

1 0.984879

(0.1361779)

1.919452 (0.0455984)

0.03565 0.88756

(0.0366012)

1.95941 (0.0219253)

0.02487

2 0.985440

(0.1367860)

1.923248 (0.0452978)

0.03575 0.88673

(0.0352889)

1.96378 (0.0215442)

0.02484

3 0.984986

(0.1362715)

1.925832 (0.0449586)

0.03571 0.88755

(0.0365944)

1.96592 (0.0215810)

0.02509

50

-0.5 0.854352

(0.020307)

1.961978 (0.014758)

0.01270

100

0.82917036 (0.0086361)

1.95773 (0.0049349)

0.00568

0 0.854345

(0.020305)

1.97006 (0.01432)

0.01293 0.82917037

(0.0086361)

1.96579 (0.0046424)

0.00577

1 0.854344

(0.020305)

1.978203 (0.01401)

0.01318 0.82916189 (0.0086332)

1.97392 (0.0044222)

0.00588

2 0.854285

(0.020273)

1.982339 (0.013892)

0.01331 0.82916838

(0.0086365)

1.97799 (0.0043442)

0.00592

3 0.854347

(0.020306)

1.984759 (0.013857)

0.01339 0.82917048

(0.0086361)

1.98046 (0.0043027)

0.00595

Fig (6-3) an illustrative overview to curve of the (GIE) distribution with real data and fuzzy data when n = 100 and and and

(14)

According to the above tables we can see that when which is leading to so which is represented in the formula (5-9) is the most efficient method to remove the fuzziness from the data and decrease efficiency methods gradually with increasing the value of which is leading to increasing the value of , when this means , so which is represented in the formula (5-13) is the least efficient method for each cases but the opposite occurs in the following:

1- For sample sizes 14 and 25 in table (6-2).

2- For sample size 50 in tables (6-1) and (6-7).

3- For sample size 100 in tables (6-5) and (6-7).

In general notes that the estimate value of scale parameter increases with increasing the value of and it didn't exceed the default value for for each cases, but for sample size 25 in table (6-3) the scale parameter did exceed the default value for by a simple amount when and

As for the estimate value of shape parameter it is irregular changing and when the follow-up the change of their value must describe each case separately. We can see that their value are exceeded the default value for in a simple amount and for each samples sizes except their value in table (6-7) for each size and in table (6-8) for sample size 50 only. It didn't exceeded but be very close to them.

REFERENCE

[1] Abouammoh A.M. and Alshingiti A.M.,(2009),'' Reliability Estimation of Generalized Inverted Exponential Distribution '' Journal of Statistical Computation and Simulation, (79)(11)(1301-1315).

[2] Bezdek J.C.,(1993),''Fuzzy Models – What Are They, and Why?'' IEEE Transactions on Fuzzy Systems, (1)( 1-6).

[3] Chen S.H., Wang S.T. and Chang S.M., (2006),''Some Properties of Graded Mean Integration Representation of L-R Type Fuzzy Numbers'' Tamsui Oxford Journal of Mathematical Sciences,(

22)( 2)(185-208).

[4] Dubois D. and Henri P.,(1980),"Fuzzy sets and system :Theory and Application" Academic preess, INC.San Diego NewYork.

[5] Full r R.,(1998),'' Fuzzy Reasoning and Fuzzy Optimization'' E tv s Lo and University, Budapest, Hungary.

[6] Gani A. N. and Abbas A.,(2013) "A New Method for Solving Intuitionistic FuzzyTransportation Problem" Applied Mathematical Sciences,( 7)( 28)(1357-1365).

[7] Hussein I.H. and Dheyab A.H.,(2015),'' A New Algorithm Using Ranking Function To Find Solution For Fuzzy Transportation Problem'' International Journal of Mathematics and Statistics Studies, (3)(3)(1-6).

[8] Khalaf W.S., (2014),"Solving Fuzzy Transportation Problem using A New Algorithm"

Journal of Applied Sciences,(4)(3)(252-258).

[9] Kumar A., Singh P. and Kaur A.,(2010),''Ranking of Generalized Exponential Fuzzy Number using Integral Value Approach'' Int. J. Advance. Soft Comput. Appl., (2) (2) (1-10).

(15)

©2016 RS Publication, [email protected] Page 338 [10] Nareshkumar S., Ghuru S.K., (2014) "Solving Fuzzy Transportation Problem Using Symmetric Triangular Fuzzy Number "International Journal of Advanced Research in Mathematics and Applications, (1)(74-83).

[11] Nasseri S.H., Taleshian F., Alizadeh Z. and Vahidi J., (2012)''A New Method for Ordering LR Fuzzy Number'' The Journal of Mathematics and Computer Science, (4)( 3)(283-294).

[12] Nguyen H.T. and Wu B.,(2006),''Fundamentals Of statistics With Fuzzy Data''Springer- verlag Berlin Heidelberg, New York,( 198).

[13] Prodanovic P., ( 2001),"Fuzzy Set Ranking Methods and Multiple Expert Decision Making"

The University of Western Ontario, (039).

[14] Potdar K.G. and D.T. Shirke, (2013),''Inference for the Parameters of Generalized Inverted Family of Distributions'' Prob.Stat.Forum, (6) (18-28).

[15] Sengupta A. and Pal T.K.,(2009),''Fuzzy Preference Ordering of Interval Numbers in Decision Problems'' Springer-verlag Berlin Heidelberg,(238).

[16] SinghU., Singh S.K. and Singh R.K., (2014)'' A Comparative Study of Traditional Estimation Methods and Maximum Product Spacings Method in Generalized Inverted Exponential Distribution'' Journal of Statistics Applications & Probability,( 3)(2)(153-169).

[17] Singh S.K., Singh U. and Kumar M., (2013)''Estimation of Parameters of Generalized Inverted Exponential Distribution for Progressive Type-II Censored Sample with Binomial Removals'' Journal of Probability and Statistics, (1-12).

[18] Tang H.C.,(2003),''Inconsistent Property Of Lee and Li Fuzzy Ranking Method'' An International Journal Computers & Mathematics with Applications, (709-713).

[19] Wang Y.-M.,(2009),''Centroid Defuzzification and The Maximizing Set and Minimizing Set Ranking Based on Alpha Level Sets'' Computers & Industrial Engineering,(1-9).

[20] Yu V.F., Dat L.Q., Quang N.H., Son T.A., Chou S. and Lin A.C., (2012),''An Extension of Fuzzy TOPSIS Approach Based on Centroid-Index Ranking Method'' Scientific Research and Essays, (7)(14)(1485-1493).