A Class of Shrinkage Testimators for the Shape Parameter of the Weibull Lifetime Model

In this paper, we propose two classes of shrinkage estimators for the shape parameter of the Weibull distribution in censored samples. The proposed estimators are studied theoretically and have been compared numerically with existing estimators. Computer intensive calculations for bias and relative efficiency show that for, different values of levels of significance and for varying constants involved in the proposed estimators, the proposed testimators fare better than classical and existing estimators


Introduction
The Weibull model (Weibull 1939(Weibull , 1951(Weibull , 1952) ) is often used in the field of life data analysis due to its flexibility.In addition, it can simulate the behavior of other statistical distributions such as the normal and the exponential.Indeed, the wide application and occurrence of the Weibull distribution in reliability engineering and in failure analysis are a wonder.Specific applications of the Weibull model are employed to represent manufacturing and delivery times in industrial engineering, to forecast weather data, to model fading channels in wireless communications, to exhibit good fit to experimental fading channel measurements, as well as in radar systems to model the dispersion of the received signals level produced by some types of clutters, etc.Other applications are studied by many other authors (see Lieblein   The estimations of the unknown parameters of the above model are quite complicated.Bain and Engelhardt (1992) 1972), has been shown to follow chi-square distribution with 2m degrees of freedom.The p.d.f. of T is given by,

Incorporating a prior value, and Shrinkage
When a life testing experimenter becomes familiar with failure data, knowledge is developed concerning the parameters of the model.The discipline of quality control deals with setting the process to a suitable average on the basis of control charts.Since the mean of the Weibull failure time depends on the shape parameter, a similar control method can be used to bring the shape parameter to some prefixed value ), ( 0 leading to improvement in the performance of an item or component, i.e., reducing the MSE of the new estimators or it may give a saving in sample size.Indeed, the prior information costs time and money; and incorporating such prior information in the estimation of the unknown parameters is also utilizes the past cost of sampling units. According to Thompson (1968), 0 is a 'natural origin' and such natural origins may arise for any one of a number of reasons, e.g., we are estimating and: (i) we believe 0 is close to true value of , or (ii) we fear that 0 may be near the true value of , i.e., something bad happens if 0 and we do not know about it.
In both cases, the value 0 is available, and in such a situation, it is natural to start with an estimator ˆof and modify it by moving it closer to 0 , so that the resulting estimator, though perhaps biased, has a smaller mean squared error than that of ˆin some interval around 0 .This method of constructing an estimator of that incorporates the prior information 0 leads to what is known as a shrunken estimator.It may be recalled that Thompson (1968) the first who proposed the shrinkage estimator, which suggests the use of a prior point guess of the parameter for improving the performance of the existing estimator .ˆ Al- Hemyari and Al-Hemyari and Ali (2010, 2012)have proposed some shrinkage testimators for the scale parameter and reliability function of the Weibull model.
The purpose of this paper is not simply to extend to extend our previous testimators (2010, 2012) to the shape parameter of the Weibull model.Rather, we assume a censored sample where the aim is to find some testimators of the shape parameter which offer some improvement over the classical and similar estimators.Assuming the scale parameter is known, two appropriate choices of exponential type shrinkage weighting functions are used and the expressions for the bias, mean squared error, and relative efficiency of the proposed testimators are derived, studied and compared numerically.

Shrinkage estimators
Define the class of Huntsberger (1955)  The shrinkage estimator of the shape parameter has been considered by several authors (Singh and Bhatkulikar 1977, Pandey 1983, Pandey, et. al. 1989, Pandey and Singh 1993, and Singh and Shukla 2000).Estimator ( 5) is also studied for the shape parameter but in different contexts (Singh et. al. 2002).It may be noted here that other authors (e.g., Kambo et. al. 1990, 1992, Parkash et. al. 2008, and Al-Hemyari et. al. 2009, 2011) have tried to develop new shrinkage estimators of the form (5) for special populations by choosing different weight functions.
It is also noted that the performance of these estimators strongly depends on the choice of ).
( If is not set in accordance with reality (i.e., large ) ( when o is close to , and small ) ( when o is away from ), it may happen that either there is no significant gain in the performance of ~or there is actually a significant loss.

Bias and MSE of
The bias of ~by definition is, is the bias of .
ˆ The mean squared error (MSE) expression of ~is given by, is the mean squared error expression of .
ˆ When 0 we have ii) Unbiasedness: Based on equation ( 6), if 1 ) ( , or 0 ˆ with probability one, the proposed estimator turns into the unbiased estimator, otherwise it is biased.Thus, we conclude the following: There does not exist any unbiased estimator of in the class of estimators } 1 ) ( 0 : { except the above undesirable cases. iii) Minimum mean squared error estimator: It is not easy with the type of the proposed testimator to establish the minimum mean squared error biased estimator, i.e., for every ) ( and every with strict inequality for at least one .But when 0 the inequality holds (see equation ( 8)), this means that by a proper choice of ), ( the proposed shrinkage estimator performs better (in the sense of smaller MSE) than ˆin the neighborhood of .0 In this section, two shrinkage estimators of the class } 1 ) ( 0 : { for the shape parameter of the Weibull distribution, when a prior guess value of the shape parameter is available from the past with known shape parameter , will be discussed.

The Shrinkage estimator 1 ~
The first proposed testimator for of the class  6) and ( 7), the bias ratio (bias/ ) and mean squared error expression of 1 ~ are given respectively by, ). / ( 0The relative efficiency of 1 ~ is denoted by Eff and given by,

The Shrinkage estimator 2 ~
Since the shrinkage estimator 1 ~ is biased, in this section, in place of unbiased estimator , ˆ1 we will use the biased estimator , )) denoting the resulting estimator by 2 ~ with the weight function Again using ( 6) and ( 7), the bias ratio (bias/ ) and mean squared error expression of 2 ~ are given respectively by The efficiency of 2 ~ relative to 2 ˆ is given by, The efficiency of 2 ~ relative to 1 ˆ is given by, are asymptotically unbiased and consistent estimators.

Preliminary Shrinkage estimators
In section 2, a class of Huntsberger type shrinkage estimator was studied, and two cases for the shape parameter with known scale parameter were discussed by using two different shrinkage weight functions and two different classical estimators.This section also deals with the estimation of the shape parameter of the Weibull distribution with known scale parameter, where we developed a preliminary test shrinkage estimator when its initial estimate 0 is given.
have the disadvantage of necessarily using the prior value in the construction of final estimators.However, it is not necessary that the prior value be close to the true value.To employ this idea in the estimation of the shape parameter of the Weibull distribution, a preliminary test is first conducted to check the closeness of 0 to before using it in a shrinkage estimator.If the preliminary test is accepted, as an estimator of ; otherwise ˆitself is taken as an estimator of .Thus, the proposed testimator is taken as one of two alternatives depending on this test.To satisfy this idea, set The class of preliminary shrinkage estimators (PSE) with this weight function is denoted by p ~and given by, where R I and R I are respectively the indicator functions of the acceptance region R and the rejection region R .The relevance of such types of shrinkage estimators lies in the fact that, though they may be biased, they have smaller MSE than ˆin some interval around .o It may be denoted here that the class of estimators ( 19) is a special case of the class (5).
It may be noted here that the class of preliminary test shrinkage estimators p ~is completely specified if the shrinkage weight factor ( cannot be guaranteed.Similarly for the choice of region R there is no unified approach.

Bias and MSE of p
The bias and mean squared error expressions of p ~are derived for any and R and given respectively by: Remark 3: From equations ( 22) and ( 23) above, it may be noted that remark 1 derived in section 2, is also valid for the shrinkage estimator p ~, i.e., the unbiasedness and minimum mean squared error estimator properties are valid when using PSE.This means that there does not exist, any unbiased estimator of in the class of estimators ), 1 ) ( 0 ( ~ p except the same undesirable cases; and for any region R with proper choice of ), ( the preliminary shrinkage estimator PSE performs better (in the sense of smaller MSE) than the classical estimator ˆwhen 0 is sufficiently close to .

Choices for region R
As was noted earlier, the performance of the class of estimators (19) ) is used, the region 2 R is given by, r is the lower 100( /2) percentile point of the chi-square distribution with m 2 degrees of freedom.In this section, two testimators for the shape parameter of the Weibull distribution, when a prior guess value of the shape parameter is available from the past with known scale parameter , will be discussed.

The PSE
By equations ( 20) and ( 21) the expressions for the bias ratio and mean squared error of 1 ~p are obtained as follows: The efficiency of ˆ is given by,

~p
In section 3.3, the SPE is studied based on .
ˆ1 In this section, in place of the unbiased estimator , ˆ1 we will study the SPE based on the biased estimator with the weight function , , , ) ( and denoting the resulting testimator by Again using equations ( 20) and( 21), the bias ratio (bias/ ) and mean squared error expression of 2 ~p are given respectively by: The efficiency of 2 ~p relative to 1 ˆ is given by,

Simulation and Numerical Results
The bias ratio and relative efficiency of were computed for different values of the constants involved in these estimators.The following numerical results and comparisons are based on these computations.

Numerical Results of : ~i
For the testimators R is used, are not reported here for space consideration.
vi) It is seen that the relative efficiency of

Comparisons
Comparing results of 2 , 1 , ~ i i given in tables 1-4 with the tables given in (Singh and Bhatkulikar 1977, Pandey 1983, Pandey, Malik, et. al. 1989, Pandey and Singh 1993, and Singh and Shukla 2000), it is seen that our proposed testimators are better both in terms of higher relative efficiency and for the wider range of for which efficiency is greater than unity.It may be noted here that the numerical results of Singh and Shukla (2000)  given in tables 7-14 with the above existing results, it is seen that our testimators compare favorably.

Conclusions
Modified shrinkage estimators in the class of Huntsberger (1955) type shrinkage estimator ~have been suggested.The performance of the proposed shrinkage estimators of the shape parameter when some prior guess value of is available have been analyzed by using the criteria of bias ratio, mean squared error and relative efficiency.The class of estimators thus obtained seems to be an improved version of the existing estimators given in subsection 4.3, subject to certain conditions.The proposed estimators lead us to formulate many interesting estimators of shrinkage type.It is identified that when the guessed value 0 coincides exactly with the true value and also when 0 is moderately far away from , we get a larger gain in efficiency over the classical estimator in the effective interval of (broader range of for which efficiency is greater than unity).Thus, even if the experimenter has less confidence in the guessed value, the efficiency of the proposed estimators can be increased considerably by suitably choosing the scalars c a, and .The suggested estimators have substantial gain in efficiency for a number of choices of c a, and , when the sample size is small i.e., for the heavy censoring ( 3 m 20 n , ).Even for large sample sizes, so far as the proper selection of scalars is concerned, all the proposed estimators are found more efficient than the classical estimator but for a smaller effective interval of .The superiority of the suggested estimators p ~over the existing estimators given in subsection 4.3 has also been recognized.
The suggested class of shrinkage estimators are therefore recommend for its use in practice.
are respectively the scale and the shape parameters.
R are specified.Consequently, the success of ~now depends upon the proper choice of ) ( and R .In general, the true value of is unknown, i.e., ) ( should not be a function of unknown and hence, a proper choice of )

1 ~p
and uses the unbiased estimator 1 ˆ (given in section 2.2) and with the shrinkage weight function , , , ) (

1 ~p
is reasonably small for all values of m, c a, and (tables13 and 14).

The Model and Classical Estimator
and Zelen 1956, Kao 1959, Berrettoni 1964, Al-Mmeida 1999, Fok et.al. 2001, Erto and Pallotta 2007, and Rinne 2009).follows an extreme value distribution (refer to Bain 1972) with the probability distribution function, e then y have proposed a simple estimation procedure of the reciprocal of the shape parameter as follows.Let Engelhardt and Bain (1973 Bain (1973 depends on a proper choice of the region R and the shrinkage function 1R simplifies to: