Record Values from the Inverse Weibull Lifetime Model: Different Methods of Estimation

In this paper, we use the lower record values from the inverse Weibull distribution (IWD) to develop and discuss different methods of estimation in two different cases, 1) when the shape parameter is known and 2) when both of the shape and scale parameters are unknown. First, we derive the best linear unbiased estimate (BLUE) of the scale parameter of the IWD. To compare the different methods of estimation, we present the results of Sultan (2007) for calculating the best linear unbiased estimates (BLUEs) of the location and scale parameters of IWD. Second, we derive the maximum likelihood estimates (MLEs) of the location and scale parameters. Further, we discuss some properties of the MLEs of the location and scale parameters. To compare the different estimates we calculate the relative efficiency between the obtained estimates. Finally, we propose some numerical illustrations by using Monte Carlo simulations and apply the findings of the paper to some simulated data.


Introduction
Record values arise naturally in many real life applications involving data relating to weather, sport, economics and life testing studies.Many authors have studied record values and the associated statistics; see, for example, [1][2][3][4][5][6][7].Reference [8] has established some recurrence relations for the moments of record values from the Gumbel distribution.Similar work has been carried out by [9,10] for the generalized extreme value and exponential distributions, respectively.Reference [11] have also discussed some inferential methods based on record values from Gumbel distribution.[12,13] have discussed inferential techniques based on Weibull and generalized Pareto distributions, respectively.Reference [14] have compared different estimates based on record values from Weibull distribution.Reference [15] has considered different loss functions to develop the Bayesian estimates of the parameters of the IWD.Let The scale form of the IWD has its density function given by while the location-scale IWD has its density function given by (1.4) Reference [16] calls the IWD as the complementary Weibull distribution, while [17] call it the reciprocal Weibull distribution.Reference [18] have discussed some useful measures for the IWD.
The IWD plays an important role in many applications, including the dynamic components of diesel engines and several data set such as the times to breakdown of an insulating fluid subject to the action of a constant tension.[19] provide an interpretation of the IWD in the context of the load-strength relationship for a component.
Reference [20] has fitted the IWD to the flood data.For more details on the IWD, see for example, [21].
The the joint pdf of where (.) f and (.) F are given in (1.1) and (1.2), respectively.
The single and product moments of record values are (see [22]) where ( )   is the gamma function.
In the following section, we derive the exact form of the BLUE scale parameter and present the BLUEs of the location-scale case of IWD.Next in Section 3, we derive the maximum likelihood estimates of the parameters of IWD.Finally, in Section 4 we discuss the relative efficiency of the obtained estimates.

The BLUEs
In this section, we derive the BLUE of the scale parameter and present the BLUEs of the location and scale parameters of the IWD.

The Scale Case
Let (1) (2) ( ) , , , denote the first n lower record from the distribution in (1.3), and let , , , . Then, the BLUE of the scale parameter  is given by and its variance is given by for details, refer to [23,6].Since the double moment , m n  can be written as , ( 1 then the covariance matrix  can be inverted analytically.Then ( , From (1.8), (2.1) and (2.4), we have the BLUE of the scale parameter  as with variance is given by  is an unbiased estimate of The cdf of T is obtained to be ( , ) where ( , ) is the incomplete gamma function.

Example
Five lower record values are simulated from IWD with = 3 c and = 1.0  as follows: 3.07586, .90607,68454, .62296,.62283,by using the coefficients of the BLUE of the scale parameter given in Table 1, we have Then 95% confidence interval for  can be calculated from (2.7) to be (0.7320149148 , 1.352569185) .

The location-Scale Case
Reference [15] has used the single and product moments of the record values from IWD.Then, he has used these moments to calculate the coefficients of BLUEs and the variances for records of size 4,5,6 and 7 and = 3, 4,5 c by using the forms (see [23]). .while Table 3 represents the variances and covariances of the BLUEs in this case.
Reference [24] has used the coefficients of BLUEs in Table 2 to construct different confidence intervals for the location and scale parameters of IWD based on Edgeworth approximation and compare them with those based on Monte Calro simulation.

The MLEs
In this section, we discuss the maximum likelihood estimates of the parameters of IWD when the available data are lower record values.We consider two different cases: 1) the scale-parameter case and 2) the location-scale parameter case.

The Scale-Parameter Case
, , , represents the first n lower record values from the scale-parameter IWD in (1.3), then the log likelihood function is given by .
Now, we discuss two cases.They are: 1) When  unknown and c known: the maximum likelihood estimate of  can be obtained from (3.1) as 2) When both of  and c are unknown: the maximum likelihood estimates of  and c can be obtained from (3.2) by solving the following two equations as 2), we see that Table 1.The coefficients of the BLUE of the scale parameter and the variance when which upon using (1.8) gives is an unbiased estimate for  in this case.
The variance of 1  is calculated to be

Lemma 1
The MLE of  given in (3.2) is asymptotically unbiased and its variance converges to zero as n   .

The Location-Scale Case
, , , represents the first n lower record values from the location-scale parameter IWD given in (1.4), then the log likelihood function is given by Now, we discuss two cases.They are: 1) When c is known: the maximum likelihood estimates of  and  can be obtained by solving the following two equations Table 4 below displays the bias and the estimated variance of the MLEs of the location and scale parameters in this case.
2) When c is unknown: the maximum likelihood estimates of  ,  and c can be obtained from (3.9) by solving the following three equations as

Relative Efficiency
To compare between the BLUEs and the MLEs obtained in Sections 2 and 3, we calculate the relative efficiency in some cases as follows: 1) For the scale case we have Table 5 given below show the relative efficiency be-  -tween the different estimates of the location and scale parameters.From Table 5, we see that, the BLUEs give more efficiency than the MLEs.
n lower record values from the IWD with density function ( 166711.Now, we calculate the lower and upper 5% percentage points of of T to be 0.025 = 0.46048001250 T a n d 0.975 = 0.8508449965 T .

 . Table 1
below shows the coefficients n

Table 5 .
The relative efficiency.