Optimal Shrinkage Estimations for the Parameters of Exponential Distribution Based on Record Values Estimación shrinkage de los parámetros de la distribución exponencial basada en valores record

This paper studies shrinkage estimation after the preliminary test for the parameters of exponential distribution based on record values. The optimal value of shrinkage coefficients is also obtained based on the minimax regret criterion. The maximum likelihood, pre-test, and shrinkage estimators are compared using a simulation study. The results to estimate the scale parameter show that the optimal shrinkage estimator is better than the maximum likelihood estimator in all cases, and when the prior guess is near the true value, the pre-test estimator is better than shrinkage estimator. The results to estimate the location parameter show that the optimal shrinkage estimator is better than maximum likelihood estimator when a prior guess is close to the true value. All estimators are illustrated by a numerical example.


Resumen
Este artículo estudia la estimación shrinkage posterior al test preliminar de los parámetros de la distribución exponencial basada en valores record.El valor óptimo de los coeficientes de shrinkage es obtenido también usando el criterio minimax regret.La máxima verosimilitud, pre-test, y los estimadores shrinkage son obtenidos usando estudios de simulación.Los resultados de la estimación del parámetro de escala muestran que el estimador shrinkage es major que el de máxima verosimilitud en todos los casos, y cuando el valor a priori es cercano del valor real, el estimador pre-test es major que el a Assistant Professor.E-mail: hzaker@yazd.ac.ir b Assistant Professor.E-mail: aajafari@yazd.ac.ir c Student.E-mail: mkarimiz68@yahoo.com

Introduction
The unknown parameters in a distribution can usually be estimated by the maximum likelihood estimator (MLE) or the uniformly minimum-variance unbiased estimator (UMVUE).These estimators are solely based on the sample information.Sometimes, prior (non-sample) information about the parameters is available from previous experience or expert knowledge.To incorporate the nonsample information into the statistical procedure, a null hypothesis regarding the information is usually formulated and tested (see e.g.Bancroft 1944, Bancroft & Han 1977, Han, Rao & Ravichandran 1988).To estimate the parameter of interest, both sample information and prior information are used in shrinkage estimation when the null hypothesis is not rejected.However, if the null hypothesis is rejected, then only the sample information is used in the estimation.
The two-parameter exponential distribution has been used widely in the field of life testing and reliability theory.The shrinkage estimators of the scale parameter have been proposed by Bhattacharya & Srivastava (1974) and Pandey (1983).Chiou & Han (1989) gave a shrinkage estimation for threshold parameter.Chiou & Han (1995) proposed a pre-test estimator and a pre-test shrinkage estimator (PTSE) for the location parameter.Chiou (1990) gave an estimation of scale parameter for two exponential distributions based on censored data.The shrinkage estimation for the difference between location parameters for two exponential distribution is given by Chiou & Miao (2005).Using a LINEX loss function, the performance of the shrinkage testimators for the scale parameter of an exponential distribution is studied by Pandey (1997) and Prakash & Singh (2008).
Record values are of interest and important in many real life applications involving data relating to meteorology, sport, economics and life testing.Let X 1 , X 2 , . . .be a sequence of independent and identically distributed random variables that has a same distribution.An observation X j will be called an upper record value if exceeds in value all of the preceding observations, i.e., if X j > X i , for every i < j.The sequence of record times T n , n ≥ 0 is defined as follows: T 1 = 1 with probability 1 and, for ≥ 1, T n = min{j : X j > X Tn−1 }.The sequence of upper record values is then defined by {X U (n) = X Tn , n = 0, 1, 2, . . .}.For details on record values and other interesting topics related to records see Ahsanullah (1995) and Arnold, Balakrishnan & Nagaraja (1998).
In this paper, the PTSE for the parameters of exponential distribution is evaluated based on record values.The optimal value of shrinkage coefficient is obtained by using the minimax regret criterion.This paper is organized as follows: In Section 2, we will give a pre-test estimation for the scale parameter of exponential distribution based on record values.Then, the optimal value of the coefficient in PTSE for this parameter is obtained.Similarly, the optimal value of the coefficient in PTSE for the location parameter in two exponential distribution is obtained in Section 3. The PTSE and MLE are compared in Section 4 using Monte Carlo simulation.These estimators are also illustrated using a numerical example in Section 5.

Optimal Shrinkage Estimation for the Scale Parameter
Suppose that we observe n upper record values X U (1) , . . ., X U (n) from an exponential model with the following probability density function (pdf): It is well-known that X U (n) is a complete sufficient statistic for θ.Also, is the MLE of θ, and 2n θ M L θ has a chi-square distribution with 2n degrees of freedom (see Arnold et al. 1998).Also, to test where χ 2 2n,γ is the γth quantile of chi-square with 2n degrees of freedom.
Assume that θ 0 is a prior guess of θ.A pre-test estimator for θ based on θ M L is given as otherwise.
(1) Zakerzadeh & Karimi (2013) showed that under the weighted square error loss L(θ; d) = ( d θ − 1) 2 the risk function of this estimator is where A j = C2δ C1δ w j g(w)dw, j = 0, 1, 2, and δ = θ0 θ , and g(.) is the pdf of chisquare random variable with 2n degrees of freedom.Note that the risk function depends on α through C 1 and C 2 .
The proposed pre-test estimator in (1), uses θ 0 to estimate θ when the preliminary test fails to reject the null hypothesis H 0 : θ = θ 0 .Instead of using θ 0 , we can use a linear combination of θ 0 and θ M L when the preliminary test fails to reject H 0 (when δ ≈ 1).This gives a shrinkage estimator which assigns suitable weights to θ 0 and θ M L .The estimator is where 0 ≤ K ≤ 1.Note that θ pt approaches θ 0 as α → 0 and it approaches θ M L as α → 1; however, θ s approaches θ pt as K → 1 and it approaches θ M L as K → 0.
The shrinkage coefficient, K is not defined explicitly as a function of the test statistic.The weighting function approach is intuitively appealing, but the mean square error of the resulting estimator usually cannot be derived unless the weighting function is in some simple form.Unfortunately, a different value of significance level (α) or a different value of shrinkage coefficient (K) results in a different estimator.The choice of these values depends on the decision criterion.Here, we obtain an optimal value of K for θ s , based on a regret function.At first we evaluate the mean square error of θ s .
Lemma 1.For a fixed value of α, the mean square error of θ s is a function of K and take the following form: where I(.) is the indicator function.Similarly the second moment of θ s is and the proof is completed.
The regret function is defined as Chiou & Han (1989) also discussion the same topic: the regret function REG(δ, K) takes a maximum value at δ L and δ U , for fixed α (see Figure 1).Thus the minimax regret criterion determines K * , optimal value of K, such that Considering n = 5 and α = 0.05, the optimal K is 0.132.The plot of risk functions for K = 0, 1, 0.132 is shown in Figure 1.To find K * , we consider two cases: Case I: The value of K * for some degrees of freedom is presented in Table 1 for α = 0.05.
Case II: The value of K * for some degrees of freedom is presented in Table 1 for α = 0.16.For the AIC optimal level of significance see Inada (1984).Note 1.In general, if we observe n upper record values X U (1) , . . ., X U (n) from location-scale exponential distribution , and 2n θ M L θ has a chi-square distribution with 2n − 2 degrees of freedom.In this case, we got the same result as with the special case, in absence of location parameter with a new statistic and degree of freedom.

Optimal Shrinkage Estimation for the Threshold Parameter
In the following discussion we always suppose that we observe n upper record values X U (1) , . . ., X U (n) from an exponential model with the pdf in (6).
It is well-known that under the hypothesis and the likelihood ratio test rejects Therefore, by assuming a prior guess such as η = η 0 , a pre-test estimator for η is given as where c = F (2,2n−2),1−α and η M L = X U (1) .The properties of η pt depends on whether η < η 0 or η > η 0 .
The pre-test estimator given in (7) uses the prior estimate η 0 when the pre-test accepts the null hypothesis.Instead of using η 0 , we can use a linear combination of η 0 and η when the pre-test accepts , this which gives a PTSE which assigns suitable weights to η 0 and η rather than assigning weight 1 to η 0 and 0 to η when the pre-test accepts.This PTSE is Lemma 2. For PTSE of η, η s , we have where f (w, t) is the joint pdf of W and T .Note that W has an exponential distribution with parameter θ and T has a gamma distribution with parameters n − 1 and θ.Also, they are independent random variables.Therefore, Revista Colombiana de Estadística 39 (2016) 33-44 and the proof is completed.
ii) The proof is similar to part i).
In this section, we study the PTSE η s following the same procedure given in Section 2. Consider the loss function L(η, η) = (η−η) 2 θ 2 .In the following lemma, the risk of η s is calculated under this loss function and is denoted by R 2 (δ, K). where Proof .The proof is obvious.
Using the regret function and similar to Section 2, we obtained the optimal K.We consider two cases: Case I: The optimal value of K for some n are presented in Table 2 for α = 0.05.
Case II: The optimal value of K for some n are presented in Table 2 for α = 0.16, which is AIC optimal level of significance (see Inada 1984).

Simulation Study
We performed a simulation study and generated record values from an exponential distribution with η = 0 and θ = 1 with n = 6, 11, 16.The simulation was repeated 2000 times, and we obtained θML , θpt , and θs , for some prior guess θ 0 of θ.We also, obtained ηML , ηpt , and ηs , for some prior guess η 0 of η.The bias and MSE of the estimators are calculated, and the results are presented in Tables 4 and 5.
From Table 4, it can be concluded that i) θs is better than θML in all cases, ii) for prior guess (θ 0 ) near the true value of θ(= 1), θpt is better than θs and iii) for prior guess (θ 0 ) far from the true value of θ(= 1), θs is better than θpt .
From Table 5, it can be concluded that i) ηML is better than ηs when η 0 (prior guess) is very far from the true value of η(= 0), otherwise ηs is better than ηML , and ii) ηs is better than ηpt for η 0 less than the true value of η, and vice versa.

Numerical Example
The following example is based on a data set discussed by Dunsmore (1983) and Balakrishnan & Chan (1994).
A rock crushing machine is kept working as long as the size of the crushed rock is larger than the rocks crushed before.Otherwise it is reset.The data given below represent the sizes of the crushed rocks up to the third reset of the machine 9. 3, 0.6, 24.4, 18.1, 6.6, 9.0, 14.3, 6.6, 13.0, 2.4, 5.6, 33.8The upper records are 9.3, 24.4, 33.8.It follows that the MLE's of θ and η are θ M L = 8.16 and η M L = 9.3, respectively.Based on these records we will compute the various estimators for the two-parameter exponential model.Table 3 present the values of the pre-test and shrinkage estimators for various choices of the prior guesses of the scale parameter (θ 0 ) and location parameter (η 0 ).
It can be seen that three estimators of θ are equal, for θ 0 is less than 5, but the null hypothesis is not rejected for θ 0 ≥ 5, and so θpt is equal to the prior guess and all estimators are different.Also, all estimators for η are different for 8.7 ≤ η 0 ≤ 9.3, and otherwise the null hypothesis is rejected, so they are equal.

Conclusion
In some cases, there is non-sample prior information about the parameter of a population.Therefore, we can use both sample and non-sample information to estimate the parameter of interest.In this paper, we considered inference about the location-scale exponential distribution when the record values are available as sample information.For the scale parameter, we proposed a PTSE based on the MLE.It is a linear combination of the prior information and MLE when the preliminary test fails to reject the null hypothesis.We evaluated the MSE of the PTSE and obtained its optimal value based on the regret function, numerically.Simulation studies showed that the optimal PTSE is better than the MLE in all cases.Similarly, we obtained the optimal PTSE for the location parameter of exponential distribution.Simulation studies showed that this estimator is better than the MLE when the prior guess is close to the true value.In the end, all estimators are illustrated by a numerical example for some prior guess.

Figure 1 :
Figure 1: The risk function of K with n = 5 and α = 0.05.

Table 1 :
Optimal value of K for some n in shrinkage estimation for the scale parameter.

Table 2 :
Optimal value of K for some n in shrinkage estimation for the location parameter.

Table 3 :
Bias and MSE of the estimators for the scale parameter.

Table 4 :
Bias and MSE of the estimators for the location parameter.

Table 5 :
ML, pre-test, and shrinkage estimations for the scale and location parameters.