# Improved Estimators of the Mean of a Normal Distribution with a Known Coefficient of Variation

Authors: Wuttichai Srisodaphol; Noppakun Tongmol
Journal: Journal of Probability and Statistics
 (2012)

---

## Abstract

This paper is to find the estimators of the meanθ for a normal distribution with mean θ and variance aθ2, a>0, θ>0. These estimators are proposed when the coefficient of variation is known. A mean square error (MSE) is a criterion to evaluate the estimators. The results show that the proposed estimators have preference for asymptotic comparisons. Moreover, the estimator based on jackknife technique has preference over others proposed estimators with some simulations studies.


---

## Body

## 1. Introduction

For the population that is distributed as normal with mean(θ) and variance (σ2), the sample mean (X-) is the unbiased and minimum variance estimator. In the situation that coefficient of variation (β) is known where β2=a=σ2/θ2 for a>0 and θ>0, Khan [1] proposed the unbiased estimator (d*) and the asymptotic variance of d* is aθ2/n(1+2a). This estimator is the linear combination between X- and sample variance (S) and the asymptotic variance of estimator d* is the Cramer-Rao bound.Arnholt and Hebert [2] improved the estimator δk*=kT where T is an unbiased estimator of θ, k=(cβ2+1)-1, and constant c are known. They found that δk* has smaller mean square error (MSE) than the estimator  T. They also gave the example for T=X- and obtained the estimator δk*=n(β2+n)-1X- and MSE(δk*)=aθ2/(a+n). Then, δk* has MSE smaller than the estimator X-.This paper focuses on improving the estimators ofθ when the coefficient of variation is known. MSE is a criterion for evaluating the estimators. The estimators are proposed by using the method of Khan [1] and Arnholt and Hebert [2]. Also, the jackknife technique [3] is used to reduce the bias of estimator. Moreover, The Bayesian estimator [4] is proposed based on noninformative prior distribution by using Jeffreys’ prior distribution.The paper is organized as follows. The improved estimators are proposed in Section2. Asymptotic comparison and simulation study results are presented in Section 3. Finally, Section 4 contains conclusions.


## 2. Improved Estimators

LetX1,X2,…,Xn be independent and distributed as normal with mean (θ) and variance (σ2), and coefficient of variation (β) is known where β2=a=σ2/θ2 for a>0 and θ>0. There are three estimators proposed as follows.(1)  LetT1 be the proposed estimator of θ based on Khan [1], and Arnholt and Hebert [2],   T1=kd* where k is a constant. T1 is a biased estimator of θ with Bias(T1)=θ(k-1) and MSE(T1)=k2Var(d*)+θ2(k-1)2 where the asymptotic variance of d* is aθ2/n(1+2a). The minimum MSE of T1 is obtained by
(2.1)MSE(T1*)=aθ2(a+n+2an),
since k=(n+2an)/(a+n+2an).(2)  LetT2 be the proposed estimator of θ based on the jackknife technique. The estimator T1* is used to construct the jackknife estimator T2 as follows. Let T1,-i* be an estimator T1* based on the sample size n-1 by deleting the ith sample. Denote
(2.2)T1,i*=nT1*-(n-1)T1,-i*,i=1,2,…,n.
The estimator T2 is given by
(2.3)T2=∑i=1nT1,i*n.
The MSE of T2 is shown in the simulation study in Section 3.(3) Bayes estimatorT3 is obtained as follows.The likelihood function ofθ given data is
(2.4)L(θ∣data)=1(2πaθ2)n/2exp{-12aθ2∑i=1n(xi-θ)2}.
The log likelihood function is
(2.5)logL(σ2,ρ∣data)=-n2ln2π-n2lnaθ2-12aθ2∑i=1n(xi-θ)2.
The Jeffreys prior distribution is
(2.6)π(θ)∝I1/2(θ),
where I(θ) is Fisher’s information.Then, the prior distribution is(2.7)π(θ)∝1+2aaθ2.
The posterior distribution, the distribution of θ given data is
(2.8)π(θ∣data)=(1+2a)/aθ2(1/(2πaθ2)n/2)exp{-(1/2aθ2)∑i=1n(xi-θ)2}∫0∞(1+2a)/aθ2(1/(2πaθ2)n/2)exp{-(1/2aθ2)∑i=1n(xi-θ)2}dθ.
Therefore, The Bayes estimator of θ, T3 is given as
(2.9)E(θ∣data)=∫0∞θπ(θ∣data)dθ.
The MSE of T3 is shown in the simulation study in Section 3.


## 3. Asymptotic Comparison and Simulation Study Results

(1)  For asymptotic comparison, the estimators are compared based on the relative efficiency (RE) of MSEs. The RE ofd* with respect to T1* is obtained by
(3.1)RE=MSE(d*)MSE(T1*)=aθ2/(n+2an)aθ2/(a+n+2an)=a+n+2ann+2an>1.
It shows that MSE(T1*) is smaller than MSE(d*).TheRE of δk* with respect to T1* is obtained by
(3.2)RE=MSE(δk*)MSE(T1*)=aθ2/(a+n)aθ2/(a+n+2an)=a+n+2ana+n>1.
It shows that MSE(T1*) is smaller than MSE(δk*).Therefore, from (3.1) and (3.2), the proposed estimator T1* has smaller MSE thand* and δk*.(2) The simulation results are shown for the comparison MSEs among the three proposed estimators,T1*, T2, and T3. Let parameters θ=5, 10, and 15, and  a=0.01, 0.09, and 0.25 with small sample size n=10, 20, and 30. The results are shown in Tables 1, 2, and 3.Table 1
MSEs of the proposed estimatorsT1*, T2, and T3 when n=10.
θ
a
MSE
(
T
1
*
)
MSE
(
T
2
)
MSE
(
T
3
)
0.01
1.061
e
-
5
1.039
e
-
5
5.362
e
-
4
5
0.09
5.831
e
-
5
4.921
e
-
5
5.183
e
-
4
0.25
9.867
e
-
3
7.116
e
-
3
1.383
e
-
2
0.01
1.277
e
-
4
1.244
e
-
4
1.785
e
-
3
10
0.09
2.016
e
-
3
1.302
e
-
3
2.026
e
-
2
0.25
5.127
e
-
2
3.014
e
-
2
3.691
e
-
1
0.01
1.446
e
-
5
1.307
e
-
5
2.748
e
-
4
15
0.09
1.260
e
-
2
1.068
e
-
2
1.598
e
-
1
0.25
1.31905
1.08686
3.91847Table 2
MSEs of the proposed estimatorsT1*, T2, and T3 when n=20.
θ
a
MSE
(
T
1
*
)
MSE
(
T
2
)
MSE
(
T
3
)
0.01
2.061
e
-
5
2.601
e
-
5
5.362
e
-
4
5
0.09
1.068
e
-
4
5.353
e
-
5
5.362
e
-
4
0.25
1.303
e
-
2
8.556
e
-
3
1.439
e
-
2
0.01
2.366
e
-
5
2.315
e
-
5
1.785
e
-
3
10
0.09
1.573
e
-
2
1.342
e
-
2
2.003
e
-
2
0.25
2.511
e
-
1
1.921
e
-
1
3.459
e
-
1
0.01
1.358
e
-
5
1.286
e
-
5
2.748
e
-
4
15
0.09
8.374
e
-
2
7.339
e
-
2
1.849
e
-
1
0.25
7.765
e
-
1
4.851
e
-
1
3.91847Table 3
MSEs of the proposed estimatorsT1*, T2, and T3 when n=30.
θ
a
MSE
(
T
1
*
)
MSE
(
T
2
)
MSE
(
T
3
)
0.01
3.303
e
-
8
3.106
e
-
8
5.362
e
-
4
5
0.09
9.253
e
-
4
7.898
e
-
4
5.362
e
-
4
0.25
3.522
e
-
3
1.705
e
-
3
1.342
e
-
2
0.01
7.008
e
-
8
6.171
e
-
8
1.785
e
-
3
10
0.09
5.700
e
-
4
2.060
e
-
4
2.003
e
-
2
0.25
1.649
e
-
1
1.034
e
-
1
3.480
e
-
1
0.01
2.259
e
-
5
2.246
e
-
5
2.748
e
-
4
15
0.09
1.606
e
-
1
1.421
e
-
1
1.849
e
-
1
0.25
1.40001
1.02217
3.91846From Tables1–3, the results show that, for small sample size n, the estimator T2 has smaller MSEs than the estimator T3. We also see that, the estimator T2 has smaller MSEs than the estimator  T1*, since T2 is constructed by using the jackknife technique to reduce bias of the biased estimator T1*. Therefore, the estimator T2 is better than the estimators T1* and T3 within the intervals of θ and a.


## 4. Conclusions

These estimatorsT1*, T2, and T3 are proposed. The estimator T1*  is improved based on the methods of Khan [1] and Arnholt and Hebert [2]. The estimator T2 is obtained by reducing bias of T1*. The estimator T3 is a Bayesian estimator for the noninformative prior distribution by using the Jeffreys prior distribution. The estimator T1* is better than the estimators d* and δk* in the asymptotic comparison. Moreover, the estimator T2 is better than the estimators T1* and T3 with some simulation studies.



---

*Source: 807045-2012-12-11.xml*