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

Abstract

This paper is to find the estimators of the mean 𝜃 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.

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 𝛿^∗_𝑘=𝑘𝑇 where 𝑇 is an unbiased estimator of 𝜃, 𝑘=(𝑐𝛽²+1)⁻¹, and constant 𝑐 are known. They found that 𝛿^∗_𝑘 has smaller mean square error (MSE) than the estimator  𝑇. They also gave the example for 𝑇=𝑋 and obtained the estimator 𝛿^∗_𝑘=𝑛(𝛽²+𝑛)⁻¹𝑋 and MSE(𝛿^∗_𝑘)=𝑎𝜃²/(𝑎+𝑛). Then, 𝛿^∗_𝑘 has MSE smaller than the estimator 𝑋.

This paper focuses on improving the estimators of 𝜃 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 Section 2. Asymptotic comparison and simulation study results are presented in Section 3. Finally, Section 4 contains conclusions.

2. Improved Estimators

Let 𝑋₁,𝑋₂,…,𝑋_𝑛 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) Let 𝑇₁ 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 MSE𝑇^∗₁=𝑎𝜃²(),𝑎+𝑛+2𝑎𝑛 (2.1) since 𝑘=(𝑛+2𝑎𝑛)/(𝑎+𝑛+2𝑎𝑛).

(2) Let 𝑇₂ be the proposed estimator of 𝜃 based on the jackknife technique. The estimator 𝑇^∗₁ is used to construct the jackknife estimator 𝑇₂ as follows. Let 𝑇^∗_1,−𝑖 be an estimator 𝑇^∗₁ based on the sample size 𝑛−1 by deleting the 𝑖th sample. Denote 𝑇^∗_1,𝑖=𝑛𝑇^∗₁−(𝑛−1)𝑇^∗_1,−𝑖,𝑖=1,2,…,𝑛. (2.2) The estimator 𝑇₂ is given by 𝑇₂=^𝑛_𝑖=1𝑇^∗_1,𝑖𝑛. (2.3) The MSE of 𝑇₂ is shown in the simulation study in Section 3.

(3) Bayes estimator 𝑇₃ is obtained as follows.

The likelihood function of 𝜃 given data is 𝐿()=1𝜃∣data2𝜋𝑎𝜃^2𝑛/2−1exp2𝑎𝜃^2𝑛_𝑖=1𝑥_𝑖−𝜃². (2.4) The log likelihood function is log𝐿𝜎²𝑛,𝜌∣data=−2𝑛ln2𝜋−2ln𝑎𝜃²−12𝑎𝜃^2𝑛_𝑖=1𝑥_𝑖−𝜃². (2.5) The Jeffreys prior distribution is 𝜋(𝜃)∝𝐼^1/2(𝜃), (2.6) where 𝐼(𝜃) is Fisher’s information.

Then, the prior distribution is 𝜋(𝜃)∝1+2𝑎𝑎𝜃². (2.7) The posterior distribution, the distribution of 𝜃 given data is 𝜋(𝜃∣data)=(1+2𝑎)/𝑎𝜃²1/2𝜋𝑎𝜃^2𝑛/2exp−1/2𝑎𝜃²∑^𝑛_𝑖=1𝑥_𝑖−𝜃²∫^∞₀(1+2𝑎)/𝑎𝜃²1/2𝜋𝑎𝜃^2𝑛/2exp−1/2𝑎𝜃²∑^𝑛_𝑖=1𝑥_𝑖−𝜃²𝑑𝜃. (2.8) Therefore, The Bayes estimator of 𝜃, 𝑇₃ is given as 𝐸(𝜃∣data)=^∞₀𝜃𝜋(𝜃∣data)𝑑𝜃. (2.9) 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 of 𝑑^∗ with respect to 𝑇^∗₁ is obtained by 𝑑RE=MSE^∗MSE𝑇^∗₁=𝑎𝜃²/(𝑛+2𝑎𝑛)𝑎𝜃²/()=𝑎+𝑛+2𝑎𝑛𝑎+𝑛+2𝑎𝑛𝑛+2𝑎𝑛>1. (3.1) It shows that MSE(𝑇^∗₁) is smaller than MSE(𝑑^∗).

The RE of 𝛿^∗_𝑘 with respect to 𝑇^∗₁ is obtained by 𝛿RE=MSE^∗_𝑘MSE𝑇^∗₁=𝑎𝜃²/(𝑎+𝑛)𝑎𝜃²/()=𝑎+𝑛+2𝑎𝑛𝑎+𝑛+2𝑎𝑛𝑎+𝑛>1. (3.2) It shows that MSE(𝑇^∗₁) is smaller than MSE(𝛿^∗_𝑘).

Therefore, from (3.1) and (3.2), the proposed estimator 𝑇^∗₁ has smaller MSE than𝑑^∗ and 𝛿^∗_𝑘.

(2) The simulation results are shown for the comparison MSEs among the three proposed estimators,𝑇^∗₁, 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 estimators 𝑇^∗₁, 𝑇₂, and 𝑇₃ when 𝑛=10.

Table 2 

MSEs of the proposed estimators 𝑇^∗₁, 𝑇₂, and 𝑇₃ when 𝑛=20.

Table 3 

MSEs of the proposed estimators 𝑇^∗₁, 𝑇₂, and 𝑇₃ when 𝑛=30.

From Tables 1–3, 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 𝑇^∗₁, since 𝑇₂ is constructed by using the jackknife technique to reduce bias of the biased estimator 𝑇^∗₁. Therefore, the estimator 𝑇₂ is better than the estimators 𝑇^∗₁ and 𝑇₃ within the intervals of 𝜃 and 𝑎.

4. Conclusions

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

Acknowledgment

The authors would like to thank Computational Science Research Group, Faculty of Science, Khon Kaen University for the financial support.