The comparative studies on stress-strength reliability for Rayleigh distribution

오지은,손중권 한국데이터정보과학회 2018 한국데이터정보과학회지 Vol.29 No.4

This paper deals with the estimation of the stress-strength parameter R = P(Y < X) with three different methods for Rayleigh distribution. We assumed that the stress and strength variables are independent. We derive a maximum likelihood estimator of R and its an asymptotic distribution.We also compute an asymptotic confidence intervals. The bootstrap estimator and confidence intervals are also derived based on maximum likelihood estimator of R. We obtained the Bayes estimator based on inversed gamma priors on scale parameters and its most plausible set for constructing the confidence interval of R which is quiet similar to classical condence interval unlike the highest posterior density region We compare the performances of three estimators using the mean squared error (MSE) of each estimator. This paper deals with the estimation of the stress-strength parameter R = P(Y <X) with three different methods for Rayleigh distribution. We assumed that the stress and strength variables are independent. We derive a maximum likelihood estimator of R and its an asymptotic distribution. We also compute an asymptotic confidence intervals. The bootstrap estimator and confidence intervals are also derived based on maximum likelihood estimator of R. We obtained the Bayes estimator based on inversed gamma priors on scale parameters and its most plausible set for constructing the confidence interval of R which is quiet similar to classical confidence interval unlike the highest posterior density region We compare the performances of three estimators using the mean squared error (MSE) of each estimator.

제임스-스타인 추정량을 이용한 부대 사격 명중률의 추정 방법

황준호,박재신,방성완 한국자료분석학회 2016 Journal of the Korean Data Analysis Society Vol.18 No.1

To make firing exercise efficient and improve unit’s shooting capability, the military commander has to know his unit’s level for accuracy of fire exactly. By this time, the sample mean, which is maximum likelihood estimator, has been used to estimate accuracy of fire for each rifle soldier. The sample mean has been widely used in point estimation and one of the most effective methods for the estimation of personal record. When estimating several rifle soldier’s accuracy of fire simultaneously, however, James-Stein estimator is more efficient than maximum likelihood estimator in terms of the mean squared error. Therefore, we suggest using James-Stein estimator in estimating military unit’s level for accuracy of fire as an alternative of the maximum likelihood estimator. In this study, we demonstrated the effectiveness of the proposed method through a simulation study and a real data analysis in terms of mean squared error. 군인에게 요구되어지는 가장 기본적인 전투기술은 사격이며, 각급 부대는 부대의 사격수준을 향상시키기 위하여 많은 시간을 사격훈련에 할당하고 있다. 사격술의 중요성과 더불어 지휘관이 부대의 사격 수준을 올바르게 파악하고 앞으로의 수준을 제대로 예측할 수 있다면 부대 운영에 큰 도움이 될 것이다. 이러한 관점에서 본 논문에서는 부대의 사격수준을 올바르게 추정하기 위한 평가방법에 대하여 연구하였다. 현재 각급 부대에서는 사수 개인의 사격 수준을 평가하기 위하여 최대우도 추정량(maximum likelihood estimator)인 표본평균(sample mean)을 사용하고 있으며, 표본평균은 각 개인의 사격 수준을 평가하는 데에는 좋은 측정 도구로 사용될 수 있다. 그러나 사격은 개인이 아닌 부대 전체의 수준에 대한 평가가 필수적이므로 부대 전체의 사격 수준을 추정하기 위한 새로운 평가방법이 필요하다. 본 논문에서는 부대의 사격수준을 평가하는데 있어 표본평균보다 더 효율적인 제임스-스타인 추정량(James-Stein estimator)을 이용한 평가방법을 제안하였으며, 모의실험 및 실제 사격자료의 분석을 통하여 제임스-스타인 추정량의 우수한 성능과 활용 가능성을 확인하였다.

확산모형에 대한 누율생성함수의 근사와 가우도 측정법

이윤동(Yoon-Dong Lee),이은경(Eun-kyung Lee) 한국경영과학회 2013 韓國經營科學會誌 Vol.38 No.1

Diffusion is a basic mathematical tool for modern financial engineering. The theory of the estimation methods for diffusion models is an important topic of the financial engineering. Many researches have been tiled to apply the likelihood estimation method for estimating diffusion models. However, the likelihood estimation method for diffusion is complicated and needs much amount of computing. In this paper we develop the estimation methods which are simple enough to be compared to the Euler approximation method, and efficient enough statistically to be compared to the likelihood estimation method. We devise pseudo-likelihood and propose the maximum pseudo-likelihood estimation methods. The pseudo-likelihoods are obtained by approximating the transition density with normal distributions. The means and the variances of the distributions are obtained from the delta expansion suggested by Lee, Song and Lee (2012). We compare the newly suggested estimators with other existing estimators by simulation study. From the simulation study we find the maximum pseudo-likelihood estimator has very similar properties with the maximum likelihood estimator. Also the maximum pseudo-likelihood estimator is easy to apply to general diffusion models, and can be obtained by simple numerical steps.

다중 점진적 중도절단에서 하프 로지스틱 분포의 추정

박성희(Seonghee Park),이경준(Kyeongjun Lee) 한국데이터정보과학회 2020 한국데이터정보과학회지 Vol.31 No.5

점진적 중도절단 방법 (progressive censoring scheme)상황에서 관측되는 시점의 자료들 사이에는 관측원의 실수 혹은 관측 기계의 오류로 인하여 또 다른 중도절단이 발생할 수 있어 다중 점진적 중도절단 방법 (multiply progressive censoring scheme) 이 새롭게 제안되었다. 본 논문은 다중 점진적 중도절단 방법 상황에서 하프 로지스틱분포의 척도 모수를 추정하였다. 이를 위해 최대우도추정량 (maximumm likelihood estimator)을 계산하였고, 테일러 급수 전개를 이용한 근사된 최대우도추정량 (approximate maximumm likelihood estimator)을 이용하여 하프 로지스틱분포의 척도 모수를 추정하였다. 또한, 다양한 다중 점진적 중도절단 상황에서 몬테카를로 모의실험을 실시하여 평균 제곱오차 및 편의를 이용하여 제안한 추정량들을 비교하였고, 사례 자료를 이용하여 제안한 추정량들을 계산하였다. Under progressive censoring scheme, some units can be failed between two points of observation with exact times of failure of these units unobserved. In this reason, multiply progressive censoring scheme was introduced. In this paper, we consider the maximum likelihood estimator of the scale parameter of the half-logistic distribution under a multiply progressive censoring scheme. And, the scale parameter of half logistic distribution is estimated by approximate maximum likelihood estimators that use Taylor series expansion. Monte Carlo simulations are conducted to compare the results among maximum likelihood estimator and approximate maximum likelihood estimators. A real data set based on the multiply progressive censoring scheme is also analyzed for illustrative purposes.

통일된 복합 중도절단에서 하프 로지스틱 분포의 모수 추정

곽재영(Jaeyoung Gwag),이경준(Kyeongjun Lee) 한국데이터정보과학회 2018 한국데이터정보과학회지 Vol.29 No.1

복합 중도절단 (hybrid censoring)은 제 1종 (type I) 복합 중도절단과 제 2종 (type II) 복합 중도절단이 있는데 두가지 방법은 각각의 장 · 단점을 가지고 있다. 이들의 단점을 보완하고 장점을 경합한 방법이 일반화된 복합 중도절단 (generalized hybrid censoring) 이다. 일반화된 복합 중도절단 역시 제 1종 복합 중도절단과 제 2종 복합 중도절단이 있고, 이들을 결합한 것이 통일된 복합 중도절단 (unified hybrid censoring)이다. 본 논문에서는 하프 로지스틱 분포의 척도 모수 (scale parameter)을 추정하기 위해 최대우도추정량 (maximum likelihood estimator)을 사용하였다. 하지만 최대우도추정량은 정확하게 계산되어지지 않아서 척도 모수에 대하여 미분된 로그 우도함수 (log-likelihood function)을 테일러 급수 방법을 이용하여 근사시킨 근사된 최대우도추정량 (approximate maximum likelihood estimator)을 사용하였다. 그리고 추정 방법들을 비교하기 위해 몬테카를로 시뮬레이션을 통해 평균제곱오차를 구하여 서로 비교하였고, 실제 예제 데이터를 이용하여 분석하였다. The mixture of type I and type II censoring schemes is known as the hybrid censoring scheme. Hybrid censoring scheme has some disadvantages. Generalized hybrid censoring schemes (type I and type II) are designed to fix the disadvantages inherent in the hybrid censoring scheme. Unified hybrid censoring combine the two types of generalized hybrid censoring scheme. In this paper, we derive maximum likelihood estimator of the scale parameter for the half logistic distribution under the unified hybrid censoring samples. Also, we derive some approximate maximum likelihood estimators of the scale parameter for the half logistic distribution under unified hybrid censored samples. The scale parameter is estimated by approximate maximum likelihood estimation method using Taylor series expansion types. We compare the estimators in the sense of the mean square error. And we present examples to illustrate all estimators.

Estimation for the Half-Triangle Distribution Based on Progressively Type-II Censored Samples

한준태,강석복 한국데이터정보과학회 2008 한국데이터정보과학회지 Vol.19 No.3

We derive some approximate maximum likelihood estimators (AMLEs) and maximum likelihood estimator (MLE) of the scale parameter in the half-triangle distribution based on progressively Type-II censored samples. We compare the proposed estimators in the sense of the mean squared error for various censored samples. We also obtain the approximate maximum likelihood estimators of the reliability function using the proposed estimators. We compare the proposed estimators in the sense of the mean squared error.

Estimation for the Half Logistic Distribution under Progressive Type-II Censoring

Kang, Suk-Bok,Cho, Young-Seuk,Han, Jun-Tae The Korean Statistical Society 2008 Communications for statistical applications and me Vol.15 No.6

In this paper, we derive the approximate maximum likelihood estimators(AMLEs) and maximum likelihood estimator of the scale parameter in a half-logistic distribution based on progressive Type-II censored samples. We compare the proposed estimators in the sense of the mean squared error for various censored samples. We also obtain the approximate maximum likelihood estimators of the reliability function using the proposed estimators. We compare the proposed estimators in the sense of the mean squared error.

Estimation of half triangle distribution under generalized progressive hybrid censored sample

Ho-Yong Kim,Suk-Bok Kang 한국데이터정보과학회 2018 한국데이터정보과학회지 Vol.29 No.3

Using some Taylor series expansions, we obtain maximum likelihood estimator(MLE) and some approximate maximum likelihood estimators (AMLEs) of the scale parameter in half triangle distribution based on generalized progressive hybrid censored samples. We also obtain some estimators of the relibility function by using the proposed estimators of the scale parameter of the half triangle distribution. Finally, Monte Carlo simulations are used to assess the validity of the proposed estimators. The proposed AMLEs are obtained explicitly with closed form and more ecient than the MLE of the scale parameter.

Estimation of the parameters of a Wishart extension on symmetric matrices

Ghorbel Emna,Kammoun Kaouthar,Louati Mahdi,Sallem Akram 한국통계학회 2022 Journal of the Korean Statistical Society Vol.51 No.4

This paper deals with the parameters of a natural extension of the Wishart distribution, that is the Riesz distribution on the space of symmetric matrices. We estimate the shape parameter using two different approaches. The first one is based on the method of moments, we give its expression and investigate some of its properties. The second represents the maximum likelihood estimator. Unfortunately, in this case we do not have an explicit formula for this estimator. This latter is expressed in terms of the digamma function and sample mean of log-gamma variables. However, we derive the strong consistency and asymptotic normality properties of this estimator. A numerical comparative study between the two estimators is carried out in order to test the performance of the proposed approaches. For the second parameter, that is the scale parameter, we prove that the distribution of the maximum likelihood estimator given by Kammoun et al. (J Statist Prob Lett 126:127–131, 2017) is related to the Riesz distribution. We examine some properties concerning this estimator and we assess its performance by a numerical study.

Maximum penalized likelihood estimation for a stress-strength reliability model using complete and incomplete data

Hassan, Marwa Khalil The Korean Statistical Society 2018 Communications for statistical applications and me Vol.25 No.4

The two parameter negative exponential distribution has many practical applications in queuing theory such as the service times of agents in system, the time it takes before your next telephone call, the time until a radioactive practical decays, the distance between mutations on a DNA strand, and the extreme values of annual snowfall or rainfall; consequently, has many applications in reliability systems. This paper considers an estimation problem of stress-strength model with two parameter negative parameter exponential distribution. We introduce a maximum penalized likelihood method, Bayes estimator using Lindley approximation to estimate stress-strength model and compare the proposed estimators with regular maximum likelihood estimator for complete data. We also introduce a maximum penalized likelihood method, Bayes estimator using a Markov chain Mote Carlo technique for incomplete data. A Monte Carlo simulation study is performed to compare stress-strength model estimates. Real data is used as a practical application of the proposed model.