Comparison of location parameters of two exponential distributions when scale parameters are different and unknown

Letx _i(1)≤x _i(2)≤…≤x _i(r_i) be the right-censored samples of sizesn _i from theith exponential distributions $\sigma _i^{ - 1} exp\{ - (x - \mu _i )\sigma _i^{ - 1} \} ,i = 1,2$ where μ_i and σ_i are the unknown location and scale parameters respectively. This paper deals with the posteriori distribution of the difference between the two location parameters, namely μ₂-μ₁, which may be represented in the form $\mu _2 - \mu _1 \mathop = \limits^\mathcal{D} x_{2(1)} - x_{1(1)} + F_1 \sin \theta - F_2 \cos \theta $ where $\mathop = \limits^\mathcal{D} $ stands for equal in distribution,F _i stands for the central F-variable with [2,2(r _i?1)] degrees of freedom and $\tan \theta = \frac{{n_2 s_{x1} }}{{n_1 s_{x2} }}, s_{x1} = (r_1 - 1)^{ - 1} \left\{ {\sum\limits_{j = 1}^{r_i - 1} {(n_i - j)(x_{i(j + 1)} - x_{i(j)} )} } \right\}$ The paper also derives the distribution of the statisticV=F ₁ sin σ?F ₂ cos σ and tables of critical values of theV-statistic are provided for the 5% level of significance and selected degrees of freedom.     

On estimating the difference of location parameters of two negative exponential distributions

Fixed-width confidence intervals for the difference of location parameters of two negative-exponential distributions have been constructed through two-stage and purely sequential schemes. The two cases when the scale parameters are equal but unknown, and unequal but unknown, have been dealt with separately. Our two-stage procedures guarantee the exact confidence coefficient to be at least the nominal prescribed level. Various second-order expansions are also considered when sequential procedures are proposed. It is noted that no new tables are needed to implement these procedures in practice.     

Testing equality of location parameters of two exponential distributions from censored samples

A modification to Tiku's (1981) test, which may be seriously biased, is proposed. The modified test is only marginally biased if at all and is substantially more powerful. A ratio test based on Tiku’s (1967) modified likelihood function is also proposed, and shown to have power comparable to the power of the ratio test based on the likelihood function. The proposed ratio test is, however, much easier from a computational viewpoint.     

Estimation of location parameters of several exponential distributions

In this paper we address the problem of simultaneous estimation of location parameters of several exponential distributions assuming that the scale parameters are unknown and possibly unequal. From a decision theoretic point of view it is shown that the standard estimators are inadmissible and the improved estimators are obtained when p, the number of populations, is more than one.     

Estimating the common hazard rate of two exponential distributions with ordered location parameters

This paper is concerned with estimating the common hazard rate of two exponential distributions with unknown and ordered location parameters under a general class of bowl-shaped scale invariant loss functions. The inadmissibility of the best affine equivariant estimator is established by deriving an improved estimator. Another estimator is obtained which improves upon the best affine equivariant estimator. A class of improving estimators is derived using the integral expression of risk difference approach of Kubokawa [A unified approach to improving equivariant estimators. Ann Statist. 1994;22(1):290–299]. These results are applied to specific loss functions. It is further shown that these estimators can be derived for four important sampling schemes: (i) complete and i.i.d. sample, (ii) record values, (iii) type-II censoring, and (iv) progressive Type-II censoring. A simulation study is carried out for numerically comparing the risk performance of these proposed estimators.     

A note on the location parameters of two exponential distributions under order restrictions

The problem of simultaneous estimation of location parameters of two independent exponential distributions is considered when location and/or scale parameters are ordered. We show that the standard estimators in the unrestricted case which use information only from the populations individually can be improved upon when various order restrictions are known to hold. The improved estimators are obtained under the quadratic loss function     

Sequential estimation of a linear function of location parameters of two negative exponential distributions

The paper considers the problem of bounded risk point estimation for a linear function of location parameters of two negative exponential distributions, including the difference in a special case, when two scale parameters are unknown. Purely sequential procedures are proposed and second order expansions of the average sample sizes and risk are given. Furthermore some simulation results are provided.     

On location and scale parameters of exponential distributions with censored observations

In this paper we first address the problem of estimating the common scale of several exponential distributions with unknown location parameters when censored samples are observed. The improved estimators are basically Stein type testimators. These testimators are then used to construct improved estimators of location parameters.     

Likelihood ratio test for testing equality of location parameters of two exponential distributions from doubly censored samples

The Likelihood Ratio (LR) test for testing equality of two exponential distributions with common unknown scale parameter is obtained. Samples are assumed to be drawn under a type II doubly censored sampling scheme. Effects of left and right censoring on the power of the test are studied. Further, the performance of the LR test is compared with the Tiku(1981) test.     

Estimation of the smaller and larger of two exponential location parameters

Assume independent random samples are drawn from two populations which are exponentially distributed with unknown location parameters and a common known scale parameter. We want to estimate the maximum and the minimum of the unknowo location paremeters. In this paper several estimators are proposed which are better than the natural estimations in terms of absolute bias and /or meaqn squared error.     

Bounded risk estimation of a linear combination of location parameters in negative exponential distributions via three-stage sampling

The paper deals with the problem of bounded risk point estimation for a linear combination of location parameters of two negative exponential distributions. Isogai and Futschik considered the situation when the location and scale parameters are all unknown. They proposed purely sequential procedures and gave second order expansions of the average sample sizes and risks. In this paper we propose three-stage procedures and derive second order expansions of the average sample sizes and risks. Further, we compare the results with those from previous work.     

Bounded risk estimation of linear combinations of the location and scale parameters in exponential distributions under two-stage sampling

Two-stage sampling is proposed for estimating linear combinations of the location and scale parameters of exponential distributions with bounded quadratic risk functions. Exact formulae for the expected values and risks of the estimators are derived, and the performance of estimators is studied. Illustrations with real data are included.     

Multi-stage estimation of the common location parameter of several exponential distributions

The problem of constructing a confidence interval of ‘preassigned width and coverage probability’ considered by Costanza/ Hamdy and Son(1986) is further analyzed. Several multi-stage estimation procedures [ like, purely sequential, accelerated sequential and three-stage procedures ] are utilized to deal with the same estimation problem. The relative advantages and disadvantages of these procedures are discussed.     

Estimation of quantiles of exponential and double exponential distributions based on two order statistics

Asymptotically best linear unbiased estimators (ABLUE) of quantiles, x^., in the two-parameter (location-scale) exponential and double exponential families are obtained as linear combinations of two suitably chosen order statistics. Exact formulae for the linear combinations are given as functions of £. The derived estimators in both cases compare favorably with the usual nonparametric estimator. Also, in the exponential case the derived estimator compares favorably with the Sarhan-Greenberg BLUE based on a complete sample     

Improved estimation of common location of two exponential populations with order restricted scale parameters using censored samples

ABSTRACT

Estimation of common location parameter of two exponential populations is considered when the scale parameters are ordered using type-II censored samples. A general inadmissibility result is proved which helps in deriving improved estimators. Further, a class of estimators dominating the MLE has been derived by an application of integrated expression of risk difference (IERD) approach of Kubokawa. A discussion regarding extending the results to a general k( ? 2) populations has been done. Finally, all the proposed estimators are compared through simulation.     

Optimal spacing of quantiles for the estimation of the mixing parameters in a mixture of two exponential distributions

Methods for estimating the mixing parameters in a mixture of two exponential distributions are proposed. The estimators proposed are consistent and BAN(best asymptotically normal). The optimal spacings for estimating these mixture parameters are calculated.     

Power of a test for equality of k exponential location parameters

Kambo and Awad (1985) defined a test statistic based on doubly censored samples to test the equality of location parameters of K exponential distributions when their common scale parameter is unknown. The power function of the test is derived in this paper and some special cases are studied.     

Generalized exponential distributions

The three-parameter gamma and three-parameter Weibull distributions are commonly used for analysing any lifetime data or skewed data. Both distributions have several desirable properties, and nice physical interpretations. Because of the scale and shape parameters, both have quite a bit of flexibility for analysing different types of lifetime data. They have increasing as well as decreasing hazard rate depending on the shape parameter. Unfortunately both distributions also have certain drawbacks. This paper considers a three-parameter distribution which is a particular case of the exponentiated Weibull distribution originally proposed by Mudholkar, Srivastava & Freimer (1995) when the location parameter is not present. The study examines different properties of this model and observes that this family has some interesting features which are quite similar to those of the gamma family and the Weibull family, and certain distinct properties also. It appears this model can be used as an alternative to the gamma model or the Weibull model in many situations. One dataset is provided where the three-parameter generalized exponential distribution fits better than the three-parameter Weibull distribution or the three-parameter gamma distribution.