Unknown Scale Parameter Research Articles

Estimating the Common Location

Suppose we have independent random samples of sizes m and n respectively from two populations characterized by a common location parameter θ and unknown scale parameters. Let ψ x and ψ y be odd location estimators of θ based on individual samples. Cohen (1976) suggested a combined estimator δ a , which under mild conditions on the density, is unbiased and improves ψ x for all a such that 0 < a ≤ a*(m, n), where a*(m, n) is a constant and m, n denote the two sample sizes. Bhattacharya (1981) enlarged this class by finding a larger upperbound A(m, n) for ‘a’. In this paper, we further improve A(m, n) and establish dominance of δ 1, over both ψ x and ψ y in cases where Bhattacharya's bound does not help. We also consider, an unbiased estimator different from δ a , and determine conditions on c to improve both ψ x and ψ y . Our results also take into account the cases not covered by Akai (1982).

Some inadmissibility results for estimating ordered uniform scale parameters

Independent random samples (of possibly unequal sizes) are drawn from k (≥2) uniform populations having unknown scale parameters μ1,…,μk. The problem of componentwise estimation of ordered parameters is investigated. The loss function is assumed to be squared error and the cases of known and unknown ordering among μ1,…,μk. are dealt with separately. Sufficient conditions for an estimator to be inadmissible are provided and as a consequence, many natural estimators are shown to be inadmissible, Better estimators are provided.

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

Letx i(1)≤x i(2)≤…≤x i(ri) 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 μi and σi 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.

Estimating the ratio of the smaller and the larger of two uniform scale parameters

Independent random samples are drawn from two uniformly distributed populations with unknown scale parameters. The problem is to estimate the ratio of the minimum and the maximum of the two scale parameters. It is important because, in ranking and selection problem the true probability of correct selection {P(CS)) is a function of this ratio. In this paper several estimators are proposed and their performances are compared empirically.

Tests for upper outliers in the two-parameter exponential distribution

The problem of detecting upper outliers in a sample from two parameter exponential distribution with unknown location and scale parameters is discussed. The block and inside-out sequential test procedures are investigated under a particular life testing situation. Two test statistics, one based on the prediction of future order statistics and the other based on score function are considered. The exact null distribution of one of the test statistic is quite involved and the simulated critical values are provided. The power performances of the proposed tests are investigated using a Monte Carlo study. The application of the above procedures for the case of Pareto sample is also indicated.

LBI tests of independence in bivariate exponential distributions

The locally best invariant test for the hypothesis of independence in bivariate distributions with exponentially distributed marginals is derived. The model consists of a family of bivariate exponential distributions with probability density function $$f_\theta (x_1 ,x_2 ;\lambda _1 ,\lambda _2 ) = \lambda _1 \lambda _2 \exp [ - (\lambda _1 x_1 + \lambda _2 x_2 )]g(\lambda _1 x_1 ,\lambda _2 x_2 ;\theta )$$ with unknown scale parameter γj (j=1, 2) and association parameter ϑ which includes the independence situation. The locally best invariant (LBI) test is derived and the asymptotic null and nonnull distributions are also derived under some regularity conditions. The results are applied to the Gumbel (1960,J. Amer. Statist. Assoc.,55, 698–707), Frank (1979,Aequationes Math.,19, 194–226, and Cook and Johnson (1981),J. Roy. Statist. Soc. Ser. B,43, 210–218) families.

Convergence rates for empirical Bayes estimation of the scale parameter in a Pareto distribution

Let f(χ∣θ) = αθ α χ α+1I (θ,∞)(χ) be the pdf of a Pareto distribution with known shape scale parameter α > 0 and unknown scale parameter θ. We study the problem of estimating the scale parameter θ under a squared-error loss through the nonparametric empirical Bayes approach. An empirical Bayes estimator is proposed and the corresponding asymptotic optimality is also investigated. It is shown that under certain weak conditions the proposed empirical Bayes estimator is asymptotically optimal and the associated rate of convergence is of order O(n − 2 3 ) .

On the admissibility of some sequential confidence intervals for the negative exponential location parameter

With reference to a general class of sequential procedures, this paper shows the admissibility of fixed-width confidence intervals, based on the smallest sample observation, for the location parameter of a negative exponential population with an unknown scale parameter.

Simultaneous confidence intervals for all distances from the worst and best populations

The population пiis characterized by an unknown location parameter θi, i = 1, …, k and scale parameter ξ. . The populations associated with θ[1] = min(θ1…,θk) and θ[k]= max(θ1…,θk) are defined to be the “worst” and the “best” populations, respectively. Simultaneous upper confidence intervals for all distances from the worst and best are derived. The cases of known and unknown scale parameter are dealth with separately and the results are applied to a exponential populations model.

Shrinkage estimators of the location parameter for certain spherically symmetric distributions

We consider estimation of a location vector for particular subclasses of spherically symmetric distributions in the presence of a known or unknown scale parameter. Specifically, for these spherically symmetric distributions we obtain slightly more general conditions and larger classes of estimators than Brandwein and Strawderman (1991,Ann. Statist.,19, 1639–1650) under which estimators of the formX +ag(X) dominateX for quadratic loss, concave functions of quadratic loss and general quadratic loss.

Bias-robustness of l-estimates of location against dependence

We assume that the distribution of a sample is the product of symmetrical distributions with unknown location and scale parameters, but the observations are actually dependent. We establish the bias-robustness of L-estimates of location against dependence. We show that the sample mean is the most robust estimate and either the sample median or the midrange is the most sensitive one. This contrasts with classical results dealing with robustness against disturbing marginal distributions.

Two stage reliability tests for new series systems

We want to do acceptance testing of newly minted series systems, all of whose component failure distributions are identical Gamma distributions with unknown shape and scale parameters. The criterion is the equilibrium mean time between failure (MTBF) of the systems. We derive a two stage acceptance testing procedure, and show that as the number of systems in the first stage test tends to infinity, the OC-Curve, derived from this two stage testing procedure, tends in probability to the nominal OC-Curve, derived for a single stage procedure assuming the shape parameter to be known. Also, we have performed a simulation study of small first stage test size properties that confirms our limiting results and gives some guidance on how to choose the number of systems for the first stage test.

Modified K-S, A-D and C-vM tests for the Pareto distribution with unknown location and scale parameters : James E. Porter IIIet al. IEEE Trans. Reliab.41, 112 (1992)

Modified K-S, A-D and C-vM tests for the Pareto distribution with unknown location and scale parameters : James E. Porter IIIet al. IEEE Trans. Reliab.41, 112 (1992)

Minimax estimators for location vectors in elliptical distributions with unknown scale parameter and its application to variance reduction in simulation

In this paper, we give an ever wider and new class of minimax estimators for the location vector of an elliptical distribution (a scale mixture of normal densities) with an unknown scale parameter. The its application to variance reduction for Monte Carlo simulation when control variates are used is considered. The results obtained thus extend (i) Berger's result concerning minimax estimation of location vectors for scale mixtures of normal densities with known scale parameter and (ii) Strawderman's result on the estimation of the normal mean with common unknown variance.

Stein estimation for non-normal spherically symmetric location families in three dimensions

We consider estimation of a location vector in the presence of known or unknown scale parameter in three dimensions. The technique of proof is Stein's integration by parts and it is used to cover several cases (e.g., non-unimodal distributions) for which previous results were known only in the cases of four and higher dimensions. Additionally, we give a necessary and sufficient condition on the shrinkage constant for improvement on the usual estimator for the spherical uniform distribution.

Multiple Comparison of Exponential Location Parameters with the Best Under Type II Censoring

SYNOPTIC ABSTRACTLet there be k independent exponential populations with unknown and unequal location parameters and with a common but unknown scale parameter. Type II censored sample of size ni is drawn from each population πi = 1, 2, …, k. Based on these k samples, multiple comparison of the location parameters with that of a control population are obtained. The results are applied to situations where there is no control population and the experimenter is interested in multiple comparison with the best (but unknown) population. Subset selection satisfying given selection goals can be achieved simultaneously.

Modified KS, AD, and C-vM tests for the Pareto distribution with unknown location and scale parameters

Standard goodness-of-fit tests based on the empirical CdF (Edf) require continuous underlying distributions with all parameters specified. Three modified Edf-type tests, the Kolmogorov-Smirnov (K-S), Anderson-Darling (A-D), and Cramer-von Mises (C-vM), are developed for the Pareto distribution with unknown parameters of location and scale and known shape parameter. The unknown parameters are estimated using best linear unbiased estimators. For each test, Monte Carlo techniques are used to generate critical values for sample sizes 5(5)30 and Pareto shape parameters 0.5

Hierarchical Bayesian reliability analysis using Erlang families of priors and hyperpriors

An hierarchical Bayes approach to reliability estimation for the exponential model with an unknown scale parameter, based on life tests that are terminated after a preassigned number of failures, is considered under the assumptions of squared error loss and Erlang distributions as the prior and hyperprior. The Bayesian estimation of reliability for the case of ‘attribute testing’ is also discussed.

Two- and multi-stage selection procedures for Weibull populations

Abstract This paper considers two- and multi-stage procedures for selecting the most reliable of several Weibull populations, with a common known shape parameter but unknown scale parameters. Under an indifference zone approach, the least favourable configuration for a multi-stage procedure has been worked out. Also, explicit comparisons are made between two- and single-stage procedures.

Estimation of the smaller and larger of two uniform scale parameters

Assume independent random samples are drawn from two populations which are uniformly distributed with unknown scale parameters. The problem is to estimate the minimum and the maximum of the two unknown scales. In this paper, several simple estimators are proposed which are better than the natural estimators in terms of standardized bias, risk under squared error loss and the Pitman nearness criterion.