TESTING THE EQUALITY OF PARAMETERS IN THE DISTRIBUTIONS WHEN THE RANGE DEPENDS UPON THE PARAMETER

The authors derive the null and non-null distributions of the test statistic v=y_min/y_max (where y_min= min x_ij, y_max= max x_ij, J=1,2, …, k) connected with testing the equality of scale parameters θ₁, θ₂, …θ_k in certain, class of density functions given by      

EQUIVARIANT ESTIMATION FOR PARAMETERS OF EXPONENTIAL DISTRIBUTIONS BASED ON TYPE-II PROGRESSIVELY CENSORED SAMPLES

ABSTRACT

In this paper, we consider exponential models and obtain minimum risk equivariant estimators of the parameters based on Type-II progressively censored samples under standardized quadratic loss function. These generalize the corresponding results for Type-II censored samples.     

A NOTE ON ESTIMATING THE LOCATION AND SCALE PARAMETERS OF THE EXPONENTIAL DISTRIBUTION FROM A CENSORED SAMPLE

   
Replacing f (x)/F (x) by α+β(x- θ)/σ in the maximum likelihood equations ∂L/∂θ and ∂L/∂σ calculated from a censored sample, a pair of estimators θ^e and σ^e, is obtained. The variances and covariances of these estimators are calculated and compared with the corresponding values for the best linear unbiassed (BLU) estimators.     

TESTS FOR EQUALITY OF TWO DISTRIBUTIONS WITH CORRELATED INTERVAL CENSORED LIFE-TIMES

ABSTRACT

Large sample properties of Life-Table estimator are discussed for interval censored bivariate survival data. We restrict our attention to the situation where response times within pairs are not distinguishable, and the univariate survival distribution is the same for any individual within any pair. The large sample properties are applied to test for equality of two distributions with correlated response times where treatments are applied to different independent sets of cohorts. Data, which can be separated into two independent sets, from an angioplasty study where more than one procedure is performed on some patients are used to illustrate this methodology.     

ESTIMATING THE PARAMETERS OF NORMAL AND LOGISTIC DISTRIBUTIONS FROM CENSORED SAMPLES1

Let g(z) be the ratio of the ordinate and the probability integral of the distribution of a variate z. The relation g(z)～+βz is used to derive (i) estimators μ_r and s?_r of the parameters of a truncated normal distribution and (ii) estimators μ_c and s?_c of the mean and standard deviation of a logistic distribution from doubly censored samples. The variances and eovariances of these estimators are obtained. They are shown to be nearly as efficient as the maximum likelihood estimators and easier to compute.     

A TEST OF THE LOCATION PARAMETER OF THE EXPONENTIAL DISTRIBUTION

In this paper, we propose a procedure for testing the location parameter of the exponential distribution for certain alternative hypotheses, which could result in the early rejection of the null hypothesis. This is a consequence of the monotone property of the test statistic which is based on the extremal quotient. The test being scale-free does not require the scale parameter to be known.     

ON ASYMPTOTIC OPTIMALITY OF SOME TESTS FOR EXPONENTIAL DISTRIBUTIONS

The probability density function (pdf) of a two parameter exponential distribution is given by f(x; p, s?) =s?^-1 exp {-(x - ρ)/s?} for x≥ρ and 0 elsewhere, where 0 < ρ < ∞ and 0 < s?∞. Suppose we have k independent random samples where the ith sample is drawn from the ith population having the pdf f(x; ρ_i, s?_i), 0 < ρ_i < ∞, 0 < s?_i < s?_i < and f(x; ρ, s?) is as given above. Let X_i1 < X_i2 <… < X_iri denote the first r_i order statistics in a random sample of size n_i, drawn from the ith population with pdf f(x; ρ_i, s?_i), i = 1, 2,…, k. In this paper we show that the well known tests of hypotheses about the parameters ρ_i, s?_i, i = 1, 2,…, k based on the above observations are asymptotically optimal in the sense of Bahadur efficiency. Our results are similar to those for normal distributions.     

A MATRIX VARIATE GENERALIZATION OF THE POWER EXPONENTIAL FAMILY OF DISTRIBUTIONS

ABSTRACT

This paper proposes a matrix variate generalization of the power exponential distribution family, which can be useful in generalizing statistical procedures in multivariate analysis and in designing robust alternatives to them. An example is added to show an application of the generalization.     

ON THE COMMON SCALE PARAMETER OF SEVERAL PARETO POPULATIONS IN CENSORED SAMPLES

The problem of estimation of an unknown common scale parameter of several Pareto distributions with unknown and possibly unequal shape parameters in censored samples is considered. A new class of estimators which includes both the maximum likelihood estimator (MLE) and the uniformly minimum variance unbiased estimator (UMVUE) is proposed and examined under a squared error loss.     

PREDICTION FOR EXPONENTIAL LIFETIMES BASED ON STEP-STRESS TESTING

ABSTRACT

This paper presents methods for constructing prediction limits for a step-stress model in accelerated life testing. An exponential life distribution with a mean that is a log-linear function of stress, and a cumulative exposure model are assumed. Two prediction problems are discussed. One concerns the prediction of the life at a design stress, and the other concerns the prediction of a future life during the step-stress testing. Both predictions require the knowledge of some model parameters. When estimates for the model parameters are available, a calibration method based on simulations is proposed for correcting the prediction intervals (regions) obtained by treating the parameter estimates as the true parameter values. Finally, a numerical example is given to illustrate the prediction procedure.     

CONDITIONAL INFERENCE PROCEDURES FOR THE PARETO AND POWER FUNCTION DISTRIBUTIONS BASED ON TYPE-II RIGHT CENSORED SAMPLES

In this paper we consider conditional inference procedures for the Pareto and power function distributions. We develop procedures for obtaining confidence intervals for the location and scale parameters as well as upper and lower n probability tolerance intervals for a proportion g, given a Type-II right censored sample from the corresponding distribution. The intervals are exact, and are obtained by conditioning on the observed values of the ancillary statistics. Since, for each distribution, the procedures assume that a shape parameter x is known, a sensitivity analysis is also carried out to see how the procedures are affected by changes in x.     

ON THE POWER BREAKDOWN OF THE NEGATIVE EXPONENTIAL DISPARITY TESTS

The generalized negative exponential disparity, discussed in Bhandari et al. (Robust inference in parametric models using the family of generalized negative exponential disparities, 2006, ANZJS, 48 , 95–114), represents an important class of disparity measures that generates efficient estimators and tests with strong robustness properties. In their paper, however, Bhandari et al. failed to provide a sharp lower bound for the power breakdown point of the corresponding tests. This was acknowledged by the authors, who indicated the possible existence of a sharper bound, but noted that they did not “have a proof at this point”. In this paper we provide an improved bound for this power breakdown point, and show with an example how this can enhance the existing results.     

PARAMETER ESTIMATION BASED ON GROUPED OR CONTINUOUS DATA FOR TRUNCATED EXPONENTIAL DISTRIBUTIONS

ABSTRACT

Parameter estimation based on truncated data is dealt with; the data are assumed to obey truncated exponential distributions with a variety of truncation time—a ₁ data are obtained by truncation time b ₁, a ₂ data are obtained by truncation time b ₂ and so on, whereas the underlying distribution is the same exponential one. The purpose of the present paper is to give existence conditions of the maximum likelihood estimators (MLEs) and to show some properties of the MLEs in two cases: 1) the grouped and truncated data are given (that is, the data each express the number of the data value falling in a corresponding subinterval), 2) the continuous and truncated data are given.     

MULTIVARLYTE ANALYSIS OF CLUSTERED SURVEY DATA: TESTING EQUALITY OF COVARLANCE MATRICES

The problem of testing the hypothesis of equality of covariance matrices in the presence of two-stage sampling is considered. Asymptotic test procedures based on linearization, grouping and jackknifing with or without transformation are proposed. The finite sample properties of these procedures are investigated in sampling experiments both from simulated known distributions and from a natural population.     

GOODNESS OF FIT STATISTICS BASED ON THE SPACINGS OF COMPLETE OR CENSORED SAMPLES

A goodness-of-fit statistic Z is defined in terms of the spacings generated by the order statistics of a complete or a censored sample from a distribution of the type (l/)f((x-μ)/), μ and unknown. The distribution of Z is studied, mostly through Monte Carlo methods. The power properties of Z for testing Exponential, Uniform, Normal, Gamma and Logistic distributions are discussed; Z is shown to be more powerful than the Smith & Bain (1976) correlation statistic, except for testing Uniform, Normal and Logistic (symmetric distributions) against symmetric alternatives. The statistic Z is generalized to test the goodness-of-fit from κ 2 independent complete or censored samples.     

A NEW TEST FOR EQUALITY OF VARIANCES FOR k NORMAL POPULATIONS

ABSTRACT

A simple test based on Gini's mean difference is proposed to test the hypothesis of equality of population variances. Using 2000 replicated samples and empirical distributions, we show that the test compares favourably with Bartlett's and Levene's test for the normal population. Also, it is more powerful than Bartlett's and Levene's tests for some alternative hypotheses for some non-normal distributions and more robust than the other two tests for large sample sizes under some alternative hypotheses. We also give an approximate distribution to the test statistic to enable one to calculate the nominal levels and P-values.     

ON THE EVALUATION OF CONTROL CHART LIMITS BASED ON PREDICTIVE DISTRIBUTIONS

ABSTRACT

We propose a Bayesian approach to obtaining control charts when there is parameter uncertainty. Our approach consists of two stages, (i) construction of the control chart where we use a predictive distribution based on a Bayesian approach to derive the rejection region, and (ii) evaluation of the control chart where we use a sampling theory approach to examine the performance of the control chart under various hypothetical specifications for the data generation model.