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.     

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.     

ON TESTING THE EQUALITY OF LOCATION PARAMETERS IN k CENSORED EXPONENTIAL DISTRIBUTIONS

Kumar and Patel (1971) have considered the problem of testing the equality of location parameters of two exponential distributions on the basis of samples censored from above, when the scale parameters are the same and unknown. The test proposed by them is shown to be biased for n₁≠n₂, while for n₁=n₂ the test possesses the property of monotonicity and is equivalent to the likelihood ratio test, which is considered by Epstein and Tsao (1953) and Dubey (1963a, 1963b). Epstein and Tsao state that the test is unbiased. We may note that when the scale parameters of k exponential distributions are unknown the problem of testing the equality of location parameters is reducible to that of testing the equality of parameters in k rectangular populations for which a test and its power function were given by Khatri (1960, 1965); Jaiswal (1969) considered similar problems in his thesis. Here we extend the problem of testing the equality of k exponential distributions on the basis of samples censored from above when the scale parameters are equal and unknown, and we establish the likelihood ratio test (LET) and the union-intersection test (UIT) procedures. Using the results previously derived by Jaiswal (1969), we obtain the power function for the LET and for k= 2 show that the test possesses the property of monotonicity. The power function of the UIT is also given.     

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.     

ONE-SIDED ESTIMATING AND TESTING PROBLEMS FOR LOCATION MODELS FROM GROUPED SAMPLES

This article deals with one-sided problems for location models from grouped samples. Suppose the support region of a density function, which does not depend on parameters, is divided into some disjointed intervals, grouped samples are the number of observations falling in each intervals respectively. The studying of grouped samples may be dated back to the beginning of the century, in which only one sample location and/or scale models is considered. This article considers one-sided estimating and testing problems for location models. Some methods for computing the maximum likelihood estimates of the parameters subject to order restrictions are proposed and a numerical example by the method is given.     

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.     

ON THE INDEPENDENCE OF THE SAMPLE MEAN AND TRANSLATION-INVARIANT STATISTICS FOR MATRIX NORMAL DISTRIBUTIONS

We present an explicit characterization of the joint dependency structure of an n×p matrix normal random matrix such that the p-dimensional sample mean vector is independent of all translation invariant statistics.     

ESTIMATING THE ANGLE BETWEEN THE MEAN DIRECTIONS OF TWO SPHERICAL DISTRIBUTIONS

This paper considers the problem of calculating a confidence interval for the angular difference between the mean directions of two spherical random variables with rotationally symmetric unimodal distributions. For large sample sizes, it is shown that the asymptotic distribution of 1 – cos α, where α is the sample angular difference, is approximately exponential if the true difference is zero, and approximately normal for a ‘large’ true difference; a scaled beta approximation is determined for the general case. For small sample sizes, a bootstrap approach is recommended. The results are applied to two sets of palaeomagnetic data.     

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.     

ESTIMATING THE COMMON MEAN OF A BIVARIATE NORMAL POPULATION

For estimating the common mean of a bivariate normal distribution, Krishnamoorthy & Rohatgi (1989) proposed some estimators which dominate the maximum likelihood estimator in a large region of the parameter space. We consider some modifications of these estimators and study their risk performance.     

THE MONTE CARLO EVALUATION OF ORTHANT PROBABILITIES FOR MULTIVARIATE NORMAL DISTRIBUTIONS

The calculation of multivariate normal orthant probabilities is practically impossible when the number of variates is greater than five or six, except in very special cases. A transformation of the integral is obtained which enables quite accurate Monte Carlo estimates to be obtained for a fairly high number of dimensions, particularly if control variates are used.     

ESTIMATION OF THE PARAMETERS OF A NORMAL DISTRIBUTION WHEN THE MEAN IS RESTRICTED TO AN INTERVAL1

Suppose it is known that the mean of a normal distribution is non-negative. Naturally one will use the sample mean truncated at zero as an estimator of the distribution mean. In this paper the properties of such an estimator are investigated.     

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.     

INFLUENCE DIAGNOSTICS FOR THE NORMAL LINEAR MODEL WITH CENSORED DATA

Methods of detecting influential observations for the normal model for censored data are proposed. These methods include one-step deletion methods, deletion of observations and the empirical influence function. Emphasis is placed on assessing the impact that a single observation has on the estimation of coefficients of the model. Functions of the coefficients such as the median lifetime are also considered. Results are compared when applied to two sets of data.     

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      

THE ASYMPTOTIC EQUIVALENCE OF THE FISHER INFORMATION MATRICES FOR TYPE I AND TYPE II CENSORED DATA FROM LOCATION-SCALE FAMILIES

Type I and Type II censored data arise frequently in controlled laboratory studies concerning time to a particular event (e.g., death of an animal or failure of a physical device). Log-location-scale distributions (e.g., Weibull, lognormal, and loglogistic) are commonly used to model the resulting data. Maximum likelihood (ML) is generally used to obtain parameter estimates when the data are censored. The Fisher information matrix can be used to obtain large-sample approximate variances and covariances of the ML estimates or to estimate these variances and covariances from data. The derivations of the Fisher information matrix proceed differently for Type I (time censoring) and Type II (failure censoring) because the number of failures is random in Type I censoring, but length of the data collection period is random in Type II censoring. Under regularity conditions (met with the above-mentioned log-location-scale distributions), we outline the different derivations and show that the Fisher information matrices for Type I and Type II censoring are asymptotically equivalent.