Precedence-type test based on Kaplan–Meier estimator of cumulative distribution function

In this paper, we introduce a precedence-type test based on Kaplan–Meier estimator of cumulative distribution function (CDF) for testing the hypothesis that two distribution functions are equal against a stochastically ordered hypothesis. This test is an alternative to the precedence life-test proposed first by Nelson (1963). After deriving the null distribution of the test statistic, we present its exact power function under the Lehmann alternative, and compare the exact power as well as simulated power (under location-shift) of the proposed test with other precedence-type tests. Next, we extend this test to the case of progressively Type-II censored data. Critical values for some combination of sample sizes and progressive censoring schemes are presented. We then examine the power properties of this test procedure and compare them to those of the weighted precedence and weighted maximal precedence tests under a location-shift alternative by means of Monte Carlo simulations. Finally, we present two examples to illustrate all the test procedures discussed here, and then make some concluding remarks.     

Double-sampling regression-cum-exponential estimator of the mean of a sensitive variable

ABSTRACT

A flexible scrambled response model using a randomization device for quantitative sensitive data is used to evaluate the protection of respondents’ privacy. A double-sampling regression-cum-exponential estimator is used to estimate the mean of a sensitive variable using the mean of a nonsensitive auxiliary variable under scrambled response. The expected bias, the expected mean square error, and the minimum mean square error of this exponential-type estimator are expressed. Simulations and empirical results show that the proposed estimator under scrambled response model has a lower mean square error and a lower bias than the ratio and the exponential estimators.     

The general expressions for the moments of lawless and wang's ordinary ridge regression estimator

In this paper, we derive the exact general expressions for the moments of an ordinary ridge regression (ORR) estimator for individual regression coefficients in a different way from Firinguetti (1987). Using the derived expressions, we evaluate numerically the first four moments of the ORR estimator, and examine its bias, mean square error, skewness and kurtosis. Further, Monte Carlo experiments are carried out in order to examine the shape of the density function of the ORR estimator.     

Estimation of reliability under ordered restriction on the parameters

This paper deals with the estimation of reliability for a strength-stress model under ordered restriction on the parameters. It is assumed that components have exponential distributions and are arranged in a parallel system and the failure of one component, results in increasing the failure rate of the remaining components. Results are derived when (i) the ordering of the means is taken into account and when (ii) the ordering of the means is ignored. Simulation studies are carried out to compare the results. It is noticed that, in almost all cases, in case (i) the estimates are closer to the true value with smaller mean squared error (MSE) and smaller’ standard deviation than in case (ii). Thus when the ordering of the means is present in the model, such information should be incorporated in the estimation of reliability.     

Linear calibration in functional regression models

This paper discusses calibration in functional regression models. Classical and inverse type estimators are considered. First order approximation to the bias and to the mean squared error (MSE) of the estimators are considered. Numerical comparisons seem to indicate that the classical estimator obtained via maximum likelihood estimation performs better than the other estimators considered.     

ON THE ESTIMATION OF SURVIVAL FUNCTION UNDER RANDOM CENSORSHIP

ABSTRACT

The paper deals with an improvement of the well-known Kaplan–Meier estimator of survival function when the censoring mechanism is random and independent of the failure times. Small sample size properties of the new estimator, as well as the original Kaplan–Meier estimator are inspected by means of Monte Carlo simulations. It follows from the simulations that the proposed estimator prevails with respect to some basic statistical characteristics.     

Integer-parameter estimation in discrete distributions

Integer-parameter restriction quite often occurs naturally in real life situations. Here we consider the problem of deriving the maximum likelihood estimators (MLE) for the case in which the parameter is restricted to a positive integer. The usual asymptotic theory for the MLE does not hold good any more and each case needs individual attention for the derivation of these results,. The estimation problem in the case of Poisson, Binomial, and Poisson-Binomial bivariate model is investigated here, A simple method of deriving the MLE and the lower bound for the variance of the integer-parameter estimator is also discussed     

Distribution-free confidence intervals for quantiles in small samples

Estimators for quantiles based on linear combinations of order statistics have been proposed by Harrell and Davis(1982) and kaigh and Lachenbruch (1982). Both estimators have been demonstrated to be at least as efficient for small sample point estimation as an ordinary sample quantile estimator based on one or two order statistics: Distribution-free confidence intervals for quantiles can be constructed using either of the two approaches. By means of a simulation study, these confidence intervals have been compared with several other methods of constructing confidence intervals for quantiles in small samples. For the median, the Kaigh and Lachenbruch method performed fairly well. For other quantiles, no method performed better than the method which uses pairs of order statistics.     

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.     

Deficiency of the mle of a smooth survival function under the proportional hazard model

The problem of estimating a smooth distribution function F at a point t is treated under the proportional hazard model of random censorship. It is shown that a certain class of properly chosen kernel type estimator of F asymptotically perform better than the maximum likelihood estimator. It is shown that the relative deficiency of the maximum likelihood estimator of F under the proportional hazard model with respect to the properly chosen kernel type estimator tends to infinity as the sample size tends to infinity.