Optimal record-based statistical procedures for the two-parameter exponential distribution

There are a number of situations in which the experimental data observed are record statistics. In this paper, optimal confidence intervals as well as uniformly most powerful (MP) tests for one-sided alternatives are developed. Since a uniformly MP test for a two-sided alternative does not exist, generalized likelihood ratio and uniformly unbiased and invariant tests are derived for the two parameters of the exponential distribution based on record data. For illustrative purposes, a data set on the times between consecutive telephone calls to a company's switchboard is analysed using the proposed procedures. Finally, some open problems in this direction are pointed out.     

Tests for symmetry of two-piece normal distribution

In this paper, classical optimum tests for symmetry of two-piece normal distribution is derived. Uniformly most powerful one-sided test for the skewness parameter is obtained when the location and scale parameters are known and is compared with sequential probability ratio test. An ad-hoc test for symmetry and likelihood ratio test for symmetry for large samples, can be found in literature for this distribution. But in this paper, we derive exact likelihood ratio test for symmetry, when location parameter is known. The exact power of the test is evaluated for different sample sizes.     

On the construction of exact lower confidence bounds on the distance between any two ranked location parameters of K populations

In this paper, an exact lower confidence bound on the distance between any two ranked location parameters of k populations is derived. This confidence bound contains the confidence bounds of Kim (1986) and of Chen, Xiong and Lam (1993) as special cases.     

A further note on testing for multisource contamination in location/scale families

Proofs of two conjectures in Gather (1989) are given.     

On marginal likelihood inference for the intra-class correlation coefficient

A p-component set of responses have been constructed by a location-scale transformation to a p-component set of error variables, the covariance matrix of the set of error variables being of intra-class covariance structure:all variances being unity, and covariance being equal [IML0001]. A sample of size n has been described as a conditional structural model, conditional on the value of the intra-class correlation coefficient ρ. The conditional technique of structural inference provides the marginal likelihood function of ρ based on the standardized residuals. For the normal case, the marginal likelihood function of ρ is seen to be dependent on the standardized residuals through the sample intra-class correlation coefficient. By the likelihood modulation technique, the nonnull distribution of the sample intra-class correlation coefficient has also been obtained.     

The generalized inference on the sign testing problem about the normal variances

For the sign testing problem about the normal variances, we develop the heuristic testing procedure based on the concept of generalized test variable and generalized p-value. A detailed simulation study is conducted to empirically investigate the performance of the proposed method. Through the simulation study, especially in small sample sizes, the proposed test not only adequately controls empirical size at the nominal level, but also uniformly more powerful than likelihood ratio test, Gutmann's test, Li and Sinha's test and Liu and Chan's test, showing that the proposed method can be recommended in practice. The proposed method is illustrated with the published data.     

On Hotelling’s T2 test in a special paired sample case

In a special paired sample case, Hotelling’s T² test based on the differences of the paired random vectors is the likelihood ratio test for testing the hypothesis that the paired random vectors have the same mean; with respect to a special group of affine linear transformations it is the uniformly most powerful invariant test for the general alternative of a difference in mean. We present an elementary straightforward proof of this result. The likelihood ratio test for testing the hypothesis that the covariance structure is of the assumed special form is derived and discussed. Applications to real data are given.     

Testing homogeneity of control and treatment populations — local optimality and related issues

Durairajan and Raman (1996 a, b) studied the robustness of Locally most powerful invariant (LMPI) tests for compound normal model in control and treatment populations. In the present paper, the Locally most powerful (LMP) tests are constructed for no contamination in normal mixture model through testing the parameter of mixture of distributions and the mixing proportion. The expected performance of LMP tests are compared using Efron's Statistical Curvature on the lines of Sen Gupta and Pal (1991). The Locally most powerful similar (LMPS) tests for the equality of control and treatment populations in the presence of nuisance parameters are also constructed. Further, the null and non-null distributions of the test statistics are derived and some power computations are made. Received: September 1, 1999; revised version: August 31, 2000     

Optimum invariant tests for random manova models

Consider the canonical-form MANOVA setup with X: n × p = (+ E, X_i n_i × p, i = 1, 2, 3, M_i: n_i × p, i = 1, 2, n₁ + n₂ + n₃) p, where E is a normally distributed error matrix with mean zero and dispersion I_n (> 0 (positive definite). Assume (in contrast with the usual case) that M₁i is normal with mean zero and dispersion I_n1) and M₂2 is either fixed or random normal with mean zero and different dispersion matrix I_n2 (being unknown. It is also assumed that M₁ E, and M₂ (if random) are all independent. For testing H₀) = 0 versus H₁: (> 0, it is shown that when either n₂ = 0 or M₂ is fixed if n₂ > 0, the trace test of Pillai (1955) is uniformly most powerful invariant (UMPI) if min(n₁, p)= 1 and locally best invariant (LBI) if min(n₁ p) > 1 underthe action of the full linear group Gl (p). When p > 1, the LBI test is also derived under a somewhat smaller group G_T(p) of p × p lower triangular matrices with positive diagonal elements. However, such results do not hold if n₂ > 0 and M₂ is random. The null, nonnull, and optimality robustness of Pillai's trace test under Gl(p) for suitable deviations from normality is pointed out.     

A note on the most powerful invariant test of Rayleigh against exponential distribution

Abstract

The problem of testing Rayleigh distribution against exponentiality, based on a random sample of observations is considered. This problem arises in survival analysis, when testing a linearly increasing hazard function against a constant hazard function. It is shown that for this problem the most powerful invariant test is equivalent to the “ratio of maximized likelihoods” (RML) test. However, since the two families are separate, the RML test statistic does not have the usual asymptotic chi-square distribution. Normal and saddlepoint approximations to the distribution of the RML test statistic are derived. Simulations show that saddlepoint approximation is more accurate than the normal approximation, especially for tail probabilities that are the main values of interest in hypothesis testing.     

Likelihood‐based Inference with Missing Data Under Missing‐at‐Random

Likelihood‐based inference with missing data is challenging because the observed log likelihood is often an (intractable) integration over the missing data distribution, which also depends on the unknown parameter. Approximating the integral by Monte Carlo sampling does not necessarily lead to a valid likelihood over the entire parameter space because the Monte Carlo samples are generated from a distribution with a fixed parameter value. We consider approximating the observed log likelihood based on importance sampling. In the proposed method, the dependency of the integral on the parameter is properly reflected through fractional weights. We discuss constructing a confidence interval using the profile likelihood ratio test. A Newton–Raphson algorithm is employed to find the interval end points. Two limited simulation studies show the advantage of the Wilks inference over the Wald inference in terms of power, parameter space conformity and computational efficiency. A real data example on salamander mating shows that our method also works well with high‐dimensional missing data.     

A testing approach to estimation

Results from the theory of uniformly most powerful invariant tests are used to develop a new parameter estimation procedure. The procedure is used to derive parameter estimators for several important distributions. Results of simulation studies comparing the performances of the new estimators and maximum likelihood estimators are presented.     

The Slippage Problem for the Circular Normal Distribution

The slippage problem occurs when an unspecified observation in a given random sample is from a distribution other than that for all the remaining observations. This paper studies the problem in terms of the 'slip' in the mean direction of a circular normal distribution. The slippage problem is first treated as a multiple decision problem with a prior which is invariant under the permutations of the hypotheses. The probabilities of accepting the various hypotheses for the Bayes rule with respect to this prior are explicitly obtained. The likelihood ratio tests for this slippage problem, for the cases when the mean directions are both known and unknown, are shown to be easily computable. The tests are illustrated through two well-known datasets. The performances of a range of tests are compared using extensive simulation.     

Confidence intervals based on L-moments for quantiles of the GP and GEV distributions with application to market-opening asset prices data

In a ground-breaking paper published in 1990 by the Journal of the Royal Statistical Society, J.R.M. Hosking defined the L-moment of a random variable as an expectation of certain linear combinations of order statistics. L-moments are an alternative to conventional moments and recently they have been used often in inferential statistics. L-moments have several advantages over the conventional moments, including robustness to the the presence of outliers, which may lead to more accurate estimates in some cases as the characteristics of distributions. In this contribution, asymptotic theory and L-moments are used to derive confidence intervals of the population parameters and quantiles of the three-parametric generalized Pareto and extreme-value distributions. Computer simulations are performed to determine the performance of confidence intervals for the population quantiles based on L-moments and to compare them to those obtained by traditional estimation techniques. The results obtained show that they perform well in comparison to the moments and maximum likelihood methods when the interest is in higher quantiles, or even best. L-moments are especially recommended when the tail of the distribution is rather heavier and the sample size is small. The derived intervals are applied to real economic data, and specifically to market-opening asset prices.     

TEST FOR TREATMENT EFFECT BASED ON BINARY DATA WITH RANDOM SAMPLE SIZES

The problem of testing for treatment effect based on binary response data is considered, assuming that the sample size for each experimental unit and treatment combination is random. It is assumed that the sample size follows a distribution that belongs to a parametric family. The uniformly most powerful unbiased tests, which are equivalent to the likelihood ratio tests, are obtained when the probability of the sample size being zero is positive. For the situation where the sample sizes are always positive, the likelihood ratio tests are derived. These test procedures, which are unconditional on the random sample sizes, are useful even when the random sample sizes are not observed. Some examples are presented as illustration.     

Reliability analysis for stress-strength model from a general family of truncated distributions under censored data

Abstract

Under progressive Type-II censoring, inference of stress-strength reliability (SSR) is studied for a general family of lower truncated distributions. When the lifetime models of the strength and stress variables have arbitrary and common parameters, maximum likelihood and pivotal quantities based generalized estimators of SSR are established, respectively. Confidence intervals are constructed based on generalized pivotal quantities and bootstrap technique under different parameter cases as well. In addition, to compare the equivalence of the strength and stress parameters, likelihood ratio testing of interested parameters is provided as a complementary. Simulation studies and two real-life data examples are provided to investigate the performance of proposed methods.     

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.     

Modified inference about the mean of the exponential distribution using moving extreme ranked set sampling

The maximum likelihood estimator (MLE) and the likelihood ratio test (LRT) will be considered for making inference about the scale parameter of the exponential distribution in case of moving extreme ranked set sampling (MERSS). The MLE and LRT can not be written in closed form. Therefore, a modification of the MLE using the technique suggested by Maharota and Nanda (Biometrika 61:601–606, 1974) will be considered and this modified estimator will be used to modify the LRT to get a test in closed form for testing a simple hypothesis against one sided alternatives. The same idea will be used to modify the most powerful test (MPT) for testing a simple hypothesis versus a simple hypothesis to get a test in closed form for testing a simple hypothesis against one sided alternatives. Then it appears that the modified estimator is a good competitor of the MLE and the modified tests are good competitors of the LRT using MERSS and simple random sampling (SRS).     

Bayesian analysis for the two-parameter Pareto distribution based on record values and times

Doostparast and Balakrishnan (Pareto record-based analysis, Statistics, under review) recently developed optimal confidence intervals as well as uniformly most powerful tests for one- and two-sided hypotheses concerning shape and scale parameters, for the two-parameter Pareto distribution based on record data. In this paper, on the basis of record values and inter-record times from the two-parameter Pareto distribution, maximum-likelihood and Bayes estimators as well as credible regions are developed for the two parameters of the Pareto distribution. For illustrative purposes, a data set on annual wages of a sample of production-line workers in a large industrial firm is analysed using the proposed procedures.