Testing the equality of two normal percentiles

Two statistics are suggested for testing the equality of two normal percentiles where population means and variances are unknown. The first is based on the generalized likelihood ratio test (LRT), the second on Cochran's statistic used in the Behrens-Fisher problem. Size and power comparisons are made by using simulation and asympototic theory.     

Linear generalizations of various matric-t distributions

The concept of a matric-t variate is extended to cases where the positive (definite) part of the variate, which is usually Wishart distributed independently of the normal part, is a linear sum of positive (definite) variates with positive coefficients. These distributions and their quadratic forms are of importance i.a, for the exact solution to the multi¬variate Behrens-Fisher problem. A few useful identities con¬cerning the invariant polynomials with matrix arguments are derived     

A note on estimating the common mean of k normal distributions and the stein problem

An unbiased estimator for the common mean of k normal distributions is suggested. A necessary and sufficient condition for the estimator Lo have a smaller variance than each sample mean is given. In the case of estimating the common mean vector of k p-variate (p ≤ 3) normal distributions a combined unbiased estimator may be used. We give a class of estimators which are better than the combined estimator when the loss is quadratic and the restriction of unbiasedness is removed.     

Bounds for the variance of the graybill-deal estimator of the common mean of two normal distributions

In this note we derive sharp lower and upper bounds for the variance of the Graybill-Deal estimator of the common mean of two normal distributions with unknown variances when the sample sizes are not necessarily equal. We also derive similar bounds for the variance of the Brown-Cohen (1974) T _a(1) class of unbiased es-timators to which the Graybill-Deal estimator belongs. Further, we illustrate the sharpness of the bounds by numerical computations in the case of the Graybill-Deal estimator.     

On two asymptotic normal distributions for the generalized wilks lambda statistic

In this paper we.present a Normal asymptotic distribution for the logarithm of the generalized Wilks Lambda statistic based on an asymptotic distribution for the determinant of a Wishart matrix. This distribution is obtained through the combined use of Taylor expansions of random variables whose exponentials have chi-square distributions and the Lindeberg-Feller version of the Central Limit Theorem, Another asymptotic Normal distribution for the logarithm of the generalized Wilks Lambda statistic for the case when at most one of the sets has an odd number of variables is derived directly from the exact distribution. Both distributions are non-degenerate and non-singular. The first Normal distribution compares favorably with other known approximations and asymptotic distributions namely for large numbers of variables and small sample sizes, while the second Normal distribution, which has a more restricted application, compares in most cases highly favorably with other known asymptotic distributions and approximations. Finally, a method to compute approximate quantiles which lay very close and converge steadily to the exact ones is presented.     

Percentage points of the statistics for testing hypotheses on mean vectors of multivariate normal distributions with missing observations

A problem of testing of hypotheses on the mean vector of a multivariate normal distribution with unknown and positive definite covariance matrix is considered when a sample with a special, though not unusual, pattern of missing observations from that population is available. The approximate percentage points of the test statistic are obtained and their accuracy has been checked by comparing them with some exact percentage points which are calculated for complete samples and some special incomplete samples. The approximate percentage points are in good agreement with exact percentage points. The above work is extended to the problem of testing the hypothesis of equality of two mean vectors of two multivariate normal distributions with the same, unknown covariance matrix     

Mean-shift outliers model in skew scale-mixtures of normal distributions

ABSTRACT

Asymmetric models have been discussed quite extensively in recent years, in situations where the normality assumption is suspected due to lack of symmetry in the data. Techniques for assessing the quality of fit and diagnostic analysis are important for model validation. This paper presents a study of the mean-shift method for the detection of outliers in regression models under skew scale-mixtures of normal distributions. Analytical solutions for the estimators of the parameters are obtained through the use of Expectation–Maximization algorithm. The observed information matrix for the calculation of standard errors is obtained for each distribution. Simulation studies and an application to the analysis of a data have been carried out, showing the efficiency of the proposed method in detecting outliers.     

Relationships between confidence intervals for the ratio of two variances of independent normal distributions

Burk at al (1984) gave a results concerning the comparison of the length of the two different confidence intervals for variance ratio, when the construction of the intervals was based on the principle of “equal tails”11. The purpose of this paper is to be solve the similar problem in case of the principle of “minimal length”.     

Testing homogeneity in a heteroscedastic contaminated normal mixture

Large-scale simultaneous hypothesis testing appears in many areas. A well-known inference method is to control the false discovery rate. One popular approach is to model the z-scores derived from the individual t-tests and then use this model to control the false discovery rate. We propose a heteroscedastic contaminated normal mixture to describe the distribution of z-scores and design an EM-test for testing homogeneity in this class of mixture models. The proposed EM-test can be used to investigate whether a collection of z-scores has arisen from a single normal distribution or whether a heteroscedastic contaminated normal mixture is more appropriate. We show that the EM-test statistic has a shifted mixture of chi-squared limiting distribution. Simulation results show that the proposed testing procedure has accurate type-I error and significantly larger power than its competitors under a variety of model specifications. A real-data example is analysed to exemplify the application of the proposed method.     

A new generalized two-sided class of distributions with an emphasis on two-sided generalized normal distribution

The ordinary-G class of distributions is defined to have the cumulative distribution function (cdf) as the value of the cdf of the ordinary distribution F whose range is the unit interval at G, that is, F(G), and it generalizes the ordinary distribution. In this work, we consider the standard two-sided power distribution to define other classes like the beta-G and the Kumaraswamy-G classes. We extend the idea of two-sidedness to other ordinary distributions like normal. After studying the basic properties of the new class in general setting, we consider the two-sided generalized normal distribution with maximum likelihood estimation procedure.     

Confidence regions for the common mean vector of several multivariate normal populations

Suppose that there are independent samples available from several multivariate normal populations with the same mean vector m? but possibly different covariance matrices. The problem of developing a confidence region for the common mean vector based on all the samples is considered. An exact confidence region centered at a generalized version of the well-known Graybill-Deal estimator of m? is developed, and a multiple comparison procedure based on this confidence region is outlined. Necessary percentile points for constructing the confidence region are given for the two-sample case. For more than two samples, a convenient method of approximating the percentile points is suggested. Also, a numerical example is presented to illustrate the methods. Further, for the bivariate case, the proposed confidence region and the ones based on individual samples are compared numerically with respect to their expected areas. The numerical results indicate that the new confidence region is preferable to the single-sample versions for practical use.     

Moments and quadratic forms of matrix variate skew normal distributions

ABSTRACT

In 2007, Domínguez-Molina et al. obtained the moment generating function (mgf) of the matrix variate closed skew normal distribution. In this paper, we use their mgf to obtain the first two moments and some additional properties of quadratic forms for the matrix variate skew normal distributions. The quadratic forms are particularly interesting because they are essentially correlation tests that introduce a new type of orthogonality condition.     

Coefficients of lee-gurland two-sample test on normal means

The Behrens-Fisher problem in comparing means of two normal populations is revisited Lee and Gurland (1975) suggested a solution to the problem and provided the set of coefficients required in computing critical values for the case α=005, where α is the nominal level of significance This solution, called the Lee-Guiland Test in this article, has proven to be practical as far as calculation is involved, and more importantly, it maintains the actual size very close to α= 0.05 for possible values of the ratio of population variances This merit has not been attained by most of the Behrens-Fisher solutions in the literature. In this article, the coefficients for other values of α, namely 0 025, 0 01 and 0.005 are provided for wider applications of the test Moreover, careful and detailed comparisons are made in terms of size and power with the other practical solution:the Welch's Approximate t I est Due to a possible drawback of the Welch's Approximate t I est in controlling the actual size, especially for small a and small sample sizes, the Lee-Gurland lest presents itself as a slightly better' alternative in testing equality of two normal population means I he coefficients mentioned above are also fitted by the functions of the reciprocals of the degrees of freedom, so that the substantial amount of table-looking can be avoided Some discussions are also made in regarding the recent “Welch vs Gosset” argument: Should the Student's t Test be dispensed off’from the routine use in testing the equality of two normal means?.     

Linear censored regression models with skew scale mixtures of normal distributions

A special source of difficulty in the statistical analysis is the possibility that some subjects may not have a complete observation of the response variable. Such incomplete observation of the response variable is called censoring. Censorship can occur for a variety of reasons, including limitations of measurement equipment, design of the experiment, and non-occurrence of the event of interest until the end of the study. In the presence of censoring, the dependence of the response variable on the explanatory variables can be explored through regression analysis. In this paper, we propose to examine the censorship problem in context of the class of asymmetric, i.e., we have proposed a linear regression model with censored responses based on skew scale mixtures of normal distributions. We develop a Monte Carlo EM (MCEM) algorithm to perform maximum likelihood inference of the parameters in the proposed linear censored regression models with skew scale mixtures of normal distributions. The MCEM algorithm has been discussed with an emphasis on the skew-normal, skew Student-t-normal, skew-slash and skew-contaminated normal distributions. To examine the performance of the proposed method, we present some simulation studies and analyze a real dataset.     

A condition for best asymptotic normality in a regular family of distributions

A theorem is presented which provides a simple sufficient condition for a weakly consistent estimator of a parameter in a regular family of distributions to be best asymptotically normal (B.A.N.). As a corollary the B.A.N. property of a maximum likelihood estimator is established under weaker conditions than those of Zacks (1971). Two examples are provided to illustrate the technique.     

Estimating the common hazard rate of two exponential distributions with ordered location parameters

This paper is concerned with estimating the common hazard rate of two exponential distributions with unknown and ordered location parameters under a general class of bowl-shaped scale invariant loss functions. The inadmissibility of the best affine equivariant estimator is established by deriving an improved estimator. Another estimator is obtained which improves upon the best affine equivariant estimator. A class of improving estimators is derived using the integral expression of risk difference approach of Kubokawa [A unified approach to improving equivariant estimators. Ann Statist. 1994;22(1):290–299]. These results are applied to specific loss functions. It is further shown that these estimators can be derived for four important sampling schemes: (i) complete and i.i.d. sample, (ii) record values, (iii) type-II censoring, and (iv) progressive Type-II censoring. A simulation study is carried out for numerically comparing the risk performance of these proposed estimators.     

Bayesian significance test for discriminating between survival distributions

An evaluation of FBST, Fully Bayesian Significance Test, restricted to survival models is the main objective of the present paper. A Survival distribution should be chosen among the tree celebrated ones, lognormal, gamma, and Weibull. For this discrimination, a linear mixture of the three distributions is an important tool: the FBST is used to test the hypotheses defined on the mixture weights space. Another feature of the paper is that all three distributions are reparametrized in that all the six parameters are written as functions of the mean and the variance of the population been studied. Some numerical results from simulations with some right-censored data are considered.     

A Test for the Equality of Parameters for Separate Regression Models in the Presence of Heteroskedasticity

Testing for the equality of regression coefficients across two regressions is a problem considered by analysts in a variety of fields. If the variances of the errors of the two regressions are not equal, then it is known that the standard large sample F-test used to test the equality of the coefficients is compromised by the fact that its actual size can differ substantially from the stated level of significance in small samples. This article addresses this problem and borrows from the literature on the Behrens-Fisher problem to provide some simple modifications of the large sample test which allows one to better control the probability of committing a Type I error. Empirical evidence is presented which indicates that the suggested modifications provide tests which are superior to well-known alternative tests over a wide range of the parameter space.     

Some robust tests of independence in symmetrical multivariate distributions

Let X =(x)^ij=(11¹, …, X,)_T, i = l, …n, be an n X random matrix having multivariate symmetrical distributions with parameters μ, Σ. The p-variate normal with mean μ and covariance matrix is a member of this family. Let be the squared multiple correlation coefficient between the first and the succeeding p₁ components, and let p² = + be the squared multiple correlation coefficient between the first and the remaining p₁ + p₂ =p – 1 components of the p-variate normal vector. We shall consider here three testing problems for multivariate symmetrical distributions. They are (A) to test p² =0 against; (B) to test against =0, 0; (C) to test against p² =0, We have shown here that for problem (A) the uniformly most powerful invariant (UMPI) and locally minimax test for the multivariate normal is UMPI and is locally minimax as p² 0 for multivariate symmetrical distributions. For problem (B) the UMPI and locally minimax test is UMPI and locally minimax as for multivariate symmetrical distributions. For problem (C) the locally best invariant (LBI) and locally minimax test for the multivariate normal is also LBI and is locally minimax as for multivariate symmetrical distributions.