Negative exponential disparity based family of goodness-of-fit tests for multinomial models

This paper investigates a new family of goodness-of-fit tests based on the negative exponential disparities. This family includes the popular Pearson's chi-square as a member and is a subclass of the general class of disparity tests (Basu and Sarkar, 1994) which also contains the family of power divergence statistics. Pitman efficiency and finite sample power comparisons between different members of this new family are made. Three asymptotic approximations of the exact null distributions of the negative exponential disparity famiiy of tests are discussed. Some numerical results on the small sample perfomance of this family of tests are presented for the symmetric null hypothesis. It is shown that the negative exponential disparity famiiy, Like the power divergence family, produces a new goodness-of-fit test statistic that can be a very attractive alternative to the Pearson's chi-square. Some numerical results suggest that, application of this test statistic, as an alternative to Pearson's chi-square, could be preferable to the I ^2/3 statistic of Cressie and Read (1984) under the use of chi-square critical values.     

Likelihood ratio test for testing order of continuous time finite markov chains

Anderson and Goodman ( 1957) have obtained the likelihood ratio tests and chi-square tests for testing the hypothesis about the order of discrete time finite Markov chains, On the similar lines we have obtained likeli¬hood ratio tests and chi-square tests (asymptotic) for testing hypotheses about the order of continuous time Markov chains (MC) with finite state space.     

Properties of doubly-truncated gamma variables

The truncated gamma distribution has been widely studied, primarily in life-testing and reliability settings. Most work has assumed an upper bound on the support of the random variable, i.e. the space of the distribution is (0,u). We consider a doubly-truncated gamma random variable restricted by both a lower (l) and upper (u) truncation point, both of which are considered known. We provide simple forms for the density, cumulative distribution function (CDF), moment generating function, cumulant generating function, characteristic function, and moments. We extend the results to describe the density, CDF, and moments of a doubly-truncated noncentral chi-square variable.     

Many-to-one comparison of nonlinear growth curves for Washington's Red Delicious apple

In this article, we are interested in comparing growth curves for the Red Delicious apple in several locations to that of a reference site. Although such multiple comparisons are common for linear models, statistical techniques for nonlinear models are not prolific. We theoretically derive a test statistic, considering the issues of sample size and design points. Under equal sample sizes and same design points, our test statistic is based on the maximum of an equi-correlated multivariate chi-square distribution. Under unequal sample sizes and design points, we derive a general correlation structure, and then utilize the multivariate normal distribution to numerically compute critical points for the maximum of the multivariate chi-square. We apply this statistical technique to compare the growth of Red Delicious apples at six locations to a reference site in the state of Washington in 2009. Finally, we perform simulations to verify the performance of our proposed procedure for Type I error and marginal power. Our proposed method performs well in regard to both.     

On the Relationship between Directional and Omnibus Statistical Tests

Abstract.  A common statistical problem involves the testing of a K -dimensional parameter vector. In both parametric and semiparametric settings, two types of directional tests – linear combination and constrained tests – are frequently used instead of omnibus tests in hopes of achieving greater power for specific alternatives. In this paper, we consider the relationship between these directional tests, as well as their relationship to omnibus tests. Every constrained directional test is shown to be asymptotically equivalent to a specific linear combination test under a sequence of contiguous alternatives and vice versa. Even when the direction of the alternative is known, the constrained test in general will not be optimal unless the objective function used to derive it is efficient. For an arbitrary alternative, insight into the power characteristics of directional tests in comparison to omnibus tests can be gained by a chi-square partition of the omnibus test.     

Estimation of proportions for multinomial contingency tables subject to marginal constraints

For a two-way table of observed proportions we consider four approaches to the problem of fitting a new set of proportions which-subject to satisfying marginal constraints on row and column totals-corresponds as closely as possible to the original table, The approaches are known to be asymptotically equivalent; but our chief interest is in establishing, at least for the 2x2 table, a hierarchy of preferences under multinomial sampling for moderate sample sizes. Implementation of minimum chi-square is described and recommended.     

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     

Umvu estimation for certain family of exponential distributions

In this paper we study certain properties of estimable and UMUV-estimable functions in a subfamily of the one-parameter exponential family of distributions for which there exists a sufficient and complete statistic following a Gamma distribution. These results are applied to the problem of estimation in the transformed chi-square family.     

Prediction intervals for the future record values from exponential distribution: comparative study

!n this paper we consider the predicf an problem of the future nth record value based an the first m (m < n) observed record values from one-parameter exponential distribution. We introduce four procedures for obtaining prediction intervals for the nth record value. The performance of the so obtained intervals is assessed through numerical and simulation studies. In these studies, we provide the means and standard errors of lower limits. upper limits and lengths of prediction intervals. Further, we check the validation of these intervals based on some point predictors.