Powerful Unbiased Test Research Articles

Comparing two poisson parameters: what to do when the optimal isn't done

Suppose two Poisson processes with rates γ1 and γ2 are observed for fixed times tl and t2. This paper considers hypothesis tests and confidence intervals for the parameter ρ = γ2/γ1. Uniformly most powerful unbiased tests and uniformly most accurate unbiased confidence intervals exist for ρ, but they require randomization and so are not used in practice. Several alternative procedures have been proposed. In the context of one-sided hypothesis tests these procedures are reviewed and compared on numerical grounds and by use of the conditionality and repeated sampling principles. It is argued that a conditional binomial test which rejects with conditional level closest to but not necessarily less than, the nominal a is the most reasonable. This test is different from the usual conditional binomial test which rejects with conditional level closeset to but less than or equal to the nominal α Numerical results indicate that an approximate procedure based on the Poisson variance stabilizing transformation has properties similar to the preferred conditional binomial test. Values for λ1 = t1λ1 required to achieve a specified power are considered. These results are also discussed in terms of test inversion to obtain confidence intervals.

Modified mantel-haenszel procedures for matched pairs

In the prospective study of a finely stratified population, one individual from each stratum is chosen at random for the “treatment” group and one for the “non-treatment” group. For each individual the probability of failure is a logistic function of parameters designating the stratum, the treatment and a covariate. Uniformly most powerful unbiased tests for the treatment effect are given. These tests are generally cumbersome but, if the covariate is dichotomous, the tests and confidence intervals are simple. Readily usable (but non-optimal) tests are also proposed for poly-tomous covariates and factorial designs. These are then adapted to retrospective studies (in which one “success” and one “failure” per stratum are sampled). Tests for retrospective studies with a continuous “treatment” score are also proposed.

Are Uniformly Most Powerful Unbiased Tests Really Best?

Abstract In teaching the development of uniformly most powerful unbiased (UMPU) tests, one rarely discusses the performance of alternative biased tests. It is shown, through the comparison of two independent Bernoulli proportions, that a biased test (the Z test) can be more powerful than the UMPU test (Fisher's exact test—randomized) in a large region of the alternative parameter space. A more general example is also given.

Are Uniformly Most Powerful Unbiased Tests Really Best?

Are Uniformly Most Powerful Unbiased Tests Really Best?

On independence of statistics in truncated exponential models

Let be the order statistics of a sample of size n from the left truncated k-parameter exponential family of distributions with unknown truncation parameter, The densities of this family are of the form . Let and gn be a vector valued function. The existence of a statistic of the form with distribution independent of X(1) and depending on θ is characterized. The case k=1 is shown to be of particular interest and an application in this case to uniformly most powerful unbiased tests is presented. Right and two-sided truncation models are also discussed.

The Relation between Score Tests and Approximate UMPU Tests in Exponential Models Common in Biometry

In several recent papers, both published and unpublished, the theory of score tests (Rao, 1973, pp. 415-420), or the equivalent C(a) test (Neyman, 1959), has been used to derive tests involving likelihoods based on exponential families which yield sufficient statistics. A recently published example is a paper by Day and Byar (1979) in which the test for combined 2 x 2 tables due to Cochran (1954) [sometimes called the Mantel-Haenszel (1959) statistic] is shown to be a 'logit score test'. Day and Byar also considered a second multiparameter test for a more general logistic model. We shall show that, in such cases, score tests can ordinarily be written down directly upon examination of the structure of the likelihood; neither differentiation of the likelihood nor matrix inversion is usually needed. Furthermore, for the case in which a single parameter is tested in the presence of nuisance parameters, the score test is simply the normal approximation to the uniformly most powerful unbiased (UMPU) test. Obviously, for one-parameter exponential families the score test is not only asymptotically locally most powerful but is also an approximation to the UMPU test (Efron, 1975).

Unbiased Estimators and Tests for Truncation and Scale Parameters

SYNOPTIC ABSTRACTBlackwell-Rao-Lehmann-Scheffe theory is used to derive the uniformly minimum variance unbiased estimators for unknown truncation and scale parameters, and the reliability function of truncation parameter distributions. Uniformly most powerful unbiased tests of hypotheses are formulated. Truncated two-parameter exponential, truncated Pareto, and two-parameter uniform distributions are considered in examples.

Optimal tests and estimators for truncated exponential families

Blackwell-Rao-Lehmann-Scheffe' theory is used to derive the minimum variance unbiased estimators for the functions of scale and truncation parameters as well as the reliability function of the truncated exponential family distribution. Uniformly most powerful unbiased tests of hypotheses are formulated. Finally, a particular model of this family, viz., the truncated exponential model is discussed.

An Exponential Subfamily which Admits UMPU Tests Based on a Single Test Statistic

Let $f(x: \theta) = a(x) \exp\{\theta_1u_1(x) + \theta_2u_2(x) + c(\theta)\}, \theta = (\theta_1, \theta_2) \in \ominus \subset R^2$, be a density with respect to the Lebesgue measure on the real line which characterizes a two-parameter exponential family of distributions. Let $(\theta_1, \eta_2)$ be the mixed parameters, where $\eta_2 = E\{u_2(X)\}$. Assume that $\theta_2$ can be represented as $\theta_2 = -\theta_1 \varphi'(\eta_2)$ where $\varphi'(\eta_2) = d\varphi(\eta_2)/d\eta_2$ for some function $\varphi(\eta_2)$. Let $(X_1, \cdots, X_n)$ be independent random variables having a common density $f(x: \theta)$ and set $T_i = \sum^n_{j=1} u_i(X_j), i = 1, 2$. It is shown that if $u_2(x)$ is a 1-1 function then the random variables $T_2$ and $Z_n = T_1 - n\varphi(T_2/n)$ are independent and that the statistic $Z_n$ is ancillary for $\theta_2$ in the presence of $\theta_1$ (i.e. the density of $Z_n$ depends on $\theta_1$ only). Furthermore, the density of $Z_n$ belongs to the one-parameter exponential family with natural parameter $\theta_1$. These results enable us to construct uniformly most powerful unbiased (UMPU) tests for various hypotheses concerning the parameter $\theta_1$ which are based on the statistic $Z_n$.

A modified power series distribution

A modification of the usual power series distribution is proposed and a statistical treatment of data generated by this probability model is indicated. It is shown that this modified power series distribution belongs to a multiparameter discrete exponential family and standard theory is then applied to obtain complete sufficient statistics of the parameters, sampling distribution of these statistics, maximum likelihood and minimum variance unbiased estimators, uniformly most powerful unbiased tests, uniformly most accurate confidence intervals and expected cover tolerance regions.

Exponential family distribution with unknown truncation parameter: two sample problem

The uniformly most powerful unbiased tests are formulated for two sample problem of a given continuous distribution belonging to the exponential family with unknown scale and truncation parameters. The two-parameter exponential and Paretc distributions are considered in examples.

Use of the Likelihood Ratio Test on the Inverse Gaussian Distribution

Abstract In the following article, the likelihood ratio test is determined for four tests of hypotheses involving the inverse Gaussian distribution. For three of the hypotheses, the test produces the same statistic as the uniformly most powerful unbiased test.

An accurate test for testing the Hardy-Weinberg equilibrium hypothesis

The paper includes uniformly most powerful unbiased test for a genetic model with two possible kinds of genes. This test is constructed on the basis of general theory of testing linear hypotheses for families of exponential distributions. An example of application for the system MN of blood-groups is given. Zbl 0465.62098; MR0549989

Construction of Optimal Unbiased Inference Procedures for the Parameters of the Gamma Distribution

Methods for constructing uniformly most powerful unbiased two-sided tests of the parameters of the gamma distribution and the associated unbiased confidence intervals are discussed. These methods make use of existing results on construction of unbiased confidence intervals for the variance of a normal distribution. An approximation is derived for the conditional distribution of the statistic W n = x/x given G n = x/β. Tabulations which are convenient in calculating confidence limits for the shape parameter are given.

Construction of Optimal Unbiased Inference Procedures for the Parameters of the Gamma Distribution

Methods for constructing uniformly most powerful unbiased two-sided tests of the parameters of the gamma distribution and the associated unbiased confidence intervals are discussed. These methods make use of existing results on construction of unbiased confidence intervals for the variance of a normal distribution. An approximation is derived for the conditional distribution of the statistic W n = x/x given G n = x/β. Tabulations which are convenient in calculating confidence limits for the shape parameter are given.

Statistical Diagnosis and Tests of Factor Hypotheses

Abstract Certain symptoms are of diagnostic value because they reflect the progress of a multistage disease. To analyze such cases, a dynamic symptom/disease model is constructed. The model is seen to be consistent with observations about skewed, dependent symptom distributions for certain diseases. The uniformly most powerful unbiased test of hypotheses about the disease factor leads to a linear discriminant function. Prior knowledge of the symptom/factor relationship can be used to correct the procedure for nuisance effects or lacking such information, the model can be structured so that the test is independent of the nuisance factors.

Statistical Diagnosis and Tests of Factor Hypotheses

Abstract Certain symptoms are of diagnostic value because they reflect the progress of a multistage disease. To analyze such cases, a dynamic symptom/disease model is constructed. The model is seen to be consistent with observations about skewed, dependent symptom distributions for certain diseases. The uniformly most powerful unbiased test of hypotheses about the disease factor leads to a linear discriminant function. Prior knowledge of the symptom/factor relationship can be used to correct the procedure for nuisance effects or lacking such information, the model can be structured so that the test is independent of the nuisance factors.

Uniformly Most Powerful Unbiased Tests on the Scale Parameter of a Gamma Distribution With a Nuisance Shape Parameter

Conditional tests on the scale parameter of the gamma distribution with an unknown nuisance shape parameter are considered. Such tests, based upon the conditional distribution of the sample mean x (or equivalently Wi , = x/x) given the geometric mean x, are uniformly most powerful unbiased tests. Percentage points of the conditional distribution are tabulated for small sample sizes and an asymptotic normal approximation is also obtained.

Optimum Test Procedures for the Mean of First Passage Time Distribution in Brownian Motion with Positive Drift (Inverse Gaussian Distribution)

In this paper we discuss testing hypotheses and interval estimation for the mean of the first passage time distribution in Brownian motion with positive drift (inverse Gaussian distribution). Optimum test procedures and confidence intervals for both one-sided and two-sided cases are derived in their exact forms, which utilize the percentage points of standard normal and Student's t distributions. In thecase of an hypothesis with a two-sided alternative, it is shown that a uniformly most powerful (UMP) unbiased test is simply a two-tailed normal test if the nuisance parameter is known, and a two-tailed Student's t test if it is unknown.

The power series distribution wite unknown truncation parameter: Two sample problem

The uniformly most powerful unbiased tests are formulated for the two sample problem of the power series distribution with unknown truncation parameter.