Power Functions Of Tests Research Articles

Accessing the Power of Tests Based on Set-Indexed Partial Sums of Multivariate Regression Residuals

The intention of the present paper is to establish an approximation method to the limiting power functions of tests conducted based on Kolmogorov-Smirnov and Cramér-von Mises functionals of set-indexed partial sums of multivariate regression residuals. The limiting powers appear as vectorial boundary crossing probabilities. Their upper and lower bounds are derived by extending some existing results for shifted univariate Gaussian process documented in the literatures. The application of multivariate Cameron-Martin translation formula on the space of high dimensional set-indexed continuous functions is demonstrated. The rate of decay of the power function to a presigned value α is also studied. Our consideration is mainly for the trend plus signal model including multivariate set-indexed Brownian sheet and pillow. The simulation shows that the approach is useful for analyzing the performance of the test.

Applications of the Likelihood Theory in Finance: Modelling and Pricing

SummaryThis paper discusses the connection between mathematical finance and statistical modelling which turns out to be more than a formal mathematical correspondence. We like to figure out how common results and notions in statistics and their meaning can be translated to the world of mathematical finance and vice versa. A lot of similarities can be expressed in terms of LeCam’s theory for statistical experiments which is the theory of the behaviour of likelihood processes. For positive prices the arbitrage free financial assets fit into statistical experiments. It is shown that they are given by filtered likelihood ratio processes. From the statistical point of view, martingale measures, completeness, and pricing formulas are revisited. The pricing formulas for various options are connected with the power functions of tests. For instance the Black–Scholes price of a European option is related to Neyman–Pearson tests and it has an interpretation as Bayes risk. Under contiguity the convergence of financial experiments and option prices are obtained. In particular, the approximation of Itô type price processes by discrete models and the convergence of associated option prices is studied. The result relies on the central limit theorem for statistical experiments, which is well known in statistics in connection with local asymptotic normal (LAN) families. As application certain continuous time option prices can be approximated by related discrete time pricing formulas.

The Power of Exceedance-Type Tests Under Lehmann Alternatives

The distribution of a rank test statistic for the two-sample problem is derived under Lehmann alternative. The test statistic is based on extreme ranks of both samples in the combined sample. The exact power function of the test is studied and compared to the power functions of several two-sample rank tests.

R-Symmetry and the Power Functions of the Tests for Inverse Gaussian Means

The two-parameter Inverse Gaussian (IG) distribution is often appropriate for modeling non negative right-skewed data due to the striking similarities with the Gaussian distribution in its basic properties and inference methods. There are about 40 such G-IG analogies developed in literature and were most recently tabulated by Mudholkar and Wang. Of these, the earliest and most commonly noted similarities are the significance tests based on student's t and F distribution for the homogeneity of one, two or several means of the IG populations. However, unlike the corresponding tests in Gaussian theory, little is known about the power function of the basic tests. In this article, we employ the IG-related root-reciprocal IG distribution and a notion of Reciprocal Symmetry to establish the monotonicity of the power function of the test of significance for the IG mean.

Hypothesis tests on the scale parameter using median ranked set sampling

In this paper, some test procedures for scale parameters of exponential and rectangular distributions involving the median ranked set sampling (MRSS) method are discussed. The necessary tables for the critical regions for the suggested tests are supplied. The power functions of the tests are compared with the power functions of the same tests suggested by Abu-Dayyeh and Muttlak (1996) using ranked set sampling (RSS) data. It is seen that the test procedures using the MRSS data have greater power than those using the RSS data when the sample is drawn from an exponential distribution.

On preferences of general two-sided tests with applications to Kolmogorov–Smirnov-type tests

Power functions of tests for Gaussian shift experiments on infinite dimensional Hilbert spaces usually can not be calculated explicitly. Therefore one analyzes the behavior of such tests in the neighborhood of the null hypothesis. Useful measures to compare the quality of different testing procedures are the gradient of a one-sided and the curvature of a two-sided test in the null hypothesis. Janssen (1995) showed that a principal component decomposition of the curvature exists based on a Hilbert–Schmidt operator. It follows that these tests have only acceptable power for a finite number of directions. In this paper we prove an even stronger general result for Gauss shifts under just mild additional assumptions. A certain optimality property of a one-sided test implicates that for a small level α the corresponding two-sided test acts only in a single direction. The results are applied to Kolmogorov–Smirnov type tests and the signal detection problem.

ISO* Property of Two-Parameter Compound Poisson Distributions with Applications

The ISO* property of noncentrality parameters is derived for the expected value of an ISO* function of independent nonnegative two-parameter compound Poisson random variables and is then applied to unbiasedness of tests and monotonicity of power functions of tests in an order-restricted hypothesis testing problem for the noncentrality parameter. The ISO* property and the Schur convexity are also studied for a class of two-parameter distributions which has the additive property (i.e., is closed under convolution) and contains the one-parameter family with the semigroup property as a special case.

A monotonicity property of the power function of multivariate tests

Let S = ∑ n k = 1 X k x l k , where the X k are independent observations from a 2-dimensional normal N( μ k , Σ) distribution, and let Λ = ∑ n k = 1 μ k μl k Σ −1 be a diagonal matrix of the form λI, where λ ≥ 0 and I is the identity matrix. It is shown that the density φ of the vector l ̃ = (l 1, l 2) of characteristic roots of S can be written as G(λ, l 1, l 2)φ 0( l ̃ ) , where G satisfies the FKG condition on R 3 +. This implies that the power function of tests with monotone acceptance region in l 1 and l 2, i.e. a region of the form { g( l 1, l 2) ≤ c} where g is nondecreasing in each argument, is nondecreasing in λ. It is also shown that the density φ of ( l 1, l 2) does not allow a decomposition φ(l 1, l 2) = G(λ, l 1, l 2)φ 0( l ̃ ) , with G satisfying the FKG condition, if Λ = diag( λ, 0) and λ > 0, implying that this approach to proving monotonicity of the power function fails in general.

A Continuous Time Approximation to the Stationary First-Order Autoregressive Model

We consider the least-squares estimator in a strictly stationary first-order autoregression without an estimated intercept. We study its continuous time asymptotic distribution based on an asymptotic framework where the sampling interval converges to zero as the sample size increases. We derive a momentgenerating function which permits the calculation of percentage points and moments of this asymptotic distribution and assess the adequacy of the approximation to the finite sample distribution. In general, the approximation is excellent for values of the autoregressive parameter near one. We also consider the behavior of the power function of tests based on the normalized leastsquares estimator. Interesting nonmonotonic properties are uncovered. This analysis extends the study of Perron [15] and helps to provide explanations for the finite sample results established by Nankervis and Savin [13].

The power functions of the likelihood ratio tests for a simply ordered trend in normal means

Likelihood ratio tests for the homogeneity of k normal means with the alternative restricted by an increasing trend are considered as well as the likelihood ratio tests of the null hypothesis that the means satisfy the trend. While the work is primarily a survey of results concerning the power functions of these tests, the extensions of some results to the case of not necessarily equal sample sizes are presented. For the case of known or unknown population variances, exact expressions are given for the power functions for k=3,4, and approximations are discussed for larger k. The topics of consistency, bias and monotonicity of the power functions are included. Also, Bartholomew's conjectures concerning minimal and maximal powers are investigated, with results of a new numerical study given.

Evaluation of the Noncentral F-Distribution by Numerical Contour Integration

The noncentral F-distribution, which yields the power function of tests of linear hypotheses as in the analysis of variance and of covariance, is computed by numerical integration of a contour integral in the complex plane. Determining the percentage points of this distribution is facilitated by approximating the integral by Laplace’s method during the initial stages of a search by the secant method.

THE NON-CENTRAL χ2- AND F-DISTRIBUTIONS AND THEIR APPLICATIONS

In the Neyman-Pearson theory of testing statistical hypotheses, the efficiency of a statistical test is to be judged by its power of detecting departures from the null hypothesis. Thus besides knowing the random sampling distribution of a given statistic T under this hypothesis, say Ho, it is also necessary to know the distribution of T under admissible hypotheses alternative to Ho. Hence the power function of the test is obtained. In the case of the wellknown tests using X2, t and F, the evaluation of their power functions involves the use of what have been called non-central distributions. For example, if we are applying the t-test to examine if a sample has come from a normal population with mean # = 0(HO), we know that under Ho, t has a 5 % chance of exceeding the 5 % point of its distribution. But in order to compute the power of the test we wish to know the chance that t exceeds this point when M has alternative values, not equal to zero. This chance is given by the non-central t-integral. This distribution has been studied by Fisher (1931), Neyman (1935), Neyman & Tokarska (1936) and Johnson & Welch (1939). In a similar way, the non-central X2and F-distributions arise in consideration of the power functions of the X2and variance-ratio tests. The power function may be used either to determine the extent of the departures from Ho in a given direction, which will be detected as significant (at a prescribed level) with a given probability, or it may be used to determine in advance the size of experiment necessary to ensure that a worth-while difference will be established as significant, if it exists. But apart from its value in this connexion, the study of non-central distributions is of considerable interest. The mathematical forms of these distributions of t, x2 and F have been long known, but their use without extensive tabling has not been easy. The present paper is therefore concerned with two lines of investigation: (a) The derivation of certain approximations to the probability integrals of (i) non-central X2, and (ii) the ratio of non-central X2 to an independent central X2, which we have termed noncentral F. These approximations, depending on tabled functions, permit easy calculation. (b) Discussion of the ways in which these distributions may be used in connexion with the power functions of statistical tests.