Approximate Percentage Points Research Articles

Locally optimal test of normality against skew-normality

Efficient Score testing procedures for testing for the skewness parameter of the one-parameter and three-parameter skew-normal distributions are investigated. Approximate percentage points for the test statistics are obtained via the Monte Carlo method. Salvan's locally optimal test is extended and simulation is used to show that the power of the extended test is higher than that of the skewness test.

TWO-STAGE ESTIMATION OF A LINEAR FUNCTION OF NORMAL MEANS WITH SECOND-ORDER APPROXIMATIONS

ABSTRACT When a confidence interval tends to be too wide, its effectiveness in bolstering any inferential statements becomes limited. Hence, an experimenter may opt to construct a confidence interval with some preassigned “small” width and preassigned “large” confidence coefficient so that any inferences drawn from this can be of some value in practice. We consider k(≥2) independent normal populations with unknown means and unknown and unequal variances. We discuss the estimation problem for a linear function of the population means with a fixed-width (=2d) confidence interval having the preassigned confidence coefficient (≥1 − α), d > 0, 0 < α < 1. But, the goal of having such a confidence interval with both preassigned width and confidence coefficient is not attainable when the sample sizes are held fixed in advance [Dantzig (1940)[8], Ghosh et al. (1997, Sec. 3.7)[13]]. Chapman (1950)[4] first gave a Stein-type two-stage procedure for the problem on hand when k = 2. It is known that in a k-sample problem, the analogous two-stage procedure requires the upper percentage points of the distribution of the sum of k independent Student's t variates. First, a Cornish-Fisher expansion of such a percentage point is derived (Theorem 2.1) in general, followed by the Tables 1 and 2 of these (approximate) percentage points which are constructed by using this expansion, with the pilot sample sizes not necessarily all equal, when k = 2, α = .05, .01 and k = 3, α = .05. Next, under the limited additional assumption that each unknown population variance has a known positive lower bound, the Chapman type two-stage estimation procedure is modified along the lines of the one-sample considerations of Mukhopadhyay and Duggan (1997)[23]. For this modified two-stage procedure, various second-order expansions for both the lower and upper bounds of the average sample sizes (Theorem 2.2) and the associated confidence coefficient (Theorem 2.3) are obtained. We may remark that the second-order expansions are meant to provide faster rates of convergence for useful approximations. Then, through extensive sets of simulations we show that the extent of over-sampling experienced by the Chapman-type procedure is significantly reduced under the new modification when k = 2(1)5, α = .05. We include examples and data to illustrate usefulness of the modified k-sample two-stage estimation technique when k = 2, 3. Additionally, the importance of the asymptotic second-order terms is highlighted with the help of data analysis (Examples 1 and 2, Sec. 3).

Testing and Locating Variance Changepoints with Application to Stock Prices

Abstract This article explores testing and locating multiple variance changepoints in a sequence of independent Gaussian random variables (assuming known and common mean). This type of problem is very common in applied economics and finance. A binary procedure combined with the Schwarz information criterion (SIC) is used to search all of the possible variance changepoints existing in the sequence. The simulated power of the proposed procedure is compared to that of the CUSUM procedure used by Inclán and Tiao to cope with variance changepoints. The SIC and unbiased SIC for this problem are derived. To obtain the percentage points of the SIC criterion, the asymptotic null distribution of a function of the SIC is obtained, and then the approximate percentage points of the SIC are tabulated. Finally, the results are applied to the weekly stock prices. The unknown but common mean case is also outlined at the end.

Testing and Locating Variance Changepoints with Application to Stock Prices

This article explores testing and locating multiple variance changepoints in a sequence of independent Gaussian random variables (assuming known and common mean). This type of problem is very common in applied economics and finance. A binary procedure combined with the Schwarz information criterion (SIC) is used to search all of the possible variance changepoints existing in the sequence. The simulated power of the proposed procedure is compared to that of the CUSUM procedure used by Inclán and Tiao to cope with variance changepoints. The SIC and unbiased SIC for this problem are derived. To obtain the percentage points of the SIC criterion, the asymptotic null distribution of a function of the SIC is obtained, and then the approximate percentage points of the SIC are tabulated. Finally, the results are applied to the weekly stock prices. The unknown but common mean case is also outlined at the end.

Additional bowman-shenton approximate percentage points for pearson distributions based on pearson type vi

The Bowman and Shenton approximate percentage points for Pearson distributions are extended to include some cases for which the skewness β1 exceeds four.Use is made of the reciprocity property existing between the Pearson Type VI distribution and the Type I distribution

UMPU test for state dependence of a parameter in a linear birth process

The transition probability function for a class of linear birth processes has been derived. An UMPU test for state dependence of the parameter has been obtained. Some approximate percentage points of the test statistic have given using a simulation technique.

Percentage Points of Likelihood Ratio Tests for Threshold Autoregression

SUMMARY Likelihood ratio tests for threshold autoregression have been considered by Chan, and Chan and Tong. However, except for the simplest case, percentage points of the (asymptotic) null distribution of the test statistic have not been tabulated. The purpose of this paper is to tabulate the approximate upper percentage points for the general case. This is done via the Poisson clumping heuristic developed by Aldous. Monte Carlo experiments show that the approximation is quite good. As an illustration, the test is then applied to two data sets.

Significance points for some tests of uniformity on the sphere†

Beran (1968) and Giné (1975) have proposed several omnibus tests for uniformity on the unit sphere in three dimensional Euclidean space. While several authors have contributed to providing approximate percentage points for the limiting distributions, no tables of the limiting distributions, percentage points thereof, or finite sample distributions or percentage points have been available. In this paper we fill this gap by: 1) finding the exact distributions of the statistics of Beran and Giné for n = 2; 2) presenting some percentage points for selected small and moderate sample sizes obtained by Monte—Carlo methods; 3) evaluating numerically the cumulative distribution functions and significance points of the limiting distributions via the Laguerre transform method (Keilsori and Nunn (1979), Keilson, Nunn and Sumita (1981), and Sumita (1981)).

Two statistics for testing for multivariate normality

Malkovich & Afifi (1973) proposed measures of multivariate skewness and kurtosis. In the present paper, distributional properties of the two statistics are examined. A method is given for calculating approximate percentage points of their distributions for all sample sizes greater than 8, for dimensions 2, 3 and 4, and, in principle, for all higher dimensions. The percentage points are derived from appropriate transformations of the asymptotic distributions. A simulation study shows the approximations to be good for sample sizes of 25 and above.

Probleme der Erweiterung der Prüfverteilungen des Maximum‐Modulus‐Testes und des DUNNETT‐Testes

AbstractSCHULZ (1976) gave formulas for the approximate computation of the percentage points needed for the comparison of sample means by the Maximum‐Modulus‐Test resp. by the DUNNETT‐Test. Our investigation shows that these formulas are also practicable if the number of the mean differences k is greater than 20. The maximum difference between the exact values and the approximate values of the percentage points amounts to 0,05 in the case of the Maximum‐Modulus‐Test with k &gt;20 and degrees of freedom v &gt; k for a level of error a = 0,05 resp. a = 0,01. In the case of the DUNNETT‐Test this maximum difference amounts to 0,02 for 20 &lt; k &lt; 50 and a = 0,05. Besides it can be here supposed that the approximate percentage points are also practicable for both k &gt;50, a = 0,05 and k &gt;20, a = 0,01.

Further approximate pearson percentage points and Cornish-Fisher

A set of 19-point rational fraction approximants for Pearson distributions at levels a = 0.025, 0.25, 0.50, 0.75, and 0.975 is added to a previous study (Biometrika, 1979). The relation to a Cornish-Fisher expansion is discussed and advice proffered on interpolation for a new deviate or new level.

Approximate percentage points for Pearson distributions

Approximate formulae for a set of percentage points of the Pearson system are given in terms of the skewness and kurtosis.

Approximations to Density Functions Using Pearson Curves

Abstract This article is an expository paper to demonstrate the usefulness of Pearson curves in density estimation especially for those unaware of this early development in statistics. It is shown how to fit the curves and how very good approximate percentage points can be obtained for intractable distributions when the first four moments (or three moments and one endpoint) are known exactly (not estimated from sample data). The effectiveness of this method in density estimation is illustrated in three somewhat disparate contexts and reference is given to others. In general, the Pearson curves give an excellent approximation to the long tail of a distribution, the tail most often needed in practical work.

Approximations to Density Functions Using Pearson Curves

Abstract This article is an expository paper to demonstrate the usefulness of Pearson curves in density estimation especially for those unaware of this early development in statistics. It is shown how to fit the curves and how very good approximate percentage points can be obtained for intractable distributions when the first four moments (or three moments and one endpoint) are known exactly (not estimated from sample data). The effectiveness of this method in density estimation is illustrated in three somewhat disparate contexts and reference is given to others. In general, the Pearson curves give an excellent approximation to the long tail of a distribution, the tail most often needed in practical work.

A test for equality of means of two correlated normal variates with missing data on both responses

AbstractWe propose a test for the equality of means of two correlated normal variates based on a sample with missing values on both variates. We assume an unknown common variance σ2 and an unknown correlation coefficient ϱ. Our test statistic, U, is a generalization of the paired t‐statistic and of Morrison's (1973) statistic for the case of missing values on one variate only. A simulation study is conducted to determine approximate percentage points of U for selected sample sizes and ϱ = =0.25, 0.50, 0.75.

A Non-Parametric Extension of a Cluster Analysis Method by Scott and Knott

Scott and Knott (1974) have used cluster analysis methods to group means in the analysis of variance. We consider an analogous distribution-free method that has been suggested by Kass (1975) for Automatic Interaction Detection (A.I.D.). Tables of the exact null distribution and approximate percentage points are given for small numbers of observations

Testing for a Single Outlier from a General Linear Regression

Several authors have considered the problem of detection of outliers from the general linear model Y = Xbeta + mu. Ellenberg [1973] among others, has advocated use of a detection method which involves examination of the set of internally standardized least squares residuals. Mickey [1974] and Snedecor and Cochran [1968], apparently concerned about the usefulness of an outlier detection method which is based on residual estimates that themselves are biassed by the presence of the outlier, have proposed two other alternatives. It is shown that the three approaches are exactly equivalent. A detection procedure is described which uses as its test statistic the maximum of the internally standardized least squares residuals, and upper and lower bounds for the percentage points of the test statistic are given by Bonferroni inequalities. The computations required to obtain these approximate percentage points are illustrated in a numerical example. Finally, a brief simulation study of the performance of the procedure illustrates that the power of the test can be influenced by the position of the outlier vis-a-vis the structure of the design matrix X.

Normalizing transformations of Student's t distribution

SUMMARY The accuracies of several normalizing transformations of a t distribution with v degrees of freedom are examined for large and small values of v. Expansions of the inverse transformations in terms of powers of v-' are produced and the first few terms compared with Fisher's general expansion. For small values of v the transformations are used to determine approximate percentage points of t which are then compared with the exact percentage points. The problem of transforming a Student's t variable with v degrees of freedom to a standard normal variable has received attention for a variety of reasons. Quenouille (1953, p. 235) was concerned with the combination of results from a series of experiments and suggested analysing the treatment effects using an inverse sinh transformation. This transformation is related to Fisher's normalizing transformation of the product moment correlation coefficient and is fairly good provided v is not too small. Anscombe (1950) proposed a modified form of this transformation which is suitable for smaller values of v. Chu (1956) examined the proportional errors in using the normal cumulative distribution function as an approximation to the cumulative distribution function of t and considered a square root of a logarithmic transformation. Modified versions of this were proposed by Wallace (1959) who constructed bounds on the deviation from the exact normal deviates. Moran (1966) used an empirical approach to develop a simple transformation of a particular percentage point of the t distribution and showed that the approximation is very good even for quite small values of v. Scott & Smith (1970) used the general expansion of the percentage points of the t distribution in terms of the corresponding normal percentage points, given by Fisher (1926), to develop a simple transformation similar in form to Moran's but suitable for any percentage point of t. In the following sections the accuracies of these transformations are compared for a selection of percentage points for large and small values of v.

On the distribution of a statistic used for testing a multinormal mean vector with incomplete data

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 missing observations from that population is available. Simple approximations to the distribution of the test statistic are derived and are used to compute the approximate percentage points. The accuracy of the approximate percentage points has been assessed by comparing them with some exact percentage points. The comparison shows that the approximate percentage points are in good agreement with the exact percentage points. The approximations to the test statistic for testing the hypothesis of equality of two mean vectors of two multivariate normal distributions are also obtained

The distribution of the sphericity test criterion

The exact distribution of Mauchly's sphericity test criterion W = | S|/[tr S/p] p, when S is the sum of product matrix from a sample of size N taken from a p-variate normal population, is obtained using contour integration and methods similar to those of Nair and Box. Tables of percentage points for p = 4(1)10, α = 0.01 and 0.05, and various values of N (including small) are given and comparisons made with approximate percentage points using methods of Box, Mauchly, Tukey and Wilks, and Davis.