Small -sample comparison of the exact and symptotic upper tail probabilitiesof chi-squared goodness -of-fit statistics: pearson's x 2,likelihood ratio,and power-divergence statistic(λ = 2/3)

The exact and asymptotic upper tail probabilities ( α= .l0, .05, .01, .001) of the three chi-squared goodness-of-fit statistics Pearson's X ², likelihood ratioG ², and power-divergence statisticD ² (λ ) , with λ = 2/3, are compared numerically for simple null hypotheses not involving parameter estimation. Three types of such hypotheses were investigated (equal cell probabilities, proportional cell probabilities, some fixed small expectations together with some increasing large expectations) for the number of cells being between 3 and 15, and for sample sizes from 10 to 40, increasing by steps of one. Rating the relative accuracy of the chi-squared approximation in terms of ±10% and ±20% intervals around α led to the following conclusions: 1. Using G ² is not recommended. 2 . At the more relevant significance levels α = .10 and α = .05X ² should be preferred over D ². Solely in case of unequal cell probabilitiesD ² is the better choice at α = .O1 and α = .001. 3 . Yarnold's (1970; Journal of the Amerin Statistical Association, 65, 864-886) rule for the minimum expectation when using X ² ("If the number of cells k is 3 or more, and if r denotes the number of expectations less than 5, then the minimum expectation may be as small as 5r/k.") generalizes to D ²; it gives a good lower limit for the expected cell frequencies, however, when the number of cells is greater than 3. For k = 3 , even sample sizes over 15 may be insufficient.     

A Note on the Estimation of Binomial Probabilities

I am concerned with the admissibility under quadratic loss of certain estimators of binomial probabilities. The minimum variance unbiased estimator is shown to be admissible for Pr(X = 0) and Pr(X = n), but it is inadmissible for Pr(X = k), where 0 < k < n. An example is given of an admissible maximum likelihood estimator (MLE). It is conjectured that the MLE is always admissible.     

The exact unconditional z-pooled test for equality of two binomial probabilities: optimal choice of the Berger and Boos confidence coefficient

Exact unconditional tests for comparing two binomial probabilities are generally more powerful than conditional tests like Fisher's exact test. Their power can be further increased by the Berger and Boos confidence interval method, where a p-value is found by restricting the common binomial probability under H ₀ to a 1?γ confidence interval. We studied the average test power for the exact unconditional z-pooled test for a wide range of cases with balanced and unbalanced sample sizes, and significance levels 0.05 and 0.01. The detailed results are available online on the web. Among the values 10^?3, 10^?4, …, 10^?10, the value γ=10^?4 gave the highest power, or close to the highest power, in all the cases we looked at, and can be given as a general recommendation as an optimal γ.     

Accuracy of Power-Divergence Statistics for Testing Independence and Homogeneity in Two-Way Contingency Tables

The small-sample accuracy of seven members of the family of power-divergence statistics for testing independence or homogeneity in contingency tables was studied via simulation. The likelihood ratio statistic G ² and Pearson's X ² statistic are among these seven members, whose behavior was studied at nominal test sizes of.01 and.05 with marginal distributions that could be uniform or skewed and with a set of sample sizes that included sparseness conditions as measured through table density (i.e., the ratio of sample size to number of cells). The likelihood ratio statistic G ² rejected the null hypothesis too often even with large table density, whereas Pearson's X ² was sufficiently accurate and only presented a minor misbehavior when table density was less than two observations/cell. None of the other five statistics outperformed Pearson's X ². A nonasymptotic variant of X ² solved the minor inaccuracies of Pearson's X ² and turned out to be the most accurate statistic for testing independence or homogeneity, even with table densities of one observation/cell. These results clearly advise against the use of the likelihood ratio statistic G ².     

Asymptotics for multisample statistics

Consider a family of square-integrable R^d-valued statistics S_k = S_k(X_1,k1; X_2,k2;…; X_m,km), where the independent samples X_i,kj respectively have k_i i.i.d. components valued in some separable metric space X_i. We prove a strong law of large numbers, a central limit theorem and a law of the iterated logarithm for the sequence {S_k}, including both the situations where the sample sizes tend to infinity while m is fixed and those where the sample sizes remain small while m tends to infinity. We also obtain two almost sure convergence results in both these contexts, under the additional assumption that S_k is symmetric in the coordinates of each sample X_i,kj. Some extensions to row-exchangeable and conditionally independent observations are provided. Applications to an estimator of the dimension of a data set and to the Henze-Schilling test statistic for equality of two densities are also presented.     

The Likelihood Ratio Test with the Box–Cox Transformation for the Normal Mixture Problem: Power and Sample Size Study

Abstract

Through simulation and regression, we study the alternative distribution of the likelihood ratio test in which the null hypothesis postulates that the data are from a normal distribution after a restricted Box–Cox transformation and the alternative hypothesis postulates that they are from a mixture of two normals after a restricted (possibly different) Box–Cox transformation. The number of observations in the sample is called N. The standardized distance between components (after transformation) is D = (μ₂ ? μ₁)/σ, where μ₁ and μ₂ are the component means and σ² is their common variance. One component contains the fraction π of observed, and the other 1 ? π. The simulation results demonstrate a dependence of power on the mixing proportion, with power decreasing as the mixing proportion differs from 0.5. The alternative distribution appears to be a non-central chi-squared with approximately 2.48 + 10N ^?0.75 degrees of freedom and non-centrality parameter 0.174N(D ? 1.4)² × [π(1 ? π)]. At least 900 observations are needed to have power 95% for a 5% test when D = 2. For fixed values of D, power, and significance level, substantially more observations are necessary when π ≥ 0.90 or π ≤ 0.10. We give the estimated powers for the alternatives studied and a table of sample sizes needed for 50%, 80%, 90%, and 95% power.     

On moments of the supremum of normed weighted averages

Let {X, X_n; n ≥ 1} be a sequence of real-valued iid random variables, 0 < r < 2 and p > 0. Let D = { A = (a_nk; 1 ≤ k ≤ n, n ≥ 1); a_nk, ? R and sup_{n, k} |a_n,k| < ∞}. Set S_n( A ) = ∑ⁿ_k=1a_{n, k}X_k for A ? D and n ≥ 1. This paper is devoted to determining conditions whereby E{sup_{n ≥ 1}, |S_n( A )|/n^1/r}^p < ∞ or E{sup_{n ≥ 2} |S_n( A )|/2n log n)^1/2}^p < ∞ for every A ? D. This generalizes some earlier results, including those of Burkholder (1962), Choi and Sung (1987), Davis (1971), Gut (1979), Klass (1974), Siegmund (1969) and Teicher (1971).     

Testing Serial Correlation in Partially Linear Single-Index Errors-in-Variables Models

In this article, we propose two test statistics for testing the underlying serial correlation in a partially linear single-index model Y = η(Z ^τα) + X ^τβ + ? when X is measured with additive error. The proposed test statistics are shown to have asymptotic normal or chi-squared distributions under the null hypothesis of no serial correlation. Monte Carlo experiments are also conducted to illustrate the finite sample performance of the proposed test statistics. The simulation results confirm that these statistics perform satisfactorily in both estimated sizes and powers.     

Small-sample comparisons for powerdivergence goodness-of-fit statistics for symmetric and skewed simple null hypotheses

Power-divergence goodness-of-fit statistics have asymptotically a chi-squared distribution. Asymptotic results may not apply in small-sample situations, and the exact significance of a goodness-of-fit statistic may potentially be over- or under-stated by the asymptotic distribution. Several correction terms have been proposed to improve the accuracy of the asymptotic distribution, but their performance has only been studied for the equiprobable case. We extend that research to skewed hypotheses. Results are presented for one-way multinomials involving k = 2 to 6 cells with sample sizes N = 20, 40, 60, 80 and 100 and nominal test sizes f = 0.1, 0.05, 0.01 and 0.001. Six power-divergence goodness-of-fit statistics were investigated, and five correction terms were included in the study. Our results show that skewness itself does not affect the accuracy of the asymptotic approximation, which depends only on the magnitude of the smallest expected frequency (whether this comes from a small sample with the equiprobable hypothesis or a large sample with a skewed hypothesis). Throughout the conditions of the study, the accuracy of the asymptotic distribution seems to be optimal for Pearson's X² statistic (the power-divergence statistic of index u = 1) when k > 3 and the smallest expected frequency is as low as between 0.1 and 1.5 (depending on the particular k, N and nominal test size), but a computationally inexpensive improvement can be obtained in these cases by using a moment-corrected h² distribution. If the smallest expected frequency is even smaller, a normal correction yields accurate tests through the log-likelihood-ratio statistic G² (the power-divergence statistic of index u = 0).     

Expected Absolute Error of the Usual Estimator of the Binomial Parameter

For X with binomial (n, p) distribution the usual measure of the error of X/n as an estimator of p is its standard error S_n(p) = √{E(X/n – p)²} = √{p(1 – p)/n}. A somewhat more natural measure is the average absolute error D_n(p) = E‖X/n – p‖. This article considers use of D_n(p) instead of S_n(p) in a student's first introduction to statistical estimation. Exact and asymptotic values of D_n(p), and the appearance of its graph, are described in detail. The same is done for the Poisson distribution.     

Charting small sample characteristics of asymptotic confidence intervals for the binomial parameterp

Small sample properties of seven confidence intervals for the binomial parameterp (based on various normal approximations) and of the Clopper-Pearson interval are compared. Coverage probabilities and expected lower and upper limits of the intervals are graphically displayed as functions of the binomial parameterp for various sample sizes.     

Comment on J. Berkson's paper “in dispraise of the exact test”

This paper rejects the preference expressed by Berkson for the heuristic test statistic T_N with standard normal distribution testing equality of two binomial probabilities in favour of Fisher's conditional exact test statistic T_E. Conditioning upon k₁ + k₂ = k shows that T_N admits too large first kind error probabilities. But also unconditionally T_N is systematically too large compared to T_E, giving too small critical levels at given experimental outcomes and power which is misleadingly too large. This is mainly due to the fact that T_N is not suitably corrected for continuity (at small sample sizes).     

Asymptotic tail behavior of a random sum with conditionally dependent subexponential summands

In this paper, we obtain some results for the asymptotic behavior of the tail probability of a random sum S_τ = ∑^τ_{k = 1}X_k, where the summands X_k, k = 1, 2, …, are conditionally dependent random variables with a common subexponential distribution F, and the random number τ is a non negative integer-valued random variable, independent of {X_k: k ? 1}.     

Concomitants of multivariate order statistics from multivariate elliptical distributions

ABSTRACT

In this article, we consider a (k + 1)n-dimensional elliptically contoured random vector (X^T₁, X₂^T, …, X^T_k, Z^T)^T = (X₁₁, …, X_1n, …, X_k1, …, X_kn, Z₁, …, Z_n)^T and derive the distribution of concomitant of multivariate order statistics arising from X₁, X₂, …, X_k. Specially, we derive a mixture representation for concomitant of bivariate order statistics. The joint distribution of the concomitant of bivariate order statistics is also obtained. Finally, the usefulness of our result is illustrated by a real-life data.     

Discretisation for inference on normal mixture models

The problem of inference in Bayesian Normal mixture models is known to be difficult. In particular, direct Bayesian inference (via quadrature) suffers from a combinatorial explosion in having to consider every possible partition of n observations into k mixture components, resulting in a computation time which is O(k ⁿ). This paper explores the use of discretised parameters and shows that for equal-variance mixture models, direct computation time can be reduced to O(D ^k n ^k), where relevant continuous parameters are each divided into D regions. As a consequence, direct inference is now possible on genuine data sets for small k, where the quality of approximation is determined by the level of discretisation. For large problems, where the computational complexity is still too great in O(D ^k n ^k) time, discretisation can provide a convergence diagnostic for a Markov chain Monte Carlo analysis.     

Precedence tests and Lehmann alternatives

G = F ^k (k > 1); G = 1 − (1−F) ^k (k < 1); G = F ^k (k < 1); and G = 1 − (1−F) ^k (k > 1), where F and G are two continuous cumulative distribution functions. If an optimal precedence test (one with the maximal power) is determined for one of these four classes, the optimal tests for the other classes of alternatives can be derived. Application of this is given using the results of Lin and Sukhatme (1992) who derived the best precedence test for testing the null hypothesis that the lifetimes of two types of items on test have the same distibution. The test has maximum power for fixed κ in the class of alternatives G = 1 − (1−F) ^k , with k < 1. Best precedence tests for the other three classes of Lehmann-type alternatives are derived using their results. Finally, a comparison of precedence tests with Wilcoxon's two-sample test is presented. Received: February 22, 1999; revised version: June 7, 2000     

Characterization of the mixtures of Rayleigh distributions by conditional expectation of order statistics

The mixture of Rayleigh random variables X ₁and X ₂ are identified in terms of relations between the conditional expectation of ( X_2:2² -X_1:2²)^r{\left( {X_{2:2}^2 -X_{1:2}^2}\right)^{r}} given X _1:2 (or X_2:2^2k{X_{2:2}^{2k}} given X_1:2,"k ￡ r){X_{1:2},\forall k\leq r)} and hazard rate function of the distribution, where X _1:2 and X _2:2 denote the corresponding order statistics, r is a positive integer. In addition, we also mention some related theorems to characterize the mixtures of Rayleigh distributions. Finally, we also give an application to Multi-Hit models of carcinogenesis (Parallel Systems) and a simulated example is used to illustrate our results.     

AN ASYMPTOTIC EXPANSION OF THE EXPECTATION OF THE ESTIMATED ERROR RATE IN DISCRIMINANT ANALYSIS1

When a sample discriminant function is computed, it is desired to estimate the error rate using this function. This is often done by computing G(-D/2), where G is the cumulative normal distribution and D² is the estimated Mahalanobis' distance. In this paper an asymptotic expansion of the expectation of G(-D/2) is derived and is compared with existing Monte Carlo estimates. The asymptotic bias of G(-D/2) is derived also and the well-known practical result that G(-D/2) gives too favourable an estimate of the true error rate     

A Comparison of Large-Sample Confidence Interval Methods for the Difference of Two Binomial Probabilities

A simulation study was done to compare seven confidence interval methods, based on the normal approximation, for the difference of two binomial probabilities. Cases considered included minimum expected cell sizes ranging from 2 to 15 and smallest group sizes (NMIN) ranging from 6 to 100. Our recommendation is to use a continuity correction of 1/(2 NMIN) combined with the use of (N ? 1) rather than N in the estimate of the standard error. For all of the cases considered with minimum expected cell size of at least 3, this method gave coverage probabilities close to or greater than the nominal 90% and 95%. The Yates method is also acceptable, but it is slightly more conservative. At the other extreme, the usual method (with no continuity correction) does not provide adequate coverage even at the larger sample sizes. For the 99% intervals, our recommended method and the Yates correction performed equally well and are reasonable for minimum expected cell sizes of at least 5. None of the methods performed consistently well for a minimum expected cell size of 2.     

Testing stability in a spatial unilateral autoregressive model

ABSTRACT

Least squares estimator of the stability parameter ? ? |α| + |β| for a spatial unilateral autoregressive process X_{k, ?} = αX_{k ? 1, ?} + βX_{k, ? ? 1} + ?_{k, ?} is investigated and asymptotic normality with a scaling factor n^5/4 is shown in the unstable case ? = 1. The result is in contrast to the unit root case of the AR(p) model X_k = α₁X_{k ? 1} + ??? + α_pX_{k ? p} + ?_k, where the limiting distribution of the least squares estimator of the unit root parameter ? ? α₁ + ??? + α_p is not normal.