Nonlocal Alternatives Research Articles

Finite Sample Lag Adjusted Critical Values of the ADF-GLS Test

Ng and Perron (Econometrica 69:1519–1554, 2001) demonstrated the merits to employing their Modified Akaike Information Criterion to select the optimal lag length in the Elliott, Rothenberg and Stock (Econometrica 64:813–836, 1996) unit rot test. Perron and Qu (Econ Lett 84:12–19, 2007) introduced an empirical method that resolved an associated power problem for non-local alternatives. While Cheung and Lai (Oxford Bull Econ Stat 57:411-419, 1995) contains response surface estimates to generate finite-sample, lag-adjusted critical five and ten percent values for use in applied work, these relate to the original Elliott et al. (Econometrica 64:813–836, 1996) test. This paper provides response surfaces estimates of critical values for both the Ng and Perron (Econometrica 69:1519–1554, 2001) and Perron and Qu (Econ Lett 84:12–19, 2007) approaches, demonstrating they are sometimes quite different, an important consideration when performing inference.

Optimal comparison of misspecified moment restriction models under a chosen measure of fit

Suppose that the econometrician is interested in comparing two misspecified moment restriction models, where the comparison is performed in terms of some chosen measure of fit. This paper is concerned with describing an optimal test of the Vuong (1989) and Rivers and Vuong (2002) type null hypothesis that the two models are equivalent under the given measure of fit (the ranking may vary for different measures). We adopt the generalized Neyman–Pearson optimality criterion, which focuses on the decay rates of the type I and II error probabilities under fixed non-local alternatives, and derive an optimal but practically infeasible test. Then, as an illustration, by considering the model comparison hypothesis defined by the weighted Euclidean norm of moment restrictions, we propose a feasible approximate test statistic to the optimal one and study its asymptotic properties. Local power properties, one-sided test, and comparison under the generalized empirical likelihood-based measure of fit are also investigated. A simulation study illustrates that our approximate test is more powerful than the Rivers–Vuong test.

Improved Approximations for the Distributions of Multinomial Goodness-of-fit Statistics Based on φ-divergence under Nonlocal Alternatives

Zografos et al. (1990) introduced the φ-divergence family of statistics Cφ to the goodness-of-fit test. The φ-divergence family of statistics Cφ includes the power divergence family of statistics proposed by Cressie and Read (Cressie and Read (1984) and Read and Cressie (1988)) as a special case. Sekiya and Taneichi (2004) derived the multivariate Edgeworth expansion assuming a continuous distribution for the distributions of power divergence statistics under a nonlocal alternative hypothesis. In this paper, we consider an expansion for the family of general φ-divergence statistics Cφ. We derive the multivariate Edgeworth expansion assuming a continuous distribution for the distribution of Cφ under a nonlocal alternative hypothesis. By using the expansion, we propose a new approximation for the power of the statistic Cφ. We numerically investigate the accuracy of the approximation when two types of concrete φ-divergence statistics are applied. By the numerical investigation, we show that the present approximation is a good approximation especially when alternative hypotheses are distant from the null hypothesis.

Bayesian Design of Single-arm Phase II Clinical Trials with Continuous Monitoring

Bayesian designs are increasingly used to conduct phase II clinical trials. However, stopping boundaries in most Bayesian designs are defined from posterior credible intervals. The use of designs based on posterior credible intervals results in a loss of efficiency when compared to formal stopping rules based on Bayesian hypothesis tests. Such designs also introduce an unnecessary element of subjectivity in the interpretation of trial results. We derive a new class of Bayesian designs based on formal hypothesis tests. The prior densities used to define the alternative hypotheses in these tests assign no mass to parameter values that are consistent with the null hypotheses and are called nonlocal alternative prior densities. We show that Bayesian designs based on hypothesis tests and nonlocal alternative prior densities are more efficient than common Bayesian designs based on posterior credible intervals and common frequentist designs. In contrast to trial designs based on Bayesian credible intervals, we demonstrate that the mis-specification of the prior densities used to describe the anticipated effect of the experimental treatment in designs based on hypothesis tests cannot increase the expected weight of evidence in favor of the trial agent. Extension of test-based designs to phase I-II designs and randomized phase II designs remains an open research question. Phase II single-arm trials designed using Bayesian hypothesis tests with nonlocal alternatives provide better operating characteristics, use fewer patients per correct decision, and provide more directly interpretable results than other commonly used Bayesian and frequentist designs. Because the mis-specification of the prior density on the effect of the experimental agent decreases the expected weight of evidence that is collected in favor of the experimental treatment, the use of Bayesian hypothesis tests to design clinical trials also eliminates a potential source of bias often associated with trials designed using posterior credible intervals.

A simple modification to improve the finite sample properties of Ng and Perron's unit root tests

The tests introduced by Ng and Perron [Ng, S., Perron, P., 20011. “Lag Length Selection and the Construction of Unit Root Tests with Good Size and Power,” Econometrica 69, 1519–1554] have the drawback that for non-local alternatives the power can be very small. The aim of this note is to point out an easy solution to this power reversal problem, which in addition leads to tests having an exact size even closer, in most cases, to nominal size. It involves using OLS instead of GLS detrended data when constructing the modified information criterion.

Evaluation of sample size and power for multi-arm survival trials allowing for non-uniform accrual, non-proportional hazards, loss to follow-up and cross-over

We present a general framework for sample size calculation in survival studies based on comparing two or more survival distributions using any one of a class of tests including the logrank test. Incorporated within this framework are the possible presence of non-uniform staggered patient entry, non-proportional hazards, loss to follow-up and treatment changes including cross-over between treatment arms. The framework is very general in nature and is based on using piecewise exponential distributions to model the survival distributions. We illustrate the use of the approach and explore its validity using simulation studies. These studies have shown that not adjusting for loss to follow-up, non-proportional hazards or cross-over can lead to significant alterations in power or equivalently, a marked effect on sample size. The approach has been implemented in the freely available program ART (for Stata). Our investigations suggest that ART is the first software to allow incorporation of all these elements. Further extensions to the methodology such as non-local alternatives for the logrank test are also considered.

Improvement of approximations for the distributions of multinomial goodness-of-fit statistics under nonlocal alternatives

Cressie and Read (J. Roy. Statist. Soc. B 46 (1984) 440–464) introduced the power divergence statistics, R a , as multinomial goodness-of-fit statistics. Each R a has a limiting noncentral chi-square distribution under a local alternative and has a limiting normal distribution under a nonlocal alternative. Taneichi et al. (J. Multivariate Anal. 81 (2002) 335–359) derived an asymptotic approximation for the distribution of R a under local alternatives. In this paper, using multivariate Edgeworth expansion for a continuous distribution, we show how the approximation based on the limiting normal distribution of R a under nonlocal alternatives can be improved. We apply the expansion to the power approximation for R a . The results of numerical investigation show that the proposed power approximation is very effective for the likelihood ratio test.

A comparison of nonnested tests for misspecified models using the method of approximate slopes

Abstract Comparisons of the Cox, Wald and J tests are conducted under a standard framework and in a misspecified setting - the data generation process includes a regressor that is not in either model. Most power comparisons have been carried out for ‘local alternatives’ but comparisons under a ‘nonlocal alternatives’ framework using the method of approximate slopes developed by Bahadur (1960, 1967) are made in this paper. Monte Carlo experiments are conducted and the Cox test is shown to have the highest small sample power and approximate slope for most of the cases considered. The results from response surface regressions provide strong evidence that the approximate slope is a good predictor of the small power of nonnested tests. It is also useful for directly comparing the small sample power of one degree of freedom tests of nonnested hypotheses.

The Danger of Extrapolating Asymptotic Local Power

IN NONLINEAR MODELS the power function is often approximated by asymptotic methods. The most common approach is to consider the asymptotic local power function. The local power function is monotonic and it has essentially the same shape as the power function in the classical normal linear regression model. However, the accuracy of the approximation can be poor at nonlocal alternatives. This note examines the exact powers of the Wald test in the case of a one parameter nonlinear regression model with normal errors. The model is based on the exponential response function f( x, O) = exp( Ox). The results show that the exact power function of the Wald statistic can be nonmonotonic. For selected designs the exact powers of the Wald test first increase and then eventually decline as the distance between the hypothesized and the true values of the parameter increases. The exponential structure appears in many nonlinear models; see Gallant (1975, 1987) and Bates and Watts (1988). This suggests that nonmonotonicity of the Wald test is a feature of a wide class of nonlinear models. Indeed, Nelson and Savin (1988) show that it arises in standard logit, probit, and Tobit models as well. The focus here on the nonlinear regression model is for expository convenience. While the existence of nonmonotonic power is not new, the surprising results are that this phenomenon occurs in very simple nonlinear models and that it can be quite severe. In such cases the asymptotic local power approximation provides a very poor guide to the performance of alternative tests.

Distribution-free tests against serial dependence: signed or unsigned ranks?

Abstract Optimal rank-based procedures were derived in Hallin et al. (1985,1987) and Hallin and Puri (1988a) for a variety of testing problems arising in time-series analysis, when the underlying innovation densities remain unspecified. Signed-rank analogues have been proposed in Hallin and Puri (1988b) for the case of symmetric unspecified innovation densities. The ARE of unsigned rank-based tests with respect to their signed counterparts however has been shown to be one (same-reference). The objective of the present paper is an investigation of the finite-sample behavior of some asymptotically optimal signed-rank tests against first-order serial dependence, as well as a finite-sample comparative study of the actual performances, under nonlocal alternatives, of signed and unsigned ranks. Exact and approximate critical values are provided for various scores (van der Waerden, Wilcoxon, Laplace, Spearman) and various series lengths. The comparative study indicates that, notwithstanding the ARE values of one, signed-rank tests may yield substantially better performances than unsigned ones.

Supremum Versions of the Log-Rank and Generalized Wilcoxon Statistics

Abstract Considerable progress has been made in generalizing to censored data hypothesis tests based on rank statistics. The most commonly used generalizations are the log-rank test statistic, which extends the Savage exponential scores statistic (Cox 1972; Mantel 1966) and the generalized Wilcoxon statistics (Gehan 1965; Peto and Peto 1972). Gill (1980) showed that in the two-sample problem the asymptotic efficiency properties of the Savage and Wilcoxon statistics are maintained in censored data by the log-rank and Peto and Peto Wilcoxon statistics, respectively, when censoring is the same in both samples. Heuristics and simulations have implied that some recently proposed supremum statistics, closely related to these two popular linear rank statistics, can be used to construct tests that provide good power against a much wider range of nonlocal alternatives (see, e.g., Fleming and Harrington 1981; Fleming, O'Brien, O'Fallon, and Harrington 1980; Gill 1980; Schumacher 1984). In this article we consider n...

Choosing the number of controls in a matched case-control study, some sample size, power and efficiency considerations.

This paper investigates the efficiency of using multiple controls in a case-control study, when there is a single binary exposure variable. Specifically, we consider the asymptotic power of the Cochran test statistic against non-local alternatives of interest. When it is desirable to take multiple controls per case, we show that the marginal return rapidly diminishes as the number of controls per case increases. The effect is as strong, if not stronger, for non-local alternatives as it is for local alternatives. Hence, it is rarely worth choosing more than three controls per case. We also provide a table of sample sizes necessary to achieve 80 per cent power for some odds ratios not equal to one. We extend the results to a special case when there are two binary exposure variables.

A Note on the Small-Sample Power Functions for Nonparametric Tests of Location in the Double Exponential Family

Abstract Eight nonparametric tests are examined in a small-sample setting. Their power functions are compared with the locally most powerful (LMP) linear rank test, primarily to see over which range the LMP test retains good power. All tests are two-sample tests, the populations are governed by the double exponential distribution, and the sample sizes are m = 5, n = 5, and m = 5, n = 4. The LMP test performs uniformly well, even under nonlocal alternatives, while its asymptotically equivalent counterpart, the median test, performs rather poorly with these small sample sizes.