ADMISSIBLE, SIMILAR TESTS: A CHARACTERIZATION

2019 ◽  
Vol 36 (2) ◽  
pp. 347-366 ◽  
Author(s):  
José Luis Montiel Olea
Keyword(s):  
Weight Function ◽  
Instrumental Variables ◽  
Structural Parameters ◽  
Average Power ◽  
Nuisance Parameter ◽  
Weighted Average ◽  
Necessary Condition ◽  
Ratio Test ◽  
Finite Sample ◽  
Characterization Result

This article studies a classical problem in statistical decision theory: a hypothesis test of a sharp null in the presence of a nuisance parameter. The main contribution of this article is a characterization of two finite-sample properties often deemed reasonable in this environment: admissibility and similarity. Admissibility means that a test cannot be improved uniformly over the parameter space. Similarity requires the null rejection probability to be unaffected by the nuisance parameter.The characterization result has two parts. The first part—established by Chernozhukov, Hansen, and Jansson (2009, Econometric Theory 25, 806–818)—states that maximizing weighted average power (WAP) subject to a similarity constraint suffices to generate admissible, similar tests. The second part—hereby established—states that constrained WAP maximization is (essentially) a necessary condition for a test to be admissible and similar. The characterization result shows that choosing an admissible, similar test is tantamount to selecting a particular weight function to report weighted average power. This result applies to full vector inference with a nuisance parameter, not to subvector inference.The article also revisits the theory of testing in the instrumental variables model. Specifically—and in light of the relevance of the weighted average power criterion in the main theoretical result—the article suggests a weight function for the structural parameters of the homoskedastic instrumental variables model, based on the priors proposed by Chamberlain (2007). The corresponding test is, by construction, admissible and similar. In addition, the test is shown to have finite- and large-sample properties comparable to those of the conditional likelihood ratio test.

scholarly journals ADMISSIBLE INVARIANT SIMILAR TESTS FOR INSTRUMENTAL VARIABLES REGRESSION

2009 ◽  
Vol 25 (3) ◽  
pp. 806-818 ◽  
Author(s):  
Victor Chernozhukov ◽  
Christian Hansen ◽  
Michael Jansson
Keyword(s):  
Likelihood Ratio ◽  
Instrumental Variables ◽  
Degrees Of Freedom ◽  
Average Power ◽  
Weighted Average ◽  
Likelihood Ratio Tests ◽  
Weak Instruments ◽  
Working Paper ◽  
Statistical Sense ◽  
Limiting Case

This paper studies a model widely used in the weak instruments literature and establishes admissibility of the weighted average power likelihood ratio tests recently derived by Andrews, Moreira, and Stock (2004, NBER Technical Working Paper 199). The class of tests covered by this admissibility result contains the Anderson and Rubin (1949, Annals of Mathematical Statistics 20, 46–63) test. Thus, there is no conventional statistical sense in which the Anderson and Rubin (1949) test “wastes degrees of freedom.” In addition, it is shown that the test proposed by Moreira (2003, Econometrica 71, 1027–1048) belongs to the closure of (i.e., can be interpreted as a limiting case of) the class of tests covered by our admissibility result.

scholarly journals SUP-TESTS FOR LINEARITY IN A GENERAL NONLINEAR AR(1) MODEL

2009 ◽  
Vol 26 (4) ◽  
pp. 965-993 ◽  
Author(s):  
Christian Francq ◽  
Lajos Horvath ◽  
Jean-Michel Zakoïan
Keyword(s):  
Time Series ◽  
Asymptotic Distribution ◽  
Null Hypothesis ◽  
Nuisance Parameter ◽  
Time Series Models ◽  
Ratio Test ◽  
Finite Sample ◽  
Chi Square ◽  
Finite Sample Properties ◽  
Linearity Testing

We consider linearity testing in a general class of nonlinear time series models of order one, involving a nonnegative nuisance parameter that (a) is not identified under the null hypothesis and (b) gives the linear model when equal to zero. This paper studies the asymptotic distribution of the likelihood ratio test and asymptotically equivalent supremum tests. The asymptotic distribution is described as a functional of chi-square processes and is obtained without imposing a positive lower bound for the nuisance parameter. The finite-sample properties of the sup-tests are studied by simulations.

scholarly journals Monitoring Persistence Change in Heavy-Tailed Observations

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 936
Author(s):  
Dan Wang
Keyword(s):  
Kernel Method ◽  
Alternative Hypothesis ◽  
Null Distribution ◽  
Real Data ◽  
Ratio Test ◽  
Finite Sample ◽  
Test Statistic ◽  
Bootstrap Approximation ◽  
Heavy Tailed ◽  
Better Than

In this paper, a ratio test based on bootstrap approximation is proposed to detect the persistence change in heavy-tailed observations. This paper focuses on the symmetry testing problems of I(1)-to-I(0) and I(0)-to-I(1). On the basis of residual CUSUM, the test statistic is constructed in a ratio form. I prove the null distribution of the test statistic. The consistency under alternative hypothesis is also discussed. However, the null distribution of the test statistic contains an unknown tail index. To address this challenge, I present a bootstrap approximation method for determining the rejection region of this test. Simulation studies of artificial data are conducted to assess the finite sample performance, which shows that our method is better than the kernel method in all listed cases. The analysis of real data also demonstrates the excellent performance of this method.

A weighted average likelihood ratio test for spatial clustering of disease

2001 ◽  
Vol 20 (19) ◽  
pp. 2977-2987 ◽  
Author(s):  
Ronald E. Gangnon ◽  
Murray K. Clayton
Keyword(s):  
Likelihood Ratio ◽  
Likelihood Ratio Test ◽  
Spatial Clustering ◽  
Weighted Average ◽  
Ratio Test

scholarly journals Universal inference

2020 ◽  
Vol 117 (29) ◽  
pp. 16880-16890 ◽  
Author(s):  
Larry Wasserman ◽  
Aaditya Ramdas ◽  
Sivaraman Balakrishnan
Keyword(s):  
Maximum Likelihood ◽  
Likelihood Ratio ◽  
Likelihood Ratio Statistic ◽  
Model Misspecification ◽  
Null Distribution ◽  
Regularity Conditions ◽  
Ratio Test ◽  
Finite Sample ◽  
Confidence Sets ◽  
Nonparametric Models

We propose a general method for constructing confidence sets and hypothesis tests that have finite-sample guarantees without regularity conditions. We refer to such procedures as “universal.” The method is very simple and is based on a modified version of the usual likelihood-ratio statistic that we call “the split likelihood-ratio test” (split LRT) statistic. The (limiting) null distribution of the classical likelihood-ratio statistic is often intractable when used to test composite null hypotheses in irregular statistical models. Our method is especially appealing for statistical inference in these complex setups. The method we suggest works for any parametric model and also for some nonparametric models, as long as computing a maximum-likelihood estimator (MLE) is feasible under the null. Canonical examples arise in mixture modeling and shape-constrained inference, for which constructing tests and confidence sets has been notoriously difficult. We also develop various extensions of our basic methods. We show that in settings when computing the MLE is hard, for the purpose of constructing valid tests and intervals, it is sufficient to upper bound the maximum likelihood. We investigate some conditions under which our methods yield valid inferences under model misspecification. Further, the split LRT can be used with profile likelihoods to deal with nuisance parameters, and it can also be run sequentially to yield anytime-valid P values and confidence sequences. Finally, when combined with the method of sieves, it can be used to perform model selection with nested model classes.

scholarly journals Thermal Drift Investigation of an SOI-Based MEMS Capacitive Sensor with an Asymmetric Structure

Sensors ◽  
2019 ◽  
Vol 19 (16) ◽  
pp. 3522 ◽  
Author(s):  
Haiwang Li ◽  
Yanxin Zhai ◽  
Zhi Tao ◽  
Yingxuan Gui ◽  
Xiao Tan
Keyword(s):  
Structural Parameters ◽  
Weighted Average ◽  
Capacitive Sensor ◽  
Expansion Ratio ◽  
Asymmetric Structure ◽  
Temperature Sensitive ◽  
Navigation Systems ◽  
Temperature Drift ◽  
Capacitive Accelerometer ◽  
Symmetric Structure

High-precision, low-temperature-sensitive microelectromechanical system (MEMS) capacitive accelerometers are widely used in aerospace, automotive, and navigation systems. An analytical study of the temperature drift of bias (TDB) and temperature drift of scale factor (TDSF) for an asymmetric comb capacitive accelerometer is presented in this paper. A five-layer model is established for the equivalent expansion ratio in the TDB and TDSF formulas, and the results calculated by the weighted average of thickness and elasticity modulus method are closest to the results of the numerical simulation. The analytical formulas of TDB and TDSF for an asymmetric structure are obtained. For an asymmetric structure, TDB is only related to thermal deformation and fabrication error. Additionally, half of the fixed electrode distance is not included in the expressions of Δ d and Δ D for asymmetric structures, thus resulting in the TDSF of the asymmetric structure being smaller compared to a symmetric structure with the same structural parameters. The TDSF of the symmetric structure is [−200.2 ppm/°C, −261.6 ppm/°C], while the results of the asymmetric structure are [−11.004 ppm/°C, −72.404 ppm/°C] under the same set of parameters. The parameters of the optimal asymmetric structure are obtained for fabrication guidance using numerical methods. In the experiment, the TDSF and TDB of the packaged structure and the non-packaged structure are compared, and a significant effect of the package on the signal output is found. The TDB is reduced from 3000 to 60 μg/°C, while the TDSF is reduced from 3000 to 140 ppm/°C.

scholarly journals Recursive Adjustment for General Deterministic Components and Improved Cointegration Rank Tests

2015 ◽  
Vol 7 (2) ◽  
Author(s):  
Benjamin Born ◽  
Matei Demetrescu
Keyword(s):  
Likelihood Ratio ◽  
Linear Trend ◽  
Generalized Least Squares ◽  
Ratio Test ◽  
Finite Sample ◽  
Vector Autoregressions ◽  
Rank Tests ◽  
Testing Procedures ◽  
Stochastic Component ◽  
Deterministic Components

AbstractThis paper discusses tests for the cointegration rank of integrated vector autoregressions when the series are recursively adjusted for deterministic components. To this end, the asymptotic properties of recursive, or adaptive, procedures for the removal of general additive deterministic components are analyzed in two different, complementary, situations. When the stochastic component of the examined time series is weakly stationary (as would be the equilibrium errors), the effect of recursive adjustment vanishes with increasing sample size. When the suitably normalized stochastic component converges weakly to some limiting continuous-time process with integrable paths (as would be the case with the common stochastic trends), recursive adjustment has a permanent effect even asymptotically: the normalized recursively adjusted process converges weakly to a recursively adjusted version of the limiting process. The null limiting distributions of the cointegration rank tests can be expressed in terms of recursively adjusted Brownian motions. Moreover, the finite-sample properties of the cointegration rank tests with recursive adjustment are examined in cases of empirical relevance: the considered deterministic components are a constant, and a constant and a linear trend, respectively. Compared to the likelihood ratio tests or the tests with generalized least squares adjustment, improvements in terms of empirical rejection frequencies under the null are found in finite samples; improvements are found under the alternative as well, with the likelihood ratio test performing increasingly better as the magnitude of the initial condition increases. Regarding rank selection, a very simple combination of the three testing procedures with different adjustments performs best.

Conditional likelihood and power: higher-order asymptotics

1992 ◽  
Vol 438 (1904) ◽  
pp. 433-446 ◽  
Keyword(s):  
Average Power ◽  
Nuisance Parameter ◽  
Second Order ◽  
Conditional Likelihood ◽  
Power Functions ◽  
Third Order ◽  
Contiguous Alternatives ◽  
Higher Order Asymptotics ◽  
Optimality Property ◽  
Lr Test

This paper considers, in the presence of a nuisance parameter, a very large class of tests that includes the conditional and the usual versions of the likelihood ratio (LR), Rao’s and Wald’s tests. Under contiguous alternatives and orthogonal parametrization, the power functions of the conditional and the usual versions of these tests have been compared and, in particular, it is seen that the power functions of the conditional versions, unlike those of the usual versions, are identical, up to the second-order, with the power functions of the corresponding tests with known nuisance parameter. An optimality property of the conditional LR test, in terms of second-order local maximinity, has been established. A test, optimal in the sense of third-order average power under contiguous alternatives, has been proposed. A weaker optimality property of Rao’s test, in terms of third-order average power, has also been indicated.

scholarly journals A POWERFUL TEST OF THE AUTOREGRESSIVE UNIT ROOT HYPOTHESIS BASED ON A TUNING PARAMETER FREE STATISTIC

2009 ◽  
Vol 25 (6) ◽  
pp. 1515-1544 ◽  
Author(s):  
Morten Ørregaard Nielsen
Keyword(s):  
Asymptotic Distribution ◽  
Unit Root ◽  
Linear Time ◽  
Nonparametric Test ◽  
Tuning Parameter ◽  
Ratio Test ◽  
Local Power ◽  
Finite Sample ◽  
Power Of The Test ◽  
The Family

This paper presents a family of simple nonparametric unit root tests indexed by one parameter,d, and containing the Breitung (2002,Journal of Econometrics108, 342–363) test as the special cased= 1. It is shown that (a) each member of the family withd> 0 is consistent, (b) the asymptotic distribution depends ondand thus reflects the parameter chosen to implement the test, and (c) because the asymptotic distribution depends ondand the test remains consistent for alld> 0, it is possible to analyze the power of the test for different values ofd. The usual Phillips–Perron and Dickey–Fuller type tests are indexed by bandwidth, lag length, etc., but have none of these three properties.It is shown that members of the family withd< 1 have higher asymptotic local power than the Breitung (2002) test, and whendis small the asymptotic local power of the proposed nonparametric test is relatively close to the parametric power envelope, particularly in the case with a linear time trend. Furthermore, generalized least squares (GLS) detrending is shown to improve power whendis small, which is not the case for the Breitung (2002) test. Simulations demonstrate that when applying a sieve bootstrap procedure, the proposed variance ratio test has very good size properties, with finite-sample power that is higher than that of the Breitung (2002) test and even rivals the (nearly) optimal parametric GLS detrended augmented Dickey–Fuller test with lag length chosen by an information criterion.

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