Moment Inequalities Research Articles

Complete convergence of the non-identically distributed pairwise NQD random sequences

ABSTRACTIn this article, we study complete convergence of the nonidentically distributed pairwise negatively quadrant dependent (NQD) random sequences by the moment inequality and terminating random variables,which extend and improve the previous relevant results.

Specification tests for partially identified models defined by moment inequalities

This paper studies the problem of specification testing in partially identified models defined by moment (in)equalities. This problem has not been directly addressed in the literature, although several papers have suggested a test based on checking whether confidence sets for the parameters of interest are empty or not, referred to as Test BP. We propose two new specification tests, denoted Test RS and Test RC, that achieve uniform asymptotic size control and dominate Test BP in terms of power in any finite sample and in the asymptotic limit.

COMPARISON OF INFERENTIAL METHODS IN PARTIALLY IDENTIFIED MODELS IN TERMS OF ERROR IN COVERAGE PROBABILITY

This paper considers the problem of coverage of the elements of the identified set in a class of partially identified econometric models with a prespecified probability. In order to conduct inference in partially identified econometric models defined by moment (in)equalities, the literature has proposed three methods: bootstrap, subsampling, and asymptotic approximation. The objective of this paper is to compare these methods in terms of the rate at which they achieve the desired coverage level, i.e., in terms of the rate at which theerror in the coverage probability(ECP) converges to zero.Under certain conditions, we show that the ECP of the bootstrap and the ECP of the asymptotic approximation converge to zero at the same rate, which is a faster rate than that of the ECP of subsampling methods. As a consequence, under these conditions, the bootstrap and the asymptotic approximation produce inference that is more precise than subsampling. A Monte Carlo simulation study confirms that these results are relevant in finite samples.

A Comparison of Inferential Methods in Partially Identified Models in Terms of Error in the Coverage Probability

This paper considers the problem of coverage of the elements of the identified set in a class of partially identified econometric models with a prespecified probability. In order to conduct inference in partially identified econometric models defined by moment (in)equalities, the literature has proposed three methods: the bootstrap, subsampling, and an asymptotic approximation. The objective of this paper is to compare these methods in terms of the rate at which they achieve the desired coverage level, i.e., in terms of the rate at which the error in the coverage probability (ECP) converges to zero.Under certain conditions, we show that the ECP of the bootstrap and the ECP of the asymptotic approximation converge to zero at the same rate, which is a faster rate than the rate of the ECP of subsampling methods. As a consequence, under these conditions, the bootstrap and the asymptotic approximation produce inference that is more precise than subsampling. A Monte Carlo simulation study confirms that these results are relevant in nite samples.

Complete qth moment convergence of weighted sums for arrays of rowwise negatively associated random variables

In this paper, the complete qth moment convergence of weighted sums for arrays of rowwise negatively associated (NA) random variables is investigated. By using moment inequality and truncation methods, some general results on complete qth moment convergence of weighted sums for arrays of rowwise NA random variables are obtained. As their applications, we not only generalize and extend the corresponding results of Baek et al. [On the complete convergence of weighted sums for arrays of negatively associated variables, J. Korean Stat. Soc. 37 (2008), pp. 73–80], Liang [Complete convergence for weighted sums of negatively associated random variables, Stat. Probab. Lett. 48 (2000), pp. 317–325 and Liang et al. [Complete moment convergence for sums of negatively associated random variables, Acta Math. Sin. English Ser. 26 (2010), pp. 419–432], but also greatly simplify their proofs.

Inequalities for positive rank and crank moments of overpartitions

In recent work, Andrews, Chan, and Kim extend a result of Garvan about even rank and crank moments of partitions to positive moments. In a similar fashion we extend a result of Mao about even rank moments of overpartitions. We investigate positive Dyson-rank, M2-rank, first residual crank, and second residual crank moments of overpartitions. In particular, we prove a conjecture of Mao which states that the positive Dyson-rank moments are larger than the positive M2-rank moments. We also prove some additional inequalities involving rank and crank moments of overpartitions, including an interlacing property.

A Simple Two-Step Method for Testing Moment Inequalities with an Application to Inference in Partially Identified Models

This paper considers the problem of testing a finite number of moment inequalities. We propose a two-step approach. In the first step, a confidence region for the moments is constructed. In the second step, this set is used to provide information about which moments are “negative.” A Bonferonni-type correction is used to account for the fact that with some probability the moments may not lie in the confidence region. It is shown that the test controls size uniformly over a large class of distributions for the observed data. An important feature of the proposal is that it remains computationally feasible, even when the number of moments is large. The finite-sample properties of the procedure are examined via a simulation study, which demonstrates, among other things, that the proposal remains competitive with existing procedures while being computationally more attractive.

The Weak Convergence of Random Sums of<i> ρ<sup>-</sup></i>-Mixing Random Sequences

Let {Xnk,n≥1,k≥1} be a ρ--mixing random sequence, by using the moment inequality and truncation method, we studied the weak convergence of random sums for ρ--mixing random sequence, which extended and improved some results in related literature.

ON THE MOMENTS OF STOCHASTIC ANNUITIES

The interest (discount) rate of annuity is a random variable, which is called stochastic. For stochastic annuities, also known as annuities under random rates of interest, we derive some moments, moments inequalities, and moments limits. When the discount rates are bounded, a strong convergence result is given showing that a stochastic annuity converges to its mean value as the number of years tend to infinity. Hence, stochastic annuities satisfy a type of strong law of large numbers.

Tripartite separability conditions exponentially violated by Gaussian states

Starting with a set of conditions for bipartite separability of arbitrary quantum states in any dimension and expressed in terms of arbitrary operators whose commutator is a $c$-number, we derive a hierarchy of conditions for tripartite separability of continuous-variable three-mode quantum states. These conditions have the form of inequalities for higher-order moments of linear combinations of the mode operators. They enable one to distinguish between all possible kinds of tripartite separability, while the strongest violation of these inequalities is a sufficient condition for genuine tripartite entanglement. We construct Gaussian states for which the violation of our conditions grows exponentially with the order of the moments of the mode operators. By going beyond second moments, our conditions are expected to be useful as well for the detection of tripartite entanglement of non-Gaussian states.

On the strong convergence for weighted sums of asymptotically almost negatively associated random variables

Applying the moment inequality of asymptotically almost negatively associated (AANA, in short) random variables which was obtained by Yuan and An (2009), some strong convergence results for weighted sums of AANA random variables are obtained without assumptions of identical distribution, which generalize and improve the corresponding ones of Zhou et al. (2011), Sung (2011, 2012) to the case of AANA random variables, respectively.

Rate of Convergence of Strong Law of Large Numbers for Positively Associated Sequences

By a maximal moment inequality for positively associated sequences, the rate of convergence of strong law of large numbers n−1/2(log n)1/2(log log n)1/ξ with any 0<ξ<2 is obtained under the moment condition that with 1<v⩽2. It extends and sharpens the corresponding ones in Yang and Chen (2007). Moreover, the existence of the moment of certain order is also deduced. Further, we obtain the relevant precise asymptotics with respect to the above rate of convergence.

Consistent estimation with many moment inequalities

In this paper, we consider estimation of the identified set when the number of moment inequalities is large relative to sample size, possibly infinite. Many applications in the recent literature on partially identified problems have this feature, including dynamic games, set-identified IV models, and parameters defined by a continuum of moment inequalities, in particular conditional moment inequalities. We provide a generic consistency result for criterion-based estimators using an increasing number of unconditional moment inequalities. We then develop more specific results for set estimation subject to conditional moment inequalities: we first derive the fastest possible rate for estimating the sharp identification region under smoothness conditions on the conditional moment functions. We also give rate conditions for inference under local alternatives.

Moment Inequalities for Maxima of Partial Sums in Probability with Applications in the Theory of Orthogonal Series

Moment Inequalities for Maxima of Partial Sums in Probability with Applications in the Theory of Orthogonal Series

The von Bahr–Esseen moment inequality for pairwise independent random variables and applications

In the paper, we generalize the von Bahr–Esseen moment inequality from independent random variables to pairwise independent random variables. As the applications, the moment convergence, the complete convergence and the strong law of large numbers are established for pairwise independent random variables.

A note on moment inequality for quadratic forms

Moment inequality for quadratic forms of random vectors is of particular interest in covariance matrix testing and estimation problems. In this paper, we prove a Rosenthal-type inequality, which exhibits new features and certain improvement beyond the unstructured Rosenthal inequality of quadratic forms when dimension of the vectors increases without bound. Applications to test the block diagonal structures and detect the sparsity in the high-dimensional covariance matrix are presented.

Equivalent Conditions of Complete Moment Convergence of Weighted Sums for ϕ-Mixing Sequence of Random Variables

In this article, the complete moment convergence of weighted sums for ϕ-mixing sequence of random variables is investigated. By applying moment inequality and truncation methods, the equivalent conditions of complete moment convergence of weighted sums for ϕ-mixing sequence of random variables are established. These results promote and improve the corresponding results obtained by Li et al. (1995) and Gut (1993) from i.i.d. to ϕ-mixing setting. Moreover, we obtain the complete moment convergence of moving average processes based on ϕ-mixing random variables, which extends the result of Kim et al. (2008) in the sense that it does not require a specific mixing rate.

Weighted KS statistics for inference on conditional moment inequalities

This paper proposes set estimators and conservative confidence regions for the identified set in conditional moment inequality models using Kolmogorov–Smirnov statistics with a truncated inverse variance weighting with increasing truncation points. The new weighting differs from those proposed in the literature in two important ways. First, this paper shows that estimators based on KS statistics with the proposed weighting function converge to the identified set at a faster rate than existing procedures based on bounded weight functions in a broad class of models. This provides a theoretical justification for inverse variance weighting in this context, and contrasts with analogous results for conditional moment equalities in which optimal weighting only affects the asymptotic variance. The results on rates of convergence of set estimators are the first such results even for the existing procedures, and involve developing the first general framework for determining consistency and rates of convergence for set estimators and confidence regions in this context. Second, the new weighting changes the asymptotic behavior, including the rate of convergence, of the KS statistic itself, requiring a new asymptotic theory in choosing the critical value. A series of examples illustrate the broad applicability of the results.

Complete moment convergence of weighted sums for processes under asymptotically almost negatively associated assumptions

For weighted sums of sequences of asymptotically almost negatively associated (AANA) random variables, we study the complete moment convergence by using the Rosenthal type moment inequalities. Our results extend the corresponding ones for sequences of independently identically distributed random variables of Chow [4].

On Properties of UBAC Class of Life Distributions

Some properties of the used better than aged in convex ordering( UBAC) class of life distributions are given. These properties include moment inequalities and moment generating functions behavior. Nonparametric estimation of UBAC survival function are discussed. In addition testing of the survival function of this class based on moment inequalities are introduced.