On the convergence rate of model selection criteria

The goal of the current paper is to compare consistent and inconsistent model selection criteria by looking at their convergence rates (to be defined in the first section). The prototypes of the two types of criteria are the AIC and BIC criterion respectively. For linear regression models with normally distributed errors, we show that the convergence rates for AIC and BIC are 0(n^-1) and 0((n log n)^-1/2) respectively. When the error distributions are unknown, the two criteria become indistinguishable, all having convergence rate O(n^-1/2). We also argue that the BIC criterion has nearly optimal convergence rate. The results partially justified some of the controversial simulation results in which inconsistent criteria seem to outperform consistent ones.     

On empirical Bayes simultaneous selection procedures for comparing normal populations with a standard

In this paper, we derive statistical selection procedures to partition k normal populations into ‘good’ or ‘bad’ ones, respectively, using the nonparametric empirical Bayes approach. The relative regret risk of a selection procedure is used as a measure of its performance. We establish the asymptotic optimality of the proposed empirical Bayes selection procedures and investigate the associated rates of convergence. Under a very mild condition, the proposed empirical Bayes selection procedures are shown to have rates of convergence of order close to O(k^−1/2) where k is the number of populations involved in the selection problem. With further strong assumptions, the empirical Bayes selection procedures have rates of convergence of order O(k^{−α(r−1)/(2r+1)}), where 1<α<2 and r is an integer greater than 2.     

Convergence rates of empirical Bayes estimation in exponential family

The convergence rates of empirical Bayes estimation in the exponential family are studied in this paper. We first develop an approach for obtaining the lower bound of empirical Bayes estimators. As an application of the approach, we demonstrate that O(n⁻¹) is the lower bound rate for priors with bounded compact support. Second, we construct an empirical Bayes estimator using kernel sequence method and show that it has a rate of convergence of $\text{O}\text{(n}{}^{\mathrm{-1}}\text{(}\text{ln}\text{n)}{}^8\text{)}$. This upper bound rate is much faster compared to the earlier results published in the literature under the same assumption.     

Convergence rates for empirical bayes estimators of parameters in linear regressin models

In this paper, the convergence rates of the EB estimators of the regression coefficients and the error variance in a linear model are obtained. The rates can approximate to O(n¹) arbitrarily. The convergency of the EB estimators of the regression coefiicients and the variance components in a variance component model is also investigated. The investigation makes use of the results concerning the convergence rates of the EB estimators of the parameters in multi-parameter exponential families.     

Super efficient frequency estimation

In this paper we propose a modified Newton-Raphson method to obtain super efficient estimators of the frequencies of a sinusoidal signal in presence of stationary noise. It is observed that if we start from an initial estimator with convergence rate O_p(n⁻¹) and use Newton-Raphson algorithm with proper step factor modification, then it produces super efficient frequency estimator in the sense that its asymptotic variance is lower than the asymptotic variance of the corresponding least squares estimator. The proposed frequency estimator is consistent and it has the same rate of convergence, namely O_p(n^−3/2), as the least squares estimator. Monte Carlo simulations are performed to observe the performance of the proposed estimator for different sample sizes and for different models. The results are quite satisfactory. One real data set has been analyzed for illustrative purpose.     

Exponential and polynomial tailbounds for change-point estimators

Let X_1n,…,X_nn be independent random elements with an unknown change point θ∈(0,1), that is X_in has a distribution ν₁ or ν₂, respectively, according to i⩽[nθ] or i>[nθ]. We propose an estimator θ_n of θ, which is defined as the maximizer of a weighted empirical process on (0,1). Finding upper bounds of polynomial and exponential type for the tails of nθ_n−[nθ], we are able to derive rates of almost sure convergence, of distributional convergence, of L_p-convergence and of convergence in the Ky-Fan- and in the Prokhorov-metric.     

Poisson approximations in selected metrics by coupling and semigroup methods with applications

Let S_n = X₁ + … + X_n, where X₁,…, X_n are independent Bernoulli random variables. In this paper, we evaluate probability metrics of the Wasserstein type between the distribution of S_n and a Poisson distribution. Our results show that, if E(S_n) = O(1) and if the individual probabilities of success of the X_i's tend uniformly to zero, then the general rate of convergence of the above mentioned metrics to zero is O(∑ⁿ_i = ₁P²_i). We also show that this rate is sharp and discuss applications of these results.     

Moments of suprema of random variables

The supremum of random variables representing a sequence of rewards is of interest in establishing the existence of optimal stopping rules. Necessary and sufficient conditions are given for existence of moments of sup_n(X_n ? c_n) and sup_n(S_n ? c_n) where X₁, X₂, … are i.i.d. random variables, S_n = X₁ + … + X_n, and c_n = (nL(n))^1/r, 0 < r < 2, L = 1, L = log, and L = log log. Following Cohn (1974), “rates of convergence” results are used in the proof.     

Estimation suroptimale de la densité par projection

Superefficiency of a projection density estimator The author constructs a projection density estimator with a data‐driven truncation index. This estimator reaches the superoptimal rates 1/n in mean integrated square error and {In ln(n/n}^1/2 in uniform almost sure convergence over a given subspace which is dense in the class of all possible densities; the rate of the estimator is quasi‐optimal everywhere else. The subspace in question may be chosen a priori by the statistician.     

Asymptotic convergence in distribution for spearman's rank correlation coefficient

The Edgeworth expansion for the distribution function of Spearman's rank correlation coefficient may be used to show that the rates of convergence for the normal and Pearson type II approximations are l/nand l/n² respectively. Using the Edgeworth expansion up to terms involving the sixth moment of the exact distribution allows an approximation with an error of order l/n³.     

Nonparametric Estimation of a Regression Function: Limiting Distribution2

Consider the regression model Y_i= g(x_i) + e_i, i = 1,…, n, where g is an unknown function defined on [0, 1], 0 = x₀ < x₁ < … < x_n≤ 1 are chosen so that max_1≤i≤n(x_i-x_{i- 1}) = 0(n^-1), and where {e_i} are i.i.d. with Ee₁= 0 and Var e₁ - s?². In a previous paper, Cheng & Lin (1979) study three estimators of g, namely, g_1n of Cheng & Lin (1979), g_2n of Clark (1977), and g_3n of Priestley & Chao (1972). Consistency results are established and rates of strong uniform convergence are obtained. In the current investigation the limiting distribution of &_in, i = 1, 2, 3, and that of the isotonic estimator g**_n are considered.     

Fitting a multiple regression function

Consider the p-dimensional unit cube [0,1]^p, p≥1. Partition [0, 1]^p into n regions, R_1,n,…,R_n,n such that the volume Δ(R_j,n) is of order n^?1,j=1,…,n. Select and fix a point in each of these regions so that we have x⁽ⁿ⁾₁,…,x⁽ⁿ⁾_n. Suppose that associated with the j-th predictor vector x⁽ⁿ⁾_j there is an observable variable Y⁽ⁿ⁾_j, j=1,…,n, satisfying the multiple regression model $\text{Y}{}^{\mathrm{(n)}}{}_{\mathrm{j}}\text{=g(}\text{x}{}^{\mathrm{(n)}}{}_{\mathrm{j}}\text{)+e}{}^{\mathrm{(n)}}{}_{\mathrm{j}}$, where g is an unknown function defined on [0, 1]^pand {e⁽ⁿ⁾_j} are independent identically distributed random variables with Ee⁽ⁿ⁾₁=0 and Var e⁽ⁿ⁾₁=σ²<∞. This paper proposes as an estimator of g(x), where k(u) is a known p-dimensional bounded density and {a_n} is a sequence of reals converging to 0 asn→∞. Weak and strong consistency of g_n(x) and rates of convergence are obtained. Asymptoticnormality of the estimator is established. Also proposed is $\text{σ}{}^2{}_{\mathrm{n}}\text{=n}{}^{\mathrm{?1}}\text{Σ}{}^{\mathrm{n}}{}_{\mathrm{j=1}}\text{(Y}{}^{\mathrm{(n)}}{}_{\mathrm{j}}\text{?g}{}_{\mathrm{n}}\text{(}\text{x}{}^{\mathrm{(n)}}{}_{\mathrm{j}}\text{))}{}^2$ as a consistent estimate of σ².     

Existence of frame SOLS of type anb1 for odd n

It has been proved that (i) FSOLS(2ⁿb¹) exist if and only if n⩾4 and n⩾1+b and (ii) FSOLS(3ⁿb¹) exist if and only if n⩾4 and n⩾1+2b/3 with 17 possible exceptions. In this article, we show that for b⩾2 and odd n, FSOLS(aⁿb¹) exists if and only if n⩾4 and n⩾1+2b/a.     

On almost sure convergence for weighted sums of pairwise negatively quadrant dependent random variables

Let {X _n , n ≥ 1} be a sequence of pairwise negatively quadrant dependent (NQD) random variables. In this study, we prove almost sure limit theorems for weighted sums of the random variables. From these results, we obtain a version of the Glivenko–Cantelli lemma for pairwise NQD random variables under some fragile conditions. Moreover, a simulation study is done to compare the convergence rates with those of Azarnoosh (Pak J Statist 19(1):15–23, 2003) and Li et al. (Bull Inst Math 1:281–305, 2006).     

The asymptotic behavior of monotone percentile regression estimates

The least-absolute-deviation estimate of a monotone regression function on an interval has been studied in the literature. If the observation points become dense in the interval, the almost sure rate of convergence has been shown to be O(n^1/4). Applying the techniques used by Brunk (1970, Nonparametric, Techniques in Statistical Inference. Cambridge Univ. Press), the asymptotic distribution of the l₁ estimator at a point is obtained. If the underlying regression function has positive slope at the point, the rate of convergence is seen to be O(n^1/3). Monotone percentile regression estimates are also considered.     

Berry–Esseen type bounds for trimmed U-statistics

Trimmed U  -statistics can be constructed in two different ways: by basing the statistic on a trimmed sample or by averaging the trimmed set of kernel values. Mild conditions are given to ensure the rate of convergence to normality is O(n^-1/2)$\mathrm{O}(n^{-1/2})$ in both cases.     

Rate of consistency of one sample tests of location

Let X₁,…,X_n be a sample from a population with continuous distribution function F(x?θ) such that F(x)+F(-x)=1 and 0<F(x)<1, x?R¹. It is shown that the power- function of a monotone test of H: θ=θ₀ against K: θ>θ₀ cannot tend to 1 as θ?θ₀ → ∞ more than n times faster than the tails of F tend to 0. Some standard as well as robust tests are considered with respect to this rate of convergence.     

Lp convergence and complete convergence for weighted sums of AANA random variables

Let {X_n, n ? 1} be a sequence of asymptotically almost negatively associated (AANA, for short) random variables which is stochastically dominated by a random variable X, and {d_ni, 1 ? i ? n, n ? 1} be a sequence of real function, which is defined on a compact set E. Under some suitable conditions, we investigate some convergence properties for weighted sums of AANA random variables, especially the L^p convergence and the complete convergence. As an application, the Marcinkiewicz–Zygmund-type strong law of large numbers for AANA random variables is obtained.     

Empirical bayes tests in a positive exponential family with partial information on priors

We investigate an empirical Bayes testing problem in a positive exponential family having pdf f{x/θ)=c(θ)u(x) exp(?x/θ), x>0, θ>0. It is assumed that θ is in some known compact interval [C₁, C₂]. The value C₁ is used in the construction of the proposed empirical Bayes test δ^* _n. The asymptotic optimality and rate of convergence of its associated Bayes risk is studied. It is shown that under the assumption that θ is in [C₁, C₂] δ^* _n is asymptotically optimal at a rate of convergence of order O(n^?1/n n). Also, δ^* _n is robust in the sense that δ^* _n still possesses the asymptotic optimality even the assumption that "C₁≦θ≦C₂ may not hold.     

On some joint distributions in fluctuation theory

For non-negative integral valued interchangeable random variables v₁, v₂,…,v_n, Takács (1967, 70) has derived the distributions of the statistics ?_n' ?¹_n' ?^(c)_n and ?^(-c)_n concerning the partial sums N_r = v₁ + v₂ + ··· + v_rr = 1,…,n. This paper deals with the joint distributions of some other statistics viz., (α^(c)_n, δ^(c)_n, Z_n), (β^(c)_n, Z_n) and (β^(-c)_n, Z_n) concerning the partial sums N_r = ε₁ + ··· + ε_rr = 1,2,…,n, of geometric random variables ε₁, ε₂,…,ε_n.