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.     

Sequence-compound estimation in scale-exponential families and speed of convergence

This paper deals with a sequence-compound estimation. The component problem is the squared error loss estimation of θ?[a,b] based on an observation X whose p.d.f. is of the form $\text{u(x)c(θ)}\text{exp}\text{(?}\text{x}\text{θ}\text{)}$. For each $\text{0<t<}\text{1}\text{2}$ a class of sequence-compound estimators $\text{ψ}\text{?}\text{=}\text{ψ}\text{?}{}_1\text{,}\text{ψ}\text{?}{}_2\text{,…)}$ is exhibited whose compound risk (average of risks) up to stage n differs from the Bayes envelope (in the component problem) w.r.t. the empiric distribution G_n of the parameters involved up to stage n by a quantity of order O(n^?δt) for a δ>0. It is also shown that at any stage i the difference of the risk of ψ?_i and the risk of the Bayes response w.r.t. G_i?1 is O(i^?δt). Examples of the above type of families are given where δ is min{1,$\text{2a}\text{b}$} and t is arbitrarily close to $\text{1}\text{2}$. Here it may be worthwhile to mention that a rate $\text{O}\text{(n}{}^{\mathrm{<mtext>?</mtext><mtext>1</mtext><mtext>2</mtext>}}\text{)}$ or better has not yet been obtained even in a very special family of densities.     

The consistency of a matching test

Let $\text{X}{}^1{}_1\text{?X}{}^1{}_2\text{???X}{}^1{}_{\mathrm{n}}$ be the order statistics of a random sample from a distribution on [0, 1]. Let A_k, the kth match, be the event that $\text{X}{}^1{}_{\mathrm{k}}\text{?(}\text{(k?1)}\text{n}\text{k}\text{n}$], and let S_n be the total number of matches. The consistency of S_n for testing uniform df, U, against df G≠U is investigated, and it is shown that S_n is consistent if the intersection of G with U has Lebesgue measure zero. It is also consistent against a sequence of alternatives approaching U at a rate less faster than $\text{n}{}^{\mathrm{<mtext>-1</mtext><mtext>2</mtext>}}$.     

A note on the estimation of the integral of ⨍2(x)

The estimation of $\text{γ=∫}{}^{\mathrm{\infty }}{}_{\mathrm{?\infty }}\text{?}{}^2\text{(x)dx}$, important in nonparametric inference, is discussed. Methods, based on either density estimators or on the lengths of nonparametric confidence intervals, are compared. We conclude that density type estimators have advantages over the estimators based on confidence intervals.     

A class of asymmetrical orthogonal resolution-IV designs

The concept of d-resolvability of orthogonal arrays of strength (d+1) is introduced. This is used to construct orthogonal resolution-IV plans of the type $\text{nt·n·2}{}^{\mathrm{n(m?1)}}\text{mn}{}^2\text{t}$. These plans are minimal and a large number of these plans are new.     

Embedding of orthogonal arrays of strength two and deficiency greater than two

Let x ≥ 0 and n ≥ 2 be integers. Suppose there exists an orthogonal array $\text{A(n, q, μ}{}^1\text{)}$ of strength 2 in n symbols with q rows and $\text{n}{}^2\text{μ}{}^1$ columns where $\text{q = q}{}^1\text{? d, q}{}^1\text{= n}{}^2\text{x + n + 1}$, $\text{μ}{}^1\text{= (n ?}\text{1}\text{)x +}\text{1}$ and d is a positive integer. Then d is called the deficiency of the orthogonal array. The question of embedding such an array into a complete array $\text{A(n, q}{}^1\text{, μ}{}^1\text{)}$ is considered for the case d ≥ 3. It is shown that for d = 3 such an embedding is always possible if n ≥ 2(d ? 1)²(2d² ? 2d + 1). Partial results are indicated if d ≥ 4 for the embedding of a related design in a corresponding balanced incomplete block design.     

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.     

Complete designs with blocks of maximal multiplicity

The set of distinct blocks of a block design is known as its support. We construct complete designs with parameters v(?7), k=3, λ=v ? 2 which contain a block of maximal multiplicity and with support size $\text{b}{}^1\text{= (}{}^{\mathrm{v}}{}_3\text{) ? 4(v ? 2)}$. Any complete design which contains such a block, and has parameters v, k, λ as above, must be supported on at most (^v₃) ? 4(v ? 2) blocks. Attention is given to complete designs because of their direct relationship to simple random sampling.     

Asymptotic distribution of hotelling's trace for two unequal covariance matrices and robustness study of test of equality of mean vectors

Asymptotic expansions for the percentiles and c.d.f., up to terms of order $\text{1}\text{n}{}^2$ of the statistic $\text{T =m}\text{T}\text{r}\text{S}{}_1\text{S}{}^{-1}{}_2$, where mS₁ and nS₂ independently distributed W(m, p, Σ₁) and W(n, p, Σ₂) respectively, are obtained using methods similar to those of Ito [4], Chattopadhyay and Pillai [2]. These expansions hold when $\text{Σ}{}_1\text{Σ}{}^{-1}{}_2\text{=}\text{I}\text{+}\text{F}$ and$\text{|}\text{Ch}{}_{\mathrm{i}}\text{(}\text{F}\text{)| < 1}$. Tables of powers of T for p = 3 and p = 4 for m = 4 and various values of n are given and comparison made with the exact powers for p = 3. These powers are useful for the study of (i) the test of equality of covariance matrices in two p-variate normal populations and (ii) robustness of test of equality of mean vectors of l normal populations against the violation of the assumption of equality of covariance matrices.     

Minimax estimation in the linear model with a relative squared error

Arnold and Stahlecker considered estimation of the regression coefficients in the linear model with a relative squared error and deterministic disturbances. They found an explicit form for a minimax linear affine solution $\text{d}{}^1$ of that problem. In the paper we generalize the result of Arnold and Stahlecker proving that the decision rule $\text{d}{}^1$ is also minimax when the class $\text{D}$ of possible estimators of the regression coefficients is unrestricted. Then we show that $\text{d}{}^1$ remains minimax in $\text{D}$ when the disturbances are random with the mean vector zero and the identity covariance matrix.     

Gaussian approximation of signed rank statistics process

We consider the signed linear rank statistics of the form

$\text{S}{}_{\mathrm{\Delta N}}\text{=}\text{∑}\text{i=1}\text{N}\text{c}{}_{\mathrm{Ni}}\text{ø(}\text{R}{}^{\mathrm{\Delta }}{}_{\mathrm{Ni}}\text{(N+1)}\text{)}\text{sgn}\text{Y}{}^{\mathrm{\Delta }}{}_{\mathrm{Ni}}$

where the c_Ni's are known real numbers, Δ∈[0,1] is an unknown real parameter,R^Δ_Ni is the rank of |Y^Δ_Ni| among |Y^Δ_Nj|, 1≤j≤N, ø is a score generating function, sgn y=1 or -1 according as y≥0 or <0, and Y^Δ_Nj, 1≤j ≤N, are independent random variables with continuous cumulative distribution functions F(y?Δd_Nj), 1≤ j≤N, respectively where the d_fNi's are known real numbers. Under suitable assumptions on the c's, d's, φ and F, it is proved that the random process {S_ΔN?S_0N?ES_ΔN, 0≤Δ≤1}, properly normalized, converges weakly to a Gaussian process, and this result is also true if ES_ΔN is replaced by Δb_N, where

$\text{b}{}_{\mathrm{N}}\text{=4}\text{∑}\text{i=1}\text{N}\text{c}{}_{\mathrm{Ni}}\text{d}{}_{\mathrm{Ni}}\text{∫}\text{0}\text{∞}\text{ø′(2F(x)?1)?}{}^2\text{(x)}\text{d}\text{x}\text{and}\text{?=F′}$

. As an application, we derive the asymptotic distribution of the properly normalized length of a confidence interval for Δ.     

Some determinant optimality aspects of main effect fold-over designs

Saha and Mohanty (1970) presented a main effect fold-over design consisting of 14 treatment combinations of the 24×33 factorial, which had the nice property of being even balanced. Calling this design DSM, this paper establishes the following specific results: (i) DSM is not d-optimal in the subclass Δe of all 14 point even balanced main effect fold-over designs of the 24×33 factorial; (ii) DSM is not d-optimal in the subclass $\text{Δ}{}^1\text{?Δ}{}^{\mathrm{e}}$ of all 14 point even and odd balanced main effect fold-over designs of the 24×33 factorial; (iii) DSM is even optimal in $\text{Δ}{}^1$ and Δe. In addition to these results two 14 point designs in $\text{Δ}{}^1$ are presented which are d-optimal and via a counter example it is shown that these designs are not odd optimal. Finally, several general matrix algebra results are given which should be useful in resolving d-optimality problems of fold-over designs of the kn11×kn22 factorial.     

Optimum designs in complete two-way layouts

In the usual two-way layout of ANOVA (interactions are admitted) let nij ? 1 be the number of observations for the factor-level combination(i, j). For testing the hypothesis that all main effects of the first factor vanish numbers $\text{n}{}^1\text{ij}$ are given such that the power function of the F-test is uniformly maximized (U-optimality), if one considers only designs (nij) for which the row-sums ni are prescribed. Furthermore, in the (larger) set of all designs for which the total number of observations is given, all D-optimum designs are constructed.     

Constrained Bayes and empirical Bayes estimation under random effects normal ANOVA model with balanced loss function

The paper develops constrained Bayes and empirical Bayes estimators in the random effects ANOVA model under balanced loss functions. In the balanced normal–normal model, estimators of the Bayes risks of the constrained Bayes and constrained empirical Bayes estimators are provided which are correct asymptotically up to O(m^-1)$\mathrm{O}(m^{-1})$, that is the remainder term is o(m^-1)$\mathrm{o}(m^{-1})$, m$m$ denoting the number of strata.     

Some aspects of the theory of estimating equations

An estimating equation for a parameter θ, based on an observation ?, is an equation g(x,θ)=0 which can be solved for θ in terms of x. An estimating equation is unbiased if the funaction g has 0 mean for every θ. For the case when the form of the frequency function p(x,θ) is completely specified up to the unknown real parameter θ, the optimality of the m.1 equation $\text{?}\text{log}\text{p}\text{?θ}\text{=0}$ in the class of all unbiased estimating equations was established by Godambe (1960). In this paper we allow the form of the frequency function p to vary assuming that $\text{x=(x}{}_1\text{,…,x}{}_{\mathrm{n}}\text{)?}\text{R}{}^{\mathrm{n}}$ and that under p, E(x_i)=θ. x₁,…, x_n are independent observations on a variate x, it is shown that among all the unbiased estimating equations for θ, $\text{x}\text{?}\text{?θ=0}$ is uniquely optimum up to a constant multiple.     

Testing of a sub-hypothesis in linear regression models with long memory errors and deterministic design

This paper considers the problem of testing a sub-hypothesis in homoscedastic linear regression models where errors form long memory moving average processes and designs are non-random. Unlike in the random design case, asymptotic null distribution of the likelihood ratio type test based on the Whittle quadratic form is shown to be non-standard and non-chi-square. Moreover, the rate of consistency of the minimum Whittle dispersion estimator of the slope parameter vector is shown to be n^-(1-α)/2$n^{-(1-\alpha )/2}$, different from the rate n^-1/2$n^{-1/2}$ obtained in the random design case, where α$\alpha$ is the rate at which the error spectral density explodes at the origin. The proposed test is shown to be consistent against fixed alternatives and has non-trivial asymptotic power against local alternatives that converge to null hypothesis at the rate n^-(1-α)/2$n^{-(1-\alpha )/2}$.     

Measures of lack of fit from tests of chi-squared type

If X² is the Pearson chi-squared statistic for testing fit, then $\text{X}{}^2\text{n}$ has long been considered an associated measure of the degree of lack of fit. Here we consider two classes of statistics of chi-squared type, each having X² as a member. The first is a class of directed divergence statistics discussed by Cressie and Read, the second consists of nonnegative definite quadratic forms in the standardized cell frequencies. We investigate the large sample behavior of $\text{T}\text{n}$, where T is any of these statistics. A number of auxiliary results on the Cressie-Read statistics are also obtained. The measures are illustrated by application to data from classical physics compiled by Stigler.     

Testing optimality of experimental designs for a regression model with random variables

Tsukanov (Theor. Probab. Appl. 26 (1981) 173–177) considers the regression model $\text{E(}\text{y}\text{|}\text{Z}\text{)=}\text{Fp}\text{+}\text{Zq}$, $\text{D(}\text{y}\text{|}\text{Z}\text{)=σ}{}^2\text{I}{}_{\mathrm{n}}$, where $\text{y}\text{(n×1)}$ is a vector of measured values,$\text{F}\text{(n×k)}$ contains the control variables, $\text{Z}\text{(n×l)}$ contains the observed values, and $\text{p}\text{(k×1)}$ and $\text{q}\text{(l×1)}$ are being estimated. Assuming that $\text{Z}\text{=}\text{FL}\text{+}\text{R}$, where $\text{L}\text{(k×l)}$ is non-random, and the rows of $\text{R}\text{(n×l)}$ are i.i.d. $\text{N(}\text{0}\text{,Σ)}$, we extend Tsukanov's results by (i) computing E(detH_p), where H_p is the covariance matrix of p?, the l.s.e. of p, (ii) considering ‘optimality in the mean’ for the largest root criterion, (iii) discussing these equations when the matrix R has a left-spherical distribution.     

Non-parametric recursive estimates of a probability density function and its derivatives

Recursive estimates of a probability density function (pdf) are known. This paper presents recursive estimates of a derivative of any desired order of a pdf. Let f be a pdf on the real line and p?0 be any desired integer. Based on a random sample of size n from f, estimators f^(p)_n of f^(p), the pth order derivatives of f, are exhibited. These estimators are of the form $\text{n}{}^{\mathrm{?1}}\text{∑n}\text{j=1}\text{δ}{}_{\mathrm{jp}}$, where δ_jp depends only on p and the jth observation in the sample, and hence can be computed recursively as the sample size increases. These estimators are shown to be asymptotically unbiased, mean square consistent and strongly consistent, both at a point and uniformly on the real line. For pointwise properties, the conditions on f^(p) have been weakened with a little stronger assumption on the kernel function.