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.     

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.     

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 $\text{n}{}^{\mathrm{<mtext>-1</mtext><mtext>2</mtext>}}$.     

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.     

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.     

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.     

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.     

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.     

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.     

Robustness of MML estimators based on censored samples and robust test statistics

We investigate the efficiences of Tiku's (1967) modified maximum likelihood estimators μ_c and σ_c (based on symmetrically censored normal samples) for estimating the location and scale parameters μ and σ of symmetric non-normal distributions. We show that μ_c and σ_c are jointly more efficient than x? and s for long-tailed distributions (kurtosis $\text{β}{}_2{}^1\text{=}\text{μ}{}_4\text{μ}{}_2{}^2\text{>4.2, β}{}_2{}^1\text{= 4.2}$ for the Logistic), and always more efficient than the trimmed mean μT and the matching sample estimate σ_T of σ. We also show that μ_c and σ_c are jointly at least as efficient as some of the more prominent “robust” estimators (Gross, 1976). We show that the statistic $\text{t}{}_{\mathrm{c}}\text{= μ}{}_{\mathrm{c}}\text{m}\text{σ}{}_{\mathrm{c}}\text{, m = n ?2r + 2rβ}$ (r is the number of observations censored on each side of the sample and β is a constant), is robust and powerful for testing an assumed value of μ. We define a statistic T_c (based on μ_c andσ_c) for testing that two symmetric distributions are identical and show that T_c is robust and generally more poweerful than the well-known nonparametric statistics (Wilcoxon, normal-score, Kolmogorov-Smirnov), against the important location-shift alternatives. We generalize the statistic T_c to test that k symmetric distibutions are identical. The asymptotic distributions of t_c and T_c are normal, under some very general regularity conditions. For small samples, the upper (lower) percentage points of t_c and T_c are shown to be closely approximated by Student's t-distributions. Besides, the statistics μ_c and σ_c (and hence t_c and T_c) are explicit and simple functions of sample observations and are easy to compute.     

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.     

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.     

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.     

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.     

New estimator for functions of the canonical correlation coefficients

Let ρ₁,…,ρ_p be the population canonical correlation coefficients from a normal distribution. This paper considers the estimation of δ₁,…,δ_p, where $\text{δ}{}_{\mathrm{i}}\text{=ρ}{}_{\mathrm{i}}{}^2\text{/(1−ρ}{}_{\mathrm{i}}{}^2\text{),}\text{i=1,…,p}$, in a decision theoretic way. Since the distribution of δ_i's is complicated, two-staged estimation has been a usual method so far; i.e., first find a good estimator of a matrix whose eigenvalues are the δ_i's, then use its eigenvalues as the estimators of δ_i's. In this paper we directly estimate δ_i's and evaluate the estimators with respect to a quadratic loss function. We propose a new class of estimators and prove its dominance over the usual estimator.