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1.
Let X1, X2,… be idd random vectors with a multivariate normal distribution N(μ, Σ). A sequence of subsets { Rn( a1, a2,…, an), n ≥ m} of the space of μ is said to be a (1 − α)-level sequence of confidence sets for μ if P(μ Rn( X1, X2,…, Xn) for every n ≥ m) ≥ 1 − α. In this note we use the ideas of Robbins Ann. Math. Statist. 41 (1970) to construct confidence sequences for the mean vector μ when Σ is either known or unknown. The constructed sequence Rn( X1, X2, …, Xn) depends on Mahalanobis'
or Hotelling's
according as Σ is known or unknown. Confidence sequences for the vector-valued parameter in the general linear model are also given. 相似文献
2.
Let
= { Ut: t > 0} be a strongly continuous one-parameter group of operators on a Banach space X and Q be any subset of a set
( X) of all probability measures on X. By
( Q;
) we denote the class of all limit measures of { Utn(μ 1 * μ 2*…*μ n)*δ xn}, where {μ n} Q, { xn} X and measures Utnμ j ( j=1, 2,…, n; N=1, 2,…) form an infinitesimal triangular array. We define classes Lm(
) as follows: L0(
)=
(
( X);
), Lm(
)=
( Lm−1(
);
) for m=1, 2,… and L∞(
)= m=0∞Lm(
). These classes are analogous to those defined earlier by Urbanik on the real line. Probability distributions from Lm(
), m=0, 1, 2,…, ∞, are described in terms of their characteristic functionals and their generalized Poisson exponents and Gaussian covariance operators. 相似文献
3.
We give a direct formulation of the invariant polynomials μGq(n)(, Δ i,;, xi,i + 1,) characterizing U( n) tensor operators p, q, …, q, 0, …, 0 in terms of the symmetric functions Sλ known as Schur functions. To this end, we show after the change of variables Δ i = γ i − δ i and xi, i + 1 = δ i − δ i + 1 that μGq(n)(,Δ i;, xi, i + 1,) becomes an integral linear combination of products of Schur functions Sα(, γ i,) · Sβ(, δ i,) in the variables {γ 1,…, γ n} and {δ 1,…, δ n}, respectively. That is, we give a direct proof that μGq(n)(,Δ i,;, xi, i + 1,) is a bisymmetric polynomial with integer coefficients in the variables {γ 1,…, γ n} and {δ 1,…, δ n}. By making further use of basic properties of Schur functions such as the Littlewood-Richardson rule, we prove several remarkable new symmetries for the yet more general bisymmetric polynomials μmGq(n)(γ 1,…, γ n; δ 1,…, δ m). These new symmetries enable us to give an explicit formula for both μmG1(n)(γ; δ) and 1G2(n)(γ; δ). In addition, we describe both algebraic and numerical integration methods for deriving general polynomial formulas for μmGq(n)(γ; δ). 相似文献
4.
By using Krasnoselskii's fixed point theorem and upper and lower solutions method, we find some sets of positive values λ determining that there exist positive T-periodic solutions to the higher-dimensional functional difference equations of the form where A( n)=diag[ a1( n), a2( n),…, am( n)], h( n)=diag[ h1( n), h2( n),…, hm( n)], aj, hj : Z→ R+, τ : Z→ Z are T -periodic, j=1,2,…, m, T1, λ>0, x : Z→ Rm, f : R+m→ R+m, where R+m={( x1,…, xm) TRm, xj0, j=1,2,…, m}, R+={ xR, x>0}. 相似文献
5.
For all integers m3 and all natural numbers a1, a2,…, am−1, let n= R( a1, a2,…, am−1) represent the least integer such that for every 2-coloring of the set {1,2,…, n} there exists a monochromatic solution to | Let t=min{a1,a2,…,am−1} and b=a1+a2++am−1−t. In this paper it is shown that whenever t=2, R(a1,a2,…,am−1)=2b2+9b+8.
It is also shown that for all values of t, R(a1,a2,…,am−1)tb2+(2t2+1)b+t3.
相似文献
6.
We consider the Tikhonov regularizer f
λ of a smooth function
f ε
H2m[0, 1], defined as the solution (see [1]) to We prove that if
f(j)(0) =
f(j)(1) = 0,
J =
m, …,
k < 2
m − 1, then ¦
f −
fλ¦
j2 Rλ
(2k − 2j + 3)/2m,
J = 0, …,
m. A detailed analysis is given of the effect of the boundary on convergence rates.
相似文献
7.
An
n-ary operation
Q:
Σn→
Σ is called an
n-ary quasigroup of order |
Σ| if in the relation
x0=
Q(
x1,…,
xn) knowledge of any
n elements of
x0,…,
xn uniquely specifies the remaining one.
Q is permutably reducible if
Q(
x1,…,
xn)=
P(
R(
xσ(1),…,
xσ(k)),
xσ(k+1),…,
xσ(n)) where
P and
R are (
n-
k+1)-ary and
k-ary quasigroups,
σ is a permutation, and 1<
k<
n. An
m-ary quasigroup
S is called a retract of
Q if it can be obtained from
Q or one of its inverses by fixing
n-
m>0 arguments. We prove that if the maximum arity of a permutably irreducible retract of an
n-ary quasigroup
Q belongs to {3,…,
n-3}, then
Q is permutably reducible.
相似文献
8.
The behavior of the posterior for a large observation is considered. Two basic situations are discussed; location vectors and natural parameters.Let
X = (
X1,
X2, …,
Xn) be an observation from a multivariate exponential distribution with that natural parameter Θ = (Θ
1, Θ
2, …, Θ
n). Let θ
x* be the posterior mode. Sufficient conditions are presented for the distribution of Θ − θ
x* given
X =
x to converge to a multivariate normal with mean vector 0 as |
x| tends to infinity. These same conditions imply that
E(Θ |
X =
x) − θ
x* converges to the zero vector as |
x| tends to infinity.The posterior for an observation
X = (
X1,
X2, …,
Xn is considered for a location vector Θ = (Θ
1, Θ
2, …, Θ
n) as
x gets large along a path, γ, in
Rn. Sufficient conditions are given for the distribution of γ(
t) − Θ given
X = γ(
t) to converge in law as
t → ∞. Slightly stronger conditions ensure that γ(
t) −
E(Θ |
X = γ(
t)) converges to the mean of the limiting distribution.These basic results about the posterior mean are extended to cover other estimators. Loss functions which are convex functions of absolute error are considered. Let δ be a Bayes estimator for a loss function of this type. Generally, if the distribution of Θ −
E(Θ |
X = γ(
t)) given
X = γ(
t) converges in law to a symmetric distribution as
t → ∞, it is shown that δ(γ(
t)) −
E(Θ |
X = γ(
t)) → 0 as
t → ∞.
相似文献
9.
We introduce a new multidimensional pattern matching problem that is a natural generalization of string matching, a well studied problem[1]. The motivation for its algorithmic study is mainly theoretical. Let
A[1:
n1,…,1:
nd] be a text matrix with
N =
n1…
ndentries and
B[1:
m1,…,1:
mr] be a pattern matrix with
M =
m1…
mrentries, where
d ≥
r ≥ 1 (the matrix entries are taken from an ordered alphabet Σ). We study the problem of checking whether some
r-dimensional submatrix of
Ais equal to
B(i.e., a
decisionquery).
Acan be preprocessed and
Bis given on-line. We define a new data structure for preprocessing
Aand propose CRCW-PRAM algorithms that build it in
O(log
N) time with
N2/
nmaxprocessors, where
nmax = max(
n1,…,
nd), such that the decision query for
Btakes
O(
M) work and
O(log
M) time. By using known techniques, we would get the same preprocessing bounds but an
O((
dr)
M) work bound for the decision query. The latter bound is undesirable since it can depend exponentially on
d; our bound, in contrast, is independent of
dand optimal. We can also answer, in optimal work, two further types of queries: (a) an
enumerationquery retrieving all the
r-dimensional submatrices of
Athat are equal to
Band (b) an
occurrencequery retrieving only the distinct positions in
Athat correspond to all of these submatrices. As a byproduct, we also derive the first efficient sequential algorithms for the new problem.
相似文献
10.
Let
denote a certain class of rational functions. For each
f ε
, consider the polynomial of degree at most
n that best approximates
f in the uniform norm. The corresponding strong unicity constant is denoted by
Mn(
f). Then there exist positive constants α and β, not depending on
n, such that
an Mn(
f) β
n,
N = 1,2,….
相似文献
11.
Let
X1,
X2, …,
Xn be random vectors that take values in a compact set in
Rd,
d ≥ 1. Let
Y1,
Y2, …,
Yn be random variables (“the responses”) which conditionally on
X1 =
x1, …,
Xn =
xn are independent with densities
f(
y |
xi, θ(
xi)),
i = 1, …,
n. Assuming that θ lives in a sup-norm compact space Θ
q,d of real valued functions, an optimal
L1-consistent estimator
of θ is constructed via empirical measures. The rate of convergence of the estimator to the true parameter θ depends on Kolmogorov's entropy of Θ
q,d.
相似文献
12.
The basic result of the paper states: Let
F1, …,
Fn,
F1′,…,
Fn′ have proportional hazard functions with λ
1 ,…, λ
n , λ
1′ ,…, λ
n′ as the constants of proportionality. Let
X(1) ≤ … ≤
X(n) (
X(1)′ ≤ … ≤
X(n)′) be the order statistics in a sample of size
n from the heterogeneous populations {
F1 ,…,
Fn}({
F1′ ,…,
Fn′}). Then (λ
1 ,…, λ
n) majorizes (λ
1′ ,…, λ
n′) implies that (
X(1) ,…,
X(n)) is stochastically larger than (
X(1)′ ,…,
X(n)′). Earlier results stochastically comparing
individual order statistics are shown to be special cases. Applications of the main result are made in the study of the robustness of standard estimates of the failure rate of the exponential distribution, when observations actually come from a set of heterogeneous exponential distributions. Further applications are made to the comparisons of linear combinations of Weibull random variables and of binomial random variables.
相似文献
13.
The zeros of the Meixner polynomial
mn(
x;
β,
c) are real, distinct, and lie in (0, ∞). Let
αn, sdenote the
sth zero of
mn(
nα;
β,
c), counted from the right; and let
αn, sdenote the
sth zero of
mn(
nα;
β,
c), counted from the left. For each fixed
s, asymptotic formulas are obtained for both
αn, sand
αn, s, as
n→∞.
相似文献
14.
Orthonormal polynomials with weight ¦τ¦
exp(−τ
4) have leading coefficients with recurrence properties which motivate the more general equations ξ
m(ξ
m − 1 + ξ
m + ξ
m + 1) = γ
m2,
M = 1, 2,…, where ξ
o is a fixed nonnegative value and γ
1, γ
2,… are positive constants. For this broader problem, the existence of a nonnegative solution is proved and criteria are found for its uniqueness. Then, for the motivating problem, an asymptotic expansion of its unique nonnegative solution is obtained and a fast computational algorithm, with error estimates, is given.
相似文献
15.
A link between Ramsey numbers for stars and matchings and the Erd
s-Ginzburg-Ziv theorem is established. Known results are generalized. Among other results we prove the following two theorems.
Theorem 5. Let m be an even integer. If
c :
e (
K2m−1)→{0, 1,…,
m−1} is a mapping of the edges of the complete graph on 2
m−1 vertices into {0, 1,…,
m−1}, then there exists a star K
1,m in
K2m−1 with edges e
1, e
2,…, e
m such that
c(
e1)+
c(
e2)++
c(
em)≡0 (mod
m).
Theorem 8. Let m be an integer. If
c :
e(
Kr(r+1)m−1)→{0, 1,…,
m−1} is a mapping of all the r-subsets of an (
r+1)
m−1 element set S into {0, 1,…,
m−1}, then there are m pairwise disjoint r-subsets
Z1,
Z2,…,
Zm of S such that
c(
Z1)+
c(
Z2)++
c(
Zm)≡0 (mod
m).
相似文献
16.
Let
X1, …,
Xn be independent random variables and define for each finite subset
I {1, …,
n} the σ-algebra
= σ{
Xi : i ε I}. In this paper
-measurable random variables
WI are considered, subject to the centering condition
E(WI
) = 0 a.s. unless
I J. A central limit theorem is proven for
d-homogeneous sums
W(
n) = Σ
I = dWI, with var
W(
n) = 1, where the summation extends over all (
nd) subsets
I {1, …,
n} of size
I =
d, under the condition that the normed fourth moment of
W(
n) tends to 3. Under some extra conditions the condition is also necessary.
相似文献
17.
Let
be a probability space and let
Pn be the empirical measure based on i.i.d. sample (
X1,…,
Xn) from
P. Let
be a class of measurable real valued functions on
For
define
Ff(
t):=
P{
ft} and
Fn,f(
t):=
Pn{
ft}. Given γ(0,1], define
n,γ(δ):=1/(
n1−γ/2δ
γ). We show that if the
L2(
Pn)-entropy of the class
grows as
−α for some α(0,2), then, for all
and all δ(0,Δ
n), Δ
n=O(
n1/2),
and
where
and
c(σ)↓1 as σ↓0 (the above inequalities hold for any fixed σ(0,1] with a high probability). Also, define
Then for all
uniformly in
and with probability 1 (for
the above ratio is bounded away from 0 and from ∞). The results are motivated by recent developments in machine learning, where they are used to bound the generalization error of learning algorithms. We also prove some more general results of similar nature, show the sharpness of the conditions and discuss the applications in learning theory.
相似文献
18.
Treated in this paper is the problem of estimating with squared error loss the generalized variance | Σ | from a Wishart random matrix
S:
p ×
p Wp(
n, Σ) and an independent normal random matrix
X:
p ×
k N(ξ, Σ
Ik) with ξ(
p ×
k) unknown. Denote the columns of
X by
X(1) ,…,
X(k) and set ψ
(0)(
S,
X) = {(
n −
p + 2)!/(
n + 2)!} |
S |, ψ
(i)(
X,
X) = min[ψ
(i−1)(
S,
X), {(
n −
p +
i + 2)!/(
n +
i + 2)!} |
S +
X(1) X′
(1) + +
X(i) X′
(i) |] and Ψ
(i)(
S,
X) = min[ψ
(0)(
S,
X), {(
n −
p +
i + 2)!/(
n +
i + 2)!}|
S +
X(1) X′
(1) + +
X(i) X′
(i) |],
i = 1,…,
k. Our result is that the minimax, best affine equivariant estimator ψ
(0)(
S,
X) is dominated by each of Ψ
(i)(
S,
X),
i = 1,…,
k and for every
i, ψ
(i)(
S,
X) is better than ψ
(i−1)(
S,
X). In particular, ψ
(k)(
S,
X) = min[{(
n −
p + 2)!/(
n + 2)!} |
S |, {(
n −
p + 2)!/(
n + 2)!} |
S +
X(1)X′
(1)|,…,| {(
n −
p +
k + 2)!/(
n +
k + 2)!} |
S +
X(1)X′
(1) + +
X(k)X′
(k)|] dominates all other ψ's. It is obtained by considering a multivariate extension of Stein's result (
Ann. Inst. Statist. Math. 16, 155–160 (1964)) on the estimation of the normal variance.
相似文献
19.
We consider the bounded integer knapsack problem (BKP)
, subject to:
, and
xj{0,1,…,
mj},
j=1,…,
n. We use proximity results between the integer and the continuous versions to obtain an
O(
n3W2) algorithm for BKP, where
W=max
j=1,…,nwj. The respective complexity of the unbounded case with
mj=
∞, for
j=1,…,
n, is
O(
n2W2). We use these results to obtain an improved strongly polynomial algorithm for the multicover problem with cyclical 1’s and uniform right-hand side.
相似文献
20.
For a fixed integer
m ≥ 0, and for
n = 1, 2, 3, ..., let λ
2m, n(
x) denote the Lebesgue function associated with (0, 1,..., 2
m) Hermite-Fejér polynomial interpolation at the Chebyshev nodes {cos[(2
k−1) π/(2
n)]:
k=1, 2, ...,
n}. We examine the Lebesgue constant
Λ2m, n max{λ
2m, n(
x): −1 ≤
x ≤ 1}, and show that
Λ2m, n = λ
m, n(1), thereby generalising a result of H. Ehlich and K. Zeller for Lagrange interpolation on the Chebyshev nodes. As well, the infinite term in the asymptotic expansion of
Λ2m, n) as
n → ∞ is obtained, and this result is extended to give a complete asymptotic expansion for
Λ2, n.
相似文献