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1.
Let X
1, X
2, ..., X
n
be independent exponential random variables such that X
i
has failure rate λ for i = 1, ..., p and X
j
has failure rate λ* for j = p + 1, ..., n, where p ≥ 1 and q = n − p ≥ 1. Denote by D
i:n
(p,q) = X
i:n
−X
i-1:n
the ith spacing of the order statistics X
1:n
≤ X
2:n
≤ ... ≤ X
n:n
, i = 1, ..., n, where X
0:n
≡ 0. The purpose of this paper is to investigate multivariate likelihood ratio orderings between spacings D
i:n
(p,q), generalizing univariate comparison results in Wen et al.(J Multivariate Anal 98:743–756, 2007). We also point out that
such multivariate likelihood ratio orderings do not hold for order statistics instead of spacings.
Supported by National Natural Science Foundation of China, the Program for New Century Excellent Talents in University (No.:
NCET-04-0569), and by the Knowledge Innovation Program of the Chinese Academy of Sciences (No.: KJCX3-SYW-S02). 相似文献
2.
Prof. Dr. A. Irle 《Metrika》1987,34(1):107-115
Summary LetX
1,X
2, ... form a sequence of martingale differences and denote byZ(a, α) = sup n (S
n
−an
α)+ the largest excess forS
n
=X
1 + ... +X
n
crossing the boundaryan
α. We give a sufficient condition for the finiteness ofEZ(a, α)β which is formulated in terms of bounds forE(X
i
+
p
andE(|X
i
|γ|X
1, ...,X
i-1), whereα, β, γ, p are suitably related. This general result is then applied to the case of independent random variables. 相似文献
3.
LetX
1,…,X
m
andY
1,…,Y
n
be two independent samples from continuous distributionsF andG respectively. Using a Hoeffding (1951) type theorem, we obtain the distributions of the vector S=(S
(1),…,S
(n)), whereS
(j)=# (X
i
’s≤Y
(j)) andY
(j) is thej-th order statistic ofY sample, under three truncation models: (a)G is a left truncation ofF orG is a right truncation ofF, (b)F is a right truncation ofH andG is a left truncation ofH, whereH is some continuous distribution function, (c)G is a two tail truncation ofF. Exploiting the relation between S and the vectorR of the ranks of the order statistics of theY-sample in the pooled sample, we can obtain exact distributions of many rank tests. We use these to compare powers of the
Hajek test (Hajek 1967), the Sidak Vondracek test (1957) and the Mann-Whitney-Wilcoxon test.
We derive some order relations between the values of the probagility-functions under each model. Hence find that the tests
based onS
(1) andS
(n) are the UMP rank tests for the alternative (a). We also find LMP rank tests under the alternatives (b) and (c). 相似文献
4.
Prof. Dr. W. Stute 《Metrika》1992,39(1):257-267
LetX
1, ...,X
n
be an i.i.d. sample from some parametric family {θ :θ (Θ} of densities. In the random censorship model one observesZ
i
=min (X
i
,Y
i
) andδ
i
=1{
x
i
≤Y
i}, whereY
i
is a censoring variable being independent ofX
i
. In this paper we investigate the strong consistency ofθ
n
maximizing the modified likelihood function based on (Z
i
,δ
i
, 1≤i≤n. The main result constitutes an extension of Wald’s theorem for complete data to censored data.
Work partially supported by the “Deutsche Forschungsgemeinschaft”. 相似文献
5.
A consistent test for multivariate normality based on the empirical characteristic function 总被引:2,自引:1,他引:1
LetX
1,X
2, …,X
n be independent identically distributed random vectors in IR
d
,d ⩾ 1, with sample mean
and sample covariance matrixS
n. We present a practicable and consistent test for the composite hypothesisH
d: the law ofX
1 is a non-degenerate normal distribution, based on a weighted integral of the squared modulus of the difference between the
empirical characteristic function of the residualsS
n
−1/2
(X
j −
) and its pointwise limit exp (−1/2|t|2) underH
d. The limiting null distribution of the test statistic is obtained, and a table with critical values for various choices ofn andd based on extensive simulations is supplied. 相似文献
6.
Michael Cramer 《Metrika》1997,46(1):187-211
The asymptotic distribution of a branching type recursion with non-stationary immigration is investigated. The recursion is
given by
, where (X
l
) is a random sequence, (L
n
−1(1)
) are iid copies ofL
n−1,K is a random number andK, (L
n
−1(1)
), {(X
l
),Y
n
} are independent.
This recursion has been studied intensively in the literature in the case thatX
l
≥0,K is nonrandom andY
n
=0 ∀n. Cramer, Rüschendorf (1996b) treat the above recursion without immigration with starting conditionL
0=1, and easy to handle cases of the recursion with stationary immigration (i.e. the distribution ofY
n
does not depend on the timen).
In this paper a general limit theorem will be deduced under natural conditions including square-integrability of the involved
random variables. The treatment of the recursion is based on a contraction method.
The conditions of the limit theorem are built upon the knowledge of the first two moments ofL
n
. In case of stationary immigration a detailed analysis of the first two moments ofL
n
leads one to consider 15 different cases. These cases are illustrated graphically and provide a straight forward means to
check the conditions and to determine the operator whose unique fixed point is the limit distribution of the normalizedL
n
. 相似文献
7.
In questo lavoro si studia il problema di ricerca della distribuzione di probabilità comune da assegnare a due numeri aleatori discreti che assumano i primin valori interi naturali in modo che la loro somma abbia moda di minima probabilità.Il problema è affrontato sia dal punto di vista teorico tramite gli strumenti della programmazione matematica, sia dal punto di vista numerico.
Pervenuto il 22-1-82 相似文献
LetX andY be two random numbers with the same distribution function; in this paper we consider the problem of finding a random numberX+Y having mode with minimal probability. In particular we have considered only the case ofX andY assuming the firstn integer values, so thatp (dimensionn) is the common distribution andq (dimension 2n–1) is the distribution ofX+Y; then the problem is to minimizem=maxq 1.In the known literature it appears that theoretical results and numerical experience have brought to various conjectures not confermed. In this paper the problem is considered from the mathematical programming point of view. Several theoretical results are obtained even if the full solution of the problem is not reached. Anyway, such results, limiting the search range of a solution, suggested extended numerical testing, also for rather large values ofn, so that non trivial conclusions can be derived.
Pervenuto il 22-1-82 相似文献
8.
LetX
1,X
2, ...,X
n
(n≥3) be a random sample on a random variableX with distribution functionF having a unique continuous inverseF
−1 over (a,b), −∞≤a<b≤∞ the support ofF. LetX
1:n
<X
2:n
<...<X
n:n
be the corresponding order statistics. Letg be a nonconstant continuous function over (a,b). Then for some functionG over (a, b) and for some positive integersr ands, 1<r+1<s≤n
相似文献
9.
10.
Herbert Vogt 《Metrika》1996,44(1):207-221
Let ζ
t
be the number of events which will be observed in the time interval [0;t] and define
as the average number of events per time unit if this limit exists. In the case of i.i.d. waiting-times between the events,E[ζ
t
] is the renewal function and it follows from well-known results of renewal theory thatA exists and is equal to 1/τ, if τ>0 is the expectation of the waiting-times.
This holds true also when τ = ∞.A may be estimate by ζ
t
/t or
where
is the mean of the firstn waiting-timesX
1,X
2, ...,X
n
. Both estimators converage with probability 1 to 1/τ if theX
i are i.i.d.; but the expectation of
may be infinite for alln and also if it is finite,
is in general a positively biased estimator ofA. For a stationary renewal process, ζ
t
/t is unbiased for eacht; if theX
i
are i.i.d. with densityf(x), then ζ
t
/t has this property only iff(x) is of the exponential type and only for this type the numbers of events in consecutive time intervals [0,t], [t, 2t], ... are i.i.d. random variables for arbitraryt > 0. 相似文献
11.
Some properties of the minimum and the maximum of random variables with joint logconcave distributions 总被引:1,自引:1,他引:0
It is shown that if (X
1, X
2, . . . , X
n
) is a random vector with a logconcave (logconvex) joint reliability function, then X
P
= min
i∈P
X
i
has increasing (decreasing) hazard rate. Analogously, it is shown that if (X
1, X
2, . . . , X
n
) has a logconcave (logconvex) joint distribution function, then X
P
= max
i∈P
X
i
has decreasing (increasing) reversed hazard rate. If the random vector is absolutely continuous with a logconcave density
function, then it has a logconcave reliability and distribution functions and hence we obtain a result given by Hu and Li
(Metrika 65:325–330, 2007). It is also shown that if (X
1, X
2, . . . , X
n
) has an exchangeable logconcave density function then both X
P
and X
P
have increasing likelihood ratio. 相似文献
12.
Let {v
n(θ)} be a sequence of statistics such that whenθ =θ
0,v
n(θ
0)
N
p(0,Σ), whereΣ is of rankp andθ εR
d. Suppose that underθ =θ
0, {Σ
n} is a sequence of consistent estimators ofΣ. Wald (1943) shows thatv
n
T
(θ
0)Σ
n
−1
v
n(θ
0)
x
2(p). It often happens thatv
n(θ
0)
N
p(0,Σ) holds butΣ is singular. Moore (1977) states that under certain assumptionsv
n
T
(θ
0)Σ
n
−
v
n(θ
0)
x
2(k), wherek = rank (Σ) andΣ
n
−
is a generalized inverse ofΣ
n. However, Moore’s result as stated is incorrect. It needs the additional assumption that rank (Σ
n) =k forn sufficiently large. In this article, we show that Moore’s result (as corrected) holds under somewhat different, but easier
to verify, assumptions.
Research partly supported by the U.S. Army Research Office through the Mathematical Sciences Institute at Cornell University. 相似文献
13.
B. M. Bennett 《Metrika》1970,15(1):6-8
Summary LetN=[n
ij
] (i=1, …,r;j=1, …,c) be the matrix of observed frequencies in anr×c contingency table fromr possibly different multinomial populations with respective probabilitiesp
i
=(p
i1, …,p
ic
).Freeman andHalton have proposed an exact conditional test for the hypothesisH
0 :p
i
=(p
1, …p
c
) of the exact test is derived. Numerical values forβ(p) were previously computed for the special case:r=3,c=2 [Bennett andNakamura, 1964]. 相似文献
14.
F. Brodeau 《Metrika》1999,49(2):85-105
This paper is devoted to the study of the least squares estimator of f for the classical, fixed design, nonlinear model X (t
i)=f(t
i)+ε(t
i), i=1,2,…,n, where the (ε(t
i))i=1,…,n are independent second order r.v.. The estimation of f is based upon a given parametric form. In Brodeau (1993) this subject has been studied in the homoscedastic case. This time
we assume that the ε(t
i) have non constant and unknown variances σ2(t
i). Our main goal is to develop two statistical tests, one for testing that f belongs to a given class of functions possibly discontinuous in their first derivative, and another for comparing two such
classes. The fundamental tool is an approximation of the elements of these classes by more regular functions, which leads
to asymptotic properties of estimators based on the least squares estimator of the unknown parameters. We point out that Neubauer
and Zwanzig (1995) have obtained interesting results for connected subjects by using the same technique of approximation.
Received: February 1996 相似文献
15.
Summary LetX andY be two random vectors with values in ℝ
k
and ℝ∝, respectively. IfZ=(X
T,Y
T)
T
is multivariate normal thenX givenY=y andY givenX=x are (multivariate) normal; the converse is wrong. In this paper simple additional conditions are stated such that the converse
is true, too. Furthermore, the case is treated that the random vectorZ=(X
1
T
, …,X
t
T
)
T
is splitted intot≥3 partsX
1, …,X
t. 相似文献
16.
LetX be ap-normal random vector with unknown mean and unknown covariance matrix and letX be partitioned asX=(X
(1)
,X
(2)
, ...,X
(r)
) whereX
(j) is a subvector of dimensionp
j such that
j=1
r
p
j
=p. We show that the tests, obtained by Dahel (1988), are locally minimax. These tests have been derived to confront Ho: =0 versusH
1: 0 on the basis of sample of sizeN, X
1, ..., XN, drawn fromX andr additional samples of sizeN
j, U
i
(j)
, i=1, ..., Nj, drawn fromX
(1), ...X
(r) respectively. We assume that the (r+1) samples are independent and thatN
j>p
j forj=0, 1, ..., r (N
oN andp
op). Whenr=2 andp=2, a Monte Carlo study is performed to compare these tests with the likelihood ratio test (LRT) given by Srivastava (1985). We also show that no locally most powerful invariant test exists for this problem. 相似文献
17.
Dr. N. Henze 《Metrika》1984,31(1):259-273
Summary For independents-variate samplesX
1, ...,X
m i.i.d.f. (.),Y
1, ...,Y
n i.i.d. g. (.), where the densitiesf (.),g (.) are assumed to be continuous on their respective sets of positivity, consider the numberT
m,n of pointsZ of the pooled sample (which are either of typeX or of typeY) such that the nearest neighbor ofZ is of the same type asZ. We show that, as
, independently of (.). An omnibus test for the two sample problem f(.)g(.) orf(.)g(.)? may be obtained by rejecting the hypothesisf(.)g(.) for large values ofT
m,n. 相似文献
18.
We propose a class of nonparametric tests for testing non-stochasticity of the regression parameterβ in the regression modely
i
=βx
i
+ɛ
i
,i=1, ...,n. We prove that the test statistics are asymptotically normally distributed both underH
0 and under contiguous alternatives. The asymptotic relative efficiencies (in the Pitman sense) with respect to the best parametric
test have also been computed and they are quite high. Some simulation studies are carried out to illustrate the results.
Research was supported by the University Grants Commission, India. 相似文献
19.
Michael Kohler 《Metrika》1998,47(1):147-163
Let (X, Y) be a pair of random variables withsupp(X)⊆[0,1]
l
andEY
2<∞. Letm
* be the best approximation of the regression function of (X, Y) by sums of functions of at mostd variables (1≤d≤l). Estimation ofm
* from i.i.d. data is considered.
For the estimation interaction least squares splines, which are defined as sums of polynomial tensor product splines of at
mostd variables, are used. The knot sequences of the tensor product splines are chosen equidistant. Complexity regularization is
used to choose the number of the knots and the degree of the splines automatically using only the given data.
Without any additional condition on the distribution of (X, Y) the weak and strongL
2-consistency of the estimate is shown. Furthermore, for everyp≥1 and every distribution of (X, Y) withsupp(X)⊆[0,1]
l
,y bounded andm
*
p-smooth, the integrated squared error of the estimate achieves up to a logarithmic factor the (optimal) rate
相似文献
20.
Suppose the observations (X
i,Y
i), i=1,…, n, are ϕ-mixing. The strong uniform convergence and convergence rate for the estimator of the regression function was studied
by serveral authors, e.g. G. Collomb (1984), L. Gy?rfi et al. (1989). But the optimal convergence rates are not reached unless
the Y
i are bounded or the E exp (a|Y
i|) are bounded for some a>0. Compared with the i.i.d. case the convergence of the Nadaraya-Watson estimator under ϕ-mixing variables needs strong moment
conditions. In this paper we study the strong uniform convergence and convergence rate for the improved kernel estimator of
the regression function which has been suggested by Cheng P. (1983). Compared with Theorem A in Y. P. Mack and B. Silverman
(1982) or Theorem 3.3.1 in L. Gy?rfi et al. (1989), we prove the convergence for this kind of estimators under weaker moment
conditions. The optimal convergence rate for the improved kernel estimator is attained under almost the same conditions of
Theorem 3.3.2 in L. Gy?rfi et al. (1989).
Received: September 1999 相似文献
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