Minimax estimation under convex loss when the parameter interval is bounded

In the present paper families of truncated distributions with a Lebesgue density forx=(x ₁,...,x _n ) ε ℝ ⁿ are considered, wheref ₀:ℝ → (0, ∞) is a known continuous function andC _n (ϑ) denotes a normalization constant. The unknown truncation parameterϑ which is assumed to belong to a bounded parameter intervalΘ=[0,d] is to be estimated under a convex loss function. It is studied whether a two point prior and a corresponding Bayes estimator form a saddle point when the parameter interval is sufficiently small.     

Berry–Esseen bounds for density estimates under NA assumption

Let {X _j } be a strictly stationary sequence of negatively associated random variables with the marginal probability density function f(x). The recursive kernel estimators of f(x) are defined by

Consistent estimation in the presence of incidental parameters

Summary Let (X,A) be a measurable space andP _ϑη |A (ϑη) ∈ Θ x H, ∥A, (θ, η) ∈ Θ×H, a parametrized family of probability measures (for short:p-measures). This paper is concerned with the problem of consistently estimatingθ from realizations governed by , where η^u ∈ H, v ∈ ℕ, are unknown.     

On Minimally-supported <Emphasis Type="Italic">D</Emphasis>-optimal Designs for Polynomial Regression with Log-concave Weight Function

This paper studies minimally-supported D-optimal designs for polynomial regression model with logarithmically concave (log-concave) weight functions. Many commonly used weight functions in the design literature are log-concave. For example, and exp(−x ²) in Theorem 2.3.2 of Fedorov (Theory of optimal experiments, 1972) are all log-concave. We show that the determinant of information matrix of minimally-supported design is a log-concave function of ordered support points and the D-optimal design is unique. Therefore, the numerically D-optimal designs can be constructed efficiently by cyclic exchange algorithm.     

A note on generalized wald’s method

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.     

Tests of significance for the average square deviation from target in a finite lot

Consider lots of discrete items 1, 2, …,N with quality characteristicsx ₁,x ₂, …,x _N . Leta be a target value for item quality. Lot quality is identified with the average square deviation from target per item in the lot (lot average square deviation from target). Under economic considerations this is an appropriate lot quality indicator if the loss respectively the profit incurred from an item is a quadratic function ofx _i −a. The present paper investigates tests of significance on the lot average square deviationz under the following assumptions: The lot is a subsequence of a process of production, storage, transport; the random quality characteristics of items resulting from this process are i.i.d. with normal distributionN(μ, σ ²); the target valuea coincides with the process meanμ.     

On characterizations of distributions by regression of adjacent generalized order statistics

Let , r ≥ 1, denote generalized order statistics, with arbitrary parameters , based on distribution function F. In this paper we characterize continuous distributions F by the regression of adjacent generalized order statistics, i.e. where are continuous and increasing functions and ψ is strictly increasing. Further we investigate in detail the case when ψ(x) = x and g is a linear function of the form g(x) = cx + d for some .     

Contributions to nonparametric generalized failure rate function estimation

For a specified distribution functionG with densityg, and unknown distribution functionF with densityf, the generalized failure rate function (x)=f(x)/gG ^–1 F(x) may be estimated by replacingf andF byf _n and , wheref _n is an empirical density function based on a sample of sizen from the distribution functionF, and . Under regularity conditions we show and, under additional restrictions whereC is a subset ofR and _n. Moreover, asymptotic normality is derived and the Berry-Esséen type bound is shown to be related to a theorem which concerns the sum of i.i.d. random variables. The order boundO(n^–1/2+c _n ^1/2 ) is established under mild conditions, wherec _n is a sequence of positive constants related tof _n and tending to 0 asn.Research was supported in part by the Army, Navy and Air Force under Office of Naval Research contract No. N00014-76-C-0608. AMS 1970 subject classifications. Primary 62G05. Secondary 60F15.     

Distributions determined by conditioning on a pair of order statistics

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

Die messende Prüfung bei Abweichung von der Normalverteilungsannahme

Summary For sampling inspection by variables in the one-sided case (item bad if variablex>a) under the usual assumption of normality with known variance ₂ the operating characteristic is given by , wherep denotes the fraction defective. If instead of a normal distribution ((·–a–)/) there is a distributionF((·–a–)/) whereF is sufficiently regular and normed like , one has the approximative operating characteristic . It is shown that for arbitrarily fixed parametersn andc the function takes the valueL _n,c ⁽⁾ (p) at the pointp _F (p)=1–F(–^–1(p)). Sufficient conditions for a simple behavior of the differencep _F (p)–p are given. In the cases of rectangular and symmetrically truncated normal distribution these conditions are shown to be fulfilled.     

Boundedly complete families which are not complete

Summary Completeness of a family of probability distributions implies its bounded completeness but not conversely. An example of a family which is boundedly complete but not complete was presented by Lehmann and Scheffe [5]. This appears to be the only such example quoted in the statistical literature. The purpose of this note is to provide further examples of this type. It is shown that any given family of power series distributions can be used to construct a class containing infinitely many boundedly complete, but not complete, families. Furthermore, it is shown that the family of continuous distributions , is boundedly complete, but not complete, whereU denotes the uniform distribution on [a, b] and {P _ϑ,ϑ ∈ IR}, is a translation family generated by a distributionP ₀ with mean value zero, which is continuous with respect to the Lebesgue measure.     

The average number of events per time unit and its estimation

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.     

A note on minimum variance

Summary Minimizing is discussed under the unbiasedness condition: and the condition (A):f _i (x) (i=1, ..., p) are linearly independent , and .     

The distribution of the maximum of a first order autoregressive process: the continuous case

We give the cumulative distribution function of M _n , the maximum of a sequence of n observations from an autoregressive process of order 1. Solutions are first given in terms of repeated integrals and then for the case, where the underlying random variables are absolutely continuous. When the correlation is positive, $$P \left( M_n \leq x \right)\ =a_{n,x},$$ where $$a_{n,x}= \sum_{j=1}^\infty \beta_{jx}\ \nu_{jx}^{n} = O \left( \nu_{1x}^{n}\right),$$ where {?? _jx } are the eigenvalues of a non-symmetric Fredholm kernel, and ?? _1x is the eigenvalue of maximum magnitude. When the correlation is negative $$P \left( M_n \leq x \right)\ =a_{n,x} +a_{n-1,x}.$$ The weights ?? _jx depend on the jth left and right eigenfunctions of the kernel. These are given formally by left and right eigenvectors of an infinite Toeplitz matrix whose eigenvalues are just {?? _jx }. These results are large deviations expansions for extremes, since the maximum need not be standardized to have a limit. In fact, such a limit need not exist. The use of the derived expansion for P(M _n ?? x) is illustrated using both simulated and real data sets.     

A consistent test for multivariate normality based on the empirical characteristic function

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.     

Reproducibility in the one-parameter exponential family

Summary LetX ₁, ...,X _n be i.i.d. random variables with common distribution an element of a linear one-parameter exponential family indexed by a natural parameter . It is proved that the distribution of is an element ofF, for all andn=1, 2, ... if and only ifF is a family of scale transformed Poisson distributions.     

Characterization of the exponential distribution function by Properties of the differenceX k+s∶n−X k∶n of order statistics

We considern independent and identically distributed random variables with common continuous distribution functionF concentrated on (0, ∞). LetX _1∶n≤X_2∶n...≤X_n∶n be the corresponding order statistics. Put $$d_s \left( x \right) = P\left( {X_{k + s:n} - X_{k:n} \geqslant x} \right) - P\left( {X_{s:n - k} \geqslant x} \right), x \geqslant 0,$$ and $$\delta _s \left( {x, \rho } \right) = P\left( {X_{k + s:n} - X_{k:n} \geqslant x} \right) - e^{ - \rho \left( {n - k} \right)x} ,\rho > 0,x \geqslant 0.$$ Fors=1 it is well known that each of the conditions d₁(x)=O ?x≥0 and δ₁ (x, p) = O ?x≥0 implies thatF is exponential; but the analytic tools in the proofs of these two statements are radically different. In contrast to this in the present paper we present a rather elementary method which permits us to derive the above conclusions for somes, 1≤n —k, using only asymptotic assumptions (either forx→0 orx→∞) ond _s(x) and δ₁ (x, p), respectively.     

Über die Anzahl von Zufallspunkten mit typ-gleichem nächsten Nachbarn und einen multivariaten Zwei-Stichproben-Test

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.