An approach to fuzzy hypothesis testing

In this paper an approach is presented how to test fuzzily formulated hypotheses with crisp data. The quantitiesα andβ, the probabilities of the errors of type I and of type II, are suitably generalized and the concept of a best test is introduced. Within the framework of a one-parameter exponential distribution family the search for a best test is considerably reduced. Furthermore, it is shown under very weak conditions thatα andβ can simultaneously be diminished by increasing the sample size even in the case of testingH ₀ against the omnibus alternativeH ₁: notH ₀, a result completely different from the case of crisp setsH ₀ andH ₁: notH ₀.     

Largest excess of boundary crossings for martingales

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.     

Queuing with balking and reneging in M|G|1 systems

Summary AM|G|1 queuing process in which units balk with a constant probability (1−β) and renege according to a negative exponential distribution has been considered. The busy period process is first investigated making use of the supplementary variable technique and discrete transforms. The expression for the joint distribution of the number of customers serviced during a busy period and the length of the busy period has been derived. FollowingGaver (1959) the general process is investigated and making use of renewal theory the ergodic properties of the general process have been studied. It has been shown that as long as reneging is permitted (α>0), the steady states always exist, but when no reneging is permitted (α=0), the steady states exist only whenλ β η<1.     

On the power function of the exact test for ther×c contingency table

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].     

On the non-null distribution ofWilcoxon's signed rank test

Summary The properties of theWilcoxon signed rank sum testW ⁺ [Wilcoxon, 1945] for the hypothesis of symmetryH _o are discussed for alternativesH toH _o. The probability generating function and cumulant generating function ofW ⁺ are derived and a limiting form of the distribution is determined.     

Ein kombiniertes test & klassifikations-problem

Summary A decision process is considered which consists of two steps: First, a nullhypothesis H₀ is to be tested. If H₀ is rejected, a decision is to be made as to which of the alternative hypotheses H₁, H₂, ... H _k is valid. This second step is called "classification". It is assumed, that in case H₀ is not valid, each of the alternative hypotheses H₁, H₂, ... H _k has the same probability. Starting with this assumption, an optimal decision process is developed which has a specified level of significance α (i.e. by which the nullhypothesis H₀ is accepted with probability α, if it is valid), and for which the probability of a correct classification is a maximum in the case where the nullhypothesis is not valid. This decision process rests on a generalisation of the fundamental lemma of Neyman and Pearson, similar to that used in discriminant analysis.     

Rank tests for testing randomness of a regression coefficient in a linear regression model

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.     

Moment consistency of estimators in partially linear models under NA samples

Consider the heteroscedastic regression model Y ^(j)(x _in , t _in ) = t _in β + g(x _in ) + σ _in e ^(j)(x _in ), 1 ≤ j ≤ m, 1 ≤ i ≤ n, where s_in²=f(u_in){\sigma_{in}^{2}=f(u_{in})}, (x _in , t _in , u _in ) are fixed design points, β is an unknown parameter, g(·) and f(·) are unknown functions, and the errors {e ^(j)(x _in )} are mean zero NA random variables. The moment consistency for least-squares estimators and weighted least-squares estimators of β is studied. In addition, the moment consistency for estimators of g(·) and f(·) is investigated.     

A further note on the asymptotic efficiency of friedman’s x r 2 -test

Summary For independently distributed error components, the asymptotic relative efficiency (A.R.E.) ofFriedman’sx _r ² -tests with respect to the classical analysis of variance test has been studied byElteren andNoether andSen [1967]. The present note extends these results to the case of correlated errors arising in some random-effects or mixed-effects models. Work supported by the U.S. Army Research Office, Durham, Grant DA-ARO-D-31-124-G 746.     

Minimax estimators andΓ-minimax estimators for a bounded normal mean under the lossl p (θ, d)=|θ-d|p

Summary Let the random variableX be normal distributed with known varianceσ ²>0. It is supposed that the unknown meanθ is an element of a bounded intervalΘ. The problem of estimatingθ under the loss functionl _p (θ, d)=|θ-d| ^p p≥2 is considered. In case the length of the intervalθ is sufficiently small the minimax estimator and theΓ(β, τ)-minimax estimator, whereΓ(β, τ) represents special vague prior information, are given.     

Two families of kurtosis measures

Two families of kurtosis measures are defined as K ₁(b)=E[ab ^−|z|] and K ₂(b)=E[a(1−|z|^b)] where z denotes the standardized variable and a is a normalizing constant chosen such that the kurtosis is equal to 3 for normal distributions. K ₂(b) is an extension of Stavig's robust kurtosis. As with Pearson's measure of kurtosis β₂=E[z ⁴], both measures are expected values of continuous functions of z that are even, convex or linear and strictly monotonic in ℜ⁻ and in ℜ⁺. In contrast to β₂, our proposed kurtosis measures give more importance to the central part of the distribution instead of the tails. Tests of normality based on these new measures are more sensitive with respect to the peak of the distribution. K ₁(b) and K ₂(b) satisfy Van Zwet's ordering and correlate highly with other kurtosis measures such as L-kurtosis and quantile kurtosis. RID="*" ID="*"  The authors thank the referees for their insightful comments that significantly improved the clarity of the article.     

On kalman filtering,posterior mode estimation and fisher scoring in dynamic exponential family regression

Summary Dynamic exponential family regression provides a framework for nonlinear regression analysis with time dependent parametersβ ₀,β ₁, …,β _t, …, dimβ _t=p. In addition to the familiar conditionally Gaussian model, it covers e.g. models for categorical or counted responses. Parameters can be estimated by extended Kalman filtering and smoothing. In this paper, further algorithms are presented. They are derived from posterior mode estimation of the whole parameter vector (β′₀, …,β′_t) by Gauss-Newton resp. Fisher scoring iterations. Factorizing the information matrix into block-bidiagonal matrices, algorithms can be given in a forward-backward recursive form where only inverses of “small”p×p-matrices occur. Approximate error covariance matrices are obtained by an inversion formula for the information matrix, which is explicit up top×p-matrices. Heinz Leo Kaufmann, my friend and coauthor for many years, died in a tragical rock climbing accident in August 1989. This paper is dedicated to his memory.     

Locally minimax test of independence in elliptically symmetrical distributions with additional observations

Summary LetX=(X _ij )=(X ₁, ...,X _n )’,X’ _i =(X _i1, ...,X _ip )’,i=1,2, ...,n be a matrix having a multivariate elliptical distribution depending on a convex functionq with parameters, 0,σ. Let ϱ²=ϱ ₂ ^-2 be the squared multiple correlation coefficient between the first and the remainingp ₂+p ₃=p−1 components of eachX _i . We have considered here the problem of testingH ₀:ϱ²=0 against the alternativesH ₁:ϱ ₁ ^-2 =0, ϱ ₂ ^-2 >0 on the basis ofX andn ₁ additional observationsY ₁ (n ₁×1) on the first component,n ₂ observationsY ₂(n ₂×p ₂) on the followingp ₂ components andn ₃ additional observationsY ₃(n ₃×p ₃) on the lastp ₃ components and we have derived here the locally minimax test ofH ₀ againstH ₁ when ϱ ₂ ^-2 →0 for a givenq. This test, in general, depends on the choice ofq of the familyQ of elliptically symmetrical distributions and it is not optimality robust forQ.     

Determination of uniformly most powerful tests in discrete sample spaces

Uniformly most powerful (UMP) tests are known to exist in one-parameter exponential families when the hypothesis H ₀ and the alternative hypothesis H ₁ are given by (i) H ₀ : θ≤θ₀, H ₁ : θ>θ₀, and (ii) H ₀ : θ≤θ₁ or θ≥θ₂, H ₁ : θ₁<θ<θ₂, where θ₁<θ₂.  Likewise, uniformly most powerful unbiased (UMPU) tests do exist when the hypotheses H ₀ and H ₁ take the form (iii) H ₀ : θ₁≤θ≤θ₂, H ₁ : θ<θ₁ or θ>θ₂, where θ₁<θ₂, and (iv) H ₀ : θ=θ₀, H ₁:θ≠θ₀.  To determine tests in case (i), only one critical value c and one randomization constant γ have to be computed. In cases (ii) through (iv) tests are determined by two critical values c ₁, c ₂ and two randomization constants γ₁, γ₂. Unlike determination of tests in case (i), computation of critical values and randomization constants in the remaining cases is rather difficult, unless distributions are symmetric. No straightforward method to determine two-sided UMP tests in discrete sample spaces seems to be known. The purpose of this note is to disclose a distribution independent principle for the determination of UMP tests in cases (ii) through (iv). Received: March 1999     

On characterizing discrete distributions via conditional expectations of weak record values

Let (W _n ,n ≥ 0) denote the sequence of weak records from a distribution with support S = { α₀,α₁,...,α _N }. In this paper, we consider regression functions of the form ψ _n (x) = E(h(W _n ) |W _n+1 = x), where h(·) is some strictly increasing function. We show that a single function ψ _n (·) determines F uniquely up to F(α₀). Then we derive an inversion formula which enables us to obtain F from knowledge of ψ _n (·), ψ _n-1(·), h(·) and F(α₀).     

Multivariate multistage methodologies for simultaneous all pairwise comparisons

We consider the problem of constructing simultaneous fixed-width confidence intervals for all pairwise treatment differences μ₁−μ _J , in the presence ofk(≥2) independent populationsN _p (μ₁,Σ), 1≤i≠j≤k. Appropriate purely sequential, accelerated sequential and three-stage sampling strategies have been developed and variousfirst-order asymptotic properties are then derived when Σ _pxp is completely unknown, but positive definite (p.d.). In the two special cases when the largest component variance in Σ is a known multiple of one of the variances or Σ=σ² H where σ(>0) is unknown, butH _pxp is known and p.d., the original multistage sampling strategies are specialized. Under such special circumstances, associatedsecond-order characteristics are then developed. It is to be noted that our present formulation and the methodologies fill important voids in the context of multivariate multiple comparisons which is a challenging area that has not yet been fully explored. Moderate sample performances of the proposed techniques were very encouraging and detailed remarks on these were included in Mukhopadhyay and Aoshima (1997).     

Testing outliers from an exponential population

Summary Likes obtains the distributions ofDixon’s statistics for an exponential population and tabulates upper 100α% points (α=0.1, 0.05, 0.01) of some of these distributions. The distribution of these statistics can be expressed in terms of finite series of beta functions and hence the probabilities of the rejection of suspected observed outliers may be easily calculated on a desk calculator. Thus we may avoid the difficult task of tabulating 100α% values of these statistics.     

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.     

A bound on the expected overshoot for some concave boundaries

LetX ₁,X ₂,… be i.i.d. with finite meanμ>0,S _n =X ₁+…+X _n . Forf(n)=n ^β ,c>0 we consider the stopping timesT _c =inf{n:S _n >c+f(n)} with overshootR _c =S _{T c} −(c+f(T _c )). For 0<β<1 we give a bound for sup _c≥0 ER _c in the spirit of Lorden’s well-known inequality forf=0.