Detlef Plachky

Boundedly complete families which are not complete

SummaryCompleteness 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 % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiWaaeaada% WcaaqaaiaaikdaaeaacaaIZaaaaiaadwfacaGGOaGaamyyaiaacYca% caWGIbGaaiykaiabgUcaRmaalaaabaGaaGymaaqaaiaaiodaaaGaam% iuamaaBaaaleaacqGHsislcaWGHbGaeyOeI0IaamOyaaqabaGccaGG% SaGaamyyaiaacYcacaWGIbGaeyicI48efv3ySLgznfgDOjdaryqr1n% gBPrginfgDObcv39gaiuaacqWFDeIuieaacaGFSaacbiGaa0xyaiaa% 9XdacaqFIbaacaGL7bGaayzFaaaaaa!57F8!

From Optimality Robustness To Sufficiency and Completeness

\left\{ {\frac{2}{3}U(a,b) + \frac{1}{3}P_{ - a - b} ,a,b \in \mathbb{R},a< b} \right\}

A Statistical Property of the Arrangements of a Finite Set

, is boundedly complete, but not complete, whereU denotes the uniform distribution on [a, b] and {Pϑ,ϑ ∈ IR}, is a translation family generated by a distributionP0 with mean value zero, which is continuous with respect to the Lebesgue measure.

Characterization of some types of completeness resp. Total completeness and their conservation under direct products

SYNOPTIC ABSTRACT A relation between the negative binomial and the negative hypergeometric distributions based on conditional distributions is proved. This relation allows to compute the critical value of an UMPU level a test concerning the comparison of probabilities of success in the one-sided case based on observed waiting times and to determine a UMVU estimator of the density of a negative binomial distribution in terms of a negative hypergeometric distribution. Moreover, some aspects of renewal theory are discussed.

On a Theorem of Polya

Let (\Omega,{\cal A},{\cal P}) stand for a statistical experiment and {\cal B},{\cal C} for some sub- σ -algebras of {\cal A} with {\cal C}\subset {\cal B} . It is shown that for any {\cal B} -measurable d\in\bigcap_{P\in {\cal P}}\,{\cal L}_{2}(\Omega,{\cal A},P) there exists some d_{1}\in\bigcap_{P\in {\cal P}}{\cal L}_{2}(\Omega,{\cal A},P) being {\cal C} -measurable and a UMVU estimator in (\Omega,{\cal A},{\cal P}) and some conditional white noise d_{2}\in\bigcap_{P\in {\cal P}}\,{\cal L}_{2}(\Omega,{\cal A},P) , i.e. E_{P}(d_{2}\vert {\cal C})=0,P\in {\cal P} , satisfying d=d_{1}+d_{2} , where d_{j},j=1,2 , are uniquely determined up to P -zero sets, if and only if {\cal C} is sufficient and complete for {\cal P}\vert {\cal B} and {\cal B} is optimality robust for {\cal P} , i.e. any {\cal B} -measurable d\in\bigcap_{P\in {\cal P}}\,{\cal L}_{2}(\Omega,{\cal A},P) being some UMVU estimator in the restricted statistical experiment (\Omega,{\cal B},{\cal P}\vert {\cal B}) is already a UMVU estimator in the original statistical experiment (\Omega,{\cal A},{\cal P}) . In particular, the special case {\cal B}={\cal A} characterizes sufficiency and completeness of {\cal C} for {\cal P} and the special case {\cal B}={\cal C} optimality robustness and completeness of {\cal C} for {\cal P} from a decomposition theoretical point of view. As an application it is shown that a σ -algebra containing a sufficient and complete sub- σ -algebra is optimality robust without being itself in general neither sufficient nor complete.

On the optimality of the sample variance

It is well-known that the region of risk for testing simple hypotheses is some closed, convex, and (1/2, 1/2)-symmetric subset of the unit square, which contains the points (0, 0) and (1, 1). It is shown that for any such subsetR of the unit square and any atomless probability measureP on some σ-algebra there exists some probability measureQ on the same σ-algebra such thatR is the corresponding region of risk for testingP againstQ. This generalizes a result of [4] and is as a first step derived here for the special case, whereP is equal to the uniform distribution on the unit interval. The corresponding distributionQ is given explicitly in this case and the general case is treated by some well-known measure-isomorphism. This method of proof shows thatQ might be chosen to be of typeQ=λQ1+(1−λ)Q2 for some λ satisfying 0≤λ≤1, whereQ1 is a probability measure, which is absolutely continuous with respect toP andQ2 is a one-point mass.

A Comparison of Two Exponential Families

It is shown that the number of non-void arrangements of {1, … k}, where is some realization of a random variable X with distribution defined by ,which occur in connection with the rencontre problem (as the probability for m fixed points of permutations of {1, … ,n} selected with probability ), is the uniquely determined unbiased estimator of . Here stands for the one-point mass at zero. Furthermore, the class of all unbiased estimators of n with respect to the family , where the case n = 0 is excluded, is determined and all locally MVU estimators at every in the model are characterized. In particular, it turns out that there does not exist a uniformly MVU resp. a locally MVU estimator for n at every in the model .

Note on sums of a random number of random variables with exponential family distributions

Some notions ofLp(μ)-completeness resp. totally Lp(μ)-completeness (1≦p≦∞) are characterized for families of probability distributions dominated by aσ-finite measureμ and their conservation with respect to direct products is proved. Furthermore, it is shown that totallyL∞(μ)-completeness does not implyL1(μ)-completeness and that there are families of probability distributions in the i.i.d. case induced by the order statistic, which are L1(μ)-complete but not totallyL∞(μ)-complete.