Mathematical Methods of Statistics
Mathematical Methods of Statistics is an international peer-reviewed journal dedicated to the mathematical foundations of statistical theory. It mostly publishes research papers with complete proofs and, occasionally, review papers on particular problems of statistics. Papers dealing with applications of statistics are also published if they contain new theoretical developments to the underlying statistical methods. The journal provides an outlet for research in advanced statistical methodology and studies where such methodology is effectively used or which stimulate its further development. The journal welcomes manuscripts from all countries.
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Том 28, № 4 (2019)
- Год: 2019
- Статей: 5
- URL: https://journal-vniispk.ru/1066-5307/issue/view/13912
Article
Admissibility of Invariant Tests for Means with Covariates
Аннотация
For a multinormal distribution with a p-dimensional mean vector θ and an arbitrary unknown dispersion matrix Σ, Rao ([8], [9]) proposed two tests for the problem of testing H0: θ1 = 0, θ2 = 0, Σ unspecified, versus H1: θ1 ≠ 0, θ2 = 0, Σ unspecified. These tests are known as Rao’s W-test and Rao’s U-test, respectively. In this paper, it is shown that Rao’s U-test is admissible while Hotelling’s T2-test is inadmissible.
243-261
On the Skewness Order of van Zwet and Oja
Аннотация
Van Zwet (1964) [16] introduced the convex transformation order between two distribution functions F and G, defined by F ≤cG if G−1 ∘ F is convex. A distribution which precedes G in this order should be seen as less right-skewed than G. Consequently, if F ≤cG, any reasonable measure of skewness should be smaller for F than for G. This property is the key property when defining any skewness measure.
In the existing literature, the treatment of the convex transformation order is restricted to the class of differentiable distribution functions with positive density on the support of F. It is the aim of this work to analyze this order in more detail. We show that several of the most well known skewness measures satisfy the key property mentioned above with very weak or no assumptions on the underlying distributions. In doing so, we conversely explore what restrictions are imposed on the underlying distributions by the requirement that F precedes G in convex transformation order.
262-278
State Occupation Probabilities in Non-Markov Models
Аннотация
The consistency of the Aalen—Johansen-derived estimator of state occupation probabilities in non-Markov multi-state settings is studied and established via a new route. This new route is based on interval functions and relies on a close connection between additive and multiplicative transforms of interval functions, which is established. Under certain assumptions, the consistency follows from explicit expressions of the additive and multiplicative transforms related to the transition probabilities as interval functions, which are obtained, in combination with certain censoring and positivity assumptions
279-290
Relative Error Prediction for Twice Censored Data
Аннотация
In this paper we consider the problem of non-parametric relative regression for twice censored data. We introduce and study a new estimate of the regression function when it is appropriate to assess performance in terms of mean squared relative error of prediction. We establish the uniform consistency with rate over a compact set and asymptotic normality of the estimator suitably normalized. The asymptotic variance is explicitly given. A Monte Carlo study is carried out to evaluate the performance of this estimate.
291-306
An Asymptotically Optimal Transform of Pearson’s Correlation Statistic
Аннотация
It is shown that for any correlation-parametrized model of dependence and any given significance level α ∈ (0, 1), there is an asymptotically optimal transform of Pearson’s correlation statistic R, for which the generally leading error term for the normal approximation vanishes for all values ρ ∈ (−1, 1) of the correlation coefficient. This general result is then applied to the bivariate normal (BVN) model of dependence and to what is referred to in this paper as the SquareV model. In the BVN model, Pearson’s R turns out to be asymptotically optimal for a rather unusual significance level α ≈ 0.240, whereas Fisher’s transform RF of R is asymptotically optimal for the limit significance level α = 0. In the SquareV model, Pearson’s R is asymptotically optimal for a still rather high significance level α ≈ 0.159, whereas Fisher’s transform RF of R is not asymptotically optimal for any α ∈ [0, 1]. Moreover, it is shown that in both the BVN model and the SquareV model, the transform optimal for a given value of α is in fact asymptotically better than R and RF in wide ranges of values of the significance level, including α itself. Extensive computer simulations for the BVN and SquareV models of dependence suggest that, for sample sizes n ≥ 100 and significance levels α ∈ {0.01, 0.05}, the mentioned asymptotically optimal transform of R generally outperforms both Pearson’s R and Fisher’s transform RF of R, the latter appearing generally much inferior to both R and the asymptotically optimal transform of R in the SquareV model.
307-318