On the asymptotic distribution of the likelihood ratio under the regularity conditions due to doob

Summary The concepts of the asymptotic maximum likelihood estimates—AMLEs in short—and their asymptotic identity are introduced in section 1. They seem to be more adequate than the usual one for uses in the large sample theory. The AMLE is a slightly weakened version of the usual maximum likelihood estimate and therefore it should have a bit wider applicability than the original one. The asymptotic normality of a consistent AMLE and Wilks’ theorem concerning the asymptotic distribution of the statistic —2 log λ, where λ is the likelihood ratio, can be obtained under the regularity conditions due to Doob in section 2. A set of conditions which assure the existence of a unique and consistent AMLE is presented in section 3 and in the final section 4 the proof of the existence of the unique and consistent AMLE under those conditions is given. This work has been motivated by the work of Ogawa, Moustafa and Roy [3].     

A remark on the incomparability of two criteria for a uniform convergence of probability measures

Summary This paper connects with Theorem 3 of the author’s paper [1], in which two criteria for type (B) _d convergence ([3]) are shown to be incomparable to each other by presenting two examples. However, the statement of the theorem is not complete. In the present paper, we shall modify the statement of the theorem and give a proof by presenting a new example.     

Swelling of nylon 6 film due to water sorption

Dimensional changes in nylon 6 film on absorption and desorption of water were studied as functions of time and the amount of absorbed water. For absorption, a plot of film expansion versus the square root of time was sigmoidal in the initial region and did not coincide with the sorption curve: the latter was linear in the same region. This behavior is very similar to that reported for nylon 6 fiber by Kunzman. A theoretical analysis based on certain assumptions about the molecular state of nylon 6 gave good agreement with experimental results. By this analysis it was shown that the characteristic swelling behavior arises from instantaneous relaxation of the polymer molecules and is not connected with a time lag between water absorption and expansion of the polymer, as Kunzman proposed.     

Atomic emission spectrometry of metal oxides using shock tube excitation

The behavior of powdered samples in the region behind the incident and reflected shock wave were investigated. When the particle size is less than 8 μm, the powdered samples are effectively vaporized and excited to emit spectral lines. It was shown that the sample particles are scattered behind the incident wave and concentrated again in a narrow portion near the observation window by the use of a reflector so that intense emission can be attained. As an excitation source, the shock tube was used for the determination of trace metals contained in copper oxide.     

General description of mechanical anisotropy of semicrystalline polymers in relation to orientation of structural units composed of crystalline and noncrystalline materials

A mathematical representation of the transformation of an orientation function between two sets of Cartesian coordinates is discussed in terms of a series expansion of the distribution function in generalized spherical harmonics. A general procedure for calculating the mechanical anisotropy of a single-phase system (a polycrystalline material) from the orientation of its structural units and the intrinsic mechanical anisotropy of the structural unit is discussed in relation to the transformation of the orientation distribution function, i.e., mutual conversion of the coefficients in the expansion of the distribution function between the two sets of Cartesian coordinates. The procedure is extended to a two-phase systems (semicrystalline polymers) containing structural units composed of crystalline and noncrystalline materials in three different geometrical arrangements.     

Necessary conditions for the convergence of kullback-leibler’s mean information

Summary Two types of necessary conditions are given for the convergence of Kullback-Leibler’s mean information, one of which is connected with an asymptotic equivalence of two sequences of probability measures, and in special cases, with convergence of a sequence of probability distributions. The other is given in terms of the generalized probability density functions.