Let Xn1,…,Xnn be the observations from a chirp type statistical model Xnt, Xnt=Acos⁡ (ωt+Δ/nt2)+Bsin⁡ ωt+Δ/nt2+ϵt, where ϵt is a stationary noise. We consider a method of estimation of parameters, A, B, ω, Δ, and ν, (where ν is the variance of ϵt’s) which is basically an approximate least-squares method. The main advantage of the proposed approach is that no assumptions are required. We make use of the three theorems which were established associated with the kernel ∑t=1neiut+vt2 and then use them to prove, under certain conditions, the consistency of the estimators.

For a Polish Sample Space with a Borel σ-field with a surjective measurable transformation, we define an equivalence relation on sample points according to their ergodic limiting averages. We show that this equivalence relation partitions the subset of sample points on measurable invariant subsets, where each limiting distribution is the unique ergodic probability measure defined on each set. The results obtained suggest some natural objects for the model of a probabilistic time-invariant phenomenon are uniquely ergodic probability spaces. As a consequence of the results gained in this paper, we propose a notion of randomness that is weaker than recent approaches to Schnorr randomness.

In a recent paper, the authors derived the exact solution for the probability mass function of the geometric distribution of orderk, expressing the roots of the associated auxiliary equation in terms of generating functions for Fuss-Catalan numbers. This paper applies the above formalism for the Fuss-Catalan numbers to treat additional problems pertaining to occurrences of success runs. New exact analytical expressions for the probability mass function and probability generating function and so forth are derived. First, we treat sequences of Bernoulli trials withr≥1occurrences of success runs of lengthkwithl-overlapping. The casel<0, where there must be a gap of at leastltrials between success runs, is also studied. Next we treat the distribution of the waiting time for therthnonoverlapping appearance of a pair of successes separated by at mostk-2failures (k≥2).

Structured spatial point patterns appear in many applications within the natural sciences. The points often record the location of key features, called landmarks, on continuous object boundaries, such as anatomical features on a human face. In other situations, the points may simply be arbitrarily spaced marks along a smooth curve, such as on handwritten numbers. This paper proposes novel exploratory methods for the identification of structure within point datasets. In particular, points are linked together to form curves which estimate the original shape from which the points are the only recorded information. Nonparametric regression methods are applied to polar coordinate variables obtained from the point locations and periodic modelling allows closed curves to be fitted even when data are available on only part of the boundary. Further, the model allows discontinuities to be identified to describe rapid changes in the curves. These generalizations are particularly important when the points represent shapes which are occluded or are intersecting. A range of real-data examples is used to motivate the modelling and to illustrate the flexibility of the approach. The method successfully identifies the underlying structure and its output could also be used as the basis for further analysis.

Thetdistribution is a useful extension of the normal distribution, which can be used for statistical modeling of data sets with heavy tails, and provides robust estimation. In this paper, in view of the advantages of Bayesian analysis, we propose a new robust coefficient estimation and variable selection method based on Bayesian adaptive Lassotregression. A Gibbs sampler is developed based on the Bayesian hierarchical model framework, where we treat thetdistribution as a mixture of normal and gamma distributions and put different penalization parameters for different regression coefficients. We also consider the Bayesiantregression with adaptive group Lasso and obtain the Gibbs sampler from the posterior distributions. Both simulation studies and real data example show that our method performs well compared with other existing methods when the error distribution has heavy tails and/or outliers.

We give an alternative duality-based proof to the solution of the expected utility maximization problem analyzed by Kim and Omberg. In so doing, we also provide an example of incomplete-market optimal investment problem for which the duality approach is conducive to an explicit solution.

This paper investigates a generalization of Fisher’s entropy type information measure under the multivariateγ-order normal distribution, related to his measure, as well as its corresponding Shannon entropy. Certain boundaries of this information measure are also proved and discussed.

In this communication, we deal with a generalized residual entropy of record values and weighted distributions. Some results on monotone behaviour of generalized residual entropy in record values are obtained. Upper and lower bounds are presented. Further, based on this measure, we study some comparison results between a random variable and its weighted version. Finally, we describe some estimation techniques to estimate the generalized residual entropy of a lifetime distribution.

For aT-variate density function, the present paper defines the point-symmetry, quasi-point-symmetry of orderk(<T), and the marginal point-symmetry of orderkand gives the theorem that the density function isT-variate point-symmetric if and only if it is quasi-point-symmetric and marginal point-symmetric of orderk. The theorem is illustrated for the multivariate normal density function.

This paper presents a technique for estimating finite population mean of the study variable in the presence of two auxiliary variables using two-phase sampling scheme when the regression line does not pass through the neighborhood of the origin. The properties of the proposed class of estimators are studied under large sample approximation. In addition, bias and efficiency comparisons are carried out to study the performances of the proposed class of estimators over the existing estimators. It has also been shown that the proposed technique has greater applicability in survey research. An empirical study is carried out to demonstrate the performance of the proposed estimators.