Analyzing Infant Mortality in Rural Bangladesh: A Frailty Modeling Approach

In practice, it may happen that data may arise from a hierarchical structure i.e., a cluster is nested within another cluster. In this case, nested frailty model is appropriate to analyze survival data to obtain optimal estimates of the parameters of interest. To identify significant determinants of infant mortality in rural Bangladesh, survival data have been extracted from Bangladesh Demographic and Health Survey (BDHS), 2014. Because of the presence of two-level clustering in data, nested frailty model has been employed for the purpose of analysis. Recommendations have been suggested based on the results obtained from the survival model to reduce the infant mortality in rural Bangladesh to a great extent. Dhaka Univ. J. Sci. 69(2): 63-69, 2021 (July)

Minimax prediction of sequences with periodically stationary increments

The problem of optimal estimation of linear functionals constructed from unobserved values of a stochastic sequence with periodically stationary increments based on its observations at points $ k<0$ is considered. For sequences with known spectral densities, we obtain formulas for calculating values of the mean square errors and the spectral characteristics of the optimal estimates of the functionals. Formulas that determine the least favourable spectral densities and minimax (robust) spectral characteristics of the optimal linear estimates of functionals are derived in the case where spectral densities of the sequence are not exactly known while some sets of admissible spectral densities are given.

MINIMAX ROOT–MEAN–SQUARE ESTIMATES OF MATRIX PARAMETERS IN LINEAR REGRESSION PROBLEMS UNDER UNCERTAINTY

The issues of parameter estimation in linear regression problems with random matrix coefficients were researched. Given that random linear functions are observed from unknown matrices with random errors that have unknown correlation matrices, the problems of guaranteed mean square estimation of linear functions of matrices were investigated. The estimates of the upper and lower guaranteed standard errors of linear estimates of observations of linear functions of matrices were obtained in the case when the sets are found, for which the unknown matrices and correlation matrices of observation errors are known. It was proved that for some partial cases such estimates are accurate. Assuming that the sets are bounded, convex and closed, more accurate two-sided estimates have been gained for guaranteed errors. The conditions when the guaranteed mean squared errors approach zero as the number of observations increases were found. The necessary and sufficient conditions for the unbiasedness of linear estimates of linear functions of matrices were provided. The notion of quasi-optimal estimates for linear functions of matrices was introduced, and it was proved that in the class of unbiased estimates, quasi-optimal estimates exist and are unique. For such estimates, the conditions of convergence to zero of the guaranteed mean-square errors were obtained. Also, for linear estimates of unknown matrices, the concept of quasi-minimax estimates was introduced and it was confirmed that they are unbiased. For special sets, which include an unknown matrix and correlation matrices of observation errors, such estimates were expressed through the solution of linear operator equations in a finite-dimensional space. For quasi-minimax estimates under certain assumptions, the form of the guaranteed mean squared error of the unknown matrix was found. It was shown that such errors are limited by the sum of traces of the known matrices. An example of finding a minimax unbiased linear estimation was given for a special type of random matrices that are included in the observation equation.

Pointwise residual method for solving primal and dual ill-posed linear programming problems with approximate data

We propose a variation of the pointwise residual method for solving primal and dual ill-posed linear programming with approximate data, sensitive to small perturbations. The method leads to an auxiliary problem, which is also a linear programming problem. Theorems of existence and convergence of approximate solutions are established and optimal estimates of approximation of initial problem solutions are achieved. doi:10.1017/S1446181120000243

Planar harmonic mappings in a family of functions convex in one direction

Let D := {z ? C : |z| < 1} be the open unit disk, and h and 1 be two analytic functions in D. Suppose that f = h + ?g is a harmonic mapping in D with the usual normalization h(0) = 0 = g(0) and h'(0) = 1. In this paper, we consider harmonic mappings f by restricting its analytic part to a family of functions convex in one direction and, in particular, starlike. Some sharp and optimal estimates for coefficient bounds, growth, covering and area bounds are investigated for the class of functions under consideration. Also, we obtain optimal radii of fully convexity, fully starlikeness, uniformly convexity, and uniformly starlikeness of functions belonging to those family.

Restoring the real world records in Men’s swimming without high-tech swimsuits

The recently concluded 2019 World Swimming Championships was another major swimming competition that witnessed some great progresses achieved by human athletes in many events. However, some world records created 10 years ago back in the era of high-tech swimsuits remained untouched. With the advancements in technical skills and training methods in the past decade, the inability to break those world records is a strong indication that records with the swimsuit bonus cannot reflect the real progressions achieved by human athletes in history. Many swimming professionals and enthusiasts are eager to know a measure of the real world records had the high-tech swimsuits never been allowed. This paper attempts to restore the real world records in Men’s swimming without high-tech swimsuits by integrating various advanced methods in probabilistic modeling and optimization. Through the modeling and separation of swimsuit bias, natural improvement, and athletes’ intrinsic performance, the result of this paper provides the optimal estimates and the 95% confidence intervals for the real world records. The proposed methodology can also be applied to a variety of similar studies with multi-factor considerations.

Maximal Fluctuations Around the Wulff Shape for Edge-Isoperimetric Sets in $$\varvec{{\mathbb {Z}}^d}$$: A Sharp Scaling Law

We derive a sharp scaling law for deviations of edge-isoperimetric sets in the lattice $${\mathbb {Z}}^d$$ Z d from the limiting Wulff shape in arbitrary dimensions. As the number n of elements diverges, we prove that the symmetric difference to the corresponding Wulff set consists of at most $$O(n^{(d-1+2^{1-d})/d})$$ O ( n ( d - 1 + 2 1 - d ) / d ) lattice points and that the exponent $$(d-1+2^{1-d})/d$$ ( d - 1 + 2 1 - d ) / d is optimal. This extends the previously found ‘$$n^{3/4}$$ n 3 / 4 laws’ for $$d=2,3$$ d = 2 , 3 to general dimensions. As a consequence we obtain optimal estimates on the rate of convergence to the limiting Wulff shape as n diverges.