Study of a unit power-logarithmic distribution

OMS-Vol. 5 (2021), Issue 1, pp. 218 – 235 Open Access Full-Text PDF
Christophe Chesneau
Abstract: This article proposes a new unit distribution based on the power-logarithmic scheme. The corresponding cumulative distribution function is defined by a special ratio of power and logarithmic functions that is dependent on one parameter. We show that this function benefits from great flexibility characterized by a large selection of convex and concave shapes. The other key functions are determined and studied. In particular, we show that the probability density function may take on different decreasing or U shapes, and the hazard rate function has a wide panel of U shapes. These functional capabilities are rare for a one-parameter unit distribution. In addition, we prove certain stochastic order results, provide the expression of the quantile function via the Lambert function, some interesting distributional results, and give simple expressions for the ordinary moments, mean, variance, skewness, kurtosis, moment generating function and incomplete moments. Subsequently, a basic statistical approach is described, to show how the new distribution can be applied in a data analysis scenario. Finally, complementary mathematical findings are presented, yielding new integrals linked to the Euler constant via some well-known moments properties.
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On some iterative methods with frozen derivatives for solving equations

OMS-Vol. 5 (2021), Issue 1, pp. 209 – 217 Open Access Full-Text PDF
Samundra Regmi, Christopher Argyros, Ioannis K. Argyros, Santhosh George
Abstract: We determine a radius of convergence for an efficient iterative method with frozen derivatives to solve Banach space defined equations. Our convergence analysis use \(\omega-\) continuity conditions only on the first derivative. Earlier studies have used hypotheses up to the seventh derivative, limiting the applicability of the method. Numerical examples complete the article.
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Dye ability of henna dye towards cotton fabrics in comparison with reactive dye by following reactive dyeing procedure

EASL-Vol. 4 (2021), Issue 2, pp. 29 – 35 Open Access Full-Text PDF
Pranay Dutta, Md. Razaya Rabbi, Mohammad Abu Sufian, Shahnaz Mahjebin
Abstract: Although synthetic dyes are commonly used, natural dyes are still being utilized and used to improve their intrinsic aesthetic properties as the main material for the body’s beauty. For example, research results have shown that henna plant leaves comprise dye together with other additives. This provides a hint that if color from henna is properly studied, it can be used not only as body decoration but may also have fiber-substrates affinity. This paper explores the dyeing possibility-the ability to dye and the fastness qualities of henna dye extracted from henna leaves on cotton fabric compared to reactive dyeing using the same dyeing technique as reactive dyeing. Also, color fastness tests have been performed according to the ISO test methods. The implications of henna dye have been shown to have poor to moderate dyeing capability towards cotton fabrics as opposed to the reactive dyes when henna dyeing is accompanied by reactive dyeing. Similarly, henna dye demonstrated satisfactory properties of fastness as opposed to reactive dye. For henna dye with 50% shade, it gives an outstanding color tone with a good level of coloration. Taken into account the ability to dye and the fastness of color, the dyeing of matured henna leaves is equally advantageous to the dyeing of cotton fabrics.
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Joint influence of measurement errors and randomized response technique on mean estimation under stratified double sampling

OMS-Vol. 5 (2021), Issue 1, pp. 192 – 199 Open Access Full-Text PDF
Ronald Onyango, Brian Oduor, Francis Odundo
Abstract: The present study proposes a generalized mean estimator for a sensitive variable using a non-sensitive auxiliary variable in the presence of measurement errors based on the Randomized Response Technique (RRT). Expressions for the bias and mean squared error for the proposed estimator are correctly derived up to the first order of approximation. Furthermore, the optimum conditions and minimum mean squared error for the proposed estimator are determined. The efficiency of the proposed estimator is studied both theoretically and numerically using simulated and real data sets. The numerical study reveals that the use of the Randomized Response Technique (RRT) in a survey contaminated with measurement errors increases the variances and mean squared errors of estimators of the finite population mean.
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Hermite-Hadamard-Fejér type inequalities for co-ordinated harmonically convex functions via Katugampola fractional integrals

EASL-Vol. 4 (2021), Issue 2, pp. 12 – 28 Open Access Full-Text PDF
Naila Mehreen, Matloob Anwar
Abstract: The aim of this paper is to establish the Hermite-Hadamard-Fejér type inequalities for co-ordinated harmonically convex functions via Katugampola fractional integral. We provide Hermite-Hadamard-Fej\’er inequalities for harmonically convex functions via Katugampola fractional integral in one dimension.
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Kolakoski sequence: links between recurrence, symmetry and limit density

ODAM-Vol. 4 (2021), Issue 1, pp. 29 – 44 Open Access Full-Text PDF
Alessandro Della Corte
Abstract: The Kolakoski sequence \(S\) is the unique element of \(\left\lbrace 1,2 \right\rbrace^{\omega}\) starting with 1 and coinciding with its own run length encoding. We use the parity of the lengths of particular subclasses of initial words of \(S\) as a unifying tool to address the links between the main open questions – recurrence, mirror/reversal invariance and asymptotic density of digits. In particular we prove that recurrence implies reversal invariance, and give sufficient conditions which would imply that the density of 1s is \(\frac{1}{2}\).
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Rate of convergence in total variation for the generalized inverse Gaussian and the Kummer distributions

OMS-Vol. 5 (2021), Issue 1, pp. 182 – 191 Open Access Full-Text PDF
Essomanda KONZOU
Abstract: The generalized inverse Gaussian distribution converges in law to the inverse gamma or the gamma distribution under certain conditions on the parameters. It is the same for the Kummer’s distribution to the gamma or beta distribution. We provide explicit upper bounds for the total variation distance between such generalized inverse Gaussian distribution and its gamma or inverse gamma limit laws, on the one hand, and between Kummer’s distribution and its gamma or beta limit laws on the other hand.
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