In this present study we have considered the Modified Lindley distribution having single parameter as a lifetime distribution. It’s hazard rate function is unimodal, reverse bathtub and becomes constant for larger values of the variate value. It is true for almost all the values of its parameter θ. We have computed Bayes estimators of θ on the basis of type II censored sample from it under various such loss functions as squared error loss function(SELF), weighted squared error loss function(WSELF), modified squared error loss function(MSELF), exponentiated squared error loss function(ESELF), precautionary loss function(PLF) and logarithmic loss function(LLF). The estimators thus obtained have been compared through their posterior risks.

KEYWORDS:Bayesian estimation, Censoring scheme, Loss function, Modified Lindley distribution, Posterior risk.

Suppose x1, . . . , xn is a random sample and it is desired to test whether the distribution is multivariate normal. Many likelihood-based methods such as mixed effects models for repeated measures are based on an assumption that the data have a multivariate normal distribution. There are many proposed goodness of fit tests for this problem. However, in many practical applications some of the components are missing for some of the vectors. There are few methods available for this very common problem. This paper describes three different methods using interpoint distances and compares their performance in several scenarios using simulation and one scenario with clinical trial data.

KEYWORDS: Missing data, energy statistic, interpoint distance, data depth.

This study models the relationship between loan-to-deposit ratio (LDR) and liquidity ratio (LR) using a conditional Beta-Normal distribution (CBND). LDR follows a Normal Distribution, while LR follows a Beta distribution with shape parameters dependent on LDR. Parameters were estimated using Maximum Likelihood Estimation (MLE). Results reveal that higher LDR values lead to more concentrated LR distributions, reflecting the nonlinear dependence between these financial indicators. A Mean Squared Error (MSE) value of 0.6922 demonstrates the model’s adequacy. The CBND framework captures skewness, nonlinearity; and asymmetry in
the joint behaviour of financial stability indicators, providing valuable insights for liquidity management and regulatory assessment in banking systems.

KEYWORDS: Beta-Normal, Conditional Distribution, Loan-to-Deposit Ratio, Liquidity Ratio, Dependence Structure, Financial-Stability.

The objective of this paper is to give saturated 4×4×m3, m3 ≥ 4 designs, minimizing the generalized variance of the main effects and the general mean. First, introductory concepts regarding saturated designs and the D-optimal criterion mentioned and useful notations, remarks, and lemmas presented. In the main results section, the upper bounds of the value of the determinant of D-optimal saturated 4 × 4 × 4 and 4 × 4 × 5 founded and the designs that achieve these bounds are presented. For the saturated 4 × 4 ×m3, m3 ≥ 6 designs, the upper bound of the determinant is given and we present a design for which the value of D-efficiency is 99, 5%.

KEYWORDS: D-optimal designs, main effect designs, saturated designs.

Background: Inconsistency in the temporal scale of exposure and outcome measurements is a major bottleneck in air pollution epidemiology, complicating causal inference from data collected by different regulatory authorities. Standard analytical methods often aggregate higher-resolution exposure data to match lower-resolution outcome data, which can obscure acute associations. In this study, we addressed this temporal incongruence by applying Functional Data Analysis models to improve the estimation of epidemiological associations.

Methods: We used data from Delhi, India (2013–2017), including daily counts of non-trauma, all-cause mortality and hourly ozone measurements. First, daily mortality was modelled using scalar-on-function regression (SoFR) on hourly ozone, adjusting for time, temperature, and relative humidity. Then, as a demonstration, daily mortality data were aggregated to monthly counts and modelled on daily average of ozone; estimates were compared against generalized additive models (GAMs) fitted to daily mortality and daily ozone.

Results: Hourly analyses revealed distinct diurnal patterns, with the strongest associations observed during the early afternoon. The largest effect was identified for the 7-day cumulative average at hour 14, corresponding to a 1.073% increase in mortality (95% CI: 0.672, 1.473) per 10 μg/m3 increase in ozone. Stratified analyses showed higher estimates for males, particularly in the 45–64 and ≥ 65 year age groups. In monthly models, the SoFR estimates fell within the 95% confidence intervals of the GAM estimates obtained from daily mortality.

Conclusions: FDA models effectively overcome information loss caused by standard methods that aggregate data to align exposure and outcome scales. Our analysis identified a critical risk window for ozone-related mortality and showed the FDA’s utility where only aggregated health data are available.

KEYWORDS:Functional Data, Scalar-on-Function Regression, Ozone exposure, Mortality.

This paper presents a semi-Markov process framework for analyzing occupational mobility within hierarchical organizations. Unlike conventional Markov models that assume memoryless transitions, the semi-Markov approach incorporates random holding times, allowing more realistic modeling of promotion delays across grades. The framework distinguishes between open and closed grades, considers multiplepromotion pathways and derives the interval transition probabilities ϕij(t) and destination probabilities γijq(l/t), which measure the likelihood that an employee starting at grade i moves through grade j and ultimately reaches grade q within time
t. A simulation study using exponential waiting times demonstrates how mobility patterns vary with entry grade, the number of required promotions and the structure of the transition matrix. Results show that higher initial grades improve chances of reaching upper positions, while paths requiring several promotions have lower probabilities.

KEYWORDS: Semi-Markov process, destination probability, exponential distribution, simulation.