108 research outputs found
More supplements to a class of logarithmically completely monotonic functions associated with the gamma function
In this article, a necessary and sufficient condition and a necessary
condition are established for a function involving the gamma function to be
logarithmically completely monotonic on . As applications of the
necessary and sufficient condition, some inequalities for bounding the psi and
polygamma functions and the ratio of two gamma functions are derived.Comment: 8 page
Sharp inequalities for polygamma functions
The main aim of this paper is to prove that the double inequality
\frac{(k-1)!}{\Bigl\{x+\Bigl[\frac{(k-1)!}{|\psi^{(k)}(1)|}\Bigr]^{1/k}\Bigr\}^k}
+\frac{k!}{x^{k+1}}<\bigl|\psi^{(k)}(x)\bigr|<\frac{(k-1)!}{\bigl(x+\frac12\bigr)^k}+\frac{k!}{x^{k+1}}
holds for and and that the constants
and are the best
possible. In passing, some related inequalities and (logarithmically) complete
monotonicity results concerning the gamma, psi and polygamma functions are
surveyed.Comment: 11 page
Bounds for the Ratio of Two Gamma Functions
By looking back at the long history of bounding the ratio Γ(x+a)/Γ(x+b) for x>-min⁡{a,b} and a,b∈ℝ, various origins of this topic are clarified, several developed courses are followed, different results are compared, useful methods are summarized, new advances are presented, some related problems are pointed out, and related references are collected
Modified-half-normal distribution and different methods to estimate average treatment effect.
This dissertation consists of three projects related to Modified-Half-Normal distribution and causal inference. In my first project, a new distribution called Modified-Half-Normal distribution was introduced. I explored a few of its distributional properties, the procedures for generating random samples based on Bayesian approaches, and the parameter estimation based on the method of moments. The second project deals with the problem of selection bias of average treatment effect (ATE) if we use the observational data. I combined the propensity score based inverse probability of treatment weighting (IPTW) method and the directed acyclic graph (DAG) to solve this problem. The third project solves the problem of bias ATE using observational data by combining the doubly robust methods with the super learner algorithm
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