The Preservation Of Log-concavity And The Bounds For Extropy

Log-concavity(log-convexity)and their various properties play an increasing-ly important role in probability,statistics,operations research and other fields.Log-concavity and strong log-concavity are defined for nonnegative functions.For nonneg-ative sequences,we consider log-concavity,ULC(n)and ULC(?).Some other con-cepts that are closely related to log-concavity are TP2,PF2 and unimodality.The paper mainly discuss two parts,the first part is the transitivity about log-concavity and its ap-plications in some associated fields,the second part is the bounds for extropy under the total variation distance constraints.Chapter 1 is the literature review for log-concavity,entropy and extropy.The rela-tionship between log-concavity and entropy,and the related concepts,such as ULC(n),ULC(?),TP2 and unimodality are also discussed.In Chapter 2,we first establish general preservation theorems of log-concavity and log-convexity under operator??Y(?,?)=E[?(X?)],??(?,)where(?)is an interval of real numbers or an interval of integers,and the random variable X? has a distribution function belonging to the family {F?,??(?)} that possessing the semigroup property.The proofs are based on the theory of stochastic comparisons and weighted distributions.The main results are applied to some special operators,for example,operators occurring in reliability,Bernstein-type operators and Beta-type operators.Some theoretical results about renewal process are proved.One important property of log-concave distributions is that the convolution of log-concave distributions is still log-concave.Based on this result,we define a set(?)F for each probability distribution F in Chapter 3.And we also investigate sufficient and necessary conditions under which(?)F(?(?)G,where F and G belong to a parametric family of distributions.Both discrete and continuous settings are considered.In Chapter 4,We study the log-concavity and TP2 property of two-parameter com-pound poisson distributions Q(x|?,v)with respect to x,? and v by exploiting the inter-relationship between log-concavity,TP2 and reproductive property.We revisit the partial monotonicity of the conditional differential entropy,including Shannon entropy,Renyi entropy and Tsallis entropy,and point out some errors in related literature in Chapter 5.The partial monotonicity of the conditional Shannon entropy under discrete setting is also established.The relationship between extropy and variational distance is studied in Chapter 6.We determine the distributions which attain the minimum or maximum extropy among these distributions under the constraints of total variation distance.The upper and lower bounds are obtained and the analytic formula for the confidence interval of extropy is established based on the bounds.