Two New Families of Local Asymptotically Minimax Lower Bounds in Parameter Estimation.

We propose two families of asymptotically local minimax lower bounds on parameter estimation performance. The first family of bounds applies to any convex, symmetric loss function that depends solely on the difference between the estimate and the true underlying parameter value (i.e.

Refinements and Extensions of Ziv's Model of Perfect Secrecy for Individual Sequences.

We refine and extend Ziv's model and results regarding perfectly secure encryption of individual sequences. According to this model, the encrypter and the legitimate decrypter share a common secret key that is not shared with the unauthorized eavesdropper. The eavesdropper is aware of the encryption scheme and has some prior knowledge concerning the individual plaintext source sequence.

Lossy Compression of Individual Sequences Revisited: Fundamental Limits of Finite-State Encoders.

We extend Ziv and Lempel's model of finite-state encoders to the realm of lossy compression of individual sequences. In particular, the model of the encoder includes a finite-state reconstruction codebook followed by an information lossless finite-state encoder that compresses the reconstruction codeword with no additional distortion. We first derive two different lower bounds to the compression ratio, which depend on the number of states of the lossless encoder.

A Universal Random Coding Ensemble for Sample-Wise Lossy Compression.

We propose a universal ensemble for the random selection of rate-distortion codes which is asymptotically optimal in a sample-wise sense. According to this ensemble, each reproduction vector, x^, is selected independently at random under the probability distribution that is proportional to 2-LZ(x^), where LZ(x^) is the code length of x^ pertaining to the 1978 version of the Lempel-Ziv (LZ) algorithm. We show that, with high probability, the resulting codebook gives rise to an asymptotically optimal variable-rate lossy compression scheme under an arbitrary distortion measure, in the sense that a matching converse theorem also holds.

Some Families of Jensen-like Inequalities with Application to Information Theory.

It is well known that the traditional Jensen inequality is proved by lower bounding the given convex function, f(x), by the tangential affine function that passes through the point (E{X},f(E{X})), where E{X} is the expectation of the random variable . While this tangential affine function yields the tightest lower bound among all lower bounds induced by affine functions that are tangential to , it turns out that when the function is just part of a more complicated expression whose expectation is to be bounded, the tightest lower bound might belong to a tangential affine function that passes through a point different than (E{X},f(E{X})). In this paper, we take advantage of this observation by optimizing the point of tangency with regard to the specific given expression in a variety of cases and thereby derive several families of inequalities, henceforth referred to as "Jensen-like" inequalities, which are new to the best knowledge of the author.