The Sample Complexity of Up-to-ε Multi-dimensional Revenue Maximization

We consider the sample complexity of revenue maximization for multiple bidders in unrestricted multi-dimensional settings. Specifically, we study the standard model of additive bidders whose values for heterogeneous items are drawn independently. For any such instance and any , we show that it is possible to learn an -Bayesian Incentive Compatible auction whose expected revenue is within of the optimal -BIC auction from only polynomially many samples. Our fully nonparametric approach is based on ideas that hold quite generally and completely sidestep the difficulty of characterizing optimal (or near-optimal) auctions for these settings. Therefore, our results easily extend to general multi-dimensional settings, including valuations that are not necessarily even subadditive , and arbitrary allocation constraints. For the cases of a single bidder and many goods, or a single parameter (good) and many bidders, our analysis yields exact incentive compatibility (and for the latter also computational efficiency). Although the single-parameter case is already well understood, our corollary for this case extends slightly the state of the art.

Supply Chain Analysis Using Distribution Requirement Planning (DRP) Based On Bullwhip Effect Parameter (Case Study: Ud. Narwastu, Surabaya)

Narwastu Firm is a company in Surabaya which produces traditional body care and aromatherapy. During the product distribution, this company encounters many mistakes such as inappropriateness between Purcashing Order (PO) from distributors/consumers and the goods delivered due to bullwhip effects. Therefore, this research aimed at reducing the bullwhip effects by implementing Collaborative, Planning, Forecasting, and Replenishment (CPFR) method and investigating a lot sizing technique for producing minimum distribution cost by Distribution Requirement Planning (DRP) method in order to regulate order and plan a certain distribution activity. The research results demonstrated that the bullwhip effect values after CPFR improvement of two products that could be repaired were 1.0 for Body Scrub Goats Milk Balibloom (BSC GM BB) 200 gr and 0.96 for Body Mist Sparkling White (BM SW) 60 ml. Meanwhile, the calculations by Distribution Requirement Planning (DRP) method with Economic Order Quantity (EOQ) method got the plans for ordering Body Scrub Lightening (BSC LIGHT) 200 gr by 182 times with total cost IDR 96,886,476 and Body Lotion Lightening (BL LIGHT) 250 ml by 235 times with total cost IDR 125,873,816.

Independent, Tough Identical Results: The Class of Tweedie on Power Variance Functions and the Class of Bar-Lev and Enis on Reproducible Natural Exponential Families

The Rao-Blackwell theorem has had a fundamental role in statistical theory. However, as opposed to what seems natural, Rao and Blackwell did not investigate and write the theorem jointly. In fact, they both published the same result independently, two years apart. Indeed, as C.R. Rao writes in Wikipedia: ”the result on one parameter case was published by Rao (1945) in the Bulletin of the Calcutta Mathematical Society and by Blackwell (1947) in The Annals of Mathematical Statistics. Only Lehmann and Sche ´e (1950) called the result as Rao-Blackwell theorem”. Forty years later, a situation very similar to the previous one seems to have happened. Tweedie (1984) in a paper published in a proceedings to a conference held in Calcutta and Bar-Lev and Enis (1986) in a paper published in The Annals of Statistics both presented for the first time, albeit two years apart, independently and in di erent contexts, the class of natural exponential families having power variance functions (NEF-PVFs). Tweedie’s results were then mentioned by Jorgensen (1987) in his fundamental paper on exponential dispersion models published in the Journal of the Royal Statistical Society, Series B. Jorgensen, however, mentioned also other researchers, including Bar-Lev and Enis, as dealt with the same problem. Nonetheless, Jorgensen (1987) stated in his paper that ”The most complete study” of NEF-PVFs was given by Tweedie (1984), a statement which has led to naming the class of NEF-PVFs as the Tweedie class. This statement of Jorgensen is entirely and utterly incorrect. Accordingly, one of the goals of this note is to 'prove' such incorrectness. Based on this 'proof' it will be evident, so I trust, that both Bar-Lev and Enis should have received the appropriate credit by re-naming the class of NEF-PVFs via the exploitation of the names of Tweedie, Bar-Lev and Enis. This would resemble the dignified and elegant manner Lehmann and Sche ´e acted on the Rao-Blackwell Theorem. Notwithstanding, the main aim of the note is to encourage young researchers to present their results with self-confidence and to get the credit they deserve.

On the Sweep Map for $\vec{k}$-Dyck Paths

Garsia and Xin gave a linear algorithm for inverting the sweep map for Fuss rational Dyck paths in $D_{m,n}$ where $m=kn\pm 1$. They introduced an intermediate family $\mathcal{T}_n^k$ of certain standard Young tableaux. Then inverting the sweep map is done by a simple walking algorithm on a $T\in \mathcal{T}_n^k$. We find their idea naturally extends for $\mathbf{k}^\pm$-Dyck paths, and also for $\mathbf{k}$-Dyck paths (reducing to $k$-Dyck paths for the equal parameter case). The intermediate object becomes a similar type of tableau in $\mathcal{T}_\mathbf{k}$ of different column lengths. This approach is independent of the Thomas-Williams algorithm for inverting the general modular sweep map.

One Parameter Chi-squared Analysis

The chi-squared method for parameter estimation and model testing is developed for the one-parameter case of a line with a slope but no intercept. Curve fitting is motivated, and several methods for curve fitting are introduced. The chi-squared method is shown to be the optimal curve fitting method whenever Gaussian distributed measurement uncertainties and a model are present. The central limit theorem, which assures Gaussian distributed measurement uncertainties for a wide range of physical experiments, is introduced. End-of-chapter problems are included (with solutions in an appendix).

Two–Parameter Chi-squared Analysis

The chi-squared parameter estimation and model testing methodology is extended to the two-parameter case of a line with a slope and an intercept. Chi-squared is shown to be a paraboloid and is displayed in a contour plot. Techniques for extracting parameter correlations and uncertainties are developed. End-of-chapter problems are included (with solutions in an appendix).

Wavenumber selection via spatial parameter jump

The Swift–Hohenberg equation describes an instability which forms finite-wavenumber patterns near onset. We study this equation posed with a spatial inhomogeneity; a jump-type parameter that renders the zero solution stable for x <0 and unstable for x >0. Using normal forms and spatial dynamics, we prove the existence of a family of steady-state solutions that represent a transition in space from a homogeneous state to a striped pattern state. The wavenumbers of these stripes are contained in a narrow band whose width grows linearly with the size of the jump. This represents a severe restriction from the usual constant-parameter case, where the allowed band grows with the square root of the parameter. We corroborate our predictions using numerical continuation and illustrate implications on stability of growing patterns in direct simulations. This article is part of the theme issue ‘Stability of nonlinear waves and patterns and related topics’.

Reliability Estimation for the Two-Parameter Exponential Family Based on Progressively Type-II Censored Data

In this paper, we deal with the problem of estimating the reliability function of the two-parameter exponential distribution. Classical Maximum likelihood and Bayes estimates for one and two parameters and the reliability function are obtained on the basis of progressively type-II censored samples. The inverted gamma conjugate prior density is assumed for the one-parameter case, whereas the joint prior density of the two-parameter case is composed of the inverted gamma and the uniform densities. A comparison between the obtained estimators is made through a Monte Carlo simulation study. A real example is used to illustrate the proposed methods.