Theoretical Statistics: Topics for a Core Course (Springer by Robert W. Keener

By Robert W. Keener

Intended because the textual content for a series of complicated classes, this publication covers significant themes in theoretical facts in a concise and rigorous type. The dialogue assumes a heritage in complicated calculus, linear algebra, likelihood, and a few research and topology. degree thought is used, however the notation and uncomplicated effects wanted are provided in an preliminary bankruptcy on likelihood, so past wisdom of those issues isn't really essential.
The presentation is designed to reveal scholars to as the various vital rules and subject matters within the self-discipline as attainable, balancing a number of techniques to inference in addition to distinct, numerical, and massive pattern equipment. relocating past extra typical fabric, the publication contains chapters introducing bootstrap equipment, nonparametric regression, equivariant estimation, empirical Bayes, and sequential layout and analysis.
The publication has a wealthy choice of routines. a number of of them illustrate how the speculation built within the booklet can be utilized in a number of functions. strategies to the various routines are integrated in an appendix.

Show description

Read or Download Theoretical Statistics: Topics for a Core Course (Springer Texts in Statistics) PDF

Similar probability & statistics books

Time Series Analysis and Forecasting by Example (Wiley Series in Probability and Statistics)

An intuition-based technique helps you to grasp time sequence research conveniently Time sequence research and Forecasting via instance presents the elemental thoughts in time sequence research utilizing quite a few examples. by means of introducing worthy conception via examples that show off the mentioned issues, the authors effectively support readers boost an intuitive knowing of probably complex time sequence versions and their implications.

Understanding Biplots

Biplots are a graphical approach for concurrently exhibiting varieties of info; generally, the variables and pattern devices defined through a multivariate info matrix or the goods labelling the rows and columns of a two-way desk. This ebook goals to popularize what's now noticeable to be an invaluable and trustworthy technique for the visualization of multidimensional information linked to, for instance, central part research, canonical variate research, multidimensional scaling, multiplicative interplay and diverse varieties of correspondence research.

Adaptive Markov Control Processes (Applied Mathematical Sciences)

This booklet is worried with a category of discrete-time stochastic keep an eye on methods often called managed Markov procedures (CMP's), often referred to as Markov choice tactics or Markov dynamic courses. beginning within the mid-1950swith Richard Bellman, many contributions to CMP's were made, and functions to engineering, records and operations study, between different components, have additionally been constructed.

Extremes in Random Fields: A Theory and Its Applications

Provides an invaluable new process for reading the extreme-value behaviour of random fields sleek technological know-how in most cases includes the research of more and more complicated information. the extraordinary values that emerge within the statistical research of complicated information are frequently of specific curiosity. This booklet makes a speciality of the analytical approximations of the statistical importance of maximum values.

Extra resources for Theoretical Statistics: Topics for a Core Course (Springer Texts in Statistics)

Example text

Suppose X ∼ Γ (α, 1/λ), with density λα xα−1 e−λx , Γ (α) x > 0. Find the cumulants of T = (X, log X) of order 3 or less. The answer will involve ψ(α) = d log Γ (α)/dα = Γ ′ (α)/Γ (α). 29. Let X1 , . . , Xn be independent random variables, and let αi and ti , i = 1, . . , n, be known constants. Suppose Xi ∼ Γ (αi , 1/λi ) with λi = θ1 + θ2 ti , i = 1, . . , n, where θ1 and θ2 are unknown parameters. Show that the joint distributions form a two-parameter exponential family. Identify the statistic T and give its mean and covariance matrix.

Are probability measures and c1 , c2 , . . are nonnegative constants, then ci µi is a measure. ) The measures µi you will want to consider are truncations of µ to sets Ai covering X with µ(Ai ) < ∞, given by µi (B) = µ(B ∩ Ai ). With the constants ci chosen properly, ci µi will be a probability measure. *25. The monotone convergence theorem states that if 0 ≤ f1 ≤ f2 · · · are measurable functions and f = lim fn , then f dµ = lim fn dµ. Use this result to prove the following assertions. a) Show that if X ∼ PX is a random variable on (E, B, P ) and f is a nonnegative measurable function, then f X(e) dP (e) = f (x) dPX (x).

This fact remains true in general and is the basis for a host of interesting and useful calculations. Integration against a conditional distributions gives a conditional expectation. Specifically, the conditional expectation of f (X, Y ) given X = x is defined as E f (X, Y ) X = x = f (x, y) dQx (y). 18) Suppose X and Y are both discrete with Y taking values in a countable set Y0 and X taking values in X0 as defined above. Then Z = X Y takes values in the countable set X0 × Y0 and is discrete with mass function pZ (z) = P (Z = z) = P (X = x, Y = y), where z = xy .

Download PDF sample

Rated 4.54 of 5 – based on 14 votes