One-sided confidence intervals for the positive linear combination of two variances

In 1971 Jones proposed an approximate procedure for finding that linear combination of scores which has maximum heritability in a twin sample. I give an exact small-sample procedure. I point out two problems: such procedures can over-optimize the heritability by capitalizing on chance, and confidence intervals and significance tests are needed. I give an approach using James-Stein shrinkage estimation and bootstrapped standard errors to address these problems. It appears that confidence intervals may be quite broad. To reduce the width of the confidence intervals, one can accept some small-sample bias in exchange for smaller sampling errors. The James-Stein approach to estimating coefficients is used to achieve reduced confidence interval width. I illustrate with a computational example using personality data.

Download Full-text

New large-sample confidence intervals for a linear combination of binomial proportions

Journal of Statistical Planning and Inference ◽

10.1016/j.jspi.2007.07.008 ◽

2008 ◽

Vol 138 (6) ◽

pp. 1884-1893 ◽

Cited By ~ 9

Author(s):

Joshua M. Tebbs ◽

Scott A. Roths

Keyword(s):

Linear Combination ◽

Confidence Intervals ◽

Large Sample ◽

Binomial Proportions

Download Full-text

Schur and $e$-Positivity of Trees and Cut Vertices

The Electronic Journal of Combinatorics ◽

10.37236/8930 ◽

2020 ◽

Vol 27 (1) ◽

Author(s):

Samantha Dahlberg ◽

Adrian She ◽

Stephanie Van Willigenburg

Keyword(s):

Linear Combination ◽

Connected Graph ◽

Symmetric Function ◽

Symmetric Functions ◽

Perfect Matching ◽

Schur Functions ◽

Connected Components ◽

Elementary Symmetric Functions ◽

Chromatic Symmetric Function ◽

Positive Linear Combination

We prove that the chromatic symmetric function of any $n$-vertex tree containing a vertex of degree $d\geqslant \log _2n +1$ is not $e$-positive, that is, not a positive linear combination of elementary symmetric functions. Generalizing this, we also prove that the chromatic symmetric function of any $n$-vertex connected graph containing a cut vertex whose deletion disconnects the graph into $d\geqslant\log _2n +1$ connected components is not $e$-positive. Furthermore we prove that any $n$-vertex bipartite graph, including all trees, containing a vertex of degree greater than $\lceil \frac{n}{2}\rceil$ is not Schur-positive, namely not a positive linear combination of Schur functions. In complete generality, we prove that if an $n$-vertex connected graph has no perfect matching (if $n$ is even) or no almost perfect matching (if $n$ is odd), then it is not $e$-positive. We hence deduce that many graphs containing the claw are not $e$-positive.

Download Full-text

Lower Confidence Limits for the Positive Linear Combination of two Variances

Biometrical Journal ◽

10.1002/bimj.4710360507 ◽

1994 ◽

Vol 36 (5) ◽

pp. 549-556

Author(s):

T. Anbupalam ◽

K. N. Ponnuswamy ◽

M. R. Srinivasan

Keyword(s):

Linear Combination ◽

Confidence Limits ◽

Lower Confidence ◽

Positive Linear Combination

Download Full-text

A Remark on Similarity Conditions

British Journal of Political Science ◽

10.1017/s0007123400008267 ◽

1975 ◽

Vol 5 (3) ◽

pp. 391-392

Author(s):

Peter Wagstaff

Keyword(s):

Linear Combination ◽

Majority Rule ◽

Single Point ◽

Utility Functions ◽

The Other ◽

Policy Space ◽

Open Convex ◽

The Family ◽

Exclusion Restrictions ◽

Positive Linear Combination

Kramer has shown how singularly restrictive are all ‘similarity’ conditions which guarantee transitivity of majority rule by limiting the family of admissible preference orderings (so-called exclusion restrictions). For suppose, plausibly, that social alternatives are points in an open convex policy space S ⊂ Rn, n ≥ 2, and that voters' preferences, {Rl)1=1…‥ l, are representable by continuously differentiable semi-strictly quasi-concave utility functions ul,. Suppose further that at a single point x ε S, any three voters' utility functions have gradients ∇ul(x), ∇uf(X), ∇uk(x), no one of which can be expressed as a positive linear combination of the other two, and no two of which are linearly dependent. Then all exclusion conditions must fail on S.

Download Full-text