## TABLE OF CONTENTS CHAPTER

### BibTeX

@MISC{O'fallon_tableof,

author = {Judith R. O'fallon},

title = {TABLE OF CONTENTS CHAPTER},

year = {}

}

### OpenURL

### Abstract

### Citations

1145 |
An introduction to multivariate statistical analysis, third ed
- Anderson
- 2003
(Show Context)
Citation Context ...where fi(~) denotes the probability density function (p.d.f.) corresponding to the i th population, 1=1,2, and a is the proportion of Population 1 in the mixture (cf. Theorem 6.3.1, p.13l of Anderson =-=[1]-=-). For the case of rectangular truncation under consideration here, let fi(~lk) denote the p.d.f. of the i th population when it is truncated to the set {~: xl ~ k}. ~en otherwise, f.(xlk) 1.t 4J(t) =... |

91 |
The statistical utilization of multiple measurements
- Fisher
- 1938
(Show Context)
Citation Context ...wever, if H(d) does have four real zeros, we have the following information about the extra two zeros, d ' and d ll , say: [9.] They can occur only when PIP2 is "near" -1; specifically, (from [6] and =-=[7]-=-): a.) -1 < P2 < -1/3 and pI! < PI < 1; or b.) 1/3 < PZ < 1 and -1 < PI < p'. [10.] Both must be located in exactly one of the intervals: a.) (-oo,nUn{d* ,-l}.) b. ) (-1, min{-p 1, -P2}) c.) (min{-PI,... |

19 |
Classification into two multivariate normal distributions with different covariance matrices
- Anderson, Bahadur
- 1962
(Show Context)
Citation Context ...where k., ;(k.), and w. and 111 Var.(Xlk) = (l-w.)E + (122(1-pZ)W.U~2 (25) 1 - 1 - 1..;;..L ~1 is the first unit vector, are defined by (3), (17), and (19), and (26) U22 is the matrix with a I-in the =-=(2,2)-=- position and zeroes elsewhere. (27) proof: Follows directly from Lemmas 2 and 3. 0 Having obtained the means and covariance matrices for the truncated distributions, we can now obtain Fisher's "best ... |

13 |
Some inequalities on Mill’s ratio and related functions
- Sampford
- 1953
(Show Context)
Citation Context ...l number and e = Var 1 (3£,) + Varz (3£,) , '1..= E1(3£,) - E2(3£,), (13) (14) provided 0 is a positive-definite matrix and L~~' proo~: From definitions (11),(13), and (14), we h~ve (n'y)2 ~ n'QnRao =-=[12]-=- states, in result (l~.l.l), of.!l attains .its supremum when!!.. =sflr. .. p.48, that such a function It is clear that the value of R 2 is unchanged when !l = ce-1.r. for any real c~O. o Now, when th... |

11 |
1812) Theorie Analytique Des Probabilities
- Laplace
(Show Context)
Citation Context ...oposed by Birnbaum [4], R(x) > 2/(x + Ix 2 +4) , we get an upper bound of unity for w(t) when t<O; i.e. w(t) < 1 for every t<O. Using the following upper bound for R(x), which was proposed by Laplace =-=[10]-=-, R(x) ~ x-I - x-3 + 3x-S , we get the following lower bound for w(t) when t<O: t 6 +7t 4 -6t 2 +9 w(t) ~ 1 - (t4_tz+3)2·, which tends to 1 as t + -~. Therefore, w(t) + 1 as t + _00. i.) To show that ... |

9 |
An inequality for Mill’s ratio
- Birnbaum
- 1942
(Show Context)
Citation Context ...e; b. ) the complement of the strip including the parallel lines, x2 = y* - dx} and x2 =yO - dx} , where y* and yO are zeros of U(y). Any additional zeros in an interval about the origin must satisfy =-=[4]-=- or [5]: [4. ] If there is an odd number of these zeros, one of them must be a local maximum or minimum of U(y). This zero does not affect the partition defined by the limiting values of U(y), since i... |

6 |
Inequalities for the Mills’ Ratio
- Boyd
- 1959
(Show Context)
Citation Context ...- o{l)] < g(t) as t ~ ~. proof: When k=O, o*>k=O for all 0>1, by Lemma 5; and so, by Lemma 4, (14) holds. Since (o*/cr) could be relatively small, let us consider bounds for R(t), t>O, stated by Boyd =-=[5]-=-: (16) In somewhat simplified notation (14) can be written X R{X) = w, (17) with { W=1-{1/(2) E (O,l) } X = 0*/0 • (18) From (17) and the lower bound of (16) it follows that < X R{X) =w, 1. e. i.e. 1.... |

3 |
Plane Truncation in Normal Populations
- Tallis
- 1965
(Show Context)
Citation Context ...appearance within each chapter. Minor results are generally designated by letters or lower-case Roman6 numerals inside parentheses. A proper name followed by a number in square brackets, e.g. Tallis =-=[13]-=-, refers to a specific book or paper listed in the Bibliography.CHAPTER II LlNEAB DISCRIMINATION BETHEEN TWO BIVARIATENO~~ POPULATIONS WITH EQUAL COVARIANCE MATRICES BASED ON A TRUNCATED SAMPLING PRO... |

2 |
Geometric interpretation of admissible linear decision boundaries for two multivariate normal distributions
- Bechtel, Gavin, et al.
- 1971
(Show Context)
Citation Context ...function of d. is This fact, together with [lJ, implies that H(d) has a unique minimum; and since H(d)<O for d=l or d = -1, there must be exactly ~o real zeros of H(d). These two are given in [2] and =-=[3]-=-. [7. ] H" (d) ~O for all d iff one of the following sets of conditions holds:a.) -1 <PZ<-1/3 and -l<PI~P"<l; b.) -1/3~' Pz ~ 1/3; c.) 1/3 < Pz < 1 and -1 < p' ~ PI < 1. proof: The zeros of H"(d) are... |

2 |
Geometry and Linear Discrimination
- Clunies-Ross, Riffenburgh
- 1960
(Show Context)
Citation Context ... 61 = Var1(YI~) = 022, 62 = Var2 (y I~) = 1, D = 62-61 = 1-022, G = C + £n(62/61) = C - £n 022 , v 2 + DG = 02 2 + (1-022 )(C - £n °22). Because this function arises again, define where C is given by =-=(6)-=-. Substitution of these values into (8) gives (12) ( (y" ,y' » = 02 ± 02 1H(a,02,02) Consequently, for the coefficient vector d' = (0,1), 0202 ± !H(a,02,02) " , «fa; , fa; » = = 1-°2 2 02 ± 021H(a,02,... |