# US 2014/0156718 A1 · Definition of Two Self-weights for Continuous Random Variable and Their Applications in Basic Statistics

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- Number: US 2014/0156718 A1
- Document ID: US20140156718A1
- Status: Published application
- Kind: A1
- Application number: 13/690,034
- Filed: 2012-11-30
- Published: 2014-06-05
- Earliest claimed: 2012-11-30
- Inventors: Ligong Chen
- Web page: https://patentgrep.com/patent/US20140156718A1

## CPC

- G06F 17/18: for evaluating statistical data {, e.g. average values, frequency distributions, probability functions, regression analysis (forecasting specially adapted for a specific administrative, business or logistic context G06Q10/04)}
- G06F: ELECTRIC DIGITAL DATA PROCESSING (computer systems based on specific computational models G06N)

## Abstract

This invention is for how to define two self-weights for continuous random variable based on sampling, how to SAC-normalize a skewed sampling distribution of a continuous random variable without changing the structure of the original measurement scale of the continuous random variable, how to infer representativeness of mathematical mean, and how to take the advantage of the two self-weights to do t-tests in a two-population comparison.

## Claims

### Claim 1

The measurements of the two self-weights, convex self-weight (denoted by C{ci}) and concave self-weight (denoted by R{ri}), defined for each random sample point xi of an original continuous random variable X to describe central tendency and dispersive tendency of the point xi in a sampling distribution, as shown in the computation framework in Table 1 based on the formulas from (7) to (19).

### Claim 2 (depends on 1)

The measurement of the convex self-weight (denoted by Z{zi}) defined for the convex self-weight C{ci} and the concave self-weight R{ri} of the original continuous random variable X{xi}, as shown in the computation framework in Table 2 modifies from the formulas from (7) to (19) in the claim 1.

### Claim 3 (depends on 1)

The estimates of expectations defined for the original random variable X{xi} and the convex self-weight C{ci} respectively in the formulas (20) and (21) with all the self-weights defined only in the claim 1.

### Claim 4 (depends on 1)

The measurements of deviations defined in the formulas (25) and (27) with all the self-weights defined only in the claim 1 for the whole distribution of the original random variable X{xi}.

### Claim 5

The measurements of sampling errors defined in the formulas (28) and (30) with all the self-weights for the expectation estimate xc of the whole distribution of the original X{xi}.

### Claim 6 (depends on 1, 10, 11)

The measurements of deviations based on the approach of “segmentation-and-combination” for data manipulation that will be claimed in the following claim 10 and claim 11 and defined in the formulas (31), (33), (34) and (36) with all the self-weights defined only in the claim 1 for the two half-distributions of the original random variable X{xi}.

### Claim 7 (depends on 1, 10)

The measurement of an asymmetrical normal range based on the approach of “segmentation-and-combination” for data manipulation that will be claimed in the following claim 10 and defined in the formulas (37) and (38) for the whole distribution of the original X{xi} with all the self-weights defined only in the claim 1 for the expectation estimate xc in the two half-distributions of the original random variable X{xi}.

### Claim 8 (depends on 1, 10)

The measurements of sampling errors based on the approach of “segmentation-and-combination” for data manipulation that will be claimed in the following claim 10 and defined in the formulas (39), (41), (42) and (44) with all the self-weights defined only in the claim 1 for the expectation estimate xc in the two half-distributions of the original random variable X{xi}.

### Claim 9 (depends on 1, 10)

The measurement of an asymmetrical confidence interval based on the approach of “segmentation-and-combination” for data manipulation that will be claimed in the following claim 10 and defined in the formulas (45) and (46) for the whole distribution of the original X{xi} with the self-weights defined only in the claim 1.

### Claim 10 (depends on 1)

The SAC normalization based on the segmentation-and-combination for data manipulation with the formulas from (47) to (54), and (56) and (57) as it has been described from the sub-section 5 to the sub-section 8 in the section DETAILED DESCRIPTION and as shown in the FIG. 7, to construct two symmetrically distributed pseudo samples and a size-doubled symmetrically distributed pseudo sample based only on the expectation estimate xc with the convex self-weights claimed in the claim 1 for the original random variable X{xi}.

### Claim 11

The SAC normalization based on the segmentation-and-combination” with the formulas from (47) to (53) and formula (56) as it has been described from the sub-section 5 to the sub-section 8 in the section DETAILED DESCRIPTION and as shown in the FIG. 7, to construct two symmetrically distributed pseudo samples and a size-doubled symmetrically distributed pseudo sample based on the expectation estimated in the arithmetical mean x to replace the xc in the formulas only mentioned in this claim for the original random variable X{xi}.

### Claim 12 (depends on 1, 2)

The statistical measurements for sample defined in formulas (24), (26), (29), (32), (35), (40), (43), (55) and (58) based on the claim 1 and claim 2.

### Claim 13 (depends on 1)

The inference of representativeness of arithmetical mean proposed in the sub-section 9 in the section DETAILED DESCRIPTION and as shown in the FIG. 9 with the convex self-weights defined in the claim 1, comprising:
(1) A symmetry test with convex self-weighted mean and concave self-weighted mean in the formulas from (59) to (62);
(2) A symmetry test with convex self-weighted deviations of two half-distributions in the formulas from (63) to (65);
(3) A probabilistic inference for representativeness of arithmetical mean in the formula (66) with the formulas from (60) and (62) for the degree of freedom calculation;
(4) A comprehensive Chi-square test for symmetry, normality and representativeness of arithmetical mean in the formulas from (67) to (70);
(5) A linear correlation description between the convex self-weight C for the convex self-weighted mean and the convex self-weight Cm to the arithmetical mean for the original random variable X{xi}, and the convex self-weight Cm is defined in the formulas from (71) to (74).

### Claim 14 (depends on 1)

The methods of plotting the statistical figures and graphics that are only related to the convex self-weights claimed in the claim 1 for the original random variable X{xi}, comprising
(1) A two-dimensional scatter plot of the original random variable X{xi} and its convex self-weight and concave self-weight in a two-dimensional space as shown in the FIG. 5 for normal distributions and in the FIG. 6 for skewed distributions;
(2) A set of two-dimensional scatter plots for describing a process of the SAC normalization based on the approach of segmentation-and-combination for data manipulation as shown in the FIG. 7;
(3) A two-dimensional scatter plot of the convex self-weight C{ci} and the convex self-weight Cm{cm,i} of the original random variable X{xi} as shown in the FIG. 8.

### Claim 15 (depends on 1, 2)

The reconstructed t-test in the four options, as described in the sub-section 10 of the section DETAILED DESCRIPTION and as shown in the FIG. 10, for comparing two populations based on sampling with the convex self-weight and the SAC normalization, comprising:
(1) The reconstructed formulas from (78) to (90) if and only if they are used in a complete t-test with any randomly variable weight obtained in any method and with/without the SAC normalization, including the self-weights defined in the claim 1 as well as in the claim 2 if the claim 2 is necessarily involved.
(2) The reconstructed formulas from (91) to (93) if and only if they are used in a complete t-test without the SAC normalization and with the self-weights defined in the claim 1 as well as in the claim 2 if the claim 2 is necessarily involved.

### Claim 16

Any equivalent transformation in a mathematical deduction that may lead to the same computation result(s) of any formula that is claimed in the above claims from 1 to 14 for the same real sample.