^{p}

^{p}

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We present evidence that one can calculate generically combinatorially expensive ^{p}^{p}^{p}^{p}^{p}^{1} and ^{1} averages) do not exist, one needs to consider using ^{p}

^{p}

^{p}

Minimization principles based on the ^{1} and ^{1} norms have recently rapidly become more common due to discovery of their important roles in sparse representation in signal and image processing [^{1} solutions are equivalent to “^{0} solutions”, that is, the sparsest solutions, an important result because it allows one to find the solution of a combinatorially expensive ^{0} maximum-sparsity minimization problem by a polynomial-time linear programming procedure for minimizing ^{1} functionals. When the data follow heavy-tailed statistical distributions and the tails of the distributions are “not too heavy,” various ^{1} minimization principles, in the form of calculation of medians and quantiles, are primary choices that are efficient and robust against the many outliers [^{1} minimization principles are applicable also to data from light-tailed distributions such as the Gaussian, but, for such distributions, are less efficient than classical procedures (calculation of standard averages and variances).

When tails of the distributions are so heavy that even ^{1} minimization principles do not exist, one needs to consider using ^{p}^{p}^{0} solutions, than ^{1} minimization principles [^{p}^{p}^{p}^{0} solution is, relative to other potential solutions, the sparsest solution does not imply that this solution is sparse to any specific degree. The sparsest solution may not be sparse in any absolute sense at all; it is just sparser than any other solution.

The approach that we will investigate in the present paper shares with compressive sensing the strategy of restricting the nature of the problem to achieve polynomial-time performance. However, we do so not by requiring sparsity to some ^{p}^{p}^{p}

The classes of ^{p}^{p}^{p}^{p}^{p}^{p}_{i}

radially strictly monotonically decreasing outwards from the mode (3a)

ψ and d^{–β} and ^{–β–1}, respectively, for given

Without loss of generality, we assume that the mode, that is, the

In a departure from the traditional use of ^{2} average) does not exist for distributions with ^{1} average) does not exist for distributions with ^{p}^{p}

In the present paper, we will investigate whether, by providing only the information that the data come from a “standard” statistical distribution that satisfies Conditions (3), the ^{p}^{p}^{p}^{p}^{p}^{p}

In Distributions 2 and 3, ^{p}^{p}^{p}^{p}

We present in ^{p}^{p}

^{p}

^{p}

^{p}

The structure of the ^{p}^{p}

One computes expressions (10) and (11) by differentiating the right sides of expressions (9) and (10), respectively, with respect to ^{2}^{2}(0) > 0, that is, there is a local minimum at ^{p}

A general analytical structure for asymmetric distributions analogous to that described above for symmetric distributions is not yet available because, for asymmetric distributions, the properties of

It is meaningful to calculate an ^{p}^{p}

^{p}

STEP 1. Sort the data _{i}_{i}

STEP 2. Choose an integer

STEP 3. Choose a point _{j}^{1} average, is generally a good choice for the initial _{j}

STEP 4. For each _{k}

STEP 5. If the _{k}_{k}_{j}_{j}^{p}_{k}_{j}

STEP 6. If convergence has not occurred within a predetermined number of iterations, stop and return an error message.

_{i}_{i}_{i}_{+1}, _{i}_{i}_{i}_{+1},_{i}_{+1})), so a minimum cannot occur there. It is sufficient, therefore, to consider only the values of _{i}_{i}_{i}^{p}_{i}_{i}_{i}

^{2}) (= the number of iterations, which cannot exceed

In computational experiments, we used samples of size I = 2000 from the symmetric heavy-tailed Distribution 2 with various α, 1 < α ≤ 3, and window sizes 2q + 1 = 7, 9, 11, . . . , 25. For comparison with _{i}, B(x_{i})) for the sample from Distribution 2 with α = 2 and p = 0.5 and 0.02. The starting point for Step 3 of the Algorithm 1 was chosen to be x_{I−2q}, a point near the end of the right tail (beyond the limited domains shown in ^{p} average and thus provides an excellent test for the robustness of Algorithm 1. Computational results for p = 0.5, 0.1 and 0.02 and for window sizes 2q + 1 = 7, 13, 19 and 25 are presented in ^{p} averages of Distribution 2, when they exist, that is, when p < α − 1, are all 0. Thus, the errors of the l^{p} averages in ^{p} averages themselves.

The entries in ^{p}^{p}^{154}, 5.02 × 10^{169}], respectively. For ^{p}^{p}^{p}^{p}^{p}^{p}^{p}^{p}^{p}

Points (_{i}_{i}

Algorithm 1 is applicable to heavy-tailed distributions in general but the rule for choosing

Sample ^{p}

α\ ^{p} |
0.5 | 0.1 | 0.02 |
---|---|---|---|

3 | 0.028 | 0.560 | 0.701 |

2 | 0.038 | 0.779 | 0.779 |

1.5 | 0.057 | 0.575 | 0.575 |

1.1 | 7.58 | 0.244 | 0.244 |

1.05 | 1.49 × 10^{30} |
0.281 | 0.476 |

1.04 | 1.14 × 10^{45} |
0.349 | 0.598 |

1.03 | 2.83 × 10^{74} |
0.466 | 0.466 |

1.02 | 1.52 × 10^{119} |
1.38 × 10^{16} |
0.516 |

Sample ^{p}

α\
^{p} |
0.5 | 0.1 | 0.02 |
---|---|---|---|

3 | 0.021 | 0.094 | 0.531 |

2 | 0.027 | 0.126 | 0.126 |

1.5 | 0.041 | 0.189 | 0.189 |

1.1 | 3.76 | 0.108 | 0.108 |

1.05 | 2.56 × 10^{29} |
0.207 | 0.207 |

1.04 | 1.14 × 10^{45} |
0.257 | 0.257 |

1.03 | 2.83 × 10^{74} |
0.341 | 0.341 |

1.02 | 1.52 × 10^{119} |
3.24 × 10^{14} |
0.516 |

Sample ^{p}

α \^{p} |
0.5 | 0.1 | 0.02 |
---|---|---|---|

3 | 0.021 | 0.015 | 0.015 |

2 | 0.021 | 0.020 | 0.020 |

1.5 | 0.031 | 0.029 | 0.029 |

1.1 | 0.902 | 0.108 | 0.108 |

1.05 | 2.56 × 10^{29} |
0.207 | 0.207 |

1.04 | 1.14 × 10^{45} |
0.257 | 0.257 |

1.03 | 2.83 × 10^{74} |
0.341 | 0.341 |

1.02 | 1.52 × 10^{119} |
1.78 × 10^{7} |
0.516 |

Sample ^{p}

α\ ^{p} |
0.5 | 0.1 | 0.02 |
---|---|---|---|

3 | 0.021 | 0.015 | 0.015 |

2 | 0.021 | 0.020 | 0.020 |

1.5 | 0.031 | 0.029 | 0.029 |

1.1 | 0.498 | 0.108 | 0.108 |

1.05 | 2.56 × 10^{29} |
0. 207 | 0.207 |

1.04 | 1.14 × 10^{45} |
0.257 | 0.257 |

1.03 | 2.83 × 10^{74} |
0.341 | 0.341 |

1.02 | 1.52 × 10^{119} |
2.37 × 10^{6} |
0.516 |

The wide-spread impression that minimization of ^{p}^{p}^{p}^{p}

Topics for future research include

Quantitative rules for using information about the underlying continuum distribution to choose the

Investigation of the advantages and disadvantages of introducing smoothing in the _{k}

Description of the class(es) of symmetric and asymmetric univariate and multivariate distributions for which radially strictly monotonic ^{p}^{p}^{p}^{p}

Investigation of convergence of the ^{p}^{p}

Investigation of the conditions under which ^{p}^{p}

Treatment of more general univariate and multivariate ^{p}^{p}^{p}^{p}

Many phenomena in human-based areas (sociology, cognitive science, psychology, economics, human networks, social media, ^{p}^{p}^{p}^{p}^{p}^{p}^{p}^{p}

The author expresses his gratitude to the referees, whose well-though-out questions and insightful comments led to significant improvements in this paper.

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