Chebyshev–Edgeworth-Type Approximations for Statistics Based on Samples with Random Sizes

Second-order Chebyshev–Edgeworth expansions are derived for various statistics from samples with random sample sizes, where the asymptotic laws are scale mixtures of the standard normal or chi-square distributions with scale mixing gamma or inverse exponential distributions. A formal construction of asymptotic expansions is developed. Therefore, the results can be applied to a whole family of asymptotically normal or chi-square statistics. The random mean, the normalized Student t-distribution and the Student t-statistic under non-normality with the normal limit law are considered. With the chi-square limit distribution, Hotelling’s generalized T02 statistics and scale mixture of chi-square distributions are used. We present the first Chebyshev–Edgeworth expansions for asymptotically chi-square statistics based on samples with random sample sizes. The statistics allow non-random, random, and mixed normalization factors. Depending on the type of normalization, we can find three different limit distributions for each of the statistics considered. Limit laws are Student t-, standard normal, inverse Pareto, generalized gamma, Laplace and generalized Laplace as well as weighted sums of generalized gamma distributions. The paper continues the authors’ studies on the approximation of statistics for randomly sized samples.

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Sebaran Peluang Acak Kontinu, Distribusi Normal, Distribusi Normal Baku, Distribusi T, Distribusi Chi Square, dan Distribusi F

10.31219/osf.io/grdnm ◽

2020 ◽

Author(s):

Ahmad Sudi Pratikno

Keyword(s):

Hypothesis Test ◽

Standard Normal Distribution ◽

Process Data ◽

Nominal Data ◽

Chi Square ◽

Stable Curve ◽

Standard Normal ◽

Research Findings ◽

F Distribution ◽

T Distribution

In statistics, there are various terms that may feel unfamiliar to researcher who is not accustomed to discussing it. However, despite all of many functions and benefits that we can get as researchers to process data, it will later be interpreted into a conclusion. And then researcher can digest and understand the research findings. The distribution of continuous random opportunities illustrates obtaining opportunities with some detection of time, weather, and other data obtained from the field. The standard normal distribution represents a stable curve with zero mean and standard deviation 1, while the t distribution is used as a statistical test in the hypothesis test. Chi square deals with the comparative test on two variables with a nominal data scale, while the f distribution is often used in the ANOVA test and regression analysis.

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On normal variance–mean mixtures as limit laws for statistics with random sample sizes

Journal of Statistical Planning and Inference ◽

10.1016/j.jspi.2015.07.007 ◽

2016 ◽

Vol 169 ◽

pp. 34-42 ◽

Cited By ~ 6

Author(s):

V.Yu. Korolev ◽

A.I. Zeifman

Keyword(s):

Random Sample ◽

Sample Sizes ◽

Limit Laws ◽

Normal Variance

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A comparison of variance estimators with known and unknown population means

SURG Journal ◽

10.21083/surg.v1i2.407 ◽

2008 ◽

Vol 1 (2) ◽

pp. 42-48

Author(s):

Jennifer Ng ◽

Julie Horrocks

Keyword(s):

Degrees Of Freedom ◽

Sample Sizes ◽

Exponential Distributions ◽

Population Variance ◽

Population Means ◽

Population Mean ◽

Variance Estimators ◽

Standard Normal ◽

T Distribution ◽

Normal Standard

Variance is a concept that is key, yet often difficult to estimate in statistics. In this paper, we consider the problem of estimating the population variance when the population mean is known. We compare two estimators, one that incorporates the known population mean and another which estimates the population mean. The standard normal, standard exponential, and t distribution with 3 degrees of freedom are considered, with sample sizes of 5, 20, 50, and 100. It is determined that both estimators are unbiased. For the normal and exponential distributions, both estimators have similar variances; however, the estimator that incorporates the known mean has marginally lower variance, and thus is recommended. For the t(3) distribution, the variances of the estimators do not exist.

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Chi-square, Student and Fisher-Snedecor Statistical Distributions and Their Application

Statistics of Ukraine ◽

10.31767/su.1(80).2018.01.02 ◽

2018 ◽

Vol 80 (1) ◽

pp. 16-23

Author(s):

F. V. Motsnyi

Keyword(s):

Probability Distributions ◽

Point Of View ◽

Experimental Results ◽

Mathematical Statistics ◽

Statistical Distributions ◽

Chi Square ◽

Chi Squared ◽

Standard Normal ◽

F Distribution ◽

T Distribution

The Chi-square distribution is the distribution of the sum of squared standard normal deviates. The degree of freedom of the distribution is equal to the number of standard normal deviates being summed. For the first time this distribution was studied by astronomer F. Helmert in connection with Gaussian low of errors in 1876. Later K. Pearson named this function by Chi-square. Therefore Chi –square distribution bears a name of Pearson’s distribution. The Student's t-distribution is any member of a family of continuous probability distributions that arises when estimating the mean of a normally distributed population in situations where the sample size is small and population standard deviation is unknown. It was developed by W. Gosset in 1908. The Fisher–Snedecor distribution or F-distribution is the ratio of two-chi-squared variates. The F-distribution provides a basis for comparing the ratios of subsetsof these variances associated with different factors. The Fisher-distribution in the analysis of variance is connected with the name of R.Fisher (1924), although Fisher himself used quantity for the dispersion proportion. The Chi-square, Student and Fisher – Snedecor statistical distributions are connected enough tight with normal one. Therefore these distributions are used very extensively in mathematical statistics for interpretation of empirical data. The paper continues ideas of the author’s works [15, 16] devoted to advanced based tools of mathematical statistics. The aim of the work is to generalize the well known theoretical and experimental results of statistical distributions of random values. The Chi-square, Student and Fisher – Snedecor distributions are analyzed from the only point of view. The application peculiarities are determined at the examination of the agree criteria of the empirical sample one with theoretical predictions of general population. The numerical characteristics of these distributions are considered. The theoretical and experimental results are generalized. It is emphasized for the corrected amplification of the Chi-square, Student and Fisher – Snedecor distributions it is necessary to have the reliable empirical and testing data with the normal distribution.

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AMSM- Sampling Distribution- Chapter Three

10.31219/osf.io/h5auc ◽

2018 ◽

Author(s):

Mohammed R. Dahman

Keyword(s):

Limit Theorem ◽

Comprehensive Analysis ◽

Sampling Distribution ◽

Sampling Strategies ◽

Population Parameters ◽

Chi Square ◽

The Central Limit Theorem ◽

F Distribution ◽

The Common ◽

T Distribution

Introduction of differences between population parameters and sample statistics are discussed. Followed with a comprehensive analysis of sampling distribution (i.e. definition, and properties). Then, we have discussed essential examiners (i.e. distribution): Z distribution, Chi square distribution, t distribution and F distribution. At the end, we have introduced the central limit theorem and the common sampling strategies.

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On the use of chi-square analyses in studies of resource utilization

Canadian Journal of Forest Research ◽

10.1139/x91-009 ◽

1991 ◽

Vol 21 (1) ◽

pp. 58-65 ◽

Cited By ~ 10

Author(s):

Dennis E. Jelinski

Keyword(s):

Resource Utilization ◽

Goodness Of Fit ◽

Small Sample ◽

Homogeneity Test ◽

Type I ◽

Sample Sizes ◽

Goodness Of Fit Test ◽

Chi Square ◽

Test Of Homogeneity ◽

Small Sample Sizes

Chi-square (χ2) tests are analytic procedures that are often used to test the hypothesis that animals use a particular food item or habitat in proportion to its availability. Unfortunately, several sources of error are common to the use of χ2 analysis in studies of resource utilization. Both the goodness-of-fit and homogeneity tests have been incorrectly used interchangeably when resource availabilities are estimated or known apriori. An empirical comparison of the two methods demonstrates that the χ2 test of homogeneity may generate results contrary to the χ2 goodness-of-fit test. Failure to recognize the conservative nature of the χ2 homogeneity test, when "expected" values are known apriori, may lead to erroneous conclusions owing to the increased possibility of committing a type II error. Conversely, proper use of the goodness-of-fit method is predicated on the availability of accurate maps of resource abundance, or on estimates of resource availability based on very large sample sizes. Where resource availabilities have been estimated from small sample sizes, the use of the χ2 goodness-of-fit test may lead to type I errors beyond the nominal level of α. Both tests require adherence to specific critical assumptions that often have been violated, and accordingly, these assumptions are reviewed here. Alternatives to the Pearson χ2 statistic are also discussed.

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A remark on the central limit question for dependent random variables

Journal of Applied Probability ◽

10.2307/3212927 ◽

1980 ◽

Vol 17 (1) ◽

pp. 94-101 ◽

Cited By ~ 14

Author(s):

Richard C. Bradley

Keyword(s):

Central Limit ◽

Random Variables ◽

Stationary Sequence ◽

Stable Limit ◽

Limit Laws ◽

Dependent Random Variables ◽

Asymptotically Normal ◽

Second Moments

Given a strictly stationary sequence {Xk, k = …, −1,0,1, …} of r.v.'s one defines for n = 1, 2, 3 …, . Here an example of {Xk} is given with finite second moments, for which Var(X1 + … + Xn)→∞ and ρ n → 0 as n→∞, but (X1 + … + Xn) fails to be asymptotically normal; instead there is partial attraction to non-stable limit laws.

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