This document discusses various statistical methods for determining the optimal number of bins and bin width in histograms. It reviews methods such as Sturges' rule, Scott's rule, Bayesian optimal binning, and others. It presents the concepts behind each method and compares their performance in simulations. The simulations involved generating data from distributions like the chi-square, normal, and uniform, and comparing the methods based on how close their histogram estimates were to the true densities. Scott's rule and Freedman-Diaconis modification generally performed best at minimizing errors. The document also discusses generalizing some methods to multivariate histograms and potential areas for further research.
This document provides an introduction to descriptive statistics and measures of central tendency, including the mean, median, and mode. It discusses how the mean can be impacted by outliers, while the median is not. The standard deviation and variance are introduced as measures of dispersion that quantify how much values vary from the mean or from each other. Finally, the document discusses different ways of organizing and graphing data, including histograms, pie charts, line graphs, and scatter plots.
This presentation describes the concept of One Sample t-test, Independent Sample t-test and Paired Sample t-test. This presentation also deals about the procedure to do the t-test through SPSS.
This document discusses identifying a research problem, which is the first step in the research process. A research problem should try to explain what, why, how, where, and who. There must be an individual to attribute the problem to, controlled and uncontrolled variables, and at least two possible outcomes. Sources of research problems include personal experience, journals, newspapers, textbooks, and previous work suggestions. When identifying a problem, over-researched topics, controversial subjects, topics that are too narrow or vague, and problems that are unfamiliar or infeasible should be avoided. The importance, researcher qualifications, and cost should also be considered. Techniques for problem identification include generally stating the problem, understanding its nature, reviewing literature, developing ideas
Social Research: Part II Types of Researchbchozinski
This document provides an overview of quantitative and qualitative research methods. Quantitative research aims to quantify data by counting and measuring variables to construct statistical models, while qualitative research seeks to understand characteristics and meanings through methods like observation and interviews. Both approaches can be used together, such as through content analysis of texts. The document also provides examples of specific research methods like surveys, experiments, field research, and ethnography.
This document provides an overview of statistics presented by five students. It defines statistics as the practice of collecting and analyzing numerical data. Descriptive statistics summarize data through parameters like the mean, while inferential statistics interpret descriptive statistics to draw conclusions. The document discusses examples of statistics, different types of charts and graphs, descriptive versus inferential statistics, and the importance and applications of statistics in fields like business, economics, and social sciences. It also covers topics like sampling methods, characteristics of sampling, probability versus non-probability sampling, and differences between the two.
Statistical Data Analysis | Data Analysis | Statistics Services | Data Collec...Stats Statswork
The present article helps the USA, the UK and the Australian students pursuing their business and marketing postgraduate degree to identify right topic in the area of marketing in business. These topics are researched in-depth at the University of Columbia, brandies, Coventry, Idaho, and many more. Stats work offers UK Dissertation stats work Topics Services in business. When you Order stats work Dissertation Services at Tutors India, we promise you the following – Plagiarism free, Always on Time, outstanding customer support, written to Standard, Unlimited Revisions support and High-quality Subject Matter Experts.
Contact Us:
Website: www.statswork.com
Email: info@statswork.com
UnitedKingdom: +44-1143520021
India: +91-4448137070
WhatsApp: +91-8754446690
This document discusses non-probability sampling methods. It describes convenience sampling as selecting sample units that are easily accessible, and volunteer sampling as using people who voluntarily participate in a study. Some advantages of non-probability sampling are that it is easy to use and less costly than probability sampling methods, though the samples may be biased and not representative of the target population. The document provides examples of both convenience sampling and volunteer sampling.
SPSS for beginners, a short course about how novices can use SPSS to analyze their research findings. With this tutorial anyone becomes able to use SPSS for basic statistical analysis. No need to be a professional to use SPSS.
This document provides an introduction to descriptive statistics and measures of central tendency, including the mean, median, and mode. It discusses how the mean can be impacted by outliers, while the median is not. The standard deviation and variance are introduced as measures of dispersion that quantify how much values vary from the mean or from each other. Finally, the document discusses different ways of organizing and graphing data, including histograms, pie charts, line graphs, and scatter plots.
This presentation describes the concept of One Sample t-test, Independent Sample t-test and Paired Sample t-test. This presentation also deals about the procedure to do the t-test through SPSS.
This document discusses identifying a research problem, which is the first step in the research process. A research problem should try to explain what, why, how, where, and who. There must be an individual to attribute the problem to, controlled and uncontrolled variables, and at least two possible outcomes. Sources of research problems include personal experience, journals, newspapers, textbooks, and previous work suggestions. When identifying a problem, over-researched topics, controversial subjects, topics that are too narrow or vague, and problems that are unfamiliar or infeasible should be avoided. The importance, researcher qualifications, and cost should also be considered. Techniques for problem identification include generally stating the problem, understanding its nature, reviewing literature, developing ideas
Social Research: Part II Types of Researchbchozinski
This document provides an overview of quantitative and qualitative research methods. Quantitative research aims to quantify data by counting and measuring variables to construct statistical models, while qualitative research seeks to understand characteristics and meanings through methods like observation and interviews. Both approaches can be used together, such as through content analysis of texts. The document also provides examples of specific research methods like surveys, experiments, field research, and ethnography.
This document provides an overview of statistics presented by five students. It defines statistics as the practice of collecting and analyzing numerical data. Descriptive statistics summarize data through parameters like the mean, while inferential statistics interpret descriptive statistics to draw conclusions. The document discusses examples of statistics, different types of charts and graphs, descriptive versus inferential statistics, and the importance and applications of statistics in fields like business, economics, and social sciences. It also covers topics like sampling methods, characteristics of sampling, probability versus non-probability sampling, and differences between the two.
Statistical Data Analysis | Data Analysis | Statistics Services | Data Collec...Stats Statswork
The present article helps the USA, the UK and the Australian students pursuing their business and marketing postgraduate degree to identify right topic in the area of marketing in business. These topics are researched in-depth at the University of Columbia, brandies, Coventry, Idaho, and many more. Stats work offers UK Dissertation stats work Topics Services in business. When you Order stats work Dissertation Services at Tutors India, we promise you the following – Plagiarism free, Always on Time, outstanding customer support, written to Standard, Unlimited Revisions support and High-quality Subject Matter Experts.
Contact Us:
Website: www.statswork.com
Email: info@statswork.com
UnitedKingdom: +44-1143520021
India: +91-4448137070
WhatsApp: +91-8754446690
This document discusses non-probability sampling methods. It describes convenience sampling as selecting sample units that are easily accessible, and volunteer sampling as using people who voluntarily participate in a study. Some advantages of non-probability sampling are that it is easy to use and less costly than probability sampling methods, though the samples may be biased and not representative of the target population. The document provides examples of both convenience sampling and volunteer sampling.
SPSS for beginners, a short course about how novices can use SPSS to analyze their research findings. With this tutorial anyone becomes able to use SPSS for basic statistical analysis. No need to be a professional to use SPSS.
Correlation & Regression Analysis using SPSSParag Shah
Concept of Correlation, Simple Linear Regression & Multiple Linear Regression and its analysis using SPSS. How it check the validity of assumptions in Regression
This document discusses various methods for summarizing data, including measures of central tendency, dispersion, and categorical data. It describes the mean, median, and mode as measures of central tendency, and how the mean can be affected by outliers while the median is not. Measures of dispersion mentioned include range, standard deviation, variance, and interquartile range. The document also discusses percentiles, standard error, and 95% confidence intervals. Key takeaways are to select appropriate summaries based on the data type and distribution.
This document provides an overview of a data analysis course that covers topics such as descriptive statistics, probability distributions, correlation, regression, hypothesis testing, clustering, and time series analysis. The course introduces descriptive statistics including measures of central tendency, dispersion, frequency distributions, and histograms. Notes are provided on calculating and interpreting mean, median, mode, range, variance, standard deviation, and other descriptive statistics.
Psychologist Stanley Smith Stevens (1946) developed the best-known classification with four levels, or scales of measurement such as Nominal, Ordinal, Interval, and Ratio. This presentation slide describes the four-level of scales with illustrations.
Descriptive research involves collecting and organizing quantitative and qualitative data to describe events or situations without looking for relationships or causes. It aims to answer questions about what currently exists, such as teachers' attitudes toward technology or how students interact with educational programs. Common descriptive research methods include surveys, interviews, observations, and collecting student portfolios. Descriptive research provides an understanding of current conditions that can help identify areas for improvement.
The document discusses quantitative research design and methodology. It describes different quantitative research methods such as surveys, interviews, and physical counts. It explains that quantitative research aims to discover how many people think, act, or feel in a certain way by using large sample sizes. The document also summarizes different quantitative research designs like descriptive, experimental, correlational, and quasi-experimental designs. It provides details on data analysis methods in quantitative research including descriptive and inferential statistics.
This document discusses ethical issues in data collection. It defines ethics and the major areas of ethical study, including meta-ethics, normative ethics, and applied ethics. The document outlines basic principles of ethical practice such as informed consent, avoiding harm, and maintaining privacy and confidentiality. Researchers should obtain informed consent, avoid deception, and not cause harm or offer excessive rewards. Overall, the document provides an overview of ethics in research and data collection and discusses principles researchers should follow to conduct ethical studies and experiments.
- The researcher analyzed survey data from 200 Muslim government officers regarding their concern for halal products. Officers were grouped by level of management: top, middle, supervisor.
- A chi-square test found no significant relationship between level of management and concern for halal products. The calculated chi-square value of 0.79 was less than the critical value of 9.49, so the null hypothesis that there is no difference was accepted.
- In summary, the level of management (top, middle, supervisor) did not have an effect on officers' reported concern for halal products based on this analysis of the survey data.
This document provides an overview of case studies as a research method. It defines a case study as an in-depth analysis of a person, group, process, situation, program, or other social unit. Case studies aim to understand the life cycle or interactions that explain the current status or development over time of the unit of study. The document lists some examples of case studies and discusses the functions and techniques used in case studies. It compares case studies to surveys and outlines some merits and demerits of the case study method.
This document discusses different types of variables:
- Categorical variables describe categories or groups like gender or subject.
- Numerical/quantitative variables can be discrete or continuous. Discrete variables have countable values while continuous variables have infinite values between any two points, like height, weight, or time.
- Examples are provided of discrete variables like number of students or apples, and continuous variables like length, area, distance, or temperature.
This document provides an outline and overview of tutorials for presenting output using the STATA data analysis software. It discusses datasets used in examples from an econometrics textbook. The tutorials cover commands for finding results, outputting regression tables, copying tables and graphs, and downloading more tutorials on additional STATA topics from the website.
Bayesian statistics uses Bayes' theorem to update probabilities as new data is acquired. The key relationship is that the posterior probability is proportional to the prior probability multiplied by the likelihood. Bayes' theorem explains that the joint probability of variables is the prior probability multiplied by the conditional probability. It can be used for problems like binomial distribution, normal distribution, and linear regression. Robust Bayesian analysis investigates the sensitivity of Bayesian results to uncertainties in the analysis.
This document provides an overview of key statistical concepts including point estimation, confidence intervals, hypothesis testing, and sample size determination. It discusses how to calculate point estimates like the sample mean. It explains how to construct confidence intervals using the normal and t-distributions. It outlines how to perform lower tail, upper tail, and two-tailed hypothesis tests on means and proportions. It also provides formulas for determining required sample sizes.
This document describes a new algorithm for dual tree kernel conditional density estimation (KCDE) that provides fast and accurate density predictions. The algorithm extends previous work on univariate KCDE to allow for multivariate labels (Y) and conditioning variables (X). It applies Gray's dual tree approach separately to the numerator and denominator of the KCDE formula, and uses error bounds to ensure the quotient estimates have bounded relative error. This new algorithm provides the fastest known method for kernel conditional density estimation for prediction tasks.
This document discusses Monte Carlo simulations in Stata to verify statistical properties like the weak law of large numbers and the central limit theorem. It introduces generating random numbers, simulations to prove the weak law using coin tosses, and simulations to demonstrate the central limit theorem using samples from a uniform distribution. The simulations allow investigating the theoretical properties of estimators by defining the data generating process.
The document discusses various methods for modeling input distributions in simulation models, including trace-driven simulation, empirical distributions, and fitting theoretical distributions to real data. It provides examples of several continuous and discrete probability distributions commonly used in simulation, including the exponential, normal, gamma, Weibull, binomial, and Poisson distributions. Key parameters and properties of each distribution are defined. Methods for selecting an appropriate input distribution based on summary statistics of real data are also presented.
1979 Optimal diffusions in a random environmentBob Marcus
This paper analyzes mathematical models of population growth and diffusion in a random environment. It presents both discrete and continuous models to determine the optimal rate of offspring dispersal to maximize population growth over time. Through analysis and examples, the paper shows that for many cases, having no dispersal (a diffusion rate of zero) is actually optimal, contrary to what numerical simulations may indicate. The optimal strategy is to not distribute the population but instead to allow it to remain concentrated, gambling on the small probability of extremely large growth.
The partitioning of an ordered prognostic factor is important in order to obtain several groups having heterogeneous survivals in medical research. For this purpose, a binary split has often been used once or recursively. We propose the use of a multi-way split in order to afford an optimal set of cut-off points. In practice, the number of groups ($K$) may not be specified in advance. Thus, we also suggest finding an optimal $K$ by a resampling technique. The algorithm was implemented into an \proglang{R} package that we called \pkg{kaps}, which can be used conveniently and freely. It was illustrated with a toy dataset, and was also applied to a real data set of colorectal cancer cases from the Surveillance Epidemiology and End Results.
Bayesian inference for mixed-effects models driven by SDEs and other stochast...Umberto Picchini
An important, and well studied, class of stochastic models is given by stochastic differential equations (SDEs). In this talk, we consider Bayesian inference based on measurements from several individuals, to provide inference at the "population level" using mixed-effects modelling. We consider the case where dynamics are expressed via SDEs or other stochastic (Markovian) models. Stochastic differential equation mixed-effects models (SDEMEMs) are flexible hierarchical models that account for (i) the intrinsic random variability in the latent states dynamics, as well as (ii) the variability between individuals, and also (iii) account for measurement error. This flexibility gives rise to methodological and computational difficulties.
Fully Bayesian inference for nonlinear SDEMEMs is complicated by the typical intractability of the observed data likelihood which motivates the use of sampling-based approaches such as Markov chain Monte Carlo. A Gibbs sampler is proposed to target the marginal posterior of all parameters of interest. The algorithm is made computationally efficient through careful use of blocking strategies, particle filters (sequential Monte Carlo) and correlated pseudo-marginal approaches. The resulting methodology is is flexible, general and is able to deal with a large class of nonlinear SDEMEMs [1]. In a more recent work [2], we also explored ways to make inference even more scalable to an increasing number of individuals, while also dealing with state-space models driven by other stochastic dynamic models than SDEs, eg Markov jump processes and nonlinear solvers typically used in systems biology.
[1] S. Wiqvist, A. Golightly, AT McLean, U. Picchini (2020). Efficient inference for stochastic differential mixed-effects models using correlated particle pseudo-marginal algorithms, CSDA, https://doi.org/10.1016/j.csda.2020.107151
[2] S. Persson, N. Welkenhuysen, S. Shashkova, S. Wiqvist, P. Reith, G. W. Schmidt, U. Picchini, M. Cvijovic (2021). PEPSDI: Scalable and flexible inference framework for stochastic dynamic single-cell models, bioRxiv doi:10.1101/2021.07.01.450748.
1) The document discusses inference for treatment regimes, which can be nonregular due to abrupt dependence on nuisance parameters.
2) It introduces a toy problem called "max of means" to illustrate challenges in inference for nonregular problems. The limiting distribution depends on whether treatment means are equal or not.
3) Local asymptotic frameworks are proposed to allow small differences between treatment means to persist as sample size increases, providing a more robust inference approach than assuming certain differences are exactly zero.
Correlation & Regression Analysis using SPSSParag Shah
Concept of Correlation, Simple Linear Regression & Multiple Linear Regression and its analysis using SPSS. How it check the validity of assumptions in Regression
This document discusses various methods for summarizing data, including measures of central tendency, dispersion, and categorical data. It describes the mean, median, and mode as measures of central tendency, and how the mean can be affected by outliers while the median is not. Measures of dispersion mentioned include range, standard deviation, variance, and interquartile range. The document also discusses percentiles, standard error, and 95% confidence intervals. Key takeaways are to select appropriate summaries based on the data type and distribution.
This document provides an overview of a data analysis course that covers topics such as descriptive statistics, probability distributions, correlation, regression, hypothesis testing, clustering, and time series analysis. The course introduces descriptive statistics including measures of central tendency, dispersion, frequency distributions, and histograms. Notes are provided on calculating and interpreting mean, median, mode, range, variance, standard deviation, and other descriptive statistics.
Psychologist Stanley Smith Stevens (1946) developed the best-known classification with four levels, or scales of measurement such as Nominal, Ordinal, Interval, and Ratio. This presentation slide describes the four-level of scales with illustrations.
Descriptive research involves collecting and organizing quantitative and qualitative data to describe events or situations without looking for relationships or causes. It aims to answer questions about what currently exists, such as teachers' attitudes toward technology or how students interact with educational programs. Common descriptive research methods include surveys, interviews, observations, and collecting student portfolios. Descriptive research provides an understanding of current conditions that can help identify areas for improvement.
The document discusses quantitative research design and methodology. It describes different quantitative research methods such as surveys, interviews, and physical counts. It explains that quantitative research aims to discover how many people think, act, or feel in a certain way by using large sample sizes. The document also summarizes different quantitative research designs like descriptive, experimental, correlational, and quasi-experimental designs. It provides details on data analysis methods in quantitative research including descriptive and inferential statistics.
This document discusses ethical issues in data collection. It defines ethics and the major areas of ethical study, including meta-ethics, normative ethics, and applied ethics. The document outlines basic principles of ethical practice such as informed consent, avoiding harm, and maintaining privacy and confidentiality. Researchers should obtain informed consent, avoid deception, and not cause harm or offer excessive rewards. Overall, the document provides an overview of ethics in research and data collection and discusses principles researchers should follow to conduct ethical studies and experiments.
- The researcher analyzed survey data from 200 Muslim government officers regarding their concern for halal products. Officers were grouped by level of management: top, middle, supervisor.
- A chi-square test found no significant relationship between level of management and concern for halal products. The calculated chi-square value of 0.79 was less than the critical value of 9.49, so the null hypothesis that there is no difference was accepted.
- In summary, the level of management (top, middle, supervisor) did not have an effect on officers' reported concern for halal products based on this analysis of the survey data.
This document provides an overview of case studies as a research method. It defines a case study as an in-depth analysis of a person, group, process, situation, program, or other social unit. Case studies aim to understand the life cycle or interactions that explain the current status or development over time of the unit of study. The document lists some examples of case studies and discusses the functions and techniques used in case studies. It compares case studies to surveys and outlines some merits and demerits of the case study method.
This document discusses different types of variables:
- Categorical variables describe categories or groups like gender or subject.
- Numerical/quantitative variables can be discrete or continuous. Discrete variables have countable values while continuous variables have infinite values between any two points, like height, weight, or time.
- Examples are provided of discrete variables like number of students or apples, and continuous variables like length, area, distance, or temperature.
This document provides an outline and overview of tutorials for presenting output using the STATA data analysis software. It discusses datasets used in examples from an econometrics textbook. The tutorials cover commands for finding results, outputting regression tables, copying tables and graphs, and downloading more tutorials on additional STATA topics from the website.
Bayesian statistics uses Bayes' theorem to update probabilities as new data is acquired. The key relationship is that the posterior probability is proportional to the prior probability multiplied by the likelihood. Bayes' theorem explains that the joint probability of variables is the prior probability multiplied by the conditional probability. It can be used for problems like binomial distribution, normal distribution, and linear regression. Robust Bayesian analysis investigates the sensitivity of Bayesian results to uncertainties in the analysis.
This document provides an overview of key statistical concepts including point estimation, confidence intervals, hypothesis testing, and sample size determination. It discusses how to calculate point estimates like the sample mean. It explains how to construct confidence intervals using the normal and t-distributions. It outlines how to perform lower tail, upper tail, and two-tailed hypothesis tests on means and proportions. It also provides formulas for determining required sample sizes.
This document describes a new algorithm for dual tree kernel conditional density estimation (KCDE) that provides fast and accurate density predictions. The algorithm extends previous work on univariate KCDE to allow for multivariate labels (Y) and conditioning variables (X). It applies Gray's dual tree approach separately to the numerator and denominator of the KCDE formula, and uses error bounds to ensure the quotient estimates have bounded relative error. This new algorithm provides the fastest known method for kernel conditional density estimation for prediction tasks.
This document discusses Monte Carlo simulations in Stata to verify statistical properties like the weak law of large numbers and the central limit theorem. It introduces generating random numbers, simulations to prove the weak law using coin tosses, and simulations to demonstrate the central limit theorem using samples from a uniform distribution. The simulations allow investigating the theoretical properties of estimators by defining the data generating process.
The document discusses various methods for modeling input distributions in simulation models, including trace-driven simulation, empirical distributions, and fitting theoretical distributions to real data. It provides examples of several continuous and discrete probability distributions commonly used in simulation, including the exponential, normal, gamma, Weibull, binomial, and Poisson distributions. Key parameters and properties of each distribution are defined. Methods for selecting an appropriate input distribution based on summary statistics of real data are also presented.
1979 Optimal diffusions in a random environmentBob Marcus
This paper analyzes mathematical models of population growth and diffusion in a random environment. It presents both discrete and continuous models to determine the optimal rate of offspring dispersal to maximize population growth over time. Through analysis and examples, the paper shows that for many cases, having no dispersal (a diffusion rate of zero) is actually optimal, contrary to what numerical simulations may indicate. The optimal strategy is to not distribute the population but instead to allow it to remain concentrated, gambling on the small probability of extremely large growth.
The partitioning of an ordered prognostic factor is important in order to obtain several groups having heterogeneous survivals in medical research. For this purpose, a binary split has often been used once or recursively. We propose the use of a multi-way split in order to afford an optimal set of cut-off points. In practice, the number of groups ($K$) may not be specified in advance. Thus, we also suggest finding an optimal $K$ by a resampling technique. The algorithm was implemented into an \proglang{R} package that we called \pkg{kaps}, which can be used conveniently and freely. It was illustrated with a toy dataset, and was also applied to a real data set of colorectal cancer cases from the Surveillance Epidemiology and End Results.
Bayesian inference for mixed-effects models driven by SDEs and other stochast...Umberto Picchini
An important, and well studied, class of stochastic models is given by stochastic differential equations (SDEs). In this talk, we consider Bayesian inference based on measurements from several individuals, to provide inference at the "population level" using mixed-effects modelling. We consider the case where dynamics are expressed via SDEs or other stochastic (Markovian) models. Stochastic differential equation mixed-effects models (SDEMEMs) are flexible hierarchical models that account for (i) the intrinsic random variability in the latent states dynamics, as well as (ii) the variability between individuals, and also (iii) account for measurement error. This flexibility gives rise to methodological and computational difficulties.
Fully Bayesian inference for nonlinear SDEMEMs is complicated by the typical intractability of the observed data likelihood which motivates the use of sampling-based approaches such as Markov chain Monte Carlo. A Gibbs sampler is proposed to target the marginal posterior of all parameters of interest. The algorithm is made computationally efficient through careful use of blocking strategies, particle filters (sequential Monte Carlo) and correlated pseudo-marginal approaches. The resulting methodology is is flexible, general and is able to deal with a large class of nonlinear SDEMEMs [1]. In a more recent work [2], we also explored ways to make inference even more scalable to an increasing number of individuals, while also dealing with state-space models driven by other stochastic dynamic models than SDEs, eg Markov jump processes and nonlinear solvers typically used in systems biology.
[1] S. Wiqvist, A. Golightly, AT McLean, U. Picchini (2020). Efficient inference for stochastic differential mixed-effects models using correlated particle pseudo-marginal algorithms, CSDA, https://doi.org/10.1016/j.csda.2020.107151
[2] S. Persson, N. Welkenhuysen, S. Shashkova, S. Wiqvist, P. Reith, G. W. Schmidt, U. Picchini, M. Cvijovic (2021). PEPSDI: Scalable and flexible inference framework for stochastic dynamic single-cell models, bioRxiv doi:10.1101/2021.07.01.450748.
1) The document discusses inference for treatment regimes, which can be nonregular due to abrupt dependence on nuisance parameters.
2) It introduces a toy problem called "max of means" to illustrate challenges in inference for nonregular problems. The limiting distribution depends on whether treatment means are equal or not.
3) Local asymptotic frameworks are proposed to allow small differences between treatment means to persist as sample size increases, providing a more robust inference approach than assuming certain differences are exactly zero.
An investigation of inference of the generalized extreme value distribution b...Alexander Decker
This document presents an investigation of parameter estimation for the generalized extreme value distribution based on record values. Maximum likelihood estimation is used to estimate the parameters β (scale parameter) and ξ (shape parameter). Likelihood equations are derived and solved numerically. Bootstrap and Markov chain Monte Carlo methods are proposed to construct confidence intervals for the parameters since intervals based on asymptotic normality may not perform well due to small sample sizes of records. Bayesian estimation of the parameters using MCMC is also investigated. An illustrative example involving simulated records is provided.
This document provides an overview of key statistical concepts taught in a statistics lab lesson, including point estimation, confidence intervals, and hypothesis testing. It defines point estimators like the sample mean that summarize a population using a sample. Confidence intervals give a range of values that the population parameter is expected to lie within. Hypothesis testing involves setting up null and alternative hypotheses and using a test statistic and critical value to reject or fail to reject the null hypothesis. Formulas for confidence intervals and hypothesis tests are presented for situations involving normal, t, and binomial distributions.
Sturges' rule for determining the number of bins in a histogram is commonly used, but its theoretical derivation is flawed. The author argues that Sturges implicitly assumed data was normally distributed when deriving the rule, but the rule does not necessarily hold for non-normal data. Alternative methods for determining the number of bins, such as Scott's rule or Freedman and Diaconis' rule, are better justified statistically and should be preferred over Sturges' rule.
law of large number and central limit theoremlovemucheca
The document provides information about the Law of Large Numbers and the Central Limit Theorem. It discusses two key concepts:
1) As the sample size increases, the sample average converges to the population average. This is known as the Law of Large Numbers and "guarantees" stable long-term results for random events.
2) Regardless of the underlying population distribution, as sample size increases, the sample mean will be approximately normally distributed around the population mean. This is the Central Limit Theorem, which allows sample means and proportions to be analyzed using normal probability models.
The document provides examples to illustrate how these concepts can be applied, such as using the Central Limit Theorem to determine the probability that a sample average
This document discusses measures of central tendency (mean, median, mode) and measures of spread (range, variance, standard deviation). It provides formulas and examples to calculate each measure. It also presents two problems, asking to calculate and compare various descriptive statistics for different data sets, such as milk yields from two cow herds and weaning weights of lambs from two breeds. A third problem asks to analyze and compare price data for rice from two markets.
This document provides information on measures of central tendency and dispersion. It discusses the mean, median, and mode as the three main measures of central tendency. It provides formulas and examples for calculating the mean, median, and mode for both ungrouped and grouped data. The document also covers measures of dispersion including range, semi-interquartile range, variance, standard deviation, and coefficient of variation. It provides formulas and examples for calculating each of these measures. Finally, the document briefly discusses chi-square tests, Pearson's correlation, and using scatterplots to examine relationships between variables.
This document provides an overview of statistical inference concepts including:
1. Best unbiased estimators, which have minimum mean squared error for a given parameter. The best unbiased estimator, if it exists, must be a function of a sufficient statistic.
2. Sufficiency and the Rao-Blackwell theorem, which states that conditioning an estimator on a sufficient statistic produces a uniformly better estimator.
3. The Cramér-Rao lower bound, which provides a lower bound on the variance of unbiased estimators. Examples are given to illustrate key concepts like when the bound may not hold.
4. Examples are worked through to find minimum variance unbiased estimators, maximum likelihood estimators, and confidence intervals for various distributions
The document discusses applying random distortion testing (RDT) in a spectral clustering context. RDT is a framework for guaranteeing a false alarm probability threshold in detecting distorted data using threshold-based tests. The document introduces RDT and spectral clustering concepts. It then proposes using the p-value from RDT as the similarity function or kernel in spectral clustering, to handle disturbed data. Experiments are conducted to compare the partitioning performance of the RDT p-value kernel to the Gaussian kernel.
Similar to Selection of bin width and bin number in histograms (20)
2. We often come across the problem when density of the variable of
interest is unknown. One popular method of estimating the unknown
density is by using the Histogram estimator.
Often the decision on bin number or bin width in a histogram is
made arbitrarily or subjectively but need not be. Here we review the
literature on various statistical procedures that have been proposed for
making the decision on optimum bin width and bin number.
3. We shall review various methods in statistical literature that
are prevalent for determining the optimal number of bins and
the bin width in a histogram.
We shall also try to present a comparative analysis so as to
determine which methods are more efficient.
The measure we use to compare the various methods of
optimal binning is sup|ĥ(x) -h(x)| where ĥ(x) is the
histogram density estimator at x and h is the true density at the
point x.
4. Proposed methods of interest for optimal binning
Sturges’ rule and Doane Modification
Scott Rule and Freedman-Diaconis modification
Bayesian optimal binning
Optimal binning by Hellinger Risk minimization
Penalized maximum log-likelihood method with
penalty A and Hogg penalty
Stochastic complexity or Kolmogorov complexity
method
5. Sturges’ Rule
If one constructs a frequency distribution with k bins , each of width 1 and
centered on the points i=0,1,. . . , k-1 . Choose the bin count of the "i" th
bin to be the Binomial coefficient . As k increases, the ideal frequency
histogram assumes the shape of a normal density with mean (k-1)/2 and
variance (k-1)/4.
According to the Sturges' rule, the optimum number of bins for the histogram
is given by,
n =
k is the number of bins to be used. This when solved for k, gives us
n=
We split the sample range into k such bins of equal length. So, the Sturges'
rule gives us a regular histogram.
6. Conceptual Fallacy of the Sturges rule
• There is conceptually a fallacy in Sturges rule derivation Instead of
choosing n= , one could have satisfied any n that satisfies
individual cell frequencies to be
• m(i)=no. of obs in “i”th cell could well have been taken to be
m(i)= n.
• So, intuitively there is no reason for choosing this particular n given
the motivation we employ in Sturges’ rule.
Doane’s law
For skewed or kurtotic distributions, additional bins may be required.
Doane proposed increasing the number of bins by log2(1 +ŷ ) where
ŷ is the standardized skewness coefficient.
7. Scott rule and Freedman Diaconis
modification
• We get an optimum band width by minimizing the asymptotic
expected L2 norm. The histogram estimator is given by
• ĥ(x)=Vk /nh where h is the bin width and n is the total no. of
observations and Vk = no. of obs lying in the “k” th bin.
• The optimum band width given by
h*(x) = [f(xk)/2γ2n] 1/3 where xk is some point lying in the “k”th
bin and γ is the Lipschitz continuity factor.
For normal density case, we observe that h*= 3.5n-1/3sd(x) for
regular case.
The Friedman Diaconis modification for non-normal data is given
by h*= 2(IQ)n-1/3
8. Hellinger risk minimization
• The Hellinger risk between the histogram density estimator ĥ(x) for a
given bin width k for a regular histogram and the true density f(x) is
defined as
H=
• We try to minimize this quantity for different choices of the bin width
or bin number .
• If the true f is known, we have no problem in dealing with this
integral. But, if the true f is not known, one may estimate f using
Bootstrapping over repeated sample from f.
9. Bayesian model for optimal binning
The likelihood of the data given the parameters M – no. of
bins and the vector tuple π ,we get
P(d/ π,M,I)= (M/V)N π1n1π2n2 ……πM- 1nM-1 πMnM where
V =Mv and v is the bin width.
Assume that the prior densities are defined as follows
P(M/I)=1/C where C= max no. of bins taken in account
P(π/M) = [π1π2 …πM ]-1/2 ᴦ (M/2)/ᴦ (1/2)M. Which is actually
a Dirichlet distribution with M parameters equal to ½ and this is
conjugate prior of multinomial distribution.
P(π, M/d,I) = k*P(π /M)P(M/)P(d/ π ,M) is obtained and
integrated over M to get the marginal distribution of M which
when maximized yields the optimal value of M.
10. Maximum penalized loglikelihood method
In this case we do maximize the loglikelihood of the multinomial
distribution corresponding to a histogram but with some penalty
function added. The penalized loglikelihood is thus of the form
Pl=log(L(ĥ, x1 , x2 ,……, xn))- penn (I) where I is the partition of
the sample range into disjoint intervals. Note that these bins need
not be of equal length i,e the histogram may be irregular.
There are various choices of the penalty , however our two
choices have been under D bins
penA=
The first penalty is applicable for both regular and irregular cases
penB(Hogg or Akaike penalty)=D-1
11. Stochastic complexity method
• This is based on the idea of encoding the data with minimum number
of bits. This is a sort of PML with no. of bits or description length as
penalty.
• If P(X|Ө) be the distribution of the data with Ө unknown and if σi
(Ө) be the standard deviation with respect to the
best estimator of “i” th co-ordiante of Ө, then the description length
is given by
- log2 (P(X|Ө))+∑ log2 ( )
We define stochastic complexity as
- log2 ∫ P(X|Ө) π(Ө)dӨ. If we take an uniform prior for Ө,
Then taking P(X|Ө) to be the multinomial distribution, we get
stochastic complexity to be
l=(m-1)ᴦ (N .N2 ….. , Nm )/(m+n-1)ǃ. Maximize wrt to m to
1
get the no. of bins
12. Simulation design
In order to compare the various methods of binning, we use
simulation experiments from 3 reference distributions namely Chi
square (2), Normal(0,1) and Uniform(1,10).
We compare the statistic T = | |ĥ(x)-f(x)|
For various methods and compare how smaller the value of T is on an
average for each of these methods.
We have simulated 1000 observations from each of the reference
distributions,computed the T statistic for each simulated run and
carry out this experiment 200 times to get a distribution of T.
13. Mean and variance of T for chi-
square(2)
Method Mean no. Mean(T) Variance(T)
of bins
Sturges 10 0.1364 0.00031
Doane 15 0.1028 0.00018
Scott 19 0.0874 0.00015
Hellinger 13 0.1151 0.00022
FD 32 0.0747 0.00023
Kolmogrov 10 0.2744 0.01288
Bayesian 12 0.1177 0.00037
Hogg 18 0.0948 0.000194
Irregular(pen 6 0.1134 0.00028
A)
15. Analysis of chi square simulation
For Chi-square(2), Freedman-diaconis and Scott’s rule
have performed very well in terms of smaller mean value of
T.
Kolmogorov’s complexity method has the maximum
spread in the t-values. The distribution of T under the
sturge’s rule dominates that under Freedman-diaconis and
Scott’s rule.
The irregular histogram method under PenA gives very
less no. of bins compared to others.
16. Mean and variance of T for N(0,1)
Method Mean Mean(T) Variance(T)
No. of
Bins
Sturges 10 0.0909 0.00013
Scott 18 0.08377 0.00025
Hellinger 20 0.08309 0.00022
FD 25 0.08687 0.00029
Kolmogrov 13 0.2243 0.0137
Bayesian 13 0.0912 0.00022
Hogg 12 0.0855 0.00011
Irregular(penA) 6 0.1984 0.00113
18. For Normal(0,1), we left out Doane's modification as it is meant for
non-normal or skewed distribution.
Sturges rule and Scott’s rule have performed very well under the
normal case, which is expected given that they are designed under
normality assumptions.
Scott, Freedman-Diaconis and Sturges rule are very close to one
another in terms of the distribution of T.
The penalized log –likelihood with penalty A has a distribution of T
that dominates the T distribution under the other methods.
The T-distribution under stochastic complexity and Hellinger
distance have maximum spread. The minimum spread is due to
Sturges rule.
19. Mean and Variance of T for U(1,10)
Method Mean no. of Mean(T) Variance(T)
Bins
Sturges 10 0.1298 0.00036
Scott 9 0.1288 0.00035
Doane 11 0.1308 0.00051
FD 9 0.1283 0.00032
Bayesian 9 0.1274 0.000361
21. Analysis under U(1,10) distribution
Most of the methods under uniform case give only
1 or 2 bins, so they cannot be compared with others
which are more stable in nature.
However, the Scott’s, Freedman Diaconis and
Sturges rule have performed well with small values
of the T and small variation in the values of T
under repeated simulations.
22. Similar to the univariate method, we try to
generalize our method for bivariate
distributions.
Here we simulate observations from bivariate
normal distribution with mean (0,0) , ρ = 0.5
and σ2 = 1.
The methods we use are the multivariate
extension of the Bayesian optimal binning
and the multivariate Scott's rule.
23. In the same vein as in univariate case, the
multivariate Scott’s rule is determined by minimizing
the asymptotic L-2 error of the expected L-2 norm.
The Multivariate Scott’s choice of bin
width is given by h*=3.5 σxk
Where d is the dimension of the
dataset and σxk the standard
deviation along “k”th co-ordinate.
24. The 3-d histogram obtained for T statistic
distribution under Scott rule
26. Bayesian optimal binning for multivariate normal
case
In this case, we select Mx bins along X axis and My
bins along the Y axis and define M= Mx My .The joint
likelihood in this case given by
h(x,y, Mx ,My )=
which is quite analogous to the univariate
case. Again taking a rectangular prior for
(Mx,My ) and dirichlet distribution of M
dimensions with each parameter ½ as prior
for ℿ.
29. We have dealt with only histogram estimators in
this paper.However,one may apply smoothing
parameter to make the estimator more efficient and
analyze the values of T-staistic for various smoothing
parameters.
We have only used Bayesian and Scott’s
multivariate extensions . However, one may try to
generalize other methods in the multivariate case .
One may use other form of penalties and observe for
which penalty, the estimator thus obtained is most
efficient.
30. From All three univariate simulation
experiments we infer that Scott’s and
Freedman-Diaconis method have been most
efficient in reducing the values of T .
No method however is uniformly best under
all scenarios.
For bivariate normal case, using Scott’s rule
and Bayesian optimal binning , we find that the
T value is smaller on an average under Scott
than under the bayesian optimal binning.