A discrete probability problem in card shuffling

Abstract

There are n cards serially numbered from 1 to n. The cards are shuffled and placed in a line one after the other on top of a table with faces up. The numbers on the faces are read from left to right. If there are consecutive numbers in increasing order of magnitude the corresponding cards are merged into one. After the merger, the cards are numbered serially from one to whatever the number of cards we now have. The cards are shuffled and placed in a line one after another on top of the table with faces up. The process continues until we have only one card left. In this paper, we develop a probabilistic recurrence relation approach to obtain the mean, variance, and distribution of the number of shuffles needed. A Markov chain formulation and its properties are discussed in the paper as well.     

VARIANCE ESTIMATION IN TWO-PHASE SAMPLING

Two‐phase sampling is often used for estimating a population total or mean when the cost per unit of collecting auxiliary variables, x, is much smaller than the cost per unit of measuring a characteristic of interest, y. In the first phase, a large sample s₁ is drawn according to a specific sampling design p(s₁) , and auxiliary data x are observed for the units i∈s₁ . Given the first‐phase sample s₁ , a second‐phase sample s₂ is selected from s₁ according to a specified sampling design {p(s₂∣s₁) } , and (y, x) is observed for the units i∈s₂ . In some cases, the population totals of some components of x may also be known. Two‐phase sampling is used for stratification at the second phase or both phases and for regression estimation. Horvitz–Thompson‐type variance estimators are used for variance estimation. However, the Horvitz–Thompson ( Horvitz & Thompson, J. Amer. Statist. Assoc. 1952 ) variance estimator in uni‐phase sampling is known to be highly unstable and may take negative values when the units are selected with unequal probabilities. On the other hand, the Sen–Yates–Grundy variance estimator is relatively stable and non‐negative for several unequal probability sampling designs with fixed sample sizes. In this paper, we extend the Sen–Yates–Grundy ( Sen , J. Ind. Soc. Agric. Statist. 1953; Yates & Grundy , J. Roy. Statist. Soc. Ser. B 1953) variance estimator to two‐phase sampling, assuming fixed first‐phase sample size and fixed second‐phase sample size given the first‐phase sample. We apply the new variance estimators to two‐phase sampling designs with stratification at the second phase or both phases. We also develop Sen–Yates–Grundy‐type variance estimators of the two‐phase regression estimators that make use of the first‐phase auxiliary data and known population totals of some of the auxiliary variables.     

Empirical likelihood confidence intervals under imputation for missing survey data from stratified simple random sampling

Missing observations due to non‐response are commonly encountered in data collected from sample surveys. The focus of this article is on item non‐response which is often handled by filling in (or imputing) missing values using the observed responses (donors). Random imputation (single or fractional) is used within homogeneous imputation classes that are formed on the basis of categorical auxiliary variables observed on all the sampled units. A uniform response rate within classes is assumed, but that rate is allowed to vary across classes. We construct confidence intervals (CIs) for a population parameter that is defined as the solution to a smooth estimating equation with data collected using stratified simple random sampling. The imputation classes are assumed to be formed across strata. Fractional imputation with a fixed number of random draws is used to obtain an imputed estimating function. An empirical likelihood inference method under the fractional imputation is proposed and its asymptotic properties are derived. Two asymptotically correct bootstrap methods are developed for constructing the desired CIs. In a simulation study, the proposed bootstrap methods are shown to outperform traditional bootstrap methods and some non‐bootstrap competitors under various simulation settings. The Canadian Journal of Statistics 47: 281–301; 2019 © 2019 Statistical Society of Canada     

Acceptance sampling based on life tests: log-logistic model

The problem of acceptance sampling when the life test is truncated at a preassigned time is considered. For various acceptance numbers, confidence levels and values of the ratio of the fixed experimental time to the specified average life, the minimum sample size necessary to ensure the specified average life, are obtained under the assumption that the lifetime variate of the test items follows a distribution belonging to Burr's family XII of distributions - called the log-logistic model. The operating characteristic values of the sampling plans and producer's risk are presented. The results are illustrated by an example.     

Decent Living Standards: Material Prerequisites for Human Wellbeing

Children whose parents experience adverse social, economic, or health-related living conditions are more likely to face similar types of disadvantage in their adult life. However, a limitation of many earlier studies is that they do not account for the multidimensionality of the concept of living conditions, and that the child generation’s life courses are targeted as static and independent from the societal context in which they are imbedded. The current investigation addressed these aspects by focusing on the complexity, duration, and timing of disadvantage with regard to how adverse circumstances in the family of origin are associated with trajectories of social, economic, and health-related living conditions across adulthood. We also examined the role of educational attainment for these associations. Analyses were based a Swedish cohort born in 1953 (n = 14,294). We first conducted sequence analysis, followed by hierarchical cluster analysis, to generate ‘outcome profiles’, i.e. trajectories of adult disadvantage. Second, several indicators of adverse circumstances in childhood were analysed by means of multinominal regression analysis, showing the odds of ending up in the different trajectories. The results indicated that individuals who grew up under adverse conditions were more likely to experience disadvantaged social, economic, and health-related trajectories. This was particularly the case for trajectories characterised by a high degree of complexity, i.e. coexisting disadvantages, and—among men only—by a longer duration of disadvantage. Educational attainment was identified as a powerful mediator, suggesting that efforts to increase equal educational opportunity may be a way of reducing the intergenerational transmission of disadvantage.     

Enacting everyday feminist collaborations: Reflexive becoming,proactive improvisation and co‐learning partnerships

We trace back our own multi‐year teaching and writing collaboration in academia to theorize feminist collaboration. Drawing from feminist theories and our autoethnographic reflections, we surface three metaphorical processes that constitute feminist collaboration. We consider feminist collaboration as: (i) reflexive becoming, that is, feminist collaborators constantly make sense of what counts as feminist as the group and context evolve; (ii) proactive improvisation, that is, feminist collaborators collectively strive for everyday transformations within situated constraints; and (iii) co‐learning partnerships, that is, feminist collaborators relate to one another in ways that uphold commitments to reflexivity, equity and care. Enacting these processes are fraught with tensions that intertwine with one another to constrain and enable feminist collaboration. We conclude the article by calling for continued theorization and engagement with feminist collaboration.     

Inference based on conditional speclfication

The first bibliography in the area of inference based on conditional specification was published in 1977, A second bibliography is compiled, and a combined subject index is given.     

Institutions and Policies for Pro‐poor Agricultural Growth

Global experience with pro‐poor growth and empirical work spanning India, Malawi and Zimbabwe demonstrates the importance of agricultural growth for poverty reduction in poor rural areas, while also pointing to the need for complementary non‐farm sector growth. Theoretical arguments, historical evidence and livelihoods modelling in poor medium‐potential rural economies suggest that, contrary to thinking dominating much of current development policy, subsidies to relieve critical seasonal credit and cash restraints and reduce market and input supply uncertainties need to help in ‘kick‐starting’ agricultural markets if increased smallholder productivity in food‐grains is to drive rural non‐farm growth. Establishing the base conditions for these to work, designing and implementing them to be effective, and then phasing them out are major challenges facing policymakers.     

Anticipated Regret,Risk Perception,or Both: Which is Most Likely Responsible for Our Intention to Gamble?

The current study investigated whether risk aversion or regret aversion could be related to a lower intention to gamble, and whether the type of gambling was a moderator of this relationship. The study took place in Macau, often called “the Las Vegas of East Asia.” A total of 373 Macau residents completed a questionnaire survey dealing with thirteen types of gambling. The results showed that risk perception and anticipated regret had a significant negative effect on the intention to gamble. This negative effect was domain-specific, varying with the type of gambling. Our findings indicated that neither risk aversion nor regret aversion can uniquely explain an individual’s risk-taking tendency consistently. Instead, which factor plays a greater role in lowering the intention to gamble—regret aversion, risk aversion, or both—is itself dependent on the type of gambling involved. The finding that not all gambles are created equal could be useful in gambling prevention and advertising appeal by providing a basis for understanding the role that cognitive and emotional factors play in different types of gambling.     

传承儒家文化充实青少年德育教育内涵

如何进行切合我国国情切合青少年特点的行之有效的德育教育？传承儒家文化的精粹，充实学校德育的内涵，以人为本，端正方向；以仁育人，修身养德；以义正人，揄善扬真。