@article{dempster1977,
  title={Maximum likelihood from incomplete data via the EM algorithm},
  author={Dempster, Arthur P and Laird, Nan M and Rubin, Donald B},
  journal={Journal of the Royal Statistical Society: Series B (Methodological)},
  volume={39},
  number={1},
  pages={1--22},
  year={1977},
  publisher={Wiley Online Library}
}


@article{Ersan2016,
author = {Ersan, Oguz},
title = {Multilayer Probability of Informed Trading},
journal={Available at SSRN 2874420},
year = {2016}
}


@article{Ersan2022a,
  title={Identifying information types in the estimation of informed trading: an improved algorithm},
  author={Ersan, Oguz and Ghachem, Montasser},
  journal={Journal of Risk and Financial Management},
  volume={17},
  number={9},
  pages={409},
  year={2024},
  publisher={MDPI}
}



@article{Ersan2022b,
  title={A methodological approach to the computational problems in the estimation of adjusted PIN model},
  author={Ersan, Oguz and Ghachem, Montasser},
  journal={Quantitative Finance},
  pages={1--13},
  year={2025},
  publisher={Taylor \& Francis}
}


@article{Ghachem2022,
  title={Estimation of the probability of informed trading models via an expectation-conditional maximization algorithm},
  author={Ghachem, Montasser and Ersan, Oguz},
  journal={Financial Innovation},
  volume={11},
  number={1},
  pages={67},
  year={2025},
  publisher={Springer}
}

@article{Ghachem2022b,
  title={PINstimation: An R Package for estimating probability of informed trading models},
  author={Ghachem, Montasser and Ersan, Oguz},
  journal={The R Journal},
  volume={15},
  number={2023},
  pages={145--168},
  year={2023}
}

@article{ChengLai2021,
author = {Tsung-Chi Cheng and Hung-Neng Lai},
title = {Improvements in estimating the probability of informed trading models},
journal = {Quantitative Finance},
volume = {21},
number = {5},
pages = {771-796},
year  = {2021},
publisher = {Routledge}
}


@article{ahn2008,
  title={Informed trading in the index option market: The case of KOSPI 200 options},
  author={Ahn, Hee-Joon and Kang, Jangkoo and Ryu, Doojin},
  journal={Journal of Futures Markets: Futures, Options, and Other Derivative Products},
  volume={28},
  number={12},
  pages={1118--1146},
  year={2008},
  publisher={Wiley Online Library}
}


@article{ward1963,
author = { Joe H.   Ward   Jr. },
title = {Hierarchical Grouping to Optimize an Objective Function},
journal = {Journal of the American Statistical Association},
volume = {58},
number = {301},
pages = {236-244},
year  = {1963},
publisher = {Taylor & Francis},
}


@article{ErsanAlici2016,
abstract = {Computational drawbacks regarding the maximum likelihood estimation (MLE) of the widely used PIN (probability of informed trading) measure (Easley et al., 1996) heavily distort the findings of a broad literature. Previously proposed methodologies are not free of computational biases mainly because involved problems are not treated accurately and in unity. Upon revealing the mistreatment in commonly used YZ algorithm (Yan and Zhang, 2012), we suggest a remedy for the problem of boundary solutions. Next, we differentiate and focus on another computational issue: "determination of powerful initial value sets". We develop a new algorithm that employs cluster analysis to assign multiple powerful sets of initial values for the MLE function. The analyses of the simulated quarterly datasets reflect that applying the algorithm outperforms the existing methods in accuracy. Most notably, none of the mean estimates on PIN and five intermediary parameters contains significant bias at 1{\%} level. Empirical evidence from BIST-30 Index constituents provides consistent and supportive results. In addition to accuracy concerns, consuming one-seventeenth of the time spent in YZ algorithm, the algorithm is highly applicable by researchers and professionals.},
author = {Ersan, Oguz and Alici, Asli},
issn = {10424431},
journal = {Journal of International Financial Markets, Institutions and Money},
keywords = {Boundary solution,Cluster analysis,Initial value determination,Market microstructure,PIN,Probability of informed trading},
month = {jul},
pages = {74--94},
publisher = {Elsevier Ltd},
title = {An unbiased computation methodology for estimating the probability of informed trading (PIN)},
volume = {43},
year = {2016}
}



@article{Gan2015,
abstract = {The probability of informed trading (PIN) is a commonly used market microstructure measure for detecting the level of information asymmetry. Estimating PIN can be problematic due to corner solutions, local maxima and floating point exceptions (FPE). Yan and Zhang [J. Bank. Finance, 2012, 36, 454-467] show that whilst factorization can solve FPE, boundary solutions appear frequently in maximum likelihood estimation for PIN. A grid search initial value algorithm is suggested to overcome this problem. We present a faster method for reducing the likelihood of boundary solutions and local maxima based on hierarchical agglomerative clustering (HAC). We show that HAC can be used to determine an accurate and fast starting value approximation for PIN. This assists the maximum likelihood estimation process in both speed and accuracy.},
author = {Gan, Quan and Wei, Wang Chun and Johnstone, David},
journal = {Quantitative Finance},
keywords = {Cluster analysis,Market microstructure,Mixture model,PIN,Skellam distribution},
month = {nov},
number = {11},
pages = {1805--1821},
publisher = {Routledge},
title = {{A faster estimation method for the probability of informed trading using hierarchical agglomerative clustering}},
volume = {15},
year = {2015}
}


@article{Duarte09,
abstract = {Recent empirical work suggests that a proxy for the probability of informed trading (PIN) is an important determinant of the cross-section of average returns. This paper examines whether PIN is priced because of information asymmetry or because of other liquidity effects that are unrelated to information asymmetry. Our starting point is a model that decomposes PIN into two components, one related to asymmetric information and one related to illiquidity. In a two-pass Fama-MacBeth [1973. Risk, return, and equilibrium: empirical tests. Journal of Political Economy 81, 607-636] regression, we show that the PIN component related to asymmetric information is not priced, while the PIN component related to illiquidity is priced. We conclude, therefore, that liquidity effects unrelated to information asymmetry explain the relation between PIN and the cross-section of expected returns. {\textcopyright} 2008 Elsevier B.V. All rights reserved.},
author = {Duarte, Jefferson and Young, Lance},
file = {::},
issn = {0304405X},
journal = {Journal of Financial Economics},
keywords = {Information asymmetry,Liquidity},
month = {feb},
number = {2},
pages = {119--138},
title = {{Why is PIN priced?}},
volume = {91},
year = {2009}
}


@article{Easley2012,
abstract = {Order flow is toxic when it adversely selects market makers, who may be unaware they are providing liquidity at a loss. We present a new procedure to estimate flow toxicity based on volume imbalance and trade intensity (the VPIN toxicity metric). VPIN is updated in volume time, making it applicable to the high-frequency world, and it does not require the intermediate estimation of non-observable parameters or the application of numerical methods. It does require trades classified as buys or sells, and we develop a new bulk volume classification procedure that we argue is more useful in high-frequency markets than standard classification procedures. We show that the VPIN metric is a useful indicator of short-term, toxicity-induced volatility. {\textcopyright} The Author 2011. Published by Oxford University Press on behalf of The Society for Financial Studies. All rights reserved.},
author = {Easley, David and {Lopez De Prado}, Marcos M. and OHara, Maureen},
journal = {Review of Financial Studies},
issn = {08939454},
month = {may},
number = {5},
pages = {1457--1493},
title = {{Flow toxicity and liquidity in a high-frequency world}},
volume = {25},
year = {2012}
}


@article{Yan2012,
abstract = {We report evidence that boundary solutions can cause a bias in the estimate of the probability of informed trading (PIN). We develop an algorithm to overcome this bias and use it to estimate PIN for nearly 80,000 stock-quarters between 1993 and 2004. We obtain two sets of PIN estimates by using the factorized likelihood functions in both Easley, Hvidkjaer, and Ohara (EHO, 2010) and Lin and Ke (LK, 2011), respectively. We find that the estimate based on the EHO factorization is systematically smaller than the estimate based on the LK factorization, meaning that there is a downward bias associated with the EHO factorization. In addition, we find that boundary solutions appear with a very high frequency when the LK factorization is used. Thus it is necessary to use the LK factorization together with the algorithm in this paper. At last, we document several interesting empirical properties of PIN. {\textcopyright} 2011 Elsevier B.V.},
author = {Yan, Yuxing and Zhang, Shaojun},
file = {::},
issn = {03784266},
journal = {Journal of Banking and Finance},
keywords = {Boundary solution,PIN,Probability of informed trading},
month = {feb},
number = {2},
pages = {454--467},
title = {{An improved estimation method and empirical properties of the probability of informed trading}},
volume = {36},
year = {2012}
}


@article{Brennan2016,
abstract = {We decompose PIN, the probability of informed trading, into good-news (PIN-G) and bad-news (PIN-B) components, which we estimate at a quarterly frequency. We first assess the validity of PIN as a measure of informed trading by calculating its association with measures of the adverse-selection component of the cost of trading. We then provide new evidence that PIN-G and PIN-B capture informed trading around earnings announcements by showing that they predict positive and negative earnings surprises, respectively. Conjecturing that investors who take long positions will be more concerned about informed selling than about informed buying since the former depresses the sale price whereas the latter raises it, we then investigate asymmetry in the pricing of private information. We find strong evidence of such asymmetry in that the effect of PIN-B on the cost of equity capital is large and highly significant, whereas the effect of PIN-G is small and statistically insignificant.},
author = {Brennan, Michael J. and Huh, Sahn Wook and Subrahmanyam, Avanidhar},
issn = {15265501},
journal = {Management Science},
keywords = {AdjPIN,Adverse-selection and noninformation components of,CAR,Cost of equity capital,Decomposition of PIN,Earnings announcements,Earnings surprises,Information asymmetry,PIN-B,PIN-G,PSOS,SUE},
month = {sep},
number = {9},
pages = {2460--2480},
publisher = {INFORMS Inst.for Operations Res.and the Management Sciences},
title = {{Asymmetric effects of informed trading on the cost of equity capital}},
volume = {62},
year = {2016}
}


@article{Ke2019,
abstract = {This study aims at the estimation of the probability of informed trading (PIN), which may fail for stocks with high levels of trading activities due to a computer's floating-point exception (FPE). In this paper, we discuss two solutions of adopting scaled trade counts and reformulating the likelihood to estimate PIN for actively traded stocks. This study shows that, although scaled data mitigates the impact of the FPE, the effectiveness of scaled data, however, appears to underperform when users adopt the unsuitable expression of the likelihood function. In contrast, the remedy of reformulating the likelihood is more stable.},
author = {Ke, Wen Chyan and Chen, Hueiling and Lin, Hsiou Wei William},
file = {::},
issn = {15446123},
journal = {Finance Research Letters},
keywords = {Floating-point exception,Maximum likelihood,PIN,Scaled trade counts},
month = {dec},
pages = {458--464},
publisher = {Elsevier Ltd},
title = {{A note of techniques that mitigate floating-point errors in PIN estimation}},
volume = {31},
year = {2019}
}


@article{WilliamLin2011,
title = {A computing bias in estimating the probability of informed trading},
journal = {Journal of Financial Markets},
volume = {14},
number = {4},
pages = {625-640},
year = {2011},
issn = {1386-4181},
author = {Hsiou-Wei {Lin} and Wen-Chyan Ke},
keywords = {Floating-point exception, Informed trading, Market microstructure},
abstract = {This study identifies a factor that leads to a bias in estimating the probability of informed trading (PIN), a widely-used microstructure measure. It is shown that, along with the numerical maximization of the likelihood function for PIN, the floating-point exception (i.e., overflow or underflow) may eliminate feasible solutions to the actual parameters in the optimization problem. Approximately 44% of PIN estimates for recent stock market data may have been subject to a downward bias that is more pronounced for active stocks than for inactive stocks. This study develops a remedy to mitigate the resulting bias.}
}



@article{Easley2010,
abstract = {We examine the potential profits of trading on a measure of private information (PIN) in a stock. A zero-investment portfolio that is size-neutral but long in high PIN stocks and short in low PIN stocks earns a significant abnormal return. The Fama-French, momentum, and liquidity factors do not explain this return. However, significant covariation in returns exists among high PIN stocks and among low PIN stocks, suggesting that PIN might proxy for an underlying factor. We create a PIN factor as the monthly return on the zero-investment portfolio above and show that it is successful in explaining returns to independent PIN-size portfolios. We also show that it is robust to inclusion of the Pstor-Stambaugh liquidity factor and the Amihud illiquidity factor. We argue that information remains an important determinant of asset returns even in the presence of these additional factors. Copyright {\textcopyright} Michael G. Foster School of Business, University of Washington 2010.},
author = {Easley, David and Hvidkjaer, Soeren and Ohara, Maureen},
issn = {00221090},
journal = {Journal of Financial and Quantitative Analysis},
month = {apr},
number = {2},
pages = {293--309},
publisher = {Cambridge University Press},
title = {{Factoring information into returns}},
volume = {45},
year = {2010}
}


@article{Easley1992,
abstract = {This paper delineates the link between the existence of information, the timing of trades, and the stochastic process of prices. We show that time affects prices, with the time between trades affecting spreads. Because the absence of trades is correlated with volume, our model predicts a testable relation between spreads and normal and unexpected volume, and demonstrates how volume affects the speed of price adjustment. Our model also demonstrates how the transaction price series will be a biased representation of the true price process, with the variance being both overstated and heteroskedastic. 1992 The American Finance Association},
author = {Easley, David and Ohara, Maureen},
issn = {15406261},
journal = {The Journal of Finance},
month = {jun},
number = {2},
pages = {577--605},
publisher = {John Wiley {\&} Sons, Ltd},
title = {{Time and the Process of Security Price Adjustment}},
volume = {47},
year = {1992}
}


@article{Easley1996,
abstract = {This article investigates whether differences in information-based trading can explain observed differences in spreads for active and infrequently traded stocks. Using a new empirical technique, we estimate the risk of information-based trading for a sample of New York Stock Exchange (NYSE) listed stocks. We use the information in trade data to determine how frequently new information occurs, the composition of trading when it does, and the depth of the market for different volume-decile stocks. Our most important empirical result is that the probability of information-based trading is lower for high volume stocks. Using regressions, we provide evidence of the economic importance of information-based trading on spreads.},
author = {Easley, David and Kiefer, Nicholas M. and Ohara, Maureen and Paperman, Joseph B.},
issn = {00221082},
journal = {Journal of Finance},
month = {sep},
number = {4},
pages = {1405--1436},
publisher = {Blackwell Publishing Inc.},
title = {Liquidity, information, and infrequently traded stocks},
volume = {51},
year = {1996}
}


@article{murtagh2014,
  title={Ward's hierarchical agglomerative clustering method: which algorithms implement Ward's criterion?},
  author={Murtagh, Fionn and Legendre, Pierre},
  journal={Journal of classification},
  volume={31},
  number={3},
  pages={274--295},
  year={2014},
  publisher={Springer}
}


@article{abad2012,
  title={From PIN to VPIN: An introduction to order flow toxicity},
  author={Abad, David and Yague, Jose},
  journal={The Spanish Review of Financial Economics},
  volume={10},
  number={2},
  pages={74--83},
  year={2012},
  publisher={Elsevier}
}


@inproceedings{lin2015,
  title={Application of Poisson Mixtures in the Estimation of Probability of Informed Trading},
  author={Lin, Emily and Lee, Cheng-Few},
  booktitle={Handbook of Financial Econometrics and Statistics},
  pages={2601--2619},
  year={2015},
  organization={Springer USA}
}


@article{LeeReady1991,
 ISSN = {00221082, 15406261},
 abstract = {This paper evaluates alternative methods for classifying individual trades as market buy or market sell orders using intraday trade and quote data. We document two potential problems with quote-based methods of trade classification: quotes may be recorded ahead of trades that triggered them, and trades inside the spread are not readily classifiable. These problems are analyzed in the context of the interaction between exchange floor agents. We then propose and test relatively simple procedures for improving trade classifications.},
 author = {Charles M. C. Lee and Mark J. Ready},
 journal = {The Journal of Finance},
 number = {2},
 pages = {733--746},
 publisher = {[American Finance Association, Wiley]},
 title = {Inferring Trade Direction from Intraday Data},
 volume = {46},
 year = {1991}
}


@article{Ellis2000,
 abstract = {Researchers are increasingly using data from the Nasdaq market to examine pricing behavior, market design, and other microstructure phenomena. The validity of any study that classifies trades as buys or sells depends on the accuracy of the classification method. Using a Nasdaq proprietary data set that identifies trade direction, we examine the validity of several trade classification algorithms. We find that the quote rule, the tick rule, and the Lee and Ready (1991) rule correctly classify 76.4%, 77.66%, and 81.05% of the trades, respectively. However, all classification rules have only a very limited success in classifying trades executed inside the quotes, introducing a bias in the accuracy of classifying large trades, trades during high volume periods, and ECN trades. We also find that extant algorithms do a mediocre job when used for calculating effective spreads. For Nasdaq trades, we propose a new and simple classification algorithm that improves over extant algorithms.},
 author = {Katrina Ellis and Roni Michaely and Maureen Ohara},
 journal = {The Journal of Financial and Quantitative Analysis},
 number = {4},
 pages = {529--551},
 publisher = {Cambridge University Press},
 title = {The Accuracy of Trade Classification Rules: Evidence from Nasdaq},
 volume = {35},
 year = {2000}
}


@article{Aktas2014,
title = {Trade classification accuracy for the BIST},
journal = {Journal of International Financial Markets, Institutions and Money},
volume = {33},
pages = {259-282},
year = {2014},
issn = {1042-4431},
author = {Osman Ulas Aktas and Lawrence Kryzanowski},
keywords = {Trade classification algorithms, Market microstructure, Developing stock market, Short sales, Agency relations},
abstract = {The accuracy of five algorithms for classifying trades as buyer- or seller-initiated is assessed for BIST-30 index constituents over a period including the Lehman collapse. The highest classification accuracy rate (over 95%) is for the one-second lagged Lee & Ready (LR) algorithm. The LR's classification accuracy is highest (lowest) for trades representing mixed agency and principal (pure principal) relations between clients and executing brokers. Unlike for U.S. markets, almost all trades are classifiable with accuracy rates of 90-plus percent for both long and short trades. As for U.S. markets, higher misclassification rates occur for trades in the first versus last 30min of the trading day, as the time between consecutive trades decreases, and for decreasing trade sizes.}
}


@article{piwowar2006,
  author =    {Michael S. Piwowar and Li Wei},
  title =     {The Sensitivity of Effective Spread Estimates to Trade-Quote Matching Algorithms},
  journal =   {Electronic Markets},
  year =      {2006},
  volume =    {16},
  number =    {2},
  pages =     {112-129},
  publisher = {Routledge},
}


@article{Easley2011,
  title={The microstructure of the \"flash crash\": flow toxicity, liquidity crashes, and the probability of informed trading},
  author={Easley, David and De Prado, Marcos M Lopez and Ohara, Maureen},
  journal={The Journal of Portfolio Management},
  volume={37},
  number={2},
  pages={118--128},
  year={2011},
  publisher={Institutional Investor Journals Umbrella}
}


@article{Brennan2018,
    author = {Brennan, Michael J. and Huh, Sahn-Wook and Subrahmanyam, Avanidhar},
    title = "{High-Frequency Measures of Informed Trading and Corporate Announcements}",
    journal = {The Review of Financial Studies},
    volume = {31},
    number = {6},
    pages = {2326-2376},
    year = {2018},
    month = {01},
    abstract = "{We explore the dynamics of informed trading around corporate announcements of merger bids, dividend initiations, SEOs, and quarterly earnings by calculating daily posterior probabilities of informed buying and selling. We find evidence of informed trading before the announcements and a significant part of the news in announcements is impounded in stock prices before the announcements by pre-event informed trading. We also find evidence of informed trading after the announcements. Most strikingly, the probability of informed trading after merger bids predicts the probability of the bid being withdrawn or met with a competing bid. For other announcements, post-announcement informed-trading probabilities predict subsequent returns.Received September 26, 2016; editorial decision December 17, 2017 by Editor Andrew Karolyi.}",
    issn = {0893-9454},
    doi = {10.1093/rfs/hhy005},
    url = {https://doi.org/10.1093/rfs/hhy005}
}


@article{Akogul2016,
  title={A comparison of information criteria in clustering based on mixture of multivariate normal distributions},
  author={Akogul, Serkan and Erisoglu, Murat},
  journal={Mathematical and Computational Applications},
  volume={21},
  number={3},
  pages={34},
  year={2016},
  publisher={Multidisciplinary Digital Publishing Institute}
}

@article{griffin2021,
  title={Estimating the probability of informed trading: A Bayesian approach},
  author={Griffin, Jim and Oberoi, Jaideep and Oduro, Samuel D},
  journal={Journal of Banking \& Finance},
  volume={125},
  pages={106045},
  year={2021},
  publisher={Elsevier}
}

@article{ke2017improved,
  title={An improved version of the volume-synchronized probability of informed trading},
  author={Ke, Wen-Chyan and Lin, Hsiou-Wei William and others},
  journal={Critical Finance Review},
  volume={6},
  number={2},
  pages={357--376},
  year={2017},
  publisher={Now Publishers, Inc.}
}
