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Medical device maker Boston Scientific is preparing to slash thousands of jobs to counter slumping sales, according to a media report Tuesday.
E.W. Scripps, a television broadcaster and newspaper publisher, said Tuesday it plans to split into two publicly traded companies.
Eight U.S. lawmakers called on the Bush administration on Monday to block a proposed buyout of Massachusetts-based technology group 3Com saying a Chinese company's role in the $2.2billion transaction "threatens the national security of the United States."
Time Warner's Internet unit AOL will eliminate 2,000 jobs as part of an ongoing restructuring to better focus on boosting online advertising, according to a memo obtained Monday by Reuters.
Citigroup CEO Charles Prince will get more time to help turn around the company's fortunes, according to CNBC On Air Editor Charles Gasparino.
Dow Chemical, the largest U.S. chemicals maker, said on Friday it has postponed a meeting with institutional investors that was scheduled for early next month.
Citigroup said it is combining its investment banking and alternative investments units into one division led by a former Morgan Stanley executive.
The chief executive of BP on Thursday outlined a plan to address industry-lagging profitability by slashing management layers, adopting consistent procedures for developing oil and gas fields and reducing "unacceptably high" costs.
Virgin Mobile USA, a venture of Sprint Nextel and Richard Branson's Virgin Group, raised $412.5 million on Wednesday, with an initial public offering that priced at the low end of expectations.
The president of the pilots' union at American Airlines ripped the company's chief executive in a letter that ended with a threat to "see you in court, in the newspapers, and on the picket line."
The charitable trust that controls Milton Hershey's fortune and The Hershey Co., the U.S. chocolate company he founded, said Tuesday it is not satisfied with the company's lackluster performance, and is working to improve it.
US Treasury debt prices were steady early Tuesday as investors looked ahead to the release of minutes from the Federal Reserve's policy meeting last month when the central bank cut benchmark interest rates.
The head of European aerospace group EADS said in remarks published on Tuesday he would propose scrapping the company's management stock option system amid an ongoing insider trading investigation.
The U.S. Food and Drug Administration is preparing to warn physicians about hazards with drugs that use ultrasound imaging machines to enhance the diagnosis of heart problems, the Wall Street Journal reported on its Web site on Sunday.
Momentum has been building toward the United Auto Workers accepting a tentative contract with General Motors, though members at some key locals backed it by narrow margins in results released Thursday.
General Electric said Thursday it will close a number of lighting plants in Brazil and the U.S., as part of a plan to restructure its consumer and industrial division, potentially cutting more than 1,400 jobs in the process.
Alcoa, the biggest U.S. aluminum producer, said Thursday it will sell its packaging and consumer and automotive castings businesses, and restructure its electrical and electronic solutions business.
Kraft Foods plans to turn Oscar Mayer meats into a separate business unit as part of a change the largest North American food company hopes will make all its businesses more accountable, a spokeswoman said Thursday.
The world's largest brokerage Merrill Lynch, which is expected to announce third-quarter losses in fixed income, said that global head of fixed income, currencies & commodities, has left the firm.
Spanish funds within a group looking to buy Spanish carrier Iberia will control both the board and hold the majority of its capital if successful, an industry source told Reuters on Thursday.


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subset: commoncrawl


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Separate names with a comma.
Discussion in 'Baptist Theology & Bible Study' started by J.D., Nov 28, 2008.
in scripture. Agree?
Now, to which Israel is the Kingdom restored?
How can a Kingdom be restored it was never a Kingdom prevously?
There were two Israel's; Spiritual Israel and national Israel. [Romans 9] Even Ryrie Dispensationalism, page127, seems to agree.
I do not buy into any form of replacement theology.
Ryrie isn't stating there are 'two' Israels nor does Ryrie state that the Church is 'spiritual Israel'.
From Ryries Basic Theology Section 13, Subsection "ll Doctrinal Characterists of Premillinnealism:
Or here in Ryries Basic Theology Section 12; subsection"I will build my Church".
Nothing about the church being spiritual Israel. We can be said to be spirtual partake WITH Israel regarding some of God's promises to it but we are distinct from Israel as the scriptures dictate.
Galatians 6:16 (KJV) And as many as walk according to this rule, peace be on them, and mercy, and upon the Israel of God.
1 Corinthians 10:18 (KJV) Behold Israel after the flesh: are not they which eat of the sacrifices partakers of the altar?
Gill again:
Sorry, but Gill's opinions are nothing more than his opinions of what the scripture is saying and not necessarily what scripture is actaully saying.
To use Gill's opinions or any other who holds to such a view means that you will acknowledge all others veiws that differ just as truthful and accurate a renderings as Gill
If by that you mean that there is genetic Israel, and that "true Israel" is a subset of genetic Israel, then yes. That is what Paul teaches in Romans 9.
This is actually a great proof that Israel and the church are distinct. You notice there are two groups here: 1) those who walk according to this rule and 2) the Israel of God. Paul is prayer for peace and mercy for both groups. Gill is incorrect on this passage.
I did not say that Ryrie said that spiritual Israel was the Church. I said that he implied there were two israel, spiritual Israel and national Israel. Did you read the page I referenced? Please do not imitate Mitchell and imply that I am lying. This seems to be a tactic of dispensationalists when they cannot defend their doctrine.
The Apostle Paul certainly teaches that Spiritual Israel and the Church are the same. Simply read the parable of the olive tree in Romans 11. Also compare the following:
Romans 2:28. For he is not a Jew, which is one outwardly; neither is that circumcision, which is outward in the flesh:
Romans 2:29. But he is a Jew, which is one inwardly; and circumcision is that of the heart, in the spirit, and not in the letter; whose praise is not of men, but of God.
Colossians 2:9. For in him dwelleth all the fulness of the Godhead bodily.
Colossians 2:10. And ye are complete in him, which is the head of all principality and power:
Colossians 2:11. In whom also ye are circumcised with the circumcision made without hands, in putting off the body of the sins of the flesh by the circumcision of Christ:
Maybe you should just keep this thread about this thread.
Is it hard for you to kick against the truth?:laugh:
And you continue to make my point.
And just what is your point?
There were two Israels: A believing remnant, spiritual israel, and carnal National Israel.
Well, okay, if all we have is opinions, let's just call the whole thing off.
But what about the other scripture I mentioned? Doesn't the fact that Paul mentions an "Israel according to the flesh" mean that there must be an "Isreal NOT according to the flesh"?
Sorry, but your opinions are nothing more than your opinions.
Sorry, but Ryries opinions are nothing more than his opinions of what the scripture is saying and not necessarily what scripture is actaully saying.
To use Ryries opinions or any other who holds to such a view means that you will acknowledge all others veiws that differ just as truthful and accurate a renderings as Ryries
Maybe you could just state your point without harassing him.


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Gap b/w front two teeth
I have a gap between front two teeth. Dentist suggested me go for braces and it will cost around 25K. I cannot effort that much as the gap is between front two teeth only.Please suggest best solution for this problem.
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Doctor Answers (4) on Gap b/w front two teeth
Dr. Abhilash Raavi
16 yrs. of exp | Hyderabad | Dentist
Hi, first best option would be braces and that's the last best option too. The cost for the braces treatment around 25k is very reasonable and justified. If You can ask the dentist for monthly payment options which will not burden you.
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Dr. Sanjay Nayak Sujir
23 yrs. of exp | Mangalore | Dentist
The closure of the gap also called a midline diastema depends upon the amount of space ...if its less u can even do it with composite fillings which do not damage teeth as much as getting caps.... If the gaps are huge then the need for braces may arise ....but by braces u will be keeping the natural look of your teeth with nothing artificial and since the treatment is for a longer period u have an option to pay in installments ....whereas putting caps may need additional treatment such as root canals and also grinding of yr natural teeth ...also the payment mah come upto almost as much as braces and one time payment .....anything artificial is not long lasting as much as natural ...hope this helps
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Dr. Ipsita kukreja
Pune | Dentist
A space in the front two teeth could be because of either a thickened frenum....so the treatment involves surgical excision of frenum{frenectomy} followed my placing the braces....which is costly and involves time...the simplest procedure would be to go for esthetic filling(composite) restoration with both the teeth which will not only save on a lot of money but will also take not more than 30-45mins...
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Dr. M. S. Sushma Susik
25 yrs. of exp | Hyderabad | Dentist
Good evening in case if you are having gaps you can close by keeping caps which is less costly &permanent
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Launching today the MyParkScotland Community Growing & Food Campaign Theme. The idea of the campaign is to highlight the great […]
We are here to connect you to your park, providing up to date information on park locations, facilities and what's on. You will also find details of exciting projects that will help to improve our parks for future generations.
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Find out about what is going on in parks across Edinburgh, Falkirk, fife and Glasgow using the MyParkScotland listings.
If you are putting on an event then please submit an event to the listing page or take a look at what is going on.
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Alternatively visit the Donate page to support your favourite park
Keep up to date with MyParkScotland by signing up to our e news.
MyParkScotland on Facebook
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MyParkScotland on Twitter
You'll find all our latest tweets here on this website but you can also follow us at MyParkScotland Twitter account for quick updates about projects and events in Edinburgh and Glasgow.
MyParkScotland on Flickr
Pictures are great for telling a story, which is why we have created a MyParkScotland Flickr account to bring together interesting and vibrant images of our parks.
I love walking though such a large and interesting park so close to the city centre.
The thing I would like to see is improved pathway along the river as it so peaceful to just stroll along the river.
Launching today the MyParkScotland Community Growing & Food Campaign Theme. The idea of the campaign is to highlight the great […]
This weekend saw the publication of the Communities and Local Government Parks Enquiry. The full report can be downloaded […]


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++ FULL-SIZE HEADPHONE RECOMMENDATIONS THREAD++ CLOSED: Please post a thread in the Introductions, Help and Advice forum
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Fallen in love with Alessandro MS1i, but are they the cans for me?
willemkempers posted in this thread in Headphones (full-size) 5/7/13 at 11:47am | 5 replies | 230 views


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Can a vegetarian person get a good meal at this restaurant?
Can a vegan person get a good meal at this restaurant?
Can a gluten free person get a good meal at this restaurant?
Is this a Malaysian restaurant?
Is this restaurant appropriate for Kids?
Is this an Indian restaurant?
Does this restaurant have waiters and waitresses?
Is this restaurant family-friendly?
Does this restaurant offer takeout or food to go?
Is this restaurant good for lunch?
Thanks for helping!
Dining alone or with friends the Roti Chennai never disappoints. Delicious veg and non veg dishes from Malaysia/South India. Don't expect a fine dining experience- just inexpensive good food.
I went to this restaurant expecting roti, and I sure got it. A delicious garlic roti and butter chicken. The butter chicken itself was probably the best butter chicken I have ever tasted, and when the menu said garlic roti, it really meant it! Powerful and filling flavour! I was super impressed.
Good voucher on GroupOn and the Roti Chenai starter was nice, however we ordered a chicken korma and butter chicken and they both tasted exactly the same! No coconut milk in the korma, and both curries were very oily. The clumps of chicken were not nice either.
Roti Chenai has been around for quite a while. It has plenty of competition these days with new Malaysian Style Restaurants opening almost every week it seems. There were three of us for Lunch. For a starter we ordered a couple of Rotis and a small bowl of mutton curry. The rotis were excellent with a crisp flaky texture. We...
We decided to eat out and Roti Chenai Cafe was nearby. We found the food to be wonderful and were interested in the lovely dishes we saw served to what looked like, some very regular customers. What a great dinner we had, with very subtle flavours of very nice food. We are looking forward to returning.
The food and service here is great. Fantastic value for money. Good variety of vegetarian and non vegetarian options. Also centrally located, if you're lucky you will get a park right outside. Definitely suggest you try it.
Met friends there for an evening out. Food quite good but had to get our glasses of water as staff seemed too busy. Value for money ok.
Very quick and efficient service. Ordered roti chenai with lamb curry. The rotis were good, however we noticed that they weren't cooked to order like they used to do which does impact on the lightness & fluffiness of them. The lamb curry was lovely as always - great flavour, great for dipping. Would recommend this place for a quick tasty...
Used to come here years back, but visited again a couple of months back. Still the same. Good for a quick, cost effective meal.
I went for lunch with a friend recently. The service was good, and it's quite nice inside, with nice touches like metal cups. My friend really enjoyed his meal, but mine was fairly standard and bland, the roti very dry and flavorless. There was nothing really wrong with it, it was acceptable.
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text: I have 2 accounts that are side by side. The problem is, the grass is growing so fast that mowing once a week just isn't enough. So I called both of them and suggested that they go to a 5 day mowing schedule until the growth settles down. I also gave them the option that I could double mow each week but it would probably be a little more money. They both said they were not interested in either option and said to just mow their lawns as is. These lawns are not highly visible, but I still take pride in my work. So I guess I am going to bite the bullet for a few weeks and double mow. Or I could just mow it once and leave it ragged..... What would you do?


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A long-acting angiotensin-converting enzyme inhibitor. It is a prodrug that is transformed in the liver to its active metabolite ramiprilat. [PubChem]
Ramipril side effect
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I don't mean someone SMOKING…I mean a mushroom/fungi SPORING!
Way to do this?
Puffballs that are ready will spore out if you tap them.
Many cup fungi will spore if you blow VERY lightly on them.
Slime molds will often 'puff' if they're in their sporing stage and you blow LIGHTLY on them.
On windy days, large groupings of mushrooms will spore up into the air (this is when back-light on the fungi REALLY works!).
Artwork and photography only. No T-shirts!
This is definately a true CHALLENGE of a challenge. Good luck!
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Currently fezoj has this honor showing his beautiful works.
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Cover Image: One Puff of Breath by Carla Wick/Jan...
Common Puffball liberating spores by dilouise was voted the most popular entry in this challenge with 14 votes.
I change now the way we choose avatar, You can now enter the image you want. No more featured i…
Voting ends 4 days from now


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This is your resource to discuss support topics with your peers, and learn from each other.
I have been given a Blackberry Curve 8900 today for my new job. I have a work's email which works fine, and I have also set it up to receive my Hotmail emails as well. However, whilst my work emails go into my works mailbox, my hotmail ones seem to go to my hotmail mailbox & my works mailbox! Obviously I would rather keep business seperate from personal so is there a way that I stop my hotmail from going to both mailboxes?
Hope someone can help.
Thanks very much,
Hi and Welcome to the Community!
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For BES users, your BES Admin can push a BES-only email icon to your device. This solution is free, but must be conducted at the BES server, and will affect all users on that BES server.
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My Reviews
Flax Oil
Arrived in fresh and fast. Tastes great and good price!
Sonne's 7 Detox Aid
Posted by Reviewer2183128 on May 02, 2011
This seems good, it is not the same color or texture as my Dr. had me on, but I do save a little money with this one.
Now Food CoQ10
Posted by Reviewer2183128 on May 02, 2011
I thought making my own capsules might save me money. I think this bottle (28g) made about 70 capsules of OO. Not too sure if it was a big savings. But it is hard to tell what this might be helping it with, I just know it is one of the supplements my Dr. wanted me to take while on a diet for yeast and lyme disease.
Vitamin B3
Posted by Reviewer2183128 on Apr 04, 2011
Just be prepared for the Niacin Flush. The first couple of weeks of taking this I got a good healthy flush out of this every time. Problem is I did not know that B3 did this to a person. Turns out it is a very good thing. You should at least research "Niacin Flush" before you take it so you are not surprised or think you are having an allergic reaction. I am sure that a person might also look to see if it has any bad side effects, but I found nothing only good remarks about how wonderful it is for your health. I recommend this product.
Kava Kava
I have not opened the bottle yet. I am not sure about it because after some research I found that I may need to be very careful of liver damage some Kava Kava can cause. My natruopathic doctor says in moderation it maybe ok to take.
Colloidal Silver Nasal Spray
Posted by Reviewer2183128 on Mar 16, 2014
I am only giving this 2 stars because I think a person should research whether silver is safe to use. Mayo Clinic website says No, don't use it. I have found the product very, very effective though for nasal congestion. It will stop my stuffed up nose within 10 minutes. So it sits in my cabinet, and I have not used it since reading about it on Mayo's website.
I have taken this about 3 weeks, and have not seen a huge difference in my focus yet. I think it is working a little. It may take a while for it to work. I do feel less anxiety though over my lack of concentration. I am doing better at day to day tasks.
Posted by Reviewer2183128 on Dec 27, 2012
love this! perfect for lemon-aid , half lemon in cup of water and pinch of sea salt over ice.
vit c
Posted by Reviewer2183128 on Dec 27, 2012
good, been taking this for a couple years
Tumeric and Bromelain
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This is the full research report for 18.104.22.168, which is an IP address.
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Intertelecom Ltd
Eugean Gnatiy
Malaya Arnautskaya Str.,100, Odessa, Ukraine, 6500
Telephone: 380487431109
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Malaya Arnautskaya Str.,100, Odessa, Ukraine, 6500
Telephone: 380487991287
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The Breakfast in Schools initiative Kickstart Breakfast that was introduced in Southland's deciles one to four schools has been hailed as a major success by East Gore principal Wes Gentle.
The programme was brought in to the lower decile schools five years ago, although not all schools eligible had introduced the breakfast.
More than 620 New Zealand schools provide breakfast.
The programme was introduced after research found that almost one fifth of children aged between 5 and 14 went to school regularly without eating or drinking anything for breakfast.
Mr Gentle said the programme at East Gore school was run by a group of volunteers and they had around 20 pupils take part each morning.
Fonterra donated milk to the initiative and Sanitarium had donated Weetbix.
"We are lucky enough to have local businesses donate jam and bread on top of what is supplied," Mr Gentle said.
Mataura Primary was the only other school in the Eastern Southland District that took part and they had been involved for the past five years.
This year would see the initiative available to all schools after the government announced in May the allocation of up to $1.9 million per year.
Fonterra sustainability and social responsibility manager Tom Newitt said the programme had been a major success and the group was now proceeding forward on what else they could do.
With the allocation of the government funding the programme had been expanded to five days a week for the decile one to four schools.
"In Southland, out of all eligible schools, 52 per cent were participating in the programme and all seemed to find great benefit," Mr Newitt said.
Nationally, 76 per cent of school principals had seen improvements in pupils' behaviour and 79 per cent had seen improvements in pupils' general health.
Although the breakfast is supplied it was up to the school to provide the equipment and people to run the breakfast clubs.
- The Southland Times
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The Wages of Fear (Le salaire de la peur) (1953)
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Charles Vanel
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A great classic
6 years, 2 months ago at Dec 30 15:12read more
In the South American jungle supplies of nitroglycerin are needed at a remote oil field. The oil company pays four men to deliver the supplies in two trucks. A tense rivalry develops between the two sets of drivers and on the rough remote roads the slightest jolt can result in death.
Release date: 16 February 1955
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Did This Teacher Cross the Line?
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Koramangala 7th Block, Bangalore, India- 560095.
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text: The Nike Air 180 trades in its customary materials for a lighter and more breathable construction as it prepares for the warmer season. Normally built from suede and mesh, here we see the Air 180 with an upper constructed from a lightweight ripstop nylon and an airy perforated leather. White is the color of choice for most of the sneaker, while subtle accents of tech grey, bright mandarine and chlorine blue are worked in elements like the Swoosh, midsole, tongue, and Air unit to give the shoes just enough of what of what they need to appeal to us. They're available now at The Good Will Out as well as other Nike Sportswear retailers.


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Negotiations between Rank and Gala Coral to Be Continued
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Judging by its latest move in that direction, it seems that Rank Group plc is still interested in acquiring Gala Coral group, even though it announced that it's abandoning the negotiations at the end of March this year, reportedly due to the obligation to take on property leases relating to casinos no longer owned by Gala.
This time, it has been reported that the two groups are back at the negotiating table, and that Ranks still appears to be interested in a possible GBP 200 million deal for the casino wing of its rival. If the deal is nailed, Rank will become the leading casino operator in the UK.
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A2 Sociology: Religion, Renewal and Choice
- Created by: LatifahCole
- Created on: 11-06-14 17:10
First 376 words of the document:
Religion, Renewal and Change
Postmodernity & Religion
Religion is changing not declining
This is because of changes in wider society such as greater individualism and
Argues against the secularisation thesis
Believing Without Belonging (Davie)
Religion is not declining but simply taking a more privatised form
People no longer attend church becos they feel it is respectable to do so
Attendance is now a matter of personal choice rather than an obligation
People now hold beliefs but don't go to church
Vicarious Religion
Davie notes a trend towards `vicarious religion'
Small number of professional clergy practise religion on behalf of a much larger
number of people, who experience it second hand
Davie compares vicarious religion to the tip of an iceberg
Beneath what appears to be only a small commitment to religion is a much wider
According to Davie, secularisation theory assumes that modernisation affects every
society in the same way, causing the decline of religion and replacement by science
Davie argues that there are multiple modernities. Eg. Britain & America are both
modern societies with very different patterns of religion
Davie believes that science and religion will continue to coexist.
× Bruce: If people no longer attend church this reflects the declining strength of their
beliefs. When people no longer believe, they no longer wish to belong.
Spiritual Shopping (Hervieu-Leger)
Agrees w/Davie and Notes there is a dramatic decline of religion in Europe
Cultural Amnesia. The idea that children used to be taught religion in the extended
family. Nowadays, we have lost the religion that used to be handed down to us.
Parents today let children decide for themselves what to believe
As a result, young people no longer inherit a fixed religious identity & they're
ignorant of traditional religion
Individual consumerism has replaced collective tradition
People have a choice as consumers of religion, they have become spiritual
Religion has become a personal spiritual journey, in which we choose the elements we want
to explore. Hervieu-Leger argues two new religious types:
Other pages in this set
Page 2
Here's a taster:
Pilgrims: Follow an individual path in a search for self-discovery
2. Converts: Join religious groups that offer a strong sense of belonging, usually based
on a shared ethnic background
As a result religion no longer acts as a source of collective identity. Religion does still have
influences on societys norms & values. Eg. Equality & human rights.
Lyon: `Jesus in Disneyland'
Agrees with Davie that believing w/o belonging is increasingly popular. Postmodern society's
features are changing the nature of religion, incl: globalisation, media and communications.…read more
Page 3
Here's a taster:
It is human nature to seek rewards and avoid costs
Religion is attractive becos it provides us w/compensators
When real rewards are unobtainable, religion compensates by promising supernatural
ones. Eg. Promising life after death
Stark and Bainbridge put forward the concept of a cycle of religious decline, revival and
renewal.…read more
Page 4
Here's a taster:
Existential Security Theory (Norris and Inglehart)
× Criticise Religious Market Theory on the grounds that it only applies to America and
fails to explain the religiosity between different societies
× Existential Security: the feeling that survival is secure enough that itt can be taken for
Religion meets a need for security in societies where people feel secure have a low level of
demand for religion:
Poor societies: people face life-threatening risks such as famine and disease have
high levels of insecurity and thus high…read more
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19 Another Look at Differentiability in Quadratic Mean
- Maud Clarke
- 1 years ago
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1 19 Aother Look at Differetiability i Quadratic Mea David Pollard 1 ABSTRACT This ote revisits the delightfully subtle itercoectios betwee three ideas: differetiability, i a L 2 sese, of the square-root of a probability desity; local asymptotic ormality; ad cotiguity A mystery The traditioal regularity coditios for maximum likelihood theory ivolve existece of two or three derivatives of the desity fuctios, together with domiatio assumptios to justify differetiatio uder itegral sigs. Le Cam (1970) oted that such coditios are uecessarily striget. He commeted: Eve if oe is ot iterested i the maximum ecoomy of assumptios oe caot escape practical statistical problems i which apparetly slight violatios of the assumptios occur. For istace the derivatives fail to exist at oe poit x which may deped o θ, or the distributios may ot be mutually absolutely cotiuous or a variety of other difficulties may occur. The existig literature is rather uclear about what may happe i these circumstaces. Note also that sice the coditios are imposed upo probability desities they may be satisfied for oe choice of such desities but ot for certai other choices. Probably Le Cam had i mid examples such as the double expoetial desity, 1 / 2 exp( x θ ), for which differetiability fails at the poit θ = x. He showed that the traditioal coditios ca be replaced by a simpler assumptio of differetiability i quadratic mea (DQM): differetiability i orm of the square root of the desity as a elemet of a L 2 space. Much asymptotic theory ca be made to work uder DQM. I particular, as Le Cam showed, it implies a quadratic approximatio property for the log-likelihoods kow as local asymptotic ormality (LAN). Le Cam s idea is simple but subtle. Whe I first ecoutered the LAN property I wrogly dismissed it as othig more tha a Taylor expasio to quadratic terms of the log-likelihood. Le Cam s DQM result showed otherwise: 1 Yale Uiversity
2 306 David Pollard oe appears to get the beefit of the quadratic expasio without payig the twice-differetiability price usually demaded by such a Taylor expasio. How ca that happe? My iitial puzzlemet was ot completely allayed by a study of several careful accouts of LAN, such as those of Le Cam (1970; 1986, Sectio 17.3), Ibragimov & Has miskii (1981, page 114), Millar (1983, page 105), Le Cam & Yag (1990, page 101), or Strasser (1985, Chapter 12). Noe of the proofs left me with the feelig that I really uderstood why secod derivatives are ot eeded. (No criticism of those authors iteded, of course.) Evetually it dawed o me that I had overlooked a vital igrediet i the proofs: the square root of a desity is ot just a elemet of a L 2 space: it is a elemet with orm 1. By rearragig some of the stadard argumets I hope to covice the getle reader of this ote that the fixed orm is the real reaso for why a assumptio of oe-times differetiability (i quadratic mea) ca covey the beefits usually associated with two-times differetiability. I claim that the Lemma i the ext Sectio is the key to uderstadig the role of DQM A lemma The cocept of differetiability makes sese for maps ito a arbitrary ormed space (L, ). For the purposes of my expositio, it suffices to cosider the case where the orm is geerated by a ier product,,. I fact, L will be L 2 (λ), the space of fuctios square-itegrable with respect to some measure λ, but that simplificatio will play o role for the momet. Amapξ from R k ito L is said to be differetiable at a poit θ 0 with derivative, ifξ(θ) = ξ(θ 0 ) + (θ θ 0 ) + r(θ) ear θ 0, where r(θ) = o( θ θ 0 ) as θ teds to θ 0. The derivative is liear; it may be idetified with a k-vector of elemets from L. For a differetiable map, the Cauchy-Schwarz iequality implies that ξ(θ 0 ), r(θ) =o( θ θ 0 ). It would usually be a bluder to assume aively that the boud must therefore be of order O( θ θ 0 2 ); typically, higher-order differetiability assumptios are eeded to derive approximatios with smaller errors. However, if ξ(θ) is costat that is, if the fuctio is costraied to take values lyig o the surface of a sphere the the aive assumptio turs out to be o bluder. Ideed, i that case, ξ(θ 0 ), r(θ) ca be writte as a quadratic i θ θ 0 plus a error of order o( θ θ 0 2 ). The sequetial form of the assertio is more coveiet for my purposes. (1) Lemma Let {δ } be a sequece of costats tedig to zero. Let ξ 0, ξ 1,...be elemets of orm oe for which ξ = ξ 0 +δ W +r, with W aæxed elemet of L ad r =o(δ ). The ξ 0, W =0 ad ξ 0, r = 1 2 δ2 W 2 + o(δ 2).
3 19. Differetiability i Quadratic Mea 307 Proof. Because both ξ ad ξ 0 have uit legth, 0 = ξ 2 ξ 0 2 = 2δ ξ 0, W order O(δ ) + 2 ξ 0, r order o(δ ) + δ 2 W 2 order O(δ 2) + 2δ W, r + r 2 order o(δ 2). O the right-had side I have idicated the order at which the various cotributios ted to zero. (The Cauchy-Schwarz iequality delivers the o(δ ) ad o(δ 2 ) terms.) The exact zero o the left-had side leaves the leadig 2δ ξ 0, W uhappily exposed as the oly O(δ ) term. It must be of smaller order, which ca happe oly if ξ 0, W =0, leavig 0 = 2 ξ 0, r +δ 2 W 2 + o(δ 2 ), as asserted. Without the fixed legth property, the ier product ξ 0, r, which iherits o(δ ) behaviour from r, might ot decrease at the O(δ 2) rate A theorem Let {P θ : θ } be a family of probability measures o a space (X, A), idexed by a subset of R k. Suppose P θ has desity f (x,θ) with respect to a sigma-fiite measure λ. Uder the classical regularity coditios twice cotiuous differetiability of log f (x,θ) with respect to θ, with a domiated secod derivative the likelihood ratio f (x i,θ) f (x i,θ 0 ) ejoys the LAN property. Write L (t) for the likelihood ratio evaluated at θ equal to θ 0 + t/. The property asserts that, if the {x i } are sampled idepedetly from P θ0, the (2) L (t) = exp ( t S 1 2 t Ɣt + o p (1) ) for each t, where Ɣ is a fixed matrix (depedig o θ 0 )ads has a cetered asymptotic ormal distributio with variace matrix Ɣ. Formally, the LAN approximatio results from the usual poitwise Taylor expasio of the log desity g(x,θ) = log f (x,θ), followig a style of argumet familiar to most graduate studets. For example, i oe dimesio, log L (θ 0 + t/ ) = ( g(xi,θ 0 + t/ ) g(x i,θ 0 ) ) = t g (x i,θ 0 ) + t 2 g (x i,θ 0 ) +..., 2
4 308 David Pollard which suggests that S be the stadardized score fuctio, 1 g (x i,θ 0 ) N ( 0, var θ0 g (x,θ 0 ) ), ad Ɣ should be the iformatio fuctio, P θ0 g (x,θ 0 ) = var θ0 g (x,θ 0 ). The dual represetatio for Ɣ allows oe to elimiate all metio of secod derivatives from the statemet of the LAN approximatio, which hits that two derivatives might ot really be eeded, as Le Cam (1970) showed. I geeral, the family of desities is said to be differetiable i quadratic mea at θ 0 if the square root ξ(x,θ)= f (x,θ) is differetiable i the L 2 (λ) sese: for some k-vector (x) of fuctios i L 2 (λ), (3) ξ(x,θ)= ξ(x,θ 0 ) + (θ θ 0 ) (x) + r(x,θ), where λ r(x,θ) 2 = o( θ θ 0 2 ) as θ θ 0. Let us abbreviate ξ(x,θ 0 ) to ξ 0 (x) ad (x)/ξ 0 (x) to D(x). From (3) oe almost gets the LAN property. (4) Theorem Assume the DQM property (3). For each Æxed t the likelihood ratio has the approximatio, uder {P,θ0 }, where L (t) = exp ( t S 1 2 t Ɣt + o p (1) ), S = 2 D(x i ) N(0, I 0 ) ad Ɣ = 1 2 I I, with I 0 = 4λ( {ξ 0 > 0}) ad I = 4λ( ). Notice the slight differece betwee Ɣ ad the limitig variace matrix for S. At least formally, 2D(x) equals the derivative of log f (x,θ): igorig problems related to divisio by zero ad distictios betwee poitwise ad L 2 (λ) differetiability, we have 2 2D(x) = f (x,θ0 ) = f (x,θ0 ) θ θ log f (x,θ 0). Also, Ɣ agai correspods to the iformatio matrix, expressed i its variace form, except for the itrusio of the idicator fuctio {ξ 0 > 0}. The extra idicator is ecessary if we wish to be careful about 0/0. Its presece is related to the property called cotiguity aother of Le Cam s great ideas as is explaied i Sectio 5.
5 19. Differetiability i Quadratic Mea 309 At first sight the derivatio of Theorem 4 from assumptio (3) agai appears to be a simple matter of a Taylor expasio to quadratic terms of the log likelihood ratio. Writig R (x) = r(x,θ 0 + t/ )/ξ 0 (x), wehave log L (t) = 2log ξ(x i,θ 0 + t/ ) ξ(x i,θ 0 ) = 2log (1 + t ) D(x i ) + R (x i ). From the Taylor expasio of log( ) about 1, the sum of logarithms ca be writte as a formal series, 2 ( ) t D(x i ) + R (x i ) ( t 2 D(x i ) + R (x i )) +... (5) = 2t D(x i ) + 2 R (x i ) 1 ( t D(x i ) ) The first sum o the right-had side gives the t S i Theorem 4. The law of large umbers gives covergece of the third term to t P θ0 DD t. Mere oe-times differetiability might ot seem eough to dispose of the secod sum. Each summad has stadard deviatio of order o(1/ ), by DQM. A sum of such terms could crudely be bouded via a triagle iequality, leavig a quatity of order o( ), which clearly would ot suffice. I fact the sum of the R (x i ) does ot go away i the limit; as a cosequece of Lemma 1, it cotributes a fixed quadratic i t. That cotributio is the surprise behid DQM A proof Let me write P to deote calculatios uder the assumptio that the observatios x 1,...,x are sampled idepedetly from P θ0. The ratio f (x i,θ 0 + t/ )/f (x i,θ 0 ) is ot well defied whe f (x i,θ 0 ) = 0, but uder P the problem ca be eglected because P { f (x i,θ 0 ) = 0 for at least oe i} =0. For other probability measures that are ot absolutely cotiuous with respect to P, oe should be more careful. It pays to be quite explicit about behaviour whe f (x i,θ 0 ) = 0 for some i, by icludig a explicit idicator fuctio {ξ 0 > 0} as a factor i ay expressios with a ξ 0 i the deomiator. Defie D i to be the radom vector (x i ){ξ 0 (x i )>0}/ξ 0 (x i ), ad, for a fixed t, defie R i, = r(ξ i,θ 0 + t/ ){ξ 0 (x i )>0}/ξ 0 (x i ). The ξ(x i,θ 0 + t/ ) {ξ 0 ( i )>0} =1 + t D i + R i,. ξ 0 (x i )
6 310 David Pollard (6) (8) The radom vector D i has expected value λ(ξ 0 ), which, by Lemma 1, is zero, eve without the traditioal regularity assumptios that justify differetiatio uder a itegral sig. It has variace 1 4 I 0. It follows by a cetral limit theorem that S = 2 D i N(0, I 0 ). Also, by a (weak) law of large umbers, 1 D i D i P (D 1 D 1 ) = 1 4 I 0 i probability. To establish rigorously the ear-lan assertio of Theorem 4, it is merely a matter of boudig the error terms i (5) ad the justifyig the treatmet of the sum of the R (x i ). Three facts are eeded. (7) Lemma Uder {P }, assumig DQM, (a) max D i =o p ( ), (b) max R i, =o p (1), (c) 2R i, 1 4 t It i probability. Let me first explai how Theorem 4 follows from Lemma 7. Together the two facts (a) ad (b) esure that with high probability log L (t) does ot ivolve ifiite values. For (t D i / ) + R i, > 1 we may the a appeal to the Taylor expasio log(1 + y) = y 1 2 y β(y), where β(y) = o(y 2 ) as y teds to zero, to deduce that log L (t) equals 2 t D i + 2 R i, ( t ) D 2 i + R i, + ( t ) D i β + R i,, which expads to t S + 2 R i, 1 (t D i ) 2 2 t D i R i, R i, 2 + o p(1) ( Di 2 ) + Ri, 2. Each of the last three sums is of order o p (1) because D i 2 / = O p (1) ad P R2 i, = λ( ξ0 2 r(x 1,θ 0 + t/ ){ξ 0 > 0}/ξ0 2 ) λ r(,θ 0 + t/ ) 2 = o(1). By virtue of (6) ad (c), the expasio simplifies to t S 1 4 t It 1 4 t I 0 t + o p (1), as asserted by Theorem 4.
7 19. Differetiability i Quadratic Mea 311 Proof of Lemma 7. Assertio (a) follows from the idetical distributios: P {max D i >ɛ } P { D i >ɛ } = P { 1 >ɛ } ɛ 2 λ 2 1 { 1 >ξ 0 ɛ } 0 by Domiated Covergece. Assertio (b) follows from (8): P {max R i, >ɛ} ɛ 2 P Ri, 2 0. Oly Assertio (c) ivolves ay subtlety. The variace of the sum is bouded by 4 P R (x i ) 2, which teds to zero. The sum of the remaiders must lie withi o p (1) of its expected value, which equals 2P θ0 R 1, = 2λ ( ξ 0 r(,θ 0 + t/ ) ), a ier product betwee two fuctios i L 2 (λ). Notice that the ξ 0 factor makes the idicator {ξ 0 > 0} redudat. It is here that the uit legth property becomes importat. Specializig Lemma 1 to the case δ = 1/, with ξ (x) = ξ(x,θ 0 + t/ ) ad W = t, we get the approximatio to the sum of expected values of the R i,, from which Assertio (c) follows. A slight geeralizatio of the LAN assertio is possible. It is ot ecessary that we cosider oly parameters of the form θ 0 + t/ for a fixed t. By arguig almost as above alog coverget subsequeces of {t } we could prove a aalog of Theorem 4 if t were replaced by a bouded sequece {t } such that θ 0 + t /. The extesio is sigificat because (Le Cam 1986, page 584) the slightly stroger result forces a form of differetiability i quadratic mea Cotiguity ad disappearace of mass For otatioal simplicity, cosider oly the oe-dimesioal case with the typical value t = 1. Let ξ 2 be the margial desity, ad Q be the joit distributio, for x 1,...,x sampled with parameter value θ 0 + 1/.As before, ξ0 2 ad P correspod to θ 0. The measure Q is absolutely cotiuous with respect to P if ad oly if it puts zero mass i the set A ={ξ 0 (x i ) = 0 for at least oe i }. Writig α for λξ 2{ξ 0 = 0}, wehave Q A = 1 ( 1 Q {ξ 0 (x i ) = 0} ) = 1 (1 α ).
8 312 David Pollard By direct calculatio, α = λ ( r + / ) 2 {ξ0 = 0} =λ 2 {ξ 0 = 0}/ + o(1/). The quatity τ = λ 2 {ξ 0 = 0} has the followig sigificace. Uder Q, the umber of observatios ladig i A has approximately a Poisso(τ) distributio; ad Q A 1 e τ. I some asymptotic sese, the measure Q becomes more early absolutely cotiuous with respect to P if ad oly if τ = 0. The precise sese is called cotiguity: the sequece of measures {Q } is said to be cotiguous with respect to {P } if Q B 0 for each sequece of sets {B } such that P B 0. Because P A = 0 for every, the coditio τ = 0 is clearly ecessary for cotiguity. It is also sufficiet. Cotiguity follows from the assertio that L, the limit i distributio uder {P } of the likelihood ratios {L (1)}, have expected value oe. ( Le Cam s first lemma see the theorem o page 20 of Le Cam ad Yag, 1990.) The argumet is simple: If PL = 1 the, to each ɛ>0 there exists a fiite costat C such that PL{L < C} > 1 ɛ. From the covergece i distributio, P L {L < C} > 1 ɛ evetually. If P B 0the Q B P B L {L < C}+Q {L C} CP B + 1 P L {L < C} < 2ɛ evetually. For the special case of the limitig exp(n(µ, σ 2 )) distributio, where µ = 1 4 I I ad σ 2 = I 0, the requiremet becomes 1 = P exp ( N(µ, σ 2 ) ) = exp ( µ σ 2). That is, cotiguity obtais whe I 0 = I (or equivaletly, λ( 2 {ξ 0 = 0}) = 0), i which case, the limitig variace of S equals Ɣ. This coclusio plays the same role as the traditioal dual represetatio for the iformatio fuctio. As Le Cam & Yag (1990, page 23) commeted, The equality... is the classical oe. Oe fids it for istace i the stadard treatmet of maximum likelihood estimatio uder Cramér s coditios. There it is derived from coditios of differetiability uder the itegral sig. The fortuitous equality is othig more tha cotiguity i disguise. From the literature oe sometimes gets the impressio that λ 2 {ξ 0 = 0} is always zero. It is ot. (9) Example Let λ be Lebesgue measure o the real lie. Defie f 0 (x) = x{0 x 1}+(2 x){1 < x 2}. For 0 θ 1 defie desities f (x,θ)= (1 θ 2 ) f 0 (x) + θ 2 f 0 (x 2). Notice that (10) λ f (x,θ) f (x, 0) θ f (x, 1) 2 = ( 1 θ 2 1) 2 = O(θ 4 ).
9 19. Differetiability i Quadratic Mea 313 The family of desities is differetiable i quadratic mea at θ = 0 with derivative (x) = f (x, 1). For this family, λ 2 {ξ 0 = 0} =1. The ear-lan assertio of Theorem 4 degeerates: I 0 = 0adI = 4, givig L (t) exp ( t 2) i probability, uder {P,θ0 }. Ideed, as Aad va der Vaart has poited out to me, the limitig experimet (i Le Cam s sese) for the models {P,t/ : 0 t } is ot the Gaussia traslatio model correspodig to the LAN coditio. Istead, the limit experimet is {Q t : t 0}, with Q t equal to the Poisso(t 2 ) distributio. That is, for each fiite set T ad each h, uder {P,h/ } the radom vectors ( dp,t/ ) : t T dp,h/ coverge i distributio to ( ) dqt : t T, dq h as a radom vector uder the Q h distributio. The couterexample would ot work if θ were allowed to take o egative values; oe would eed (x) = f (x, 1) to get the aalog of (10) for egative θ. The failure of cotiguity is directly related to the fact that θ = 0 lies o boudary of the parameter iterval. I geeral, λ {ξ 0 = 0} must be zero at all iterior poits of the parameter space where DQM holds. O the set {ξ 0 = 0} we have 0 ξ(x,θ 0 +t/ ) = t + r, where r 0. Alog a subsequece, r 0, leavig the coclusio that t 0 almost everywhere o the set {ξ 0 = 0}. At a iterior poit, t ca rage over all directios, which forces = 0 almost everywhere o {ξ = 0}; at a iterior poit, {ξ = 0} =0 almost everywhere. More geerally, oe eeds oly to be able to approach θ 0 from eough differet directios to force = 0o{ξ 0 = 0} as i the cocept of a cotiget i Le Cam & Yag (1990, Sectio 6.2). The assumptio that θ 0 lies i the iterior of the parameter space is ot always easy to spot i the literature. Some authors, such as Le Cam & Yag (1990, page 101), prefer to dispese with the domiatig measure λ, by recastig differetiability i quadratic mea as a property of the desities dp θ /dp θ0, whose square roots correspod to the ratios ξ(x,θ){ξ 0 > 0}/ξ 0 (x). With that approach, the behaviour of o the set {ξ 0 = 0} must be specified explicitly. The cotiguity requiremet that P θ puts, at worst, mass of order o( θ θ 0 2 ) i the set {ξ 0 = 0} is the made part of the defiitio of differetiability i quadratic mea Refereces Ibragimov, I. A. & Has miskii, R. Z. (1981), Statistical Estimatio: Asymptotic Theory, Spriger-Verlag, New York.
10 314 David Pollard Le Cam, L. (1970), O the assumptios used to prove asymptotic ormality of maximum likelihood estimators, Aals of Mathematical Statistics 41, Le Cam, L. (1986), Asymptotic Methods i Statistical Decisio Theory, Spriger-Verlag, New York. Le Cam, L. & Yag, G. L. (1990), Asymptotics i Statistics: Some Basic Cocepts, Spriger-Verlag. Millar, P. W. (1983), The miimax priciple i asymptotic statistical theory, Spriger Lecture Notes i Mathematics pp Strasser, H. (1985), Mathematical Theory of Statistics: Statistical Experimets ad Asymptotic Decisio Theory, De Gruyter, Berli.
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